Dynamics of soliton-like solutions for slowly varying, generalized gKdV equations: refraction vs. reflection
DDYNAMICS OF SOLITON-LIKE SOLUTIONS FOR SLOWLY VARYING,GENERALIZED GKDV EQUATIONS: REFRACTION VS. REFLECTION
CLAUDIO MU ˜NOZ C.
Abstract.
In this work we continue our study of the description of the soliton-like solutions ofthe variable coefficients, subcritical gKdV equation u t + ( u xx − λu + a ( εx ) u m ) x = 0 , in R t × R x , m = 2 , , with 0 ≤ λ <
1, 1 < a ( · ) < ε small enough. In [34] we proved the existence (and uniqueness in most of the cases) of a pure soliton-like solution u ( t ) satisfyinglim t →−∞ (cid:107) u ( t ) − Q ( · − (1 − λ ) t ) (cid:107) H ( R ) = 0 , ≤ λ < , provided ε small enough. Here R ( t, x ) := Q c ( x − ( c − λ ) t ) is the standard H -soliton solutionof R t + ( R xx − λR + R m ) x = 0. In addition, this solution is global in time and satisfies, for all0 < λ ≤ − mm +3 , sup t (cid:29) ε (cid:107) u ( t ) − − / ( m − Q c ∞ ( · − ρ ( t )) (cid:107) H ( R ) ≤ Kε / , (0.1)for suitable scaling and translation parameters c ∞ ( λ ) ≥ ρ (cid:48) ( t ) ∼ ( c ∞ − λ ), and K >
0. Inthe cubic case, m = 3, this result also holds for λ = 0.The purpose of this paper is the following. We give an almost complete description of theremaining case − mm +3 < λ <
1. Surprisingly, there exists a fixed, positive number ˜ λ ∈ ( − mm +3 , ε , such that the following alternative holds:(1) Refraction . For all − mm +3 < λ < ˜ λ , the soliton solution behaves as in [34], and satisfies(0.1), but now λ < c ∞ <
1, and ρ (cid:48) ( t ) ∼ c ∞ − λ > Reflection . If ˜ λ < λ <
1, then the soliton-like solution is reflected by the potential and itsatisfies sup t (cid:29) ε (cid:107) u ( t ) − Q c ∞ ( · − ρ ( t )) (cid:107) H ( R ) ≤ Kε / . with 0 < c ∞ < λ , and ρ (cid:48) ( t ) ∼ c ∞ − λ <
0. This last is a completely new type of soliton-likesolution for gKdV equations, also present in the NLS case [33].Moreover, for any 0 < λ <
1, with λ (cid:54) = λ , the solution is not pure as t → + ∞ , in the sensethat lim sup t → + ∞ (cid:107) u ( t ) − κ ( λ ) Q c ∞ ( · − ρ ( t )) (cid:107) H ( R ) > , with κ ( λ ) depending on λ . Introduction and Main Results
In this work we continue our study of the dynamics of a soliton for some generalized Korteweg-de Vries equations (gKdV), started in [34]. In that paper the objective was the study of the globalbehavior of a generalized soliton solution for the following subcritical, variable coefficients gKdVequation: u t + ( u xx − λu + a ( εx ) u m ) x = 0 , in R t × R x , m = 2 , . (1.1)Here u = u ( t, x ) is a real-valued function, ε > λ ≥ potential a ( · ) a smooth, positive function satisfying some specific properties, see (1.5) below. Date : September, 2010.2000
Mathematics Subject Classification.
Primary 35Q51, 35Q53; Secondary 37K10, 37K40.
Key words and phrases. gKdV equations, integrability theory, soliton dynamics, slowly varying medium, reflec-tion, refraction.This research was supported in part by a CONICYT-Chile and an
Allocation de Recherche grants. a r X i v : . [ m a t h . A P ] S e p Dynamics of soliton solutions for perturbed gKdV equations
The above equation represents in some sense a simplified model of long dispersive waves in achannel with variable depth , which takes in account large variations in the shape of the solitarywave. The primary physical model, and the dynamics of a generalized soliton-solution, was for-mally described by Karpman-Maslov, Kaup-Newell, Asano, and Ko-Kuehl [19, 20, 1, 21], withfurther results by Grimshaw [10], and Lochak [23]. From a mathematical point of view, an addi-tional objective was the study of perturbations of integrable systems, in this case the KdV equation( m = 2). See [34, 36] and references therein for a detailed physical introduction to this model.The main novelty in the works above cited was the discovery of a dispersive tail behind thesoliton, with small height but large width, as a consequence of the lack of conserved quantitiessuch as mass or energy. However, no mathematically rigorous proof of this phenomenon was given.In addition, from the mathematical point of view, equation (1.1) is a variable coefficients versionof the gKdV equation u t + ( u xx − λu + u m ) x = 0 , in R t × R x ; m ≥ . (1.2)This last equation is important due to the existence of localized, exponentially decaying andsmooth solutions called solitons . Given real numbers x (=the translation parameter), and c > u ( t, x ) := Q c ( x − x − ( c − λ ) t ) , with Q c ( s ) := c m − Q ( c / s ) , (1.3)and where Q is the unique –up to translations– function satisfying the second order nonlinearordinary differential equation Q (cid:48)(cid:48) − Q + Q m = 0 , Q > , Q ∈ H ( R ) . (1.4)In this case, this solution belongs to the Schwartz class and it is explicitly given by the formula Q ( x ) = (cid:104) m + 12 cosh ( ( m − x ) (cid:105) m − . In particular, if c > λ the solution (1.3) represents a solitary wave , of scaling c and velocity ( c − λ ), defined for all time moving to the right without any change in shape, velocity, etc. Inother words, a soliton represents a pure , traveling wave solution with invariant profile . In addition,this equation allows soliton solutions with negative velocities, moving to the left direction, provided c < λ . Finally, for the case c = λ , one has a stationary soliton solution, Q λ ( x ). These two lastsolutions do not exist in the standard model of gKdV (namely when λ = 0.) In this sense, thedynamics of (1.2) is richer than the usual inviscid gKdV equation.Coming back to (1.1), the corresponding Cauchy problem has been considered in [34]; in par-ticular, we showed global well-posedness for H ( R ) initial data, even in the absence of somestandard conserved quantities. The proof of this result is an adaptation of the fundamental workof Kenig, Ponce and Vega [22], in addition to the introduction of some new monotone quantities.See Proposition 2.1 below for more details.One fundamental question related to (1.2) is how to generalize a soliton-like solution to morecomplicated models. In [3], the existence of soliton solutions for generalized KdV equations withsuitable autonomous nonlinearities was established. However, very little is known in the case ofan inhomogeneous nonlinearity, as in the case of (1.1). In a general situation, no elliptic, time-independent ODE can be associated to the soliton solution, unlike the standard autonomous casestudied in [3]. Other methods are needed.Concerning some time dependent, generalized KdV and mKdV equations ( m = 2 and m = 3),Dejak-Jonsson, and Dejak-Sigal [6, 7] studied the dynamics of a soliton for not too large times, of O ( ε − ). Recently, Holmer [13] has improved some of the Dejak-Sigal results in the KdV case, upto the Ehrenfest time O ( | log ε | ε − ). In their model, the perturbation is of linear type, which donot allow large variations on the soliton shape, different to the scaling itself. In this paper we will not make any distinction between soliton and solitary wave, unlike in the mathematical-physics literature. laudio Mu˜noz 3
Finally, in [34] we described the soliton dynamics for all time in the case of the time independent,perturbed gKdV equation (1.1). In order to state this last result, and our present main results,let us first describe the framework that we have considered for the potential a ( · ) in (1.1). Setting and hypotheses on a ( · ). Concerning the function a in (1.1), we assume that a ∈ C ( R )and there exist constants K, γ > < a ( r ) < , a (cid:48) ( r ) > , | a ( k ) ( r ) | ≤ Ke − γ | r | , for all r ∈ R , < a ( r ) − ≤ Ke γr , for all r ≤ , and0 < − a ( r ) ≤ Ke − γr for all r ≥ . (1.5)In particular, lim r →−∞ a ( r ) = 1 and lim r → + ∞ a ( r ) = 2. The choice (1 and 2) here do not implya loss of generality, it just simplifies the computations. In addition, we assume the followinghypothesis: there exists K > m = 2 , | ( a /m ) (3) ( s ) | ≤ K ( a /m ) (cid:48) ( s ) , for all s ∈ R . (1.6)This condition is generally satisfied, however a (cid:48) ( · ) must not be a compactly supported function.In addition, note that (1.1) formally behaves as a gKdV equation (1.2), with constant coefficients1 and 2, as x → ±∞ .Let us remark some important facts about (1.1) (see [34] for more details.) First, this equationis not invariant anymore under scaling and spatial translations. Moreover, a nonzero solution of(1.1) might lose or gain some mass , depending on the sign of u , in the sense that, at least formally,the quantity M [ u ]( t ) := 12 (cid:90) R u ( t, x ) dx (= mass ) (1.7)satisfies the identity ∂ t M [ u ]( t ) = − εm + 1 (cid:90) R a (cid:48) ( εx ) u m +1 ( t, x ) dx. (1.8)On the other hand, the energy E a [ u ]( t ) := 12 (cid:90) R u x ( t, x ) dx + λ (cid:90) R u ( t, x ) dx − m + 1 (cid:90) R a ( εx ) u m +1 ( t, x ) dx (1.9)remains formally constant for all time. Recall that these quantities are conserved for H -solutionsof (1.2), inside the corresponding interval of existence.In addition, there exists another conservation law, valid only for solutions with enough decayat infinity: (cid:90) R u ( t, x ) dx = constant . (1.10)Now let us describe what we mean by a soliton-like solution of (1.1). Indeed, in [34] weintroduced the concept of pure generalized soliton-solution for (1.1), of size c = 1 and velocity1 − λ > Definition 1.1 (Pure generalized soliton-solution for (1.1), [34]) . Let 0 ≤ λ < pure generalized soliton-likesolution (of scaling equals 1 and initial velocity equals 1 − λ >
0) if there exist a C real valuedfunction ρ = ρ ( t ), defined for all large time, and a global in time H ( R ) solution u ( t ) of (1.22)such that lim t →−∞ (cid:107) u ( t ) − Q ( · − (1 − λ ) t ) (cid:107) H ( R ) = 0 , (1.11)lim t → + ∞ (cid:13)(cid:13) u ( t ) − − / ( m − Q c ∞ ( · − ρ ( t )) (cid:13)(cid:13) H ( R ) = 0 , (1.12)with lim t → + ∞ ρ ( t ) = + ∞ , and where c ∞ = c ∞ ( λ ) > suggested by the energyconservation law . Dynamics of soliton solutions for perturbed gKdV equations
Remark . The above definition describes a soliton-like solution being completely pure at both t → ±∞ . Note e.g. that the standard soliton Q ( x − (1 − λ ) t ) is a pure soliton solution of (1.2),with invariant profile and no dispersive behavior. The coefficient 2 − / ( m − in front the solitonsolution in (1.12) comes from the fact that (1.1) behaves like the standard gKdV equation u t + ( u xx − λu + 2 u m ) x = 0 , as x → + ∞ .However, in this definition we do not consider a possible case of a reflected soliton,lim t → + ∞ (cid:107) u ( t ) − Q c ∞ ( · − ρ ( t )) (cid:107) H ( R ) = 0 , lim t → + ∞ ρ ( t ) = −∞ . Remark c ∞ ) . Let us explain in more detail the main argument –based in theenergy conservation law–, to determine the scaling c ∞ ( λ ). Let u ( t ) be a pure soliton solution, asin Definition 1.1. Then one has E a [ u ]( −∞ ) = ( λ − λ ) M [ Q ] , with λ given by λ := 5 − mm + 3 ∈ (0 , , (1.13)(cf. Appendix C.1 for the details.) On the other hand, one has E a [ u ](+ ∞ ) = c m − − ∞ ( λ )2 m − ( λ − c ∞ ( λ ) λ ) M [ Q ];for c ∞ = c ∞ ( λ ). From the energy conservation law one obtains c m − − ∞ ( λ )2 m − ( λ − c ∞ ( λ ) λ ) = λ − λ , that is, c λ ∞ ( c ∞ − λλ ) − λ = 2 m +3 (1 − λλ ) − λ . (1.14)In [34] we proved the existence of a unique solution c ∞ ( λ ) ≥ ≤ λ ≤ λ (see Lemma 3.1 for more details.) Moreover, the application λ (cid:55)→ c ∞ ( λ ) isa smooth decreasing map with c ∞ (0) = 2 / ( m +3) and c ∞ ( λ ) = 1. However, note that (1.14) isvalid only under the assumptions (1.11)-(1.12). In particular, if there exists a reflected soliton, itshould obey a different scaling law. Remark . Note that a pure generalized soliton-like solution may loss almostone half of its mass during the interaction. Indeed, a simple computation shows that the mass atinfinity is given by M [ u ]( −∞ ) = M [ Q ] , M [ u ](+ ∞ ) = 2 − / ( m − c / ( m − − / ∞ M [ Q ] . Since c ∞ ( λ ) is a decreasing map in λ (see preceding remark), one has e.g. M [ u ](+ ∞ ) | λ =0 = 2 m +3 − m − M [ Q ] ,M [ u ](+ ∞ ) | λ = λ = 2 − m − M [ Q ] . Description of the dynamics.
Let us be more precise. By assuming the validity of (1.5) and(1.6), we proved, among other things, the following result.
Theorem 1.1 (Dynamics of solitons for gKdV under slowly varying medium, see [34]) . Suppose m = 2 , and , and let ≤ λ < be a fixed number. Consider λ as in (1.13), and c ∞ ( λ ) satisfying (1.14). There exists a small constant ε > such that for all < ε < ε thefollowing holds. laudio Mu˜noz 5 (1) Existence of a soliton-like solution .There exists a solution u ∈ C ( R , H ( R )) of (1.1), global in time, such that lim t →−∞ (cid:107) u ( t ) − Q ( · − (1 − λ ) t ) (cid:107) H ( R ) = 0 , (1.15) with conserved energy E a [ u ]( t ) = ( λ − λ ) M [ Q ] . This solution is unique in the followingcases: ( i ) m = 3 ; and ( ii ) m = 2 , , provided λ > . (2) Interaction soliton-potential and refraction .Suppose now in addition that < λ ≤ λ for the cases m = 2 , , and ≤ λ ≤ λ if m = 3 . There exist constants
K, T, c + > and a C -function ρ ( t ) , defined in [ T, + ∞ ) ,such that w + ( t, · ) := u ( t, · ) − − / ( m − Q c + ( · − ρ ( t )) satisfies (a) Stability and asymptotic stability . For any t ≥ T , (cid:107) w + ( t ) (cid:107) H ( R ) + | ρ (cid:48) ( t ) − ( c ∞ ( λ ) − λ ) | + | c + − c ∞ ( λ ) | ≤ Kε / ; (1.16) and for some fixed < β < ( c ∞ − λ ) , depending on ε , lim t → + ∞ (cid:107) w + ( t ) (cid:107) H ( x>βt ) = 0 . (1.17)(b) Bounds on the scaling parameter . Define θ := m − − > . One has, for all λ > , K lim sup t → + ∞ (cid:107) w + ( t ) (cid:107) H ( R ) ≤ (cid:0) c + c ∞ (cid:1) θ − ≤ K lim inf t → + ∞ (cid:107) w + ( t ) (cid:107) H ( R ) . (1.18)The proof of this result, and in particular of (1.16), requires the introduction of an approximatesolution, up to first order in ε . Roughly speaking, the solution u ( t ) behaves like a well modulatedsoliton-solution, plus a small order term, namely u ( t, x ) ∼ µ ( t ) Q c ( t ) ( x − ρ ( t )) + εν ( t ) A c ( t ) ( x − ρ ( t )) , (1.19)where c ( t ) , ρ ( t ) are the scaling and position parameters, and µ ( t ) , ν ( t ) A c are unknown functions,to be found. In [34] we proved that this description is a good approximation of the dynamics,provided ( c, ρ ) follow a well defined dynamical system, of the form (cf. Lemma 2.6 for moredetails): (cid:40) c (cid:48) ( t ) ∼ εf ( t ) , c ( − T ε ) ∼ ,ρ (cid:48) ( t ) ∼ c ( t ) − λ, ρ ( − T ε ) ∼ − (1 − λ ) T ε , (1.20)for a given function f ( t ) > T ε (cid:29) ε (see (1.30) for a precisedefinition.) Therefore, the infinite dimensional dynamics reduces to a simple finite dimensionalproblem, which describes the main properties of the soliton solution. Once this system is wellunderstood, the main problem reduces to an advanced form of stability argument, in the spirit ofWeinstein, and Martel-Merle [42, 26]. Remark . Note that u ( t ) behaves like an almost pure soliton solution, in the sense of Definition 1.1, up toan error of order ε / in H ( R ). A first sight, the order of magnitude of this term may appearsomehow strange. However, it can explained by the existence of a dispersive tail behind thesoliton, formally found by physicists in [20]. This tail is mathematically described by the function A c in (1.19). Indeed, one can see (cf. Proposition 4.2), that A c is an almost flat function, withsupport of size O ( ε − ). From this fact, it is clear that (cid:107) εA c ( · − ρ ( t )) (cid:107) H ( R ) ≤ Kε / , (cid:107) εA c ( · − ρ ( t )) (cid:107) L ( R ) = O (1) . Note that this bound holds even for the cubic case, m = 3, which makes a big difference with themodel studied in [6]. In that paper the authors found an upper bound of order ε . We believe This is the case of nonpositive energy. Recall that c ∞ ( λ ) ≥ ≤ λ ≤ λ . Dynamics of soliton solutions for perturbed gKdV equations that our upper bound is bigger due to the shape variation experienced by the soliton, which is notpresent in the theory developed by [6].
Remark . Stability (1.16) and asymptotic stability (1.17) of solitary waves for gKdV equationsas stated in the above Theorem have been widely studied since the ’80s. The main ideas of ourproof are classical in the literature. For more details, see e.g. [2, 4, 30, 37].In addition, by using a contradiction argument and the L -conservation law (1.10), it wasproved that no soliton-like solution exist in this regime: Theorem 1.2 (Non-existence of pure soliton-like solution for 1.1, [34]) . Under the context of Theorems 1.1, suppose m = 2 , , with < λ ≤ λ . There exists ε > such that for all < ε < ε , lim sup t → + ∞ (cid:107) w + ( t ) (cid:107) H ( R ) > . (1.21) Remark . Let us explain in some words the proof of this last theorem. The proof it is mainlybased in an argument introduced in [31], in a completely different context. We suppose thatlim t → + ∞ (cid:107) w + ( t ) (cid:107) H ( R ) = 0. Using a monotonicity argument, one can show that, for any λ > t (cid:29) ε − (cid:107) w + ( t ) (cid:107) H ( R ) ≤ Ke − εγt , up to a small modulation parameter in the space variable. This time decay can be traducedin space decay via a new monotonicity formula, which allows to define the integral of w + ( t ) as t → + ∞ , and proves that it is small. Using the L -conservation law, and comparing the resultobtained at both t ∼ ±∞ , we obtain the desired contradiction.However, this argument does not give a quantitative lower bound on the size of the defect w + ( t ),as t → + ∞ . Remark . From the proof of this result in [34], we emphasize that the same conclusion inTheorem 1.2 holds for any < λ < t largeenough, after some minor modifications (cf. Section 5.)Summarizing, Theorems 1.1 and 1.2 can be represented in the following figure: ↑ t → xQ − / ( m − Q c + c + > c ∞ (Thm. 1.2) non-zero defect a ≡ a ≡ O H ( ε / ) t ∼ t → + ∞ [Thm. 1.1 (2)][Thm. 1.1 (1)] t → −∞ A first important question left open in [34] was the behavior of the solution u ( t ) from Definition1.1 in the case of positive energy , namely λ < λ <
1. The analysis in this case requires moreattention due the fact that the scaling of the soliton solution decreases as long as the interactionsoliton-potential takes place. This behavior is in part a consequence of the competition betweenthe strength of the potential and the initial kinetic energy. In this paper our first objective is todescribe in detail that case. Indeed, in the next paragraphs we will state the following surprisingresult: given a fixed λ close to 1, for any small ε > reflected by the potential laudio Mu˜noz 7 a ( ε · ). This result is basically a consequence of the fact that, given 0 < λ < c > c < λ , the small soliton Q c ( · − ( c − λ ) t ), solution of u t + ( u xx − λu + u m ) x = 0 , in R t × R x , moves towards the left . Main Results.
Let us recall the setting of our problem. Let 0 < λ < (cid:40) u t + ( u xx − λu + a ( εx ) u m ) x = 0 in R t × R x ,m = 2 , < ε ≤ ε ; a ( ε · ) satisfying (1 . . . (1.22)Here ε > λ = ˜ λ ( m ) be the uniquesolution of the algebraic equation˜ λ ( 1 − λ ˜ λ − λ ) − λ = 2 m +3 , λ < ˜ λ < , λ given by (1 . . (1.23)(See Lemma 3.1 for more details.) We claim that this number represents a sort of equilibrium between the energy of the solitary wave and the strength of the potential. Indeed, first we provethat the dynamics in the case λ < λ < ˜ λ is similar to that of [34]. Theorem 1.3 (Interaction soliton-potential and refraction, case λ < λ < ˜ λ ) . Suppose λ < λ < ˜ λ . There exists ε > such that for all < ε < ε the following holds. Thereexist constants K, ˜ T , c + , c ∞ ( λ ) > , with λ < c ∞ ( λ ) < ; and a smooth function ρ ( t ) ∈ R suchthat the function w + := u ( t ) − − m − Q c + ( · − ρ ( t )) satisfies for all t ≥ ˜ T , (cid:107) w + ( t ) (cid:107) H ( R ) + | ρ (cid:48) ( t ) − c ∞ ( λ ) + λ | + | c + − c ∞ | ≤ Kε / , (1.24) and lim t → + ∞ (cid:107) w + ( t ) (cid:107) H ( x>βt ) = 0 , for a fixed < β < ( c ∞ ( λ ) − λ ) . Moreover, for θ := m − − , K lim sup t → + ∞ (cid:107) w + ( t ) (cid:107) H ( R ) ≤ (cid:0) c + c ∞ (cid:1) θ − ≤ Kε. (1.25)Note that this generalized soliton solution behaves, as t → + ∞ , as a solitary wave with velocity ∼ c ∞ − λ >
0, but smaller than the initial one (= 1 − λ ).Now we consider the case ˜ λ < λ <
1. Here a completely new behavior is present. The solitonsolution is, in this case, a reflected solitary wave.
Theorem 1.4 (Interaction soliton-potential and reflection, case ˜ λ < λ < . Suppose now ˜ λ < λ < , with ε > small enough. Then there exist constants K, ˜ T , c + , c ∞ ( λ ) > , with < c ∞ ( λ ) < λ ; and a smooth function ρ ( t ) ∈ R such that w + := u ( t ) − Q c + ( · − ρ ( t )) satisfies for all t ≥ ˜ T , (cid:107) w + ( t ) (cid:107) H ( R ) + | ρ (cid:48) ( t ) − c ∞ ( λ ) + λ | + | c + − c ∞ | ≤ Kε / , (1.26) and lim t → + ∞ (cid:107) w + ( t ) (cid:107) H ( x>βt ) = 0 , for a fixed β ∈ ( − λ, c ∞ ( λ ) − λ ) . Finally, K lim sup t → + ∞ (cid:107) w + ( t ) (cid:107) H ( R ) ≤ (cid:0) c ∞ c + (cid:1) θ − ≤ Kε. (1.27)
Dynamics of soliton solutions for perturbed gKdV equations
Some few remarks are in order.
Remark . Note that in (1.27) the final scaling c + is smaller or equal than c ∞ ( λ ). This is a bigsurprise, present in the case of a reflected soliton. In particular, it differs from the results foundin the recent literature (compare with the results found in [26, 27, 28, 33].)
Remark . We believe that Theorem 1.4 is the first completely rigorousresult showing the existence and global description of a reflected solitary wave under a slowlyvarying potential; in this case for gKdV equations. Preliminary, formal results in this directioncan be found in [18, 17, 5, 41, 40, 7, 13].
Remark . With a slight abuse of notation, we have denoted by w + ( t ), ρ ( t ), c + , etc.some different functions or parameters (cf. Theorems 1.1, 1.3 and 1.4.) However, since the rangeof validity of each definition depends on λ , and each region of validity in λ is pairwise disjoint, wehave chosen this method, in order to simplify the notation. Remark . Note that the coefficients in front of Q c + in Theorems 1.3 and 1.4 are different sincethe potential a ( · ) behaves in a different way depending on x → ±∞ . Remark . In the case ˜ λ < λ <
1, andcompared with the case 0 < λ < ˜ λ , the equation for the parameter 0 < c ∞ ( λ ) < λ − λ = c m − − ∞ ( λ )( λ − λ c ∞ ( λ )) , that is, c λ ∞ ( λλ − c ∞ ) − λ = ( λλ − − λ (compare with (1.14), and see also Lemma 3.3 for more details.) In addition the final mass isgiven now by the quantity M [ u ](+ ∞ ) = c m − − ∞ ( λ ) M [ Q ] < λ m − − M [ Q ] , modulo an error of order at most ε . Remark λ = ˜ λ ) . The behavior of the solution in the case λ = ˜ λ remains an interestingopen problem. It seems that in the case λ = ˜ λ the solution u ( t ) behaves asymptotically at infinityas an almost bounded state of the form 2 − / ( m − Q ˜ λ ( x − ρ ( t )), for some ρ (cid:48) ( t ) small and close tozero. See Lemma 3.2 and Remark 3.3 for more details. Main ideas in the proof of Theorems 1.3 and 1.4.
Similar to [34], the proof of this resultis based in a detailed description of the behavior of a finite dimensional dynamical system , for thecase λ < λ <
1, which leads to the different behaviors above mentioned. Indeed, from [34], onehas that the scaling c ( t ) and the translation ρ ( t ) associated to the soliton solution, satisfy, at thefirst order in ε , the dynamical system (1.20), now for a given function f ( t ) <
0. Since c (cid:48) ( t ) < t ≥ − T ε , the scaling is a decreasing quantity in time. The key point is then the following: anecessary condition to obtain a reflected soliton is that ρ (cid:48) ( t ) < c ( t ) < λ . Therefore we need to check the values of λ for which the scaling c ( t ) satisfies c ( t ) > λ for all t ≥ − T ε , or c ( t ) = λ for some t > − T ε . After some computations, it turns out that thesharp parameter deciding between these two regimes is given by ˜ λ in (1.23) (see Remark 3.1 formore details.) In addition, we have to prove that c ( t ) remains far from zero for all time, which isnot direct since c ( t ) is always a decreasing quantity. Remark . From the above results we do not discard the existence of small solitary wavestraveling to the left (since a small soliton moves to the left), at least for the case m = 2. In thecubic and quartic cases, we believe there are no such soliton solutions.Finally, we prove that there is no pure soliton solution at both sides of time. In Theorem 1.5 we will prove that it is actually smaller. laudio Mu˜noz 9
Theorem 1.5 (Inelastic character of the interaction soliton-potential) . Suppose λ < λ < , with λ (cid:54) = ˜ λ. Then one has lim sup t → + ∞ (cid:107) w + ( t ) (cid:107) H ( R ) > , in particular c + > c ∞ ( λ ) for < λ < ˜ λ , and c + < c ∞ ( λ ) in the case ˜ λ < λ < . From Remark 1.7, the proof of this result is a consequence of estimates (1.24)-(1.26), and thesame argument developed for the proof of Theorem 1.2 in [34].The following figure illustrates the behavior stated in Theorems 1.3, 1.4 and 1.5. H -defect > (Thm. 1.5) Q − / ( m − Q c + , λ < c + < (Thm. 1.3) Q c + , 0 < c + < λ (Thm. 1.4) t → −∞ case λ < λ < ˜ λ case ˜ λ < λ < t ∼ t → + ∞ O H ( ε / ) a ≡ a ≡ ↑ t → x A second important open question from [34] and this paper is to establish a lower bound onthe defect w + ( t ) as the time goes to infinity, at least in the case 0 < λ < λ (cid:54) = ˜ λ (the cases λ = 0and λ = ˜ λ seem harder.) We expect to treat this problem in a forthcoming paper (see [35].) Forthe moment, and based in some formal computations (cf. Proposition 4.2 and Remark 4.4) weclaim that lim inf t → + ∞ (cid:107) w + ( t ) (cid:107) H ( R ) ≥ Kε p m , with p = p = 1 , p = 2 . (1.28) Remark . The interaction soliton-potential has be also considered inthe case of the nonlinear Schr¨odinger equation with a slowly varying potential, or a soliton-defectinteraction. See e.g. Gustafson et al. [11, 12], Gang and Sigal [8], Gang and Weinstein [9], andHolmer, Marzuola and Zworski [14, 15, 16], for more details. See also our recent work [33] onsoliton dynamics for a modified NLS equation.
Notation.
In this paper, both
K, γ > ε , and possiblychanging from one line to another. Let us define, for m = 2 , µ = µ ( λ ) := 99100 (1 − λ λ ) − λ λ . (1.29) Since λ < λ <
1, this number is always a positive quantity, less than 1. In addition, let us define,for ε > T ε := ε − − − λ > . (1.30)Third, we consider the unperturbed energy E [ u ]( t ) := 12 (cid:90) R u x ( t ) + λ (cid:90) R u ( t ) − m + 1 (cid:90) R u m +1 ( t ) , (1.31)namely E [ u ] = E a ≡ [ u ].Finally, we denote by Y the set of C ∞ functions f such that for all j ∈ N there exist K j , r j > x ∈ R we have | f ( j ) ( x ) | ≤ K j (1 + | x | ) r j e − µ | x | . (1.32) Plan of this work.
Let us explain the organization of this paper. First, in Section 2 we introduce some basic toolsto study the interaction problem, and state several important asymptotic results. In Section 3we study a finite-dimensional dynamical system which describes the dynamics in a approximativeway. Next, in section 4 we describe the interaction soliton-potential, based in the construction ofan approximate solution (see Appendix B for that computation). Finally in Section 5 we provethe main results of this article.
Acknowledgments . I wish to thank Y. Martel and F. Merle for presenting me this problemand for their continuous encouragement and support, during the elaboration of this work; and theDIM members at Universidad de Chile for their kind hospitality, and where part of this work waswritten. 2.
Preliminaries
The purpose of this section is to recall some important properties needed through this paper.For more details or the proof of these results, see Section 2 and 3 in [34].2.1.
The Cauchy problem.
First we recall the following local well-posedness result for theCauchy problem associated to (1.22).Let u ∈ H s ( R ), s ≥ λ >
0. We consider the following initial value problem (cid:40) u t + ( u xx − λu + a ( εx ) u m ) x = 0 in R t × R x u ( t = 0) = u , (2.1)where m = 2 , Proposition 2.1 (Local and global well-posedness, see [22] and Proposition 2.1 in [34]) . (1) Local well posedness in H s ( R ) .Suppose u ∈ H s ( R ) , s ≥ . Then there exist a maximal interval of existence I ( with ∈ I ) , and a unique (in a certain sense) solution u ∈ C ( I, H s ( R )) of (2.1). In addition,for any t ∈ I the energy E a [ u ]( t ) from (1.9) remains constant, and the mass M [ u ]( t ) defined in (1.7) satisfies (1.8). (2) Global existence in H ( R ), λ > .Suppose now u ∈ H ( R ) , and λ > . Then I is of the form I = (˜ t , + ∞ ) , for some −∞ ≤ ˜ t < ; and there exists ε > small such that sup t ≥ (cid:107) u ( t ) (cid:107) H ( R ) ≤ K. Finally, suppose u ∈ L ( R ) ∩ H ( R ) . Then (1.10) is well defined and remains constantfor all t ∈ I . laudio Mu˜noz 11 Remark . In order to prove item (2) in the above result, we introduced in [34] a modified mass, decreasing in time. Indeed, consider for all t ∈ I , m = 2 , M [ u ]( t ) := 12 (cid:90) R a /m ( εx ) u ( t, x ) dx. (2.2)Then for any m = 2 , t ∈ I we have ∂ t ˆ M [ u ]( t ) = − ε (cid:90) R ( a /m ) (cid:48) ( εx ) u x − ε (cid:90) R [ λ ( a /m ) (cid:48) − ε ( a /m ) (3) ]( εx ) u . (2.3)In conclusion, from (1.6) there exists ε > < ε ≤ ε and for all t ≥
0, one hasˆ M [ u ]( t ) ≤ ˆ M [ u ](0) . (2.4)The global existence follows from the subcritical nature of the nonlinearity ( m < Spectral properties of the linear gKdV operator.
In this paragraph we consider someimportant properties concerning the linearized KdV operator associated to (1.22). Fix c > m = 2 , L w ( y ) := − w yy + cw − mQ m − c ( y ) w, where Q c ( y ) := c m − Q ( √ cy ) . (2.5)Here w = w ( y ). We also denote L := L c =1 . Lemma 2.2 (Spectral properties of L , see [27]) . The operator L defined (on L ( R ) ) by (2.5) has domain H ( R ) , it is self-adjoint and satisfiesthe following properties: (1) First eigenvalue . There exist a unique λ m > such that L Q m +12 c = − λ m Q m +12 c . (2) The kernel of L is spawned by Q (cid:48) c . Moreover, Λ Q c := ∂ c (cid:48) Q c (cid:48) (cid:12)(cid:12) c (cid:48) = c = 1 c (cid:104) m − Q c + 12 xQ (cid:48) c (cid:105) , (2.6) satisfies L (Λ Q c ) = − Q c . Finally, the continuous spectrum of L is given by σ cont ( L ) =[ c, + ∞ ) . (3) Inverse . For all h = h ( x ) polynomially growing function such that (cid:82) R hQ (cid:48) c = 0 , thereexists a unique polynomially growing function ˆ h such that (cid:82) R ˆ hQ (cid:48) c = 0 and L ˆ h = h .Moreover, if h is even (resp. odd), then ˆ h is even (resp. odd). (4) Regularity in the Schwartz space S . For h ∈ H ( R ) , L h ∈ S implies h ∈ S . (5) Coercivity . (a) There exists
K, σ c > such that for all w ∈ H ( R ) B [ w, w ] := (cid:90) R ( w y + cw − mQ m − c w ) ≥ σ c (cid:90) R w − K (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) R wQ c (cid:12)(cid:12)(cid:12)(cid:12) − K (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) R wQ (cid:48) c (cid:12)(cid:12)(cid:12)(cid:12) . In particular, if (cid:90) R wQ c = (cid:90) R wQ (cid:48) c = 0 , then the functional B [ w, w ] is positive definitein H ( R ) . (b) Now suppose that (cid:90) R wQ c = (cid:90) R wyQ c = 0 . Then the same conclusion as above holds. Construction of a soliton-like solution.
Let us recall the following result of existence anduniqueness of a pure soliton-like solution for (1.22) for t → −∞ , valid for any fixed ≤ λ < Proposition 2.3 (Existence and uniqueness of a pure soliton-like solution, [34]) . Suppose ≤ λ < fixed. There exists ε > small enough such that the following holds forany < ε < ε . There is a solution u ∈ C ( R , H ( R )) of (1.22) such that lim t →−∞ (cid:107) v ( t ) − Q ( · − (1 − λ ) t ) (cid:107) H ( R ) = 0 , (2.7) and energy E a [ u ]( t ) = ( λ − λ ) M [ Q ] . Moreover, there exist constants
K, γ > such that for alltime t ≤ − T ε , , (cid:107) u ( t ) − Q ( · − (1 − λ ) t ) (cid:107) H ( R ) ≤ Kε − e εγt . (2.8) In particular, (cid:107) u ( − T ε ) − Q ( · + (1 − λ ) T ε ) (cid:107) H ( R ) ≤ Kε − e − γε − ≤ Kε , (2.9) provided < ε < ε small enough.Finally, this solution is unique for all λ > , and in the case λ = 0 , m = 3 . Remark . Note that the energy identity above follows directly from (2.7), Appendix C.1 andthe energy conservation law from Proposition 2.1.The proof of this Proposition is standard and follows the work of Martel [24], where the exis-tence of a unique N -soliton solution for gKdV equations was established. Although there existpossible different proofs of this result, the method employed in [24] has the advantage of giving anexplicit uniform bound in time (cf. (2.8)). This bound is indeed consequence of some compactnessproperties.2.4. Stability and asymptotic stability results for large time.
In order to prove the stabilityproperties contained in Theorems 1.3 and 1.4, we recall the following result, proved in [34] for thecase 0 < λ ≤ λ , but still valid for any fixed 0 < λ < c ∞ >
0, satisfying λ < c ∞ . Proposition 2.4 (Stability and asymptotic stability in H , see [34]) . Let m = 2 , and , and let < λ < , c ∞ > λ . Let < β < ( c ∞ − λ ) be a fixed number.There exists ε > (depending on β ) such that if < ε < ε the following hold.Suppose that for some time t ≥ T ε and t ≤ X ≤ t , one has (cid:13)(cid:13) u ( t ) − Q c ∞ ( x − X ) (cid:13)(cid:13) H ( R ) ≤ ε / , (2.10) where u ( t ) is a H -solution of (1.22). Then u ( t ) is defined for every t ≥ t and there exists K, c + > and a C -function ρ ( t ) defined in [ t , + ∞ ) such that (1) Stability . sup t ≥ t (cid:13)(cid:13) u ( t ) − − / ( m − Q c ∞ ( · − ρ ( t )) (cid:13)(cid:13) H ( R ) ≤ Kε / , (2.11) where | ρ ( t ) + X | ≤ Kε / , and for all t ≥ t , | ρ (cid:48) ( t ) − c ∞ + λ | ≤ Kε / . (2) Asymptotic stability . One has lim t → + ∞ (cid:13)(cid:13) u ( t ) − Q c + ( · − ρ ( t )) (cid:13)(cid:13) H ( x>βt ) = 0 . (2.12) In addition, lim t → + ∞ ρ (cid:48) ( t ) = c + − λ, | c + − c ∞ | ≤ Kε / . (2.13) Remark . In other words, the above result states that once the soliton has crossed the interactionregion, it behaves like a standard soliton of a gKdV equation, and stability and asymptotic stabilityhold. Let us recall that, from [34], this result is valid for any m = 2 , < λ < c ∞ > λ . In addition, it is still valid for m = 3 and λ = 0. with T ε defined in (1.30) laudio Mu˜noz 13 Remark . Let us recall that the hypothesis c ∞ > λ is essential; otherwise the soliton shouldhave negative velocity (= c ∞ − λ ) and it would return to the interaction region. Indeed, theoriginal proof in [34] falls to be correct since the quantity ( c ∞ ( λ ) − λ ) ˆ M [ u ]( t ) in the Weinsteinfunctional is no longer decreasing. Later, in Lemma 3.1, we will see that c ∞ ( λ ), as introduced inTheorem 1.3, satisfies c ∞ ( λ ) > λ for any 0 ≤ λ < ˜ λ (cf. (1.23).)In order to prove the global stability result of Theorem 1.4, we will need a version of the aboveProposition for the case of a reflected soliton , namely when ˜ λ < λ <
1. Let us recall that, given0 < λ < c > c < λ , the small soliton Q c ( · − ( c − λ ) t ), solution of u t + ( u xx − λu + u m ) x = 0 , in R t × R x , moves towards the left. Proposition 2.5 (Stability and asymptotic stability in H ( R ), reflection case) . Suppose m = 2 , or . Let < λ < and c ∞ > be such that c ∞ < λ . Let − λ < β < c ∞ − λ .There exists ε > such that if < ε < ε the following hold.Suppose that for some time t ≥ KT ε and t ≤ X ≤ t (cid:13)(cid:13) u ( t ) − Q c ∞ ( x + X ) (cid:13)(cid:13) H ( R ) ≤ ε / . (2.14) where u ( t ) is a H -solution of (1.22). Then u ( t ) is defined for every t ≥ t , and there exists K > and a C -function ρ ( t ) , defined in [ t , + ∞ ) , such that for all t ≥ t , (1) Stability . (cid:107) u ( t ) − Q c ∞ ( · − ρ ( t )) (cid:107) H ( R ) + | ρ ( t ) + X | + | ρ (cid:48) ( t ) − c ∞ + λ | ≤ Kε / . (2.15)(2) Asymptotic stability . There exists c + > such that lim t → + ∞ (cid:13)(cid:13) u ( t ) − Q c + ( · − ρ ( t )) (cid:13)(cid:13) H ( x>βt ) = 0 . (2.16) In addition, lim t → + ∞ ρ (cid:48) ( t ) = c + − λ, | c + − c ∞ | ≤ Kε / . (2.17)The proof of this statement requires several new ideas, in particular, the introduction of amodified mass, almost increasing in time. It turns out that these requirements are satisfied e.g.by the quantity M [ u ]( t ) := (cid:90) R u ( t, x )2 a ( εx ) dx. (2.18)Summarizing, the stability theory requires the introduction of two different, almost monotonemasses, depending on c ∞ . Indeed, if (cid:40) c ∞ > λ = ⇒ we use ˆ M [ u ]( t ) , (cf. (2.2)) ,c ∞ < λ = ⇒ we use M [ u ]( t ) . Proof of Proposition 2.5.
See Appendix A. (cid:3)
Let us finish this section with a result concerning the dynamical system associated to theparameters of the soliton solution.
Existence of approximate dynamical parameters.
In this paragraph we recall the ex-istence of a unique solution to the dynamical system found in [34], and involving the evolutionof the first order scaling and translation parameters of the soliton solution, ( C ( t ) , P ( t )), in theinteraction region. The behavior of this solution is essential to understand the actual dynamics ofthe soliton solution inside this region.Let us fix some notation. Denote, for C > P ∈ R given, f ( C, P ) := p C ( C − λλ ) a (cid:48) ( εP ) a ( εP ) , p := 4 m + 3 , (2.19)with λ as in (1.13). Lemma 2.6 (Existence and basic properties of dynamical parameters, see [34]) . Suppose m = 2 , or , λ , a ( · ) , T ε be as in Theorem 1.1, (1.5) and (1.30). Let ≤ λ ≤ λ .There exists a unique solution ( C ( t ) , U ( t )) , with C bounded positive, monotone increasing, definedfor all t ≥ − T ε , of the following system (cid:40) C (cid:48) ( t ) = εf ( C ( t ) , P ( t )) , C ( − T ε ) = 1 ,P (cid:48) ( t ) = C ( t ) − λ, P ( − T ε ) = − (1 − λ ) T ε . (2.20) In addition, one has ≤ C ( t ) < C (+ ∞ ) = c ∞ (1 + O ( ε )) , and (1 − λ ) t ≤ P ( t ) ≤ ( c ∞ − λ ) t ,with c ∞ = c ∞ ( λ ) being the unique positive solution of c λ ∞ ( c ∞ − λλ ) − λ = 2 p (1 − λλ ) − λ , c ∞ ( λ ) ≥ . (2.21) In particular, one has c ∞ ( λ = 0) = 2 p > and c ∞ ( λ = λ ) = 1 .Remark . Let us explain the importance of this result. The above lemma formally describesthe dynamics of the soliton solution by means of some approximate, finite dimensional system ofthe variables C ( t ) and P ( t ). In other words, the dynamics in the case ε ∼ C ( t ) , P ( t )).In the next section, our objective is to extend this result to the full range λ < λ < . In thiscase, from (2.19), (2.20), and the initial condition C ( − T ε ) = 1, the scaling C ( t ) is a decreasingfunction in time . In this direction, a first key property to prove is that C ( t ) remains far fromzero independently of ε . Moreover, a new behavior is possible if there exists some time t suchthat C ( t ) = λ . In that case, the soliton should be formally reflected by the potential.3. Study of a dynamical system revisited
This section is devoted to the study of the approximate dynamical system describing the evolu-tion of the first order scaling and translation parameters ( C ( t ) , P ( t )), inside the interaction region,in the case λ < λ <
1. This system shares many properties with the nonlinear system consideredin [34] for 0 ≤ λ ≤ λ , that is Lemma 2.6; however, the large time behavior in the case λ < λ < λ . Lemma 3.1 (Existence of dynamical parameters, case λ < λ < . Suppose m = 2 , or . Let λ , a ( · ) , p and f be as in (1.13), (1.5) and (2.19). Then thereexists ε > small such that, for all < ε < ε , the following holds. (1) Existence .There exists a unique solution ( C ( t ) , P ( t )) , with C ( t ) bounded, positive and monotonedecreasing, defined for all t ≥ − T ε , of the following nonlinear system (cid:40) C (cid:48) ( t ) = εf ( C ( t ) , P ( t )) , C ( − T ε ) = 1 ,P (cid:48) ( t ) = C ( t ) − λ, P ( − T ε ) = − (1 − λ ) T ε . (3.1) laudio Mu˜noz 15 In addition for all t ≥ − T ε one has < C ( t ) ≤ and C λ ( t )( λλ − C ( t )) − λ = ( λλ − − λ a p ( εP ( t )) a p ( − ε − / ) . (3.2) Moreover, lim t → + ∞ C ( t ) exists and satisfies lim t → + ∞ C ( t ) > µ ( λ ) > , for all λ < λ < (cf. (1.29).) (2) Asymptotic behavior .Let λ < ˜ λ < be the unique number satisfying ˜ λ ( 1 − λ ˜ λ − λ ) − λ = 2 p . (3.3) Then, (a)
For all λ < λ ≤ ˜ λ , one has lim t → + ∞ C ( t ) > λ and lim t → + ∞ P ( t ) = + ∞ . (b) For all ˜ λ < λ < , there exists a unique t ∈ ( − T ε , + ∞ ) such that C ( t ) = λ , with lim t → + ∞ C ( t ) < λ . Moreover, lim t → + ∞ P ( t ) = −∞ . Finally, one has the bound − T ε < t ≤ K ( λ ) T ε , for a positive constant K ( λ ) , independent of ε . Before the proofs, some remark are in order.
Remark λ ) . Let us say some words about where theparameter ˜ λ comes from. Indeed, since this parameter decides whether the soliton is reflected ornot, a formal necessary condition is then the existence of t ≥ − T ε such that C ( t ) = λ , for λ > ˜ λ .Let us suppose this property. Replacing in (3.2), we get λ ( 1 − λ λ − λ ) − λ = a p ( εP ( t )) a p ( − ε − / ) ;(recall that λ < λ < P ( t ). Since 1 < a ( · ) <
2, we have thatif λ ( 1 − λ λ − λ ) − λ > p a p ( − ε − / )then there is no solution for the above equation. So, since the left hand side above does not dependon ε , in order to ensure the existence of a point t , a necessary condition is that λ ( 1 − λ λ − λ ) − λ ≤ p . Finally, we define ˜ λ to be the worst possible case, such that the equality is reached in the aboveinequality. Proof of Lemma 3.1.1.
The local existence of a solution (
C, P ) of (3.1) is a direct consequence of the Cauchy-Lipschitz-Picard theorem. In addition, C ≡ , λλ are constant solutions. Since C ( − T ε ) = 1 and λ > λ , wehave C globally defined, strictly decreasing and satisfying 0 < C ( t ) < λλ for all t ≥ − T ε . Now we use (3.1)-(2.19) to obtain some a priori estimates on the solution C . Note that( C ( t ) − λ ) C ( t )( λλ − C ( t )) C (cid:48) ( t ) = − εp ( C ( t ) − λ ) a (cid:48) a ( εP ( t )) = − εpP (cid:48) ( t ) a (cid:48) a ( εP ( t )) . In particular, (1 − λ ) ∂ t log( λλ − C ( t )) + λ ∂ t log C ( t ) = p∂ t log a ( εP ( t )) . By integration on [ − T ε , t ], and by using C ( − T ε ) = 1, we obtain (3.2).Since 1 ≤ a ≤ C is bounded we have P bounded on compact sets and consequently weobtain global existence. Using C > ε small C λ ( t ) ≥ − λ λ ) − λ = ⇒ C ( t ) ≥ µ ( λ ) . (cf. (1.29)). (3.4) Moreover, lim t → + ∞ C ( t ) exists and it is always far from zero , independent of ε , as long as λ < λ <
1. This proves the first part of the Lemma. Now, given λ < λ <
1, we study the existence of a point t > − T ε such that C ( t ) = λ . Apriori, replacing this condition in (3.2), we have λ ( 1 λ − − λ = ( λλ − − λ a p ( εP ( t )) a p ( − ε − / ) . (3.5)By choosing λ := λ (1 + δ ), with δ > t does not exist if λ = λ (1 + δ ), with δ > λ ∈ ( λ ,
1) be the unique solution of (3.3). Since the function λ ∈ ( λ , (cid:55)→ f ( λ ) := λ ( 1 − λ λ − λ ) − λ ∈ (0 , + ∞ )is strictly decreasing , we have f ( λ ) ≥ p provided λ < λ ≤ ˜ λ . Therefore, from (3.5) we have2 p ≤ f ( λ ) = a p ( εP ( t )) a p ( − ε − / ) < p . In conclusion, since f ( λ ) is independent of ε , there is no t ∈ R such that C ( t ) = λ. Thus, bycontinuity we have C ( t ) > λ for all t ≥ − T ε and lim + ∞ C ( · ) ≥ λ . Moreover, if lim + ∞ C ( · ) = λ ,we have from (3.2), after passing to the limit, f ( λ ) ≤ lim sup t → + ∞ a p ( εP ( t )) a p ( − ε − / ) < p , λ ≤ ˜ λ, a contradiction. Therefore, lim + ∞ C ( · ) > λ . Moreover, from the equation for P in (3.1) one has,for all t ≥ P ( t ) = P ( − T ε ) + (cid:90) − T ε ( C ( s ) − λ ) ds + (cid:90) t ( C ( s ) − λ ) ds ≥ P ( − T ε ) + ( C (0) − λ ) t ;and thus lim t → + ∞ P ( t ) = + ∞ . Now, let us prove that for all λ ∈ (˜ λ,
1) there exists t ∈ R such that C ( t ) = λ (andtherefore lim + ∞ C ( · ) < λ .) By contradiction, let us suppose C ( t ) > λ for all t ≥ − T ε , with˜ c ∞ := lim + ∞ C ( · ) ≥ λ. First, let us suppose ˜ c ∞ > λ . Thus lim + ∞ P ( · ) = + ∞ and from (3.2) we have˜ c λ ∞ ( λλ − ˜ c ∞ ) − λ = ( λλ − − λ p a p ( − ε − / ) . (3.6)Since ˜ c ∞ > λ one has˜ c λ ∞ ( λ − λ ˜ c ∞ λ − λ ) − λ ≤ max r ∈ (0 , r λ ( λ − λ rλ − λ ) − λ = f ( λ ) < p , a contradiction with (3.6) for small ε .Now we suppose ˜ c ∞ = λ . Here we have two possibilities: either P ∞ := lim t → + ∞ P ( t ) = + ∞ ,or P ∞ < + ∞ . For the first case, by following the preceding analysis, we have˜ c λ ∞ ( λ − λ ˜ c ∞ λ − λ ) − λ = f ( λ ) < p , a contradiction with (3.6), for small ε . Otherwise, from the equation of C (cid:48) ( t ) in (3.1), one haslim t → + ∞ C (cid:48) ( t ) = lim t → + ∞ εf ( C ( t ) , P ( t )) = pελ (1 − λ ) a (cid:48) ( εP ∞ ) a ( εP ∞ ) (cid:54) = 0;for all m = 2 , t → + ∞ C (cid:48) ( t ) = lim + ∞ C ( t ) t = 0. More precisely, one has f (cid:48) ( λ ) = − (1 − λ )(1 − λ ) − λ ( λ − λ ) − λ . laudio Mu˜noz 17 In conclusion, we have that there exists at least one t > − T ε such that C ( t ) = λ . From C (cid:48) < t is unique. We finally prove some properties of P ( t ) in the case ˜ λ < λ <
1. From (3.2), one has f ( λ ) = a p ( εP ( t )) a p ( − ε − / ) . Since f ( λ ) ∈ (1 , p ) for fixed λ ∈ (˜ λ, ε , one has, for small ε , | εP ( t ) | ≤ K ( λ ); (3.7)(the constant K becomes singular as λ approaches ˜ λ or 1.) Therefore, from (3.1) one has C (cid:48) ( t ) = − εpλ ( 1 λ − a (cid:48) ( εP ( t )) a ( εP ( t )) ≤ − κ ( λ ) ε, κ ( λ ) > α > ε ), since C (cid:48)(cid:48) ( t ) = O L ∞ ( ε ), C ( t − αε ) ≥ λ + κ ( λ ) α + O ( α ) ≥ λ + 910 κ ( λ ) α. (3.8)We use this identity to obtain P ( t ) = − (1 − λ ) T ε + (cid:90) t − αε − T ε ( C ( s ) − λ ) ds + (cid:90) t t − αε ( C ( s ) − λ ) ds ≥ − (1 − λ ) T ε + 910 κ ( λ ) α ( t − αε + T ε ) − Kαε , and therefore t ≤ K ( λ ) T ε .Finally, note that P ( t ) is strictly decreasing for all t > t . Therefore, for all t ≥ t + 1 one has C ( t + 1) < λ and P ( t ) = P ( t ) + (cid:90) t +1 t ( C ( s ) − λ ) ds + (cid:90) tt +1 ( C ( s ) − λ ) ds ≤ P ( t ) + ( C ( t + 1) − λ )( t − t − t → + ∞ P ( t ) = −∞ . The proof is complete. (cid:3) Some of the properties found in the above Lemma allow to introduce the following definition.
Definition 3.1 (Escape time) . Suppose λ < λ ≤ ˜ λ . Let us define the escape time ˜ T ε > − T ε such that P ( ˜ T ε ) := − P ( − T ε ) =(1 − λ ) T ε . Otherwise, if ˜ λ < λ <
1, let us consider ˜ T ε > t such that P ( ˜ T ε ) := P ( − T ε ) = − (1 − λ ) T ε .The next result states that in the interval λ < λ < ˜ λ the soliton leaves the interaction zone bythe right hand side, with a well determined scaling c ∞ ( λ ) ∈ ( λ, K ( λ ) T ε , with K becoming unbounded as λ approaches ˜ λ . Lemma 3.2 (Asymptotic behavior, case λ < λ < ˜ λ ) . Suppose λ < λ < ˜ λ , m = 2 , or . (1) There exists a unique solution c ∞ = c ∞ ( λ ) of the following algebraic equation c λ ∞ ( λ − λ c ∞ λ − λ ) − λ = 2 p , λ < c ∞ < . (3.9) In addition, λ (cid:55)→ c ∞ ( λ ) is a strictly decreasing map with c ∞ ( λ ) = 1 and c ∞ ( λ ) > c ∞ (˜ λ ) =˜ λ . (2) Let ( C ( t ) , P ( t )) be the solution of (3.1). Then C ( ˜ T ε ) = c ∞ ( λ ) , and ˜ T ε ≤ K ( λ ) T ε , with K ( λ ) ∼ ( c ∞ ( λ ) − λ ) − .Remark . Note that the condition c ∞ > λ is essential, because there exists another minimalbranch of solutions c ∗∞ ( λ ) < λ increasing in λ with c ∗∞ ( λ ) = 0 and c ∗∞ (˜ λ ) = ˜ λ . Proof.
The proof of existence and uniqueness of a solution c ∞ ( λ ) of (3.9) is similar to Lemma 4.4in [34]. We skip the details.Let ˜ c ∞ ( λ, ε ) := lim + ∞ C . From (3.2) and lim + ∞ P = + ∞ one has˜ c λ ∞ ( λ − λ ˜ c ∞ λ − λ ) − λ = 2 p a p ( − ε − / ) , λ < ˜ c ∞ < . (3.10)Now let us define for r ∈ (0 , g ( r ) := r λ ( λ − λ rλ − λ ) − λ . Note that g ( r ) is strictly decreasing in the interval ( λ, c ∞ < ˜ c ∞ . Moreover, from the behavior of a in (1.5) we have ˜ c ∞ = c ∞ + O ( ε ), for all ε small. This implies that ˜ c ∞ ( λ, ε ) − λ > c ∞ ( λ ) − λ > , (3.11)uniformly for all ε small enough. On the other hand, at time t = ˜ T ε one has C ( ˜ T ε ) λ ( λ − λ C ( ˜ T ε ) λ − λ ) − λ = a p ( ε − / ) a p ( − ε − / ) , < C ( ˜ T ε ) < λ, therefore C ( ˜ T ε ) = c ∞ ( λ ) + O ( ε ). Moreover,(1 − λ ) T ε = P ( ˜ T ε ) = P ( − T ε ) + (cid:90) ˜ T ε − T ε ( C ( s ) − λ ) ds ≥ − (1 − λ ) T ε + (˜ c ∞ ( λ, ε ) − λ )( ˜ T ε + T ε ) . From this inequality and (3.11) we obtain, for all λ < λ < ˜ λ , the upper bound ˜ T ε ≤ K ( λ ) T ε ,with K ( λ ) ∼ ( c ∞ ( λ ) − λ ) − . Note that K ( λ ) becomes singular as λ ↑ ˜ λ . (cid:3) Remark . Note that c ∞ (˜ λ ) = ˜ λ and therefore in the last inequality above one has, for λ = ˜ λ ,(1 − ˜ λ ) T ε ≥ − (1 − ˜ λ ) T ε + (˜ c ∞ (˜ λ, ε ) − c ∞ (˜ λ ))( ˜ T ε + T ε ) . Since ˜ c ∞ (˜ λ, ε ) − c ∞ (˜ λ ) = O ( ε ) for ε small, we cannot obtain any reasonable upper bound of thetime ˜ T ε in this case. Further developments are probably necessary.Now we consider the case ˜ λ < λ <
1. Here we obtain the following striking result: the soliton isfinally reflected by the potential. The final scaling is given by a modified parameter 0 < c ∞ ( λ ) < λ ∈ (˜ λ, Lemma 3.3 (Asymptotic behavior, case ˜ λ < λ < . Suppose ˜ λ < λ < . There exists a unique solution c ∞ ( λ ) of the following algebraic equation c λ ∞ ( λ − λ c ∞ λ − λ ) − λ = 1 , < c ∞ < λ. (3.12) In addition, the map λ (cid:55)→ c ∞ ( λ ) is strictly increasing with c ∞ ( λ ) ≥ c ∞ (˜ λ ) > µ (˜ λ ) , and lim λ ↑ c ∞ ( λ ) =1 . Finally, one has C ( ˜ T ε ) = c ∞ ( λ ) , and ˜ T ε ≤ K ( λ ) T ε .Proof. The proof of existence and uniqueness of a solution c ∞ ( λ ) of (3.12) is similar to Lemma4.4 in [34]. We skip the details.Let ˜ c ∞ ( λ, ε ) := lim + ∞ C . From (3.2) and lim + ∞ P = −∞ one has˜ c λ ∞ ( λ − λ ˜ c ∞ λ − λ ) − λ = 1 a p ( − ε − / ) . with 0 < ˜ c ∞ < λ . From the behavior of a in (1.5) we have ˜ c ∞ = c ∞ ( λ ) + O ( ε ), for all ε small.This implies that λ − ˜ c ∞ ( λ, ε ) ≥ λ − c ∞ ( λ )) > , laudio Mu˜noz 19 uniformly for all ε small enough. On the other hand, at time t = ˜ T ε one has C ( ˜ T ε ) λ ( λ − λ C ( ˜ T ε ) λ − λ ) − λ = a p ( − ε (1 − λ ) T ε ) a p ( − ε − / ) = 1 , < C ( ˜ T ε ) < λ, therefore by uniqueness C ( ˜ T ε ) = c ∞ ( λ ).Finally, we prove the upper bound on ˜ T ε . We have P ( − T ε ) = − (1 − λ ) T ε = − (1 − λ ) T ε + (cid:90) ˜ T ε − T ε ( C ( s ) − λ ) ds. From here we have for β >
00 = (cid:90) t − βε − T ε ( C ( s ) − λ ) ds + (cid:90) t + βε t − βε ( C ( s ) − λ ) ds − (cid:90) ˜ T ε t + βε ( λ − C ( s )) ds ≤ (1 − λ )( t + βε + T ε ) + Kβε − (cid:90) ˜ T ε t + βε ( λ − C ( s )) ds. Similarly to estimate (3.8), one has for β > ε , C ( t + βε ) ≤ λ − ν ( λ ) β + O ( β ) , ν ( λ ) > . (3.13)Inserting this estimate above, and using the estimate on t , one has˜ T ε ≤ K ( λ ) T ε , as desired. (cid:3) Remark . In [34], from a simple study of the dynamical system in the case 0 ≤ λ ≤ λ , wefound that the soliton leaves the interaction region at time t = T ε . However, since the dynamics is repulsive in the case λ < λ <
1, the soliton takes more time to exit this region, either by the lefthand side or the right one. Fortunately, in the case of an asymptotically flat potential, the escapetime is of the same order as T ε . Therefore, in what follows, ˜ T ε will denote the correspondingescape time, for all ≤ λ < λ (cid:54) = ˜ λ . Moreover, we know that ˜ T ε ∼ T ε .4. Description of the interaction soliton-potential
This is the main section of this paper. Here we will describe in detail (see also [34] for thecase 0 ≤ λ ≤ λ ), the dynamics of the soliton-like solution, inside the interaction region, for times t ∈ [ − T ε , ˜ T ε ], and λ < λ <
1, still avoiding the more difficult case λ = ˜ λ . In order to obtainthis result, we need to construct some modulation parameters ( c ( t ) , ρ ( t )) satisfying, up to order ε / , the dynamical system given in Lemma 3.1. Since we understand the formal behavior of thenonlinear problem for ( C ( t ) , P ( t )), the rigorous description is reduced to the use of an advancedform of Weinstein functional, as in [26, 33, 34, 28] (compare with Theorem 4.1 in [34].)Let us recall that, from (2.9), and for all ε small enough, we have (cid:107) u ( − T ε ) − Q ( · + (1 − λ ) T ε ) (cid:107) H ( R ) ≤ Kε , (4.1)with u ( t ) the solution constructed in Proposition 2.3. Proposition 4.1 (Dynamics in the interaction region, case 0 ≤ λ < λ (cid:54) = ˜ λ ) . Suppose ≤ λ < , with λ (cid:54) = ˜ λ , cf. (3.3). There exists a constant ε > such that the followingholds for any < ε < ε .Let u = u ( t ) be a globally defined H solution of (1.22) satisfying (4.1). Then one has (1) Case 0 ≤ λ ≤ λ . (cf. [34]) There exist a number K > , a final scaling c ∞ ( λ ) ≥ and ρ ε ∈ R such that (cid:107) u ( T ε ) − − / ( m − Q c ∞ ( · − ρ ε ) (cid:107) H ( R ) ≤ K ε / . (4.2) In addition, lim λ ↑ λ c ∞ ( λ ) = 1 . Moreover, we have the bounds (1 − λ ) T ε ≤ ρ ε ≤ ( c ∞ ( λ ) − λ ) T ε , (4.3) valid for ε sufficiently small. (2) Case λ < λ < ˜ λ .There exists K > , a final scaling λ < c ∞ ( λ ) < and ρ ε ∈ R such that (cid:107) u ( ˜ T ε ) − − / ( m − Q c ∞ ( · − ρ ε ) (cid:107) H ( R ) ≤ K ε / . (4.4) In addition, lim λ ↓ λ c ∞ ( λ ) = 1 , lim λ ↑ ˜ λ c ∞ ( λ ) = ˜ λ . Moreover, we have the bounds ( c ∞ ( λ ) − λ ) T ε ≤ ρ ε ≤ (1 − λ ) T ε . (4.5)(3) Case ˜ λ < λ < .Now there exists a constant K > , a final scaling µ ( λ ) < c ∞ ( λ ) < λ and ˆ ρ ε ∈ R suchthat (cid:107) u ( ˜ T ε ) − Q c ∞ ( · − ˆ ρ ε ) (cid:107) H ( R ) ≤ K ε / . (4.6) In addition, lim λ ↑ c ∞ ( λ ) = 1 . Finally, we have the bounds − K ( λ ) T ε ≤ ˆ ρ ε ≤ − K ( λ ) T ε , (4.7) valid for ε sufficiently small and some K , K > .Remark . The first part of the above Proposition (namely, the case 0 ≤ λ ≤ λ ), was proven in[34]. Now we give a different proof which allows us to find, at least formally, a lower bound on thedefect of the soliton-like solution. The proof of the two cases involved in the region λ < λ < Remark . Estimate (4.7) on ˆ ρ ε shows that the soliton solution is, at time ˜ T ε ( ∼ T ε ), outside theinteraction region; moreover, it is on the left hand side. In other words, this estimate proves thatthe soliton is reflected by the potential.4.1. Construction of an approximate solution describing the interaction.
We look for˜ u ( t, x ), an approximate solution of (1.22), carrying out a specific structure. In particular, weconstruct ˜ u as a suitable modulation of the soliton Q ( x − (1 − λ ) t ), solution of the following gKdVequation: u t + ( u xx − λu + u m ) x = 0 . (4.8)Let t ∈ [ − T ε , ˜ T ε ], c = c ( t ) > ρ ( t ) ∈ R be bounded functions to be chosen later, and y := x − ρ ( t ) and R ( t, x ) := Q c ( t ) ( y )˜ a ( ερ ( t )) , (4.9)where ˜ a ( s ) := a m − ( s ) . The parameter ˜ a describes the shape variation of the soliton through theinteraction. Concerning the parameters c ( t ) and ρ ( t ), we will assume that for all t ∈ [ − T ε , ˜ T ε ], | c ( t ) − C ( t ) | + | ρ (cid:48) ( t ) − P (cid:48) ( t ) | ≤ ε / . (4.10)with ( C ( t ) , P ( t )) given from Lemmas 2.6 and 3.1. Later we will improve these constraints byconstructing parameters ( c ( t ) , ρ ( t )) with better estimates.As in [34], the form of ˜ u ( t, x ) will be the sum of the soliton plus a correction term:˜ u ( t, x ) := R ( t, x ) + w ( t, x ) , (4.11) laudio Mu˜noz 21 where w is given by w ( t, x ) := (cid:40) εd ( t ) A c ( y ) , if m = 2 , ,εd ( t ) A c ( y ) + ε B c ( t, y ) , if m = 3 , (4.12)and d ( t ) := a (cid:48) ˜ a m ( ερ ( t )) . (4.13)Here A c ( y ) and B c ( t, y ) are unknown functions. Remark . In [34] we looked for an approximate solution of the form w ( t ) = εd ( t ) A c ( y ), forall m = 2 , ε B c inthe cubic case, in order to improve the quality of our approximation. In the other cases, namely m = 2 and 4, we just need to consider a unique, special choice of A c to obtain a difference withthe dynamics of our solution from a hypothetical, completely pure soliton solution.We want to measure the size of the error produced by inserting ˜ u as defined in (4.12) in theequation (1.22). For this, let S [˜ u ]( t, x ) := ˜ u t + (˜ u xx − λ ˜ u + a ( εx )˜ u m ) x . (4.14)Our first result is the following Proposition 4.2 (Improved decomposition of S [˜ u ], see also [34]) . Suppose ( c ( t ) , ρ ( t )) satisfying (4.10). There exists γ > independent of ε small, and an ap-proximate solution ˜ u of the form (4.11)-(4.12)-(4.13), such that for all t ∈ [ − T ε , ˜ T ε ] , (1) The error term (4.14) is given by S [˜ u ]( t, x ) = ( c (cid:48) ( t ) − εf ( t ) − ε δ m, f ( t )) ∂ c ˜ u + ( ρ (cid:48) ( t ) − c ( t ) + λ − εf ( t ) − ε δ m, f ( t )) ∂ ρ ˜ u + ˜ S [˜ u ]( t, x ) , (4.15) with ∂ ρ ˜ u := ∂ ρ R − w y + O ( ε e − εγ | ρ ( t ) | A c ) , and δ m, being the Kronecker’s symbol ( δ , =1 , δ , = δ , = 0) . (2) A c , B c satisfy A c , ∂ c A c ∈ L ∞ ( R ) , A (cid:48) c ∈ Y , | A c ( y ) | ≤ Ke − γy as y → + ∞ , lim −∞ A c (cid:54) = 0 , (cid:90) R Q c ( y ) A c ( y ) = (cid:90) R yQ c ( y ) A c ( y ) = 0; (4.16) and for m = 3 , B (cid:48) c ( t, · ) ∈ L ∞ ( R ) , | B c ( t, y ) | ≤ Ke − γy e − εγ | ρ ( t ) | as y → + ∞ , | B c ( t, y ) | + | ∂ c B c ( t, y ) | ≤ K | y | e − εγ | ρ ( t ) | , as y → −∞ , (cid:90) R Q c ( y ) B c ( y ) = (cid:90) R yQ c ( y ) B c ( y ) = 0 . (4.17)(3) In addition, f ( t ) = f ( c ( t ) , ρ ( t )) is given by (2.19), f ( t ) = f ( c ( t ) , ρ ( t )) := − ξ m (cid:112) c ( t ) ( λ − λ c ( t )) a (cid:48) a ( ερ ( t )) , ξ m := (3 − m )(5 − m ) ( (cid:82) R Q ) (cid:82) R Q ; (4.18) f ( t ) = f ( c ( t ) , ρ ( t )) := ˜ ξ (cid:112) c ( t ) ( c ( t ) − λ ) a (cid:48) a ( ερ ( t )) , ˜ ξ := λ (cid:82) R Q ) (cid:82) R Q , (4.19) and f ( t ) satisfies the decomposition f ( t ) := f ( t ) a (cid:48) a ( ερ ( t )) + f ( t ) a (cid:48)(cid:48) a ( ερ ( t )) , | f i ( t ) | ≤ K. (4.20) (4) Finally, ˜ S [˜ u ]( t, · ) is a polynomially growing function as y → −∞ , and exponentially de-caying as y → + ∞ . It satisfies (cid:107) ˜ S [˜ u ]( t, · ) (cid:107) L ( y ≥− ε ) + (cid:107) ∂ x ˜ S [˜ u ]( t, · ) (cid:107) L ( y ≥− ε ) ≤ Kε / e − εγ | ρ ( t ) | + Kε . (4.21) Moreover, one has the improved estimates (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) R Q c ˜ S [˜ u ] (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) R yQ c ˜ S [˜ u ] (cid:12)(cid:12)(cid:12)(cid:12) ≤ Kε e − εγ | ρ ( t ) | + Kε , (4.22) for the quadratic and quartic cases, and (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) R Q c ˜ S [˜ u ] (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) R yQ c ˜ S [˜ u ] (cid:12)(cid:12)(cid:12)(cid:12) ≤ Kε e − εγ | ρ ( t ) | + Kε , (4.23) in the cubic case.Remark . A formal lower bound in the defect of the soliton solution can be seen as a consequenceof the fact that f ( t ) (cid:54) = 0, and f ( t ) (cid:54) = 0 for m = 3. These perturbations of the dynamical system(3.1) imply the lower bounds suggested in (1.28). That is the reason because we perform a secondorder improvement of the solution in the cubic case. Proof.
A similar proof is contained in [34]. Now we improve our result by adding the terms f ( t ) , f ( t ) and f ( t ) above, which will be of great importance to quantify the lower bound on thedefect. For the sake of clarity we include the proof in Appendix B. (cid:3) Since ˜ u (cid:54)∈ L ( R ), we need to perform a correction in our approximate solution, in order toobtain a valid L solution.4.2. Correction to the solution ˜ u . The next results are contained in [34]. However, we needsome new estimates. Consider a cutoff function η ∈ C ∞ ( R ) satisfying the following properties: (cid:40) ≤ η ( s ) ≤ , ≤ η (cid:48) ( s ) ≤ , for any s ∈ R ; η ( s ) ≡ s ≤ − , η ( s ) ≡ s ≥ . (4.24)Define η ε ( y ) := η ( εy + 2) , (4.25)and for w = w ( t, y ) the first order correction constructed in Lemma B.4, redefine ˜ u ( t, x ) := η ε ( y )˜ u ( t, x ) = η ε ( y )( R ( t, x ) + w ( t, x )) , (4.26)and similarly for R ( t ) and w ( t ). Note that, by definition,˜ u ( t, x ) = 0 for all y ≤ − ε . (4.27)The following Proposition deals with the error associated to this cut-off function, and the newapproximate solution ˜ u . Proposition 4.3 (Final approximate solution for (1.22), [34]) . There exist constants ε , γ, K > such that for all < ε < ε the following holds. (1) Consider the localized function ˜ u ( t ) = R ( t ) + w ( t ) defined in (4.25)-(4.26). Then we have (a) L -solution . For all t ∈ [ − T ε , ˜ T ε ] , w ( t, · ) ∈ H ( R ) , with (cid:107) w ( t, · ) (cid:107) H ( R ) ≤ Kε / e − γε | ρ ( t ) | . (4.28)(b) Almost orthogonality . For all t ∈ [ − T ε , ˜ T ε ] one has (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) R w ( t, x ) Q c ( y ) dx (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) R yw ( t, x ) Q c ( y ) dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ Kε . (4.29) See Step 7 in Appendix B for a precise description. laudio Mu˜noz 23 (2) Almost solution . The error associated to the new function ˜ u ( t ) satisfies S [˜ u ] = ( c (cid:48) ( t ) − εf ( t ) − ε δ m, f ( t )) ∂ c ˜ u +( ρ (cid:48) ( t ) − c ( t ) + λ − εf ( t ) − ε δ m, f ( t ))( ∂ ρ ˜ u + εη (cid:48) ε ˜ u ) + ˜ S [˜ u ]( t ) , with (cid:107) εη (cid:48) ε ˜ u (cid:107) H ( R ) ≤ Kε / e − εγ | ρ ( t ) | , and (cid:107) ˜ S [˜ u ]( t ) (cid:107) H ( R ) ≤ Kε / e − γε | ρ ( t ) | . (4.30) Finally, estimates (4.22)-(4.23) remains unchanged.Proof.
The proof of (4.28) follows from a direct computation. Indeed, (cid:107) w ( t ) η ε (cid:107) H ( R ) ≤ K (cid:107) w ( t ) (cid:107) H ( y ≥− ε ) , but from (4.16)-(4.17), (cid:107) εd ( t ) A c ( y ) + ε B c ( t, y ) (cid:107) H ( y ≥− ε ) ≤ Kε / e − εγ | ρ ( t ) | . Let us consider now (4.29). Here we have, using (4.16), (cid:90) R w ( t, x ) η ε ( y ) Q c ( y ) = (cid:90) R w ( t, x )( η ( εy + 2) − Q c ( y ) . Note that η ( εy + 2) − ≡ y ≥ − ε . Using the exponential decay of Q c ( y ), we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) R w ( t, x ) η ε ( y ) Q c ( y ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ K (cid:90) y ≤− ε ε | y | e √ cy + K (cid:90) y ∈ ( − ε , − ε ) ε | y | e − ( εy +2) e √ cy ≤ Ke − γ/ε ≤ Kε . The proofs for yA c , B c and yB c are similar. We skip the details.For the proof of (4.30), we proceed as follows. First of all, a simple computation shows that S [ η ε ˜ u ] = η ε S [˜ u ] + ( η ε ) t ˜ u + 3 εη (cid:48) ε ˜ u xx + 3 ε η (2) ε ˜ u x + ε η (3) ε ˜ u − λεη (cid:48) ε ˜ u + εη (cid:48) ε a ( εx )˜ u m . Since supp η ( k ) ε ⊆ [ − ε , − ε ] for k = 1 , εη (cid:48) ε ˜ u xx + 3 ε η (2) ε ˜ u x + ε η (3) ε ˜ u − λεη (cid:48) ε ˜ u + εη (cid:48) ε a ( εx )˜ u m == O H ( R ) ( ε / e − εγ | ρ ( t ) | ) + O H ( R ) ( ε ) . Similarly, from the definition of ρ (cid:48) ( t ) and (4.10)( η ε ) t ˜ u = − ρ (cid:48) ( t ) εη (cid:48) ε ˜ u = O H ( R ) ( ε / e − εγ | ρ ( t ) | ) + O H ( R ) ( ε ) . Collecting the above terms, we have S [ η ε ˜ u ] = η ε S [˜ u ] + O H ( R ) ( ε / e − εγ | ρ ( t ) | ) + O H ( R ) ( ε ) . Finally, from the decomposition (4.15), one has S [˜ u ] = dynamical system + ˜ S [˜ u ], with η ε ˜ S [˜ u ] = O H ( R ) ( ε / e − εγ | ρ ( t ) | + ε ) . Indeed, we have, from (4.21), (4.13) and (4.16), (cid:107) η ε ˜ S [˜ u ] (cid:107) H ( R ) ≤ Kε / e − εγ | ρ ( t ) | + Kε . Finally, one has η ε (cid:104) ( c (cid:48) ( t ) − εf ( t ) − ε δ m, f ( t )) ∂ c ˜ u + ( ρ (cid:48) ( t ) − c ( t ) + λ − εf ( t ) − ε δ m, f ( t )) ∂ ρ ˜ u (cid:105) = ( c (cid:48) ( t ) − εf ( t ) − ε δ m, f ( t )) ∂ c ( η ε ˜ u ) + ( ρ (cid:48) ( t ) − c ( t ) + λ − εf ( t ) − ε δ m, f ( t )) ∂ ρ ( η ε ˜ u )+ ε ( ρ (cid:48) ( t ) − c ( t ) + λ − εf ( t ) − ε δ m, f ( t )) η (cid:48) ε ˜ u. Since εη (cid:48) ε ˜ u = O H ( R ) ( ε / e − εγ | ρ ( t ) | ), from this last estimate, we get the final conclusion. (cid:3) Remark . Note that, even under a second order term (= ε B c ) in our approximate solution ˜ u , we have no chance of improving the associated error, and we obtain the same result as in [34],namely O ( ε / ). We believe that this phenomenon is a consequence of the fact that, since A c is not localized, we have lost most of the accuracy of ˜ u , in the standard energy space. Furtherimprovements should consider e.g. a new, more accurate description of the correction term w ( t )in H ( R ).4.3. H -estimates. In this subsection we recall some estimates concerning our approximate so-lution.
Lemma 4.4 (First estimates on ˜ u ,[34]) . (1) Decay away from zero . Suppose f = f ( y ) ∈ Y , with y = x − ρ ( t ) . Then there exist K, γ > constants such that for all t ∈ [ − T ε , ˜ T ε ] (cid:107) a (cid:48) ( εx ) f ( y ) (cid:107) H ( R ) ≤ Ke − γε | ρ ( t ) | . (4.31)(2) Almost soliton solution . The following estimates hold for all t ∈ [ − T ε , ˜ T ε ] : (cid:107) ˜ u t + ρ (cid:48) ˜ u x − c (cid:48) ∂ c ˜ u (cid:107) H ( R ) ≤ Kεe − γε | ρ ( t ) | , (4.32)˜ u xx − λ ˜ u + a ( εx )˜ u m = 1˜ a ( c − λ ) Q c + O L ( R ) ( εe − γε | ρ ( t ) | ) , (4.33) and (cid:107) (˜ u xx − c ˜ u + a ( εx )˜ u m ) x (cid:107) H ( R ) ≤ Kεe − γε | ρ ( t ) | + Kε . (4.34)In addition, we have the following result. Claim t = − T ε , [34]) . Let (
C, P ) be the unique solution of the dynamical systems (2.20) and (3.1), for any 0 ≤ λ < λ (cid:54) = ˜ λ . There exist constants K, ε > < ε < ε the approximate solution ˜ u constructed in Proposition 4.3 satisfies (cid:107) ˜ u ( − T ε , C ( − T ε ) , P ( − T ε )) − Q ( · + (1 − λ ) T ε ) (cid:107) H ( R ) ≤ Kε . (4.35)In concluding this section, we have constructed and approximate solution ˜ u ( t ) describing, atleast formally, the interaction soliton-potential. In the next section we will show that the solution u constructed in Theorem 2.3 actually behaves like ˜ u inside the interaction region [ − T ε , ˜ T ε ].4.4. Stability.
In this section our objective is to prove that the approximate solution ˜ u ( t ) de-scribes the dynamics of interaction of the solution u ( t ), inside the interval [ − T ε , ˜ T ε ]. Recall thatfrom (4.1) and (4.35), one has (cid:107) u ( − T ε ) − ˜ u ( − T ε , C ( − T ε ) , P ( − T ε ))) (cid:107) H ( R ) ≤ Kε . (4.36)In addition, from (4.30) one has (cid:107) ˜ S [˜ u ]( t ) (cid:107) H ( R ) ≤ Kε / e − γε | ρ ( t ) | , (4.37)for some K, γ >
0, and λ (cid:54) = ˜ λ . Proposition 4.5 (Exact solution close to the approximate solution ˜ u ) . Suppose λ ∈ (0 , , λ (cid:54) = ˜ λ . There exists ε > such that the following holds for any < ε < ε .There exist K > independent of ε and unique C functions c, ρ : [ − T ε , ˜ T ε ] → R such that, forall t ∈ [ − T ε , ˜ T ε ] , (cid:107) u ( t ) − ˜ u ( t, c ( t ) , ρ ( t )) (cid:107) H ( R ) ≤ K ε / , (4.38) and | ρ (cid:48) ( t ) − c ( t ) + λ − εf ( t ) − ε δ m, f ( t ) | + ε − / | c (cid:48) ( t ) − εf ( t ) − ε δ m, f ( t ) | + | c ( t ) − C ( t ) | ≤ K ε / . (4.39) laudio Mu˜noz 25 Finally, one has | c ( − T ε ) − | + | ρ ( − T ε ) + (1 − λ ) T ε | ≤ Kε , (4.40) with K > independent of K . Before the proof of this result, let us finish the proof of Proposition 4.1.4.5.
Proof of Proposition 4.1.
We are now in position to give a direct proof of Proposition 4.1.Indeed, since (4.36) is satisfied, we have (4.38) for all time t ∈ [ − T ε , ˜ T ε ]; in particular, at t = ˜ T ε one has (cid:107) u ( ˜ T ε ) − ˜ u ( ˜ T ε , c ( ˜ T ε ) , ρ ( ˜ T ε )) (cid:107) H ( R ) ≤ K ε / , with | c ( ˜ T ε ) − C ( ˜ T ε ) | ≤ K ε / . Furthermore, after integration in time of (4.39) | ρ ( ˜ T ε ) − P ( ˜ T ε ) | ≤ K ε − / − / . (4.41)Finally, from (4.11), (4.12), (4.13) and Proposition 4.2, one has (cid:107) ˜ u ( ˜ T ε , c ( ˜ T ε ) , ρ ( ˜ T ε )) − − / ( m − Q c ∞ ( λ ) ( · − ρ ( ˜ T ε )) (cid:107) H ( R ) ≤ K ε / , ≤ λ < ˜ λ, and (cid:107) ˜ u ( ˜ T ε , c ( ˜ T ε ) , ρ ( ˜ T ε )) − Q c ∞ ( λ ) ( · − ρ ( ˜ T ε )) (cid:107) H ( R ) ≤ K ε / , ≤ ˜ λ < λ < . By defining ρ ε := ρ ( ˜ T ε ), using (4.41) and using the triangle inequality, the conclusion follows,provided Proposition 4.5 holds. Remark . For the sake of clarity in the forthcoming computations, let us denote c (cid:48) := c (cid:48) − εf − ε δ m, f , and ρ (cid:48) := ρ (cid:48) − c + λ − εf − ε δ m, f . Proof of Proposition 4.5.
Let K ∗ > H ( R ) of the flow,there exists − T ε < T ∗ ≤ ˜ T ε with T ∗ := sup (cid:8) T ∈ [ − T ε , ˜ T ε ] , such that for all t ∈ [ − T ε , T ] , there exists a smooth r ( t ) ∈ R , such that (cid:107) u ( t ) − ˜ u ( · ; C ( t ) , r ( t )) (cid:107) H ( R ) ≤ K ∗ ε / (cid:9) . (4.42)The objective is to prove that T ∗ = T ε for K ∗ large enough and α > T ∗ < T ε and reaching a contradiction with the definitionof T ∗ by proving some independent estimates for (cid:107) u ( t ) − ˜ u ( · ; C ( t ) , r ( t )) (cid:107) H ( R ) . Lemma 4.6 (Modulation) . Assume < ε < ε ( K ∗ ) small enough. There exist K > and unique C functions c ( t ) , ρ ( t ) such that, for all t ∈ [ − T ε , T ∗ ] , z ( t ) = u ( t ) − ˜ u ( t, c ( t ) , ρ ( t )) satisfies (cid:90) R z ( t, x ) yQ c ( y ) dx = (cid:90) R z ( t, x ) Q c ( y ) dx = 0 . (4.43) Moreover, we have, for all t ∈ [ − T ε , T ∗ ] , (cid:107) z ( − T ε ) (cid:107) H ( R ) + | c ( − T ε ) − C ( − T ε ) | ≤ Kε / , (cid:107) z ( t ) (cid:107) H ( R ) + | c ( t ) − C ( t ) | ≤ KK ∗ ε / . (4.44) In addition, z ( t ) satisfies the following equation z t + (cid:8) z xx − λz + a ( εx )[(˜ u + z ) m − ˜ u m ] (cid:9) x + ˜ S [˜ u ] + c (cid:48) ( t ) ∂ c ˜ u + ρ (cid:48) ( t ) ∂ ρ ˜ u = 0 . (4.45) Finally, there exists γ > independent of K ∗ such that for every t ∈ [ − T ε , T ∗ ] , | ρ (cid:48) ( t ) | ≤ K (cid:104) ( m − εe − γε | ρ ( t ) | ) (cid:104) (cid:90) R z e − γ √ c | y | (cid:105) / + (cid:90) R e − γ √ c | y | z ( t ) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) R yQ c ˜ S [˜ u ] (cid:12)(cid:12)(cid:12)(cid:12) (cid:105) , (4.46) and | c (cid:48) ( t ) | ≤ K (cid:104) (cid:90) R e − γ √ c | y | z ( t ) + εe − γε | ρ ( t ) | (cid:104) (cid:90) R e − γ √ c | y | z ( t ) (cid:105) / + (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) R Q c ˜ S [˜ u ] (cid:12)(cid:12)(cid:12)(cid:12) (cid:105) . (4.47) Proof.
The proof of (4.43)-(4.44) is a standard consequence of the Implicit function theorem,applied for each time t ∈ [ − T ε , T ∗ ]. Similarly, (4.45) is a direct computation.Let us prove (4.46) and (4.47). Let us recall that f ≡ yQ c to obtain ∂ t (cid:90) R yQ c z − (cid:90) R ( yQ c ) t z − (cid:90) R ( yQ c ) x (cid:8) z xx − λz + a ( εx )[(˜ u + z ) m − ˜ u m ] (cid:9) + (cid:90) R yQ c ˜ S [˜ u ] + c (cid:48) (cid:90) R yQ c ∂ c ˜ u + ρ (cid:48) (cid:90) R yQ c ∂ ρ ˜ u = 0 . Therefore, ρ (cid:48) (cid:90) R yQ c ∂ ρ ˜ u = − (cid:90) R yQ c ˜ S [˜ u ] − c (cid:48) (cid:90) R yQ c ∂ c ˜ u + (cid:90) R ( yQ c ) y L z − ρ (cid:48) (cid:90) R ( yQ c ) y z + c (cid:48) (cid:90) R y Λ Q c z − ε ( f + εδ m, f ) (cid:90) R ( yQ c ) y z + ε ( f + εδ m, f ) (cid:90) R y Λ Q c z + (cid:90) R ( yQ c ) y a ( εx )[(˜ u + z ) m − ˜ u m − m ˜ u m − z ]+ m (cid:90) R ( yQ c ) y [ a ( εx )˜ u m − − Q m − c ] z. Note that L{ ( yQ c ) y } = − ( m − Q mc − cQ c . From here and (4.43) one has | ρ (cid:48) | ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) R yQ c ˜ S [˜ u ] (cid:12)(cid:12)(cid:12)(cid:12) + K ( m − εe − εγ | ρ ( t ) | ) (cid:2) (cid:90) R e − γ √ c | y | z (cid:3) / + K | c (cid:48) | ( εe εγ | ρ ( t ) | + (cid:107) z ( t ) (cid:107) L ( R ) ) + K (cid:90) R e − γ √ c | y | z . We consider now (4.47). We integrate (4.45) against Q c to obtain ∂ t (cid:90) R Q c z − (cid:90) R ( Q c ) t z + (cid:90) R Q c (cid:8) z xx − λz + a ( εx )[(˜ u + z ) m − ˜ u m ] (cid:9) x + (cid:90) R Q c ˜ S [˜ u ] + c (cid:48) (cid:90) R Q c ∂ c ˜ u + ρ (cid:48) (cid:90) R Q c ∂ ρ ˜ u = 0 . So we have c (cid:48) (cid:90) R Q c ∂ c ˜ u = − (cid:90) R Q c ˜ S [˜ u ] − ρ (cid:48) (cid:90) R Q c ∂ ρ ˜ u + (cid:90) R Q (cid:48) c L z − ρ (cid:48) (cid:90) R Q (cid:48) c z + c (cid:48) (cid:90) R Λ Q c z − ε ( f + εf ) (cid:90) R Q (cid:48) c z + ε ( f + εf ) (cid:90) R Λ Q c z + (cid:90) R Q (cid:48) c a ( εx )[(˜ u + z ) m − ˜ u m − m ˜ u m − z ] + m (cid:90) R Q (cid:48) c [ a ( εx )˜ u m − − Q m − c ] z. After a similar computation to the recently performed, one gets | c (cid:48) | ≤ K | ρ (cid:48) | (cid:104) (cid:90) R e − γ √ c | y | z ( t ) (cid:105) / + Kε | ρ (cid:48) | e − εγ | ρ ( t ) | + Kεe − εγ | ρ ( t ) | (cid:104) (cid:90) R e − γ √ c | y | z ( t ) (cid:105) / + (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) R Q c ˜ S [˜ u ] (cid:12)(cid:12)(cid:12)(cid:12) + K (cid:90) R e − γ √ c | y | z ≤ Kεe − εγ | ρ ( t ) | (cid:104) (cid:90) R e − γ √ c | y | z ( t ) (cid:105) / + KK ∗ ε / (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) R yQ c ˜ S [˜ u ] (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) R Q c ˜ S [˜ u ] (cid:12)(cid:12)(cid:12)(cid:12) + K (cid:90) R e − γ √ c | y | z . Using (4.22)-(4.23), we obtain the final result. The proof is complete. (cid:3) laudio Mu˜noz 27
Virial estimate.
A better understanding of the estimate on the scaling parameter (4.47) needsthe introduction of a viriel estimate, in the spirit of [34] (see Lemma 6.4). See also [13] for asimilar result, in the context of a different gKdV equation.First of all, we define some auxiliary functions. Let φ ∈ C ∞ ( R ) be an even function satisfyingthe following properties (cid:40) φ (cid:48) ≤ , + ∞ ); φ ( x ) = 1 on [0 , ,φ ( x ) = e − x on [2 , + ∞ ) and e − x ≤ φ ( x ) ≤ e − x on [0 , + ∞ ) . (4.48)Now, set ψ ( x ) := (cid:82) x φ . It is clear that ψ an odd function. Moreover, for | x | ≥ ψ (+ ∞ ) − ψ ( | x | ) = e −| x | . (4.49)Finally, for A >
0, denote ψ A ( x ) := A ( ψ (+ ∞ ) + ψ ( xA )) > e −| x | /A ≤ ψ (cid:48) A ( x ) ≤ e −| x | /A . (4.50)Note that lim x →−∞ ψ ( x ) = 0. We claim the following Lemma 4.7 (Sharp Virial-type estimate) . There exist
K, A , δ > such that for all t ∈ [ − T ε , T ∗ ] and for some γ = γ ( A ) > , ∂ t (cid:90) R z ( t, x ) ψ A ( y ) ≤ − δ (cid:90) R ( z x + z )( t, x ) e − A | y | + KA K ∗ ε / e − εγ | ρ ( t ) | . (4.51) Proof.
Let t ∈ [ − T ε , T ∗ ]. Replacing the value of z t given by (4.45), we have ∂ t (cid:90) R z ψ A ( y ) = 2 (cid:90) R zz t ψ A ( y ) − ρ (cid:48) ( t ) (cid:90) R z ψ (cid:48) A ( y )= 2 (cid:90) R ( zψ A ( y )) x ( z xx − λz + ma ( εx ) ˜ R m − z ) (4.52) − ( c − λ + εf + ε δ m, f )( t ) (cid:90) R z ψ (cid:48) A − ρ (cid:48) ( t ) (cid:90) R z∂ ρ ˜ uψ A (4.53)+2 (cid:90) R ( zψ A ( y )) x a ( εx )[(˜ u + z ) m − ˜ u m − m ˜ u m − z ] (4.54) − c (cid:48) ( t ) (cid:90) R z∂ c ˜ uψ A − ρ (cid:48) ( t ) (cid:90) R z ψ (cid:48) A (4.55)+2 m (cid:90) R z ( zψ A ( y )) x a ( εx )(˜ u m − − ˜ R m − ) − (cid:90) R zψ A ˜ S [˜ u ] . (4.56)First of all, note that | (4 . | ≤ K (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) R z x ψ A ( y ) a ( εx )[(˜ u + z ) m − ˜ u m − m ˜ u m − z ] (cid:12)(cid:12)(cid:12)(cid:12) + K (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) R ψ (cid:48) A ( y ) a ( εx ) z [(˜ u + z ) m − ˜ u m − m ˜ u m − z ] (cid:12)(cid:12)(cid:12)(cid:12) ≤ KA K ∗ ε / (cid:90) R z ( t ) e − γ √ c | y | + KK ∗ ε / (cid:90) R z ( t ) e − A | y | + K (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) R z m +1 ( ψ A ( y ) a ( εx )) x (cid:12)(cid:12)(cid:12)(cid:12) ≤ KK ∗ A ε / (cid:90) R z ( t ) e − A | y | + KA ε (cid:107) z ( t ) (cid:107) m +1 H ( R ) ≤ KK ∗ A ε / (cid:90) R z ( t ) e − A | y | + K ( K ∗ ) m +1 A ε ( m +3) / . for A large, but independent of ε . Now, by using (4.46) and (4.47) it is easy to check that for A large enough, and some constants δ , ε small, one has | (4 . | ≤ | c (cid:48) ( t ) | (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) R z∂ c ˜ uψ A (cid:12)(cid:12)(cid:12)(cid:12) + KK ∗ ε / (cid:90) R z ( t ) e − A | y | ≤ δ (cid:90) R z ( t ) e − A | y | + KK ∗ A ε / e − εγ | ρ ( t ) | . On the other hand, the terms (4.52) and (4.53) goes similarly to the terms B and B in AppendixB of [25]. Indeed, we have(4 .
52) + (4 .
53) = − (cid:90) R ψ (cid:48) A (3 z x + cz − mQ m − c z ) − m (cid:90) R ( Q m − c ) (cid:48) z ψ A + (cid:90) R z ψ (3) A − ρ (cid:48) ( t ) (cid:90) R z∂ ρ ˜ uψ A +2 m (cid:90) R ( zψ A ) x z ( a ˜ R m − − Q m − c ) − ε ( f + εδ m, f ) (cid:90) R z ψ (cid:48) A . We finally get, taking ε small, depending on A ,(4 .
52) + (4 . ≤ − δ (cid:90) R ( z x + z )( t ) e − A | y | . Finally, the term (4.56) can be estimated as follows | (4 . | ≤ K (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) R z ( zψ A ( y )) x a ( εx )(˜ u m − − ˜ R m − ) (cid:12)(cid:12)(cid:12)(cid:12) + K (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) R zψ A ˜ S [˜ u ] (cid:12)(cid:12)(cid:12)(cid:12) ≤ K (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) R z ψ (cid:48) A ( y ) a ( εx )(˜ u m − − ˜ R m − ) (cid:12)(cid:12)(cid:12)(cid:12) + K (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) R zz x ψ A ( y ) a ( εx )(˜ u m − − ˜ R m − ) (cid:12)(cid:12)(cid:12)(cid:12) + KA K ∗ ε e − εγ | ρ ( t ) | ≤ KA ε (cid:90) R ( z ( t ) + z x ( t )) e − A | y | + KA K ∗ ε / e − εγ | ρ ( t ) | + KA K ∗ ε / . Collecting these estimates, we finally get (4.51). (cid:3)
A simple but very important conclusion of the last estimate, is the following. One has, from(4.47) and (4.51), (cid:90) t − T ε | c (cid:48) ( s ) | ds ≤ KK ∗ ε, (4.57)for all t ∈ [ − T ε , T ∗ ], by taking A large enough, independent of ε and K ∗ . In other words, weimprove the estimate on the integral of | c (cid:48) ( t ) | (a crude integration of (4.47) gives (cid:82) t − T ε | c (cid:48) ( s ) ds | ≤ Kε − .)4.6. Energy functional for z . Consider the functional F defined as follows F ( t ) := 12 (cid:90) R ( z x + c ( t ) z ) − m + 1 (cid:90) R a ( εx )[(˜ u + z ) m +1 − ˜ u m +1 − ( m + 1)˜ u m z ] . (4.58)Similary to [34], and thanks to Lemma 2.2, we have the following coercivity property: thereexist K, ν >
0, independent of K ∗ and ε such that for every t ∈ [ − T ε , T ∗ ] F ( t ) ≥ ν (cid:107) z ( t ) (cid:107) H ( R ) − K ( εe − γε | ρ ( t ) | + ε ) (cid:107) z ( t ) (cid:107) L ( R ) − K (cid:107) z ( t ) (cid:107) L ( R ) . (4.59)The next step is to obtain independent estimates on ˜ F ( T ∗ ). We follow [34], but now estimate4.57 is the key element to close the argument. Lemma 4.8 (Estimates on F ( t )) . The following properties hold for any t ∈ [ − T ε , T ∗ ] . laudio Mu˜noz 29 (1) First time derivative. F (cid:48) ( t ) = − (cid:90) R z t (cid:8) z xx − cz + a ( εx )[(˜ u + z ) m − ˜ u m ] (cid:9) + 12 c (cid:48) (cid:90) R z − (cid:90) R a ( εx )˜ u t [(˜ u + z ) m − ˜ u m − m ˜ u m − z ] . (4.60)(2) Integration in time. There exist constants
K, γ > such that F ( t ) − F ( − T ε ) ≤ K ( K ∗ ) ε − + K ( K ∗ ) ε − + KK ∗ ε + K (cid:90) t − T ε εe − εγ | ρ ( s ) | (cid:107) z ( s ) (cid:107) H ( R ) ds. (4.61) Proof.
First of all, (4.60) is a simple computation. Let us consider (4.61). Replacing (4.45) in(4.60) we get F (cid:48) ( t ) = ( c ( t ) − λ ) (cid:90) R a ( εx )[(˜ u + z ) m − ˜ u m ] z x (4.62)+ ρ (cid:48) ( t ) (cid:90) R ∂ ρ ˜ u (cid:8) z xx − cz + a ( εx )[(˜ u + z ) m − ˜ u m ] (cid:9) (4.63)+ c (cid:48) ( t ) (cid:90) R ∂ c ˜ u (cid:8) z xx − cz + a ( εx )[(˜ u + z ) m − ˜ u m ] (cid:9) (4.64)+ (cid:90) R ˜ S [˜ u ] (cid:8) z xx − cz + a ( εx )[(˜ u + z ) m − ˜ u m ] (cid:9) (4.65)+ 12 c (cid:48) ( t ) (cid:90) R z + 12 ε ( f + εδ m, f )( t ) (cid:90) R z (4.66) − (cid:90) R a ( εx )˜ u t [(˜ u + z ) m − ˜ u m − m ˜ u m − z ] . (4.67)Now we consider the case m = 2, the other cases being similar (see [34] for more details.) First ofall, note that 12 εf ( t ) (cid:90) R z ≤ Kεe − εγ | ρ ( t ) | (cid:107) z ( t ) (cid:107) L ( R ) . Next, after some simplifications, we get(4 .
62) = ( c − λ ) (cid:90) R a ( εx )[2˜ uz + z ] z x = − ( c − λ ) (cid:90) R [ a ( εx )˜ u x z + εa (cid:48) ( εx )˜ uz + 13 εa (cid:48) ( εx ) z ] . From this, using (4.31),(4 .
62) + ( c − λ ) (cid:90) R a ( εx )˜ u x z ≤ Kεe − γε | ρ ( t ) | (cid:107) z ( t ) (cid:107) L ( R ) + Kε (cid:107) z ( t ) (cid:107) H ( R ) . (4.68)Now we estimate (4.63). Since ∂ ρ ˜ u = ∂ ρ R + O ( η ε w y ) + O H ( R ) ( ε / e − εγ | ρ ( t ) | ) (cf. Proposition4.3), one has (4 .
63) = ρ (cid:48) (cid:90) R ∂ ρ ˜ u (cid:8) z xx − cz + a ( εx )[2˜ uz + z ] (cid:9) = − ρ (cid:48) (cid:90) R a ( εx )˜ u x z + O ( εe − εγ | ρ ( t ) | (cid:107) z ( t ) (cid:107) L ( R ) ) . (4.69)Similarly, we have from (4.43)(4 .
64) = c (cid:48) (cid:90) R ∂ c ˜ u (cid:8) z xx − cz + a ( εx )[2˜ uz + z ] (cid:9) = c (cid:48) (cid:90) R a ( εx ) ∂ c ˜ uz + O ( εe − εγ | t | (cid:107) z ( t ) (cid:107) L ( R ) ) . (4.70) On the one hand, we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) R ˜ S [˜ u ] (cid:8) z xx − cz + a ( εx )[2˜ uz + z ] (cid:9)(cid:12)(cid:12)(cid:12)(cid:12) ≤≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) R ∂ x ˜ S [˜ u ] z x (cid:12)(cid:12)(cid:12)(cid:12) + K (1 + K ∗ ε / ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) R ˜ S [˜ u ] z (cid:12)(cid:12)(cid:12)(cid:12) + K (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) R ˜ S [˜ u ]˜ uz (cid:12)(cid:12)(cid:12)(cid:12) ≤ Kε ( e − εγ | ρ ( t ) | + ε ) (cid:107) z ( t ) (cid:107) H ( R ) + K (1 + K ∗ ε / ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) R ˜ S [˜ u ] z (cid:12)(cid:12)(cid:12)(cid:12) ≤ KK ∗ ε e − εγ | ρ ( t ) | . (4.71)Finally, (4 .
67) = − (cid:90) R a ( εx )(˜ u t + ρ (cid:48) ˜ u x − c (cid:48) ∂ c ˜ u ) z + ρ (cid:48) (cid:90) R a ( εx )˜ u x z − c (cid:48) (cid:90) R a ( εx ) ∂ c ˜ uz + O ( εe − εγ | ρ ( t ) | (cid:107) z ( t ) (cid:107) L ( R ) ) . (4.72)We get then from (4.32) and (4.68)-(4.72) F (cid:48) ( t ) ≤ c (cid:48) (cid:107) z ( t ) (cid:107) L ( R ) + Kεe − γε | ρ ( t ) | (cid:107) z ( t ) (cid:107) L ( R ) + Kε (cid:107) z ( t ) (cid:107) H ( R ) . Collecting the above estimates and (4.46), and using (4.57), after an integration, we finally get F ( t ) − F ( − T ε ) ≤ K ( K ∗ ) ε − + KK ∗ ε + K (cid:90) t − T ε εe − γε | ρ ( s ) | (cid:107) z ( s ) (cid:107) H ( R ) ds, as desired. The cases m = 3 and 4 are similar. (cid:3) We are finally in position to show that T ∗ < T ε leads to a contradiction. End of proof of Proposition 4.5
Since from Lemma 4.6, F ( − T ε ) ≤ Kε, using (4.59) andLemma (4.61) we get (cid:107) z ( t ) (cid:107) L ( R ) ≤ K (cid:104) ε + ( K ∗ ) ε − + ( K ∗ ) ε − + K ∗ ε + (cid:90) t − T ε εe − γε | ρ ( s ) | (cid:107) z ( s ) (cid:107) H ( R ) ds (cid:105) . Now, by Gronwall’s inequality (see e.g. [34] for a detailed proof), there exists a large constant
K >
0, but independent of K ∗ and ε , such that (cid:107) z ( t ) (cid:107) H ( R ) ≤ Kε + K ( K ∗ ) ε − . (4.73)Indeed, we just need to justify that (cid:12)(cid:12)(cid:12)(cid:82) t − T ε εe − γε | ρ ( s ) | ds (cid:12)(cid:12)(cid:12) ≤ K , independent of ε and K ∗ . It is clearthat this estimate holds in the case 0 ≤ λ < ˜ λ , since ρ (cid:48) ( s ) ≥ ( c ( s ) − λ ) ≥ ( c ∞ ( λ ) − λ ) > λ < λ < P (cid:48) ( t ) = 0 . To overcome this difficulty, we split theproof into three parts, arguing similarly to the proof of Lemma 3.3. First, we suppose t ≤ t − αε ,for α > ε . It is clear that (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) t − T ε εe − γε | ρ ( s ) | ds (cid:12)(cid:12)(cid:12)(cid:12) ≤ Kα , since ρ (cid:48) ( t ) ∼ c ( t ) − λ ∼ α (see (3.8).) Let us suppose t − αε ≤ t ≤ t + αε . In this case one has (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) t − T ε εe − γε | ρ ( s ) | ds (cid:12)(cid:12)(cid:12)(cid:12) ≤ Kα + Kα.
Finally, the remaining case t ≥ t + αε is similar to the first case. Since each estimate is independentof K ∗ and ε , provided ε small, we get the final conclusion.Let us come back to the main proof. From estimate (4.73), and taking ε small, and K ∗ largeenough, we obtain that for all t ∈ [ − T ε , T ∗ ], (cid:107) z ( t ) (cid:107) H ( R ) ≤
14 ( K ∗ ) ε. (4.74)Therefore, we improve the estimate on z ( t ) stated in (4.44). laudio Mu˜noz 31 Next, we prove that (cid:107) u ( T ∗ ) − ˜ u ( · ; C ( T ∗ ) , ρ ( T ∗ )) (cid:107) H ( R ) ≤ K ∗ ε / . (4.75)Indeed, expanding the definition of the energy (1.9), E a [˜ u ( · , c, ρ ) + z ]( t ) = E a [˜ u ]( t ) − (cid:90) R z (˜ u xx − λ ˜ u + a ( εx )˜ u m ) − m + 1 (cid:90) R a ( εx )[(˜ u + z ) m +1 − ˜ u m +1 − ( m + 1)˜ u m z ] . Now we use (4.33), the definition of ˜ u given in (4.11) and the orthogonality condition (4.43); weget E a [˜ u ( · , c, ρ ) + z ]( t ) = E a [ R + w ]( t ) + O ( εe − εγ | ρ ( t ) | (cid:107) z ( t ) (cid:107) H ( R ) ) + O ( (cid:107) z ( t ) (cid:107) H ( R ) )= E a [ R + w ]( t ) + O ( K ∗ ε / e − εγ | ρ ( t ) | ) + O (( K ∗ ) ε ) . On the other hand, a simple computation shows that E a [ R + w ]( t ) = E a [ R ]( t ) − (cid:90) R w ( R xx − λR + a ( ερ ) R m ) − (cid:90) R w ( a ( εx ) − a ( ερ )) R m − m + 1 (cid:90) R a ( εx )[( R + w ) m +1 − R m +1 − ( m + 1) R m w ]= E a [ R ]( t ) − a ( c − λ ) (cid:90) R wQ c + O ( ε (cid:107) w ( t ) (cid:107) H ( R ) ) + O ( (cid:107) w ( t ) (cid:107) H ( R ) )= E a [ R ]( t ) + O ( ε e − εγ | ρ ( t ) | ) + O ( ε ) . Note that in the last line we have used (4.28) and (4.29). Finally, E a [ R ]( t ) = 1˜ a ( ερ ) (cid:104) (cid:90) R Q (cid:48) c + λ (cid:90) R Q c − m + 1 (cid:90) R Q m +1 c (cid:105) − m + 1)˜ a ( ερ ) (cid:90) R (cid:104) a ( εx ) a ( ερ ) − (cid:105) Q m +1 c = 1˜ a ( ερ ) E [ Q c ] + O ( ε ) = c θ ˜ a ( ερ ) ( λ − λ c ) M [ Q ] + O ( ε ) , (for the last identity, see Lemma C.1.)Now we invoke the energy conservation law. We have, for all t ∈ [ − T ε , T ∗ ], E a [˜ u ( · , c, ρ ) + z ]( t ) = E a [˜ u ( · , c, ρ ) + z ]( − T ε ) . Therefore, c θ ( t )˜ a ( ερ ( t )) ( λ − λ c ( t )) M [ Q ] (cid:12)(cid:12)(cid:12) t − T ε = O (( K ∗ ) ε ) + O ( K ∗ ε / ) + O ( ε ) . We finally get | c ( T ∗ ) − C ( T ∗ ) | ≤ Kε / + KK ∗ ε + K ( K ∗ ) ε. Using this estimate, (4.74), and the triangle inequality, we get finally (4.75), provided K ∗ is largeenough. This estimate contradicts the definition of T ∗ given in (4.42), and concludes the proof ofProposition 4.5. (cid:3) Proof of the Main Theorems
In this small section we prove the main results, namely Theorems 1.3, 1.4, and 1.5. It turnsout that Theorems 1.3 and 1.4 are of similar structure.
Proof of Theorems 1.3 and 1.4.
Let us consider u ( t ) be the solution of (1.22) satisfying (1.15).Then, from Proposition 2.3, one has (2.9). Therefore, Proposition 4.1 implies that u ( t ) satisfieseither (4.4) , or (4.6), depending on λ ∈ ( λ , ˜ λ ) or λ ∈ (˜ λ, c ∞ ( λ ) > λ . For the proof of (1.27), we need to be careful. Indeed, from the energyconservation law, one has, for all t ≥ t , E a [ u ]( −∞ ) = E a [ Q c + ( · − ρ ( t )) + w + ( t )]In particular, from the property of asymptotic stability, and Appendix C.1 we have as t → + ∞ ( λ − λ ) M [ Q ] = ( c + ) θ ( λ − λ c + ) M [ Q ] + E + . (5.1)From this identity E + := lim t → + ∞ E a [ w + ]( t ) is well defined. Next, note that from the stabilityresult (2.11) and the Morrey embedding we have that, for any λ > E [ w + ]( t ) = 12 (cid:90) R ( w + x ) ( t ) + λ (cid:90) R ( w + ) ( t ) − m + 1 (cid:90) R a ( εx )( w + ) m +1 ( t ) ≥ (cid:90) R ( w + x ) ( t ) + λ (cid:90) R ( w + ) ( t ) − Kε ( m − / (cid:90) R a ( εx )( w + ) ( t ) ≥ ν (cid:107) w + ( t ) (cid:107) H ( R ) for some ν = ν ( λ ) >
0. Passing to the limit, we obtain lim sup t → + ∞ E [ w + ]( t ) ≤ E + .On the one hand, note that after an algebraic manipulation the equation for c ∞ in (3.12) canbe written in the following form: c θ ∞ ( λ c ∞ − λ ) M [ Q ] = ( λ − λ ) M [ Q ] . On the other hand, note that from (5.1) and the preceding inequality, we have ν lim sup t → + ∞ (cid:107) w + ( t ) (cid:107) H ( R ) ≤ ( c + ) θ ( λ c + − λ ) M [ Q ] − ( λ − λ ) M [ Q ] . Putting together both estimates, we get˜ ν lim sup t → + ∞ (cid:107) w + ( t ) (cid:107) H ( R ) ≤ ( c + ) θ +1 − c θ +1 ∞ − λλ (( c + ) θ − c θ ∞ ) , for some ˜ ν >
0. Using a similar argument as in Lemma A.3 we have˜ ν lim sup t → + ∞ (cid:107) w + ( t ) (cid:107) H ( R ) ≤ λ ( λ − c ∞ )( c θ ∞ − ( c + ) θ ) + O ( (cid:12)(cid:12) ( c + ) θ − c θ ∞ (cid:12)(cid:12) ) . From this inequality and the bound | c + − c ∞ | ≤ Kε we get (cid:0) c ∞ c + (cid:1) θ − ≥ ˜ ν lim sup t → + ∞ (cid:107) w + ( t ) (cid:107) H ( R ) , as desired. Proof of Theorem 1.5.
Since we have the validity of the stability and asymptotic stabilityproperties, from Remark 1.7 we can apply almost the same proof as in [34] to conclude Theorem1.5. Indeed, let us follow the proof of Theorem 1.3 in [34]. It is clear that the proof adapts withoutmodifications in the case λ < λ < ˜ λ , which is the case where c ∞ ( λ ) > λ . The case ˜ λ < λ < E a [ v ]( t ) + ( c ∞ ( λ ) − λ ) ˆ M [ v ]( t ) , laudio Mu˜noz 33 with ˆ M [ v ]( t ) defined in (2.2). Lemma 7.3 holds with the assumption − λ < σ < ( c ∞ ( λ ) − λ ).On the other hand, Lemma 7.4 is valid with the assumption ˜ σ > ( c ∞ ( λ ) − λ ). Finally, in theconclusion of the proof we use that c ∞ ( λ ) < λ < Appendix A. Proof of Proposition 2.5
In this section we sketch the proof of the stability and asymptotic stability result in the case ofa reflected soliton. Note that in this case we have c ∞ ( λ ) < λ . For a detailed proof concerning thecase c ∞ ( λ ) > λ , see e.g. Theorem 6.1 in [34]. Proof of the Stability result.
Let us recall that the main difference between Propositions 2.5and Theorem 6.1 in [34] is in the modified mass introduced to construct a Weinstein functional.In the former, we have worked with ˆ M [ u ]( t ) (cf. (2.2)), and now we will use M [ u ]( t ), defined in(2.18).Let us assume that for some K > t ≥ ˜ T ε , (cid:107) u ( t ) − Q c ∞ ( · − X ) (cid:107) H ( R ) ≤ Kε / . (A.1)From the local and global Cauchy theory exposed in Proposition 2.1, we know that the solution u is well defined for all t ≥ t .Let D > K be a large number to be chosen later, and set T ∗ := sup (cid:110) t ≥ t | ∀ t (cid:48) ∈ [ t , t ) , ∃ ˜ ρ ( t (cid:48) ) ∈ R smooth, such that | ˜ ρ (cid:48) ( t (cid:48) ) − c ∞ + λ | ≤ , | ˜ ρ ( t ) − X | ≤ , and (cid:107) u ( t (cid:48) ) − Q c ∞ ( · − ˜ ρ ( t (cid:48) )) (cid:107) H ( R ) ≤ D ε / (cid:111) . (A.2)Observe that T ∗ > t is well-defined since D > K , (A.1) and the continuity of t (cid:55)→ u ( t ) in H ( R ).The objective is to prove T ∗ = + ∞ , and thus (2.11). Therefore, for the sake of contradiction, inwhat follows we shall suppose T ∗ < + ∞ .The first step to reach a contradiction is to decompose the solution u ( t ) in two parts: solitonplus an error term, on the interval [ t , T ∗ ], using standard modulation theory around the soliton.In particular, we will find a special ρ ( t ) satisfying the hypotheses in (A.2), but withsup t ∈ [ t ,T ∗ ] (cid:107) u ( t ) − Q c ∞ ( · − ρ ( t )) (cid:107) H ( R ) ≤ D ε / , (A.3)a contradiction with the definition of T ∗ . Lemma A.1 (Modulated decomposition) . For ε > small enough, independent of T ∗ , there exist C functions ρ , c , defined on [ t , T ∗ ] ,with c ( t ) > and such that the function z ( t ) given by z ( t, x ) := u ( t, x ) − R ( t, x ) , (A.4) where R ( t, x ) := Q c ( t ) ( x − ρ ( t )) , satisfies for all t ∈ [ t , T ∗ ] , (cid:90) R R ( t, x ) z ( t, x ) dx = (cid:90) R ( x − ρ ( t )) R ( t, x ) z ( t, x ) dx = 0 , (A.5) (cid:107) z ( t ) (cid:107) H ( R ) + | c ( t ) − c ∞ | ≤ KD ε / , and (A.6) (cid:107) z ( t ) (cid:107) H ( R ) + | ρ ( t ) − X | + | c ( t ) − c ∞ | ≤ Kε / , (A.7) where K is not depending on D . In addition, z ( t ) now satisfies the following modified gKdVequation z t + (cid:8) z xx − λz + a ( εx )[( R + z ) m − R m ] + ( a ( εx ) − Q mc (cid:9) x + c (cid:48) ( t )Λ Q c + ( c − λ − ρ (cid:48) )( t ) Q (cid:48) c = 0 . (A.8) Furthermore, for some constant γ > independent of ε , we have the improved estimates: | ρ (cid:48) ( t ) + λ − c ( t ) | ≤ K ( m − (cid:104) (cid:90) R e − γ | x − ρ ( t ) | z ( t, x ) dx (cid:105) + K (cid:90) R e − γ | x − ρ ( t ) | z ( t, x ) dx + Ke − γεt ; (A.9) and | c (cid:48) ( t ) | c ( t ) ≤ K (cid:90) R e − γ | x − ρ ( t ) | z ( t, x ) dx + Ke − γεt (cid:107) z ( t ) (cid:107) H ( R ) + Kεe − εγt . (A.10) Remark
A.1 . Note that from (A.6) and taking ε small enough we have an improved the bound on ρ ( t ). Indeed, for all t ∈ [ t , T ∗ ], | ρ (cid:48) ( t ) − c ∞ + λ | + | ρ ( t ) − X | ≤ D ε / . Thus, in order to reach a contradiction, we only need to show (A.3). Note that for any t ≥ t , ρ ( t ) ≤
110 ( c ∞ ( λ ) − λ ) t . (A.11)This inequality implies that the soliton position is far away from the potential interaction region. Proof of Lemma A.1.
See [34]. (cid:3)
Almost conserved quantities and monotonicity
By using the decomposition proved in Lemma A.1, we have the following mass and energymonotonicity.
Lemma A.2 (Monotonicity of mass backwards in time, see Lemma 7.1 in [34]) . Suppose < λ < . Consider the mass M [ u ]( t ) introduced in (2.18). Then there exists ε > such that for all < ε < ε one has, M [ u ]( t (cid:48) ) − M [ u ]( t ) ≥ − Ke − εγt , (A.12) that for all t, t (cid:48) ≥ t , with t (cid:48) ≥ t .Remark A.2 . Note that the above identity is valid only in the case λ >
0, and it is a consequenceof (1.6) and the following identity ∂ t (cid:90) R u a ( εx ) = 2 ε (cid:90) R a (cid:48) a ( εx ) u x + ε (cid:90) R u (cid:2) λ a (cid:48) a ( εx ) − ε ( a (cid:48) a ) (cid:48)(cid:48) ( εx ) (cid:3) − ε (cid:90) R a (cid:48) a ( εx ) u m +1 . Lemma A.3 (Almost conservation of modified mass and energy) . Consider M = M [ R ] and E a = E a [ R ] the modified mass and energy of the soliton R (cf.(A.4)). Then for all t ∈ [ t , T ∗ ] we have M [ R ]( t ) = 12 c θ ( t ) (cid:90) R Q + O ( e − εγt ); (A.13) E a [ R ]( t ) = 12 c θ ( t )( λ − λ c ( t )) (cid:90) R Q + O ( e − εγt ) . (A.14) Furthermore, we have the bound | E a [ R ]( t ) − E a [ R ]( t ) + ( c ( t ) − λ )( M [ R ]( t ) − M [ R ]( t )) |≤ K (cid:12)(cid:12)(cid:12)(cid:12)(cid:104) c ( t ) c ( t ) (cid:105) θ − (cid:12)(cid:12)(cid:12)(cid:12) + Ke − εγt . (A.15) Proof.
We start by showing the first identity, namely (A.13). First of all, note that from (2.2), M [ R ]( t ) = 12 (cid:90) R a R = 12 c θ ( t ) (cid:90) R Q + 12 (cid:90) R ( 1 a ( εx ) − R . laudio Mu˜noz 35 From (A.11), (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) R ( 1 a ( εx ) − R (cid:12)(cid:12)(cid:12)(cid:12) ≤ Ke − γεt , (A.16)for some constants K, γ >
0. Now we consider (A.14). Here we have E a [ R ]( t ) = 12 (cid:90) R R x + λ (cid:90) R R − m + 1 (cid:90) R a ( εx ) R m +1 = c θ ( t ) (cid:104) c ( t )( 12 (cid:90) R Q (cid:48) − m + 1 (cid:90) R Q m +1 ) + λ (cid:90) R Q (cid:105) + 1 m + 1 (cid:90) R (1 − a ( εx )) R m +1 . Similarly to a recent computation, we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) R (1 − a ( εx )) R m +1 (cid:12)(cid:12)(cid:12)(cid:12) ≤ Ke − γεt , for some constants K, γ >
0. On the other hand, from Appendix C we have that (cid:82) R Q (cid:48) − m +1 (cid:82) R Q m +1 = − λ (cid:82) R Q , λ = − mm +3 , and thus E a [ R ]( t ) = 12 c θ ( t )( λ − λ c ( t )) (cid:90) R Q + O ( e − γεt ) . Adding both identities we have E a [ R ]( t ) + ( c ( t ) − λ ) M [ R ]( t ) = c θ ( t )( c ( t ) − λ c ( t )) M [ Q ] + O ( e − εγt ) . In particular, E a [ R ]( t ) − E a [ R ]( t ) + ( c ( t ) − λ )( M [ R ]( t ) − M [ R ]( t )) == λ M [ Q ] (cid:104) c θ +12 ( t ) − c θ +12 ( t ) − c ( t ) λ [ c θ ( t ) − c θ ( t )] (cid:105) + O ( e − εγt ) . To obtain the last estimate (A.15) we perform a Taylor development up to the second order(around y = y ) of the function g ( y ) := y θ +12 θ ; and where y := c θ ( t ) and y := c θ ( t ). Note that θ +12 θ = λ and y / θ = c ( t ). The conclusion follows at once. (cid:3) Now our objective is to estimate the quadratic term involved in (A.15). Following [30], weshould use a “mass conservation” identity. However, since the mass is not conserved, we need tocombine (2.4)-(2.18) in order to obtain the desired estimate.
Lemma A.4 (Quadratic control on the variation of c ( t )) . | E a [ R ]( t ) − E a [ R ]( t ) + ( c ( t ) − λ )( M [ R ]( t ) − M [ R ]( t )) |≤ K (cid:107) z ( t ) (cid:107) H ( R ) + K (cid:107) z ( t ) (cid:107) H ( R ) + Ke − εγt . (A.17) Proof.
From (2.4) and (A.5) we have for all t ∈ [ t , T ∗ ],ˆ M [ R ]( t ) − ˆ M [ R ]( t ) ≤ ˆ M [ z ]( t ) − ˆ M [ z ]( t ) + Ke − εγt ( (cid:107) z ( t ) (cid:107) L ( R ) + (cid:107) z ( t ) (cid:107) L ( R ) ) , namely c θ ( t ) − c θ ( t ) ≤ K (cid:107) z ( t ) (cid:107) L + K (cid:107) z ( t ) (cid:107) L + K (1 + D ε / ) e − εγt . On the other hand, from (A.12) one has c θ ( t ) − c θ ( t ) ≥ − Ke − εγt (1 + D ε / ) − K (cid:107) z ( t ) (cid:107) L − K (cid:107) z ( t ) (cid:107) L . Combining both inequalities, we obtain (cid:12)(cid:12)(cid:12)(cid:12)(cid:104) c ( t ) c ( t ) (cid:105) θ − (cid:12)(cid:12)(cid:12)(cid:12) ≤ K (cid:107) z ( t ) (cid:107) L ( R ) + K (cid:107) z ( t ) (cid:107) L ( R ) + K (1 + D ε / ) e − γεt . Plugin this estimate in (A.15) and taking ε even smaller, we get the conclusion. (cid:3) A.0.1.
Energy estimates.
Let us now introduce the second order functional F ( t ) := 12 (cid:90) R (cid:8) z x + [ λ + ( c ( t ) − λ ) 1 a ( εx ) ] z (cid:9) − m + 1 (cid:90) R a ( εx )[( R + z ) m +1 − R m +1 − ( m + 1) R m z ] . This functional, related to the Weinstein functional, have the following properties.
Lemma A.5 (Energy expansion) . Consider E a [ u ] and M [ u ] the energy and mass defined in (1.9)-(2.18). Then we have for all t ∈ [ t , T ∗ ] , E a [ u ]( t ) + ( c ( t ) − λ ) M [ u ]( t ) = E a [ R ] + ( c ( t ) − λ ) M [ R ] + F ( t ) + O ( e − γεt (cid:107) z ( t ) (cid:107) H ( R ) ) . Proof.
Using the orthogonality condition (A.5), we have E a [ u ]( t ) = E a [ R ] − (cid:90) R z ( a ( εx ) − R m + 12 (cid:90) R z x + λ (cid:90) R z − m + 1 (cid:90) R a ( εx )[( R + z ) m +1 − R m +1 − ( m + 1) R m z ] . Moreover, following (A.16), we easily get (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) R z ( a ( εx ) − R m (cid:12)(cid:12)(cid:12)(cid:12) ≤ Ke − γεt (cid:107) z ( t ) (cid:107) H ( R ) . Similarly, M [ u ]( t ) = M [ R ] + M [ z ] + (cid:90) R ( 1 a ( εx ) − Rz = M [ R ] + M [ z ] + O ( e − εγt (cid:107) z ( t ) (cid:107) H ( R ) ) . Collecting the above estimates, we have E a [ u ]( t ) + ( c ( t ) − λ ) M [ u ]( t ) = E a [ R ] + ( c ( t ) − λ ) M [ R ] + 12 (cid:90) R (cid:110) z x + [ ( c ( t ) − λ ) a ( εx ) + λ ] z (cid:111) − (cid:90) R a ( εx ) m + 1 [( R + z ) m +1 − R m +1 − ( m + 1) R m z ] + O ( e − γεt (cid:107) z ( t ) (cid:107) H ( R ) ) . This concludes the proof. (cid:3)
Lemma A.6 (Modified coercivity for F ) . There exists ε > such that for all < ε < ε the following hold. There exist K, ν > ,independent of K ∗ such that for every t ∈ [ t , T ∗ ] F ( t ) ≥ ν (cid:107) z ( t ) (cid:107) H ( R ) − Kεe − γεt (cid:107) z ( t ) (cid:107) L ( R ) + O ( (cid:107) z ( t ) (cid:107) L ( R ) ) . (A.18) Proof.
First of all, note that F ( t ) = 12 (cid:90) R (cid:110) z x + [ ( c ( t ) − λ ) a ( εx ) + λ ] z − mQ m − c z (cid:111) + O ( (cid:107) z ( t ) (cid:107) H ( R ) ) + O ( e − γεt (cid:107) z ( t ) (cid:107) H ( R ) ) . Since ( c ( t ) − λ ) a ( εx ) + λ ≥ c ( t ) for all x ∈ R , we have F ( t ) = 12 (cid:90) R ( z x + c ( t ) z − mQ m − c z ) + O ( (cid:107) z ( t ) (cid:107) H ( R ) ) + O ( e − γεt (cid:107) z ( t ) (cid:107) H ( R ) ) . From Lemma 2.2 and (A.5)-(A.6) we finally obtain (A.18). (cid:3) laudio Mu˜noz 37
A.0.2.
Conclusion of the proof.
Now we prove that our assumption T ∗ < + ∞ leads inevitablyto a contradiction. Indeed, from Lemmas A.5 and A.6, we have for all t ∈ [ t , T ∗ ] and for someconstant K > , K (cid:107) z ( t ) (cid:107) H ( R ) ≤ E a [ u ]( t ) − E a [ u ]( t ) + ( c ( t ) − λ )[ M [ u ]( t ) − M [ u ]( t )]+ E a [ R ]( t ) − E a [ R ]( t ) + ( c ( t ) − λ )[ M [ R ]( t ) − M [ R ]( t )]+ K F ( t ) + Kε sup t ∈ [ t ,T ∗ ] e − γεt (cid:107) z ( t ) (cid:107) L ( R ) + K sup t ∈ [ t ,T ∗ ] (cid:107) z ( t ) (cid:107) L ( R ) . From Lemmas A.1 and A.3, Corollary A.4 and the energy conservation we have (cid:107) z ( t ) (cid:107) H ( R ) ≤ Kε + ( c ( t ) − λ )[ M [ u ]( t ) − M [ u ]( t )]+ K sup t ∈ [ t ,T ∗ ] (cid:107) z ( t ) (cid:107) H ( R ) + Ke − εγt (1 + D ε / ) + KD ε / . Finally, from (2.2) we have M [ u ]( t ) − M [ u ]( t ) ≥ − Ke − γεt . Collecting the preceding estimateswe have for ε > D = D ( K ) large enough (cid:107) z ( t ) (cid:107) H ( R ) ≤ D ε, which contradicts the definition of T ∗ . The conclusion is thatsup t ≥ t (cid:107) u ( t ) − Q c ( t ) ( · − ρ ( t )) (cid:107) H ( R ) ≤ Kε / . Using (A.6), we finally get (2.11). This finishes the proof.
Proof of the asymptotic stability result.
In this paragraph we sketch the proof of asymptoticstability property in the case c ∞ ( λ ) < λ , namely ˜ λ < λ <
1, which is the case of the reflectedsolitary wave. A detailed proof for the case 0 < λ < λ can be found in [34], which adapts withoutmodifications to the case λ < λ < ˜ λ .Let us consider the remaining case, ˜ λ < λ <
1. We continue with the notation introduced inthe proof of the stability property (2.15). From the above mentioned stability result, it is easyto check that the decomposition (A.4) showed in Lemma A.1 and all its conclusions hold for alltime t ≥ t .Consider − λ < β < ( c ∞ ( λ ) − λ ), and let us follow the proof described in [34]. First of all,the Virial estimate (cf. Lemma 6.4 in [34]) holds with no important modifications.Second, Lemma 6.8, about monotonicity for mass and energy, needs some modifications. Indeed,for x > t, t ≥ t , and ˜ y ( x ) := x − ( ρ ( t ) + σ ( t − t ) + x ), the modifiedquantities I x ,t ( t ) := (cid:90) R a /m ( εx ) u ( t, x ) φ (˜ y ( x )) dx, ˜ I x ,t ( t ) := (cid:90) R a /m ( εx ) u ( t, x ) φ (˜ y ( − x )) dx, (A.19)and J x ,t ( t ) := (cid:90) R [ u x + a /m ( εx ) u − a ( εx ) m + 1 u m +1 ]( t, x ) φ (˜ y ( x )) dx, with φ ( x ) := π arctan( e x/K ) . Here σ ∈ ( − λ, ( c ∞ ( λ ) − λ )) is a fixed quantity, to be chosen later. First of all, note that the equivalent of estimate (6.32) is a consequence of the following in-equality, valid for K > ε small enough:12 ∂ t (cid:90) R a /m ( εx ) φ (˜ y ( x )) u = − (cid:90) R a /m φ (cid:48) u x + mm + 1 (cid:90) R a /m +1 ( εx ) φ (cid:48) u m +1 + 12 (cid:90) R u a /m ( εx ) (cid:2) − ( σ + λ ) φ (cid:48) + φ (3) (cid:3) − ε (cid:90) R ( a /m ) (cid:48) ( εx ) φu x − ε (cid:90) R u [ λ ( a /m ) (cid:48) − ε ( a /m ) (3) ]( εx ) φ + 32 ε (cid:90) R u (cid:2) ε ( a /m ) (2) ( εx ) φ (cid:48) + ( a /m ) (cid:48) ( εx ) φ (cid:48)(cid:48) (cid:3) . (A.20)In this last computation we have six terms. Let us see each one in detail. In what follows we usethe decomposition (A.4). First of all, one has (cid:90) R φ (cid:48) a /m u x = (cid:90) R φ (cid:48) a /m ( R x + 2 R x z x + z x ) . Recall that R ( t ) is exponentially decreasing in x − ρ ( t ). On the other hand, φ (cid:48) (˜ y ) is exponentiallydecreasing away from zero. Therefore, one has, for K large, (cid:90) R a /m φ (cid:48) u x = (cid:90) R a /m φ (cid:48) z x + O ( e − x /K e − ( t − t ) /K ) . Similarly, (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) R a /m +1 φ (cid:48) u m +1 (cid:12)(cid:12)(cid:12)(cid:12) ≤ Ke − ( t − t ) /K e − x /K + Kε ( m − / (cid:90) R a /m φ (cid:48) z . On the other hand, since σ + λ > , (cid:90) R a /m u (cid:2) − ( σ + λ ) φ (cid:48) + φ (3) (cid:3) = −
12 ( σ + λ ) (cid:90) R a /m φ (cid:48) z + O ( e − x /K e − ( t − t ) /K ) , and − ε (cid:90) R ( a /m ) (cid:48) ( εx ) φu x − ε (cid:90) R u [ λ ( a /m ) (cid:48) − ε ( a /m ) (3) ]( εx ) φ ≤ , provided ε is small. Finally, (cid:12)(cid:12)(cid:12)(cid:12) ε (cid:90) R u (cid:2) ε ( a /m ) (2) ( εx ) φ (cid:48) + ( a /m ) (cid:48) ( εx ) φ (cid:48)(cid:48) (cid:3)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Kεe − ( t − t ) /K e − x /K + Kε (cid:90) R a /m z φ (cid:48) . After these estimates, it is easy to see that12 ∂ t (cid:90) R a /m ( εx ) φ ( y ) u ≤ Ke − ( t − t ) /K e − x /K . The conclusion follows after integration in time: one has, for all 0 < ε < ε and for all t, t ≥ t with t ≥ t , I x ,t ( t ) − I x ,t ( t ) ≤ Ke − x /K . (A.21)This estimate is an improved version of (6.32) in [34]. On the other hand, to obtain (6.33), weperform a similar computation. Therefore, if t ≥ t , one has˜ I x ,t ( t ) − ˜ I x ,t ( t ) ≤ Ke − x /K . (A.22)Finally if t ≥ t , after a similar computation as performed in [34], J x ,t ( t ) − J x ,t ( t ) ≤ Ke − x /K . (A.23)From these estimates, the Virial identity and the decomposition above mentioned, one has (6.35)-(6.39). The rest of the proof is direct, and no deep modifications are needed. The proof iscomplete. laudio Mu˜noz 39 Appendix B. Proof of Proposition 4.2
This section is an improvement of the Appendix A in [34]. Now we suppose that the parameters( c ( t ) , ρ ( t )) are not fixed, but satisfy (4.10). Step 0. Preliminaries.
From (4.14), we easily have that S [˜ u ] = I + II + III , (B.1)where (we omit the dependence on t, x ) I := S [ R ] , II = II ( w ) := w t + ( w xx − λw + m a ( εx ) R m − w ) x , (B.2)and for m = 2 , III := (cid:8) a ( εx )[( R + w ) m − R m − mR m − w ] (cid:9) x . (B.3)Recall that w is given by (4.12). Since w varies, depending on m = 3 or m (cid:54) = 3, we have toconsider two different cases in our computations.In the next results, we expand the terms in (B.1). Note that ˜ a = a m − , and R ( t, x ) = Q c ( t ) ( y )˜ a ( ερ ( t )) , y = x − ρ ( t ) . Step 1. Computation of I.Lemma B.1. (1)
Suppose m = 2 or . One has I = F I ( t, y ) + εF I ( t, y ) + ε F I c ( t, y ) , (B.4) where F I ( t, y ) := ( c (cid:48) ( t ) − εf ( t )) ∂ c R ( t ) + ( ρ (cid:48) ( t ) − c ( t ) + λ − εf ( t )) ∂ ρ R ( t ) , (B.5) f ( t ) and f ( t ) are given by (2.19)-(4.18), and F I ( t ; y ) := f ( t ) Λ Q c ( y )˜ a ( ερ ( t )) − ˜ a (cid:48) ˜ a ( ερ ( t ))( c ( t ) − λ ) Q c ( y )+ a (cid:48) ˜ a m ( ερ ( t ))( yQ mc ( y )) y − f ( t ) Q (cid:48) c ( y )˜ a ( ερ ( t )) . (B.6) Finally, for all t ∈ [ − T ε , ˜ T ε ] , one has (cid:107) F I c ( t, · ) (cid:107) H ( R ) ≤ K ( e − εγ | ρ ( t ) | + ε ) . (2) Suppose now m = 3 . Then one has I = F I ( t, y ) + εF I ( t, y ) + ε F I ( t, y ) + ε F I c ( t, y ) , (B.7) with F I given by F I ( t, y ) := ( c (cid:48) ( t ) − εf ( t ) − ε f ( t )) ∂ c R ( t ) + ( ρ (cid:48) ( t ) − c ( t ) + λ − ε f ( t )) ∂ ρ R ( t ) , (B.8) and f ( t ) , f ( t ) and f ( t ) given by (2.19), (4.19) and (4.20) respectively. In addition, F I is given by (B.6) (with f ≡ ), and F I ( t, y ) := f ( t ) a / ( ερ ( t )) Λ Q c ( y ) − f ( t ) a / ( ερ ( t )) Q (cid:48) c ( y ) + a (cid:48)(cid:48) a / ( ερ ( t ))( y Q c ( y )) y . (B.9) Finally, for all t ∈ [ − T ε , ˜ T ε ] , one has (cid:107) F I c ( t, · ) (cid:107) H ( R ) ≤ K ( e − εγ | ρ ( t ) | + ε ) . Proof of Lemma B.1.
We compute (from now on, and for the sake of simplicity, we avoid the explicit dependence intime t and space y in the computations): I = R t + ( R xx − λR + a ( εx ) R m ) x = c (cid:48) ˜ a Λ Q c − ρ (cid:48) ˜ a Q (cid:48) c − ε ˜ a (cid:48) ρ (cid:48) ˜ a Q c + 1˜ a Q (3) c − λ ˜ a Q (cid:48) c + 1˜ a m ( a ( εx ) Q mc ) x . Note that via a Taylor expansion,( a ( εx ) Q mc ) x = a ( ερ )( Q mc ) y + εa (cid:48) ( ερ )( yQ mc ) y + 12 ε a (cid:48)(cid:48) ( ερ )( y Q mc ) y + O H ( R ) ( ε ) . Therefore, using the equation satisfied by Q c , namely, Q (cid:48)(cid:48) c − cQ c + Q mc = 0, one has I = c (cid:48) ˜ a Λ Q c − ρ (cid:48) ˜ a Q (cid:48) c − εm − a (cid:48) ρ (cid:48) ˜ a m Q c + 1˜ a Q (3) c − λ ˜ a Q (cid:48) c + 1˜ a ( Q mc ) (cid:48) + εa (cid:48) ˜ a m ( yQ mc ) y + ε a (cid:48)(cid:48) a m ( y Q mc ) x + O H ( R ) ( ε )= 1˜ a ( Q (cid:48)(cid:48) c − cQ c + Q mc ) (cid:48) + c (cid:48) ˜ a Λ Q c − ( ρ (cid:48) − c + λ ) Q (cid:48) c ˜ a − ε ˜ a (cid:48) ˜ a ( ρ (cid:48) − c + λ ) Q c − ε ˜ a (cid:48) ˜ a ( c − λ ) Q c + εa (cid:48) ˜ a m ( yQ mc ) y + ε a (cid:48)(cid:48) a m ( y Q mc ) y + O H ( R ) ( ε )= ( c (cid:48) − εf − δ m, ε f ) Λ Q c ˜ a − ( ρ (cid:48) − c + λ − εf − δ m, ε f )( Q (cid:48) c ˜ a + ε ˜ a (cid:48) ˜ a Q c )+ ε (cid:2) f ˜ a Λ Q c − ˜ a (cid:48) ˜ a ( c − λ ) Q c + a (cid:48) ˜ a m ( yQ mc ) y − f ˜ a Q (cid:48) c (cid:3) + ε F I ( t, y ) + O H ( ε e − εγ | ρ ( t ) | + ε ) , with F I given by (B.9), and δ m, the Kronecker delta symbol. Moreover F I ( t, y ) ∈ Y for all t ∈ [ − T ε , ˜ T ε ] and (cid:107) F I ( t, · ) (cid:107) H ( R ) ≤ Ke − εγ | ρ ( t ) | . From the last identity above, we define F I and F I as above mentioned (cf. (B.5)-(B.8)-(B.6).)Moreover, depending on the value of m , we define F I c as the rest term of quadratic or cubic orderin ε . Indeed, for m = 3 we have F I c ( t, · ) = O H ( ε e − εγ | ρ ( t ) | + ε ), and for m = 2 or 4, we have F I c = ε F I ( t, y ) + O H ( ε e − εγ | ρ ( t ) | + ε ). In both cases, the corresponding estimates, and thedecompositions (B.4)-(B.7) are straightforward. The proof is complete. (cid:3) Step 2. Computation of II.Lemma B.2 (Decomposition of II ) . Suppose that ( A c , B c ) satisfy (4.16)-(4.17). Let w given by (4.12). The following expansionshold: (1) Case m = 2 , . We have II = ( c (cid:48) − εf ) ∂ c w − ( ρ (cid:48) − c + λ − εf ) w y − ( L w ) y + ε (cid:2) ε d (cid:48) A c + f d ∂ c A c (cid:3) + ε F II c ( t ; y ) , with F II c ( t ; · ) ∈ Y , uniformly in time. In addition, (cid:107) F II c ( t ; y ) (cid:107) H ( R ) ≤ Ke − γε | ρ ( t ) | . We assume these properties in order to simplify the computations. Later, we will prove that this is indeed thecase. laudio Mu˜noz 41 (2) Case m = 3. Here one has II = ( c (cid:48) − εf − ε f ) ∂ c w − ( ρ (cid:48) − c + λ − ε f ) w y − ( L w ) y + ε (cid:2) ε d (cid:48) A c + f d ∂ c A c + 3 d a (cid:48) ( ερ ) a ( ερ ) ( yQ c A c ) y (cid:3) + ε (cid:2) d f ∂ c A c + f ∂ c B c − f ( B c ) y (cid:3) + ε f ∂ c B c + ε F II c ( t ; y ) , with F II c ( t ; · ) ∈ Y , uniformly in time. In addition, (cid:107) F II c ( t ; y ) (cid:107) H ( R ) ≤ Ke − γε | ρ ( t ) | . Proof.
Let D := D c ( t, y ), y = x − ρ ( t ), be a general, smooth function. We compute II ( D ) := D t + ( D xx − λD + m a ( εx ) R m − D ) x . We have II ( D ) = c (cid:48) ( t ) ∂ c D + D t − ρ (cid:48) ( t ) D y + (cid:2) D yy − λD + a ( εx ) a ( ερ ) mQ m − c D (cid:3) x = D t − ( L D ) y + ( c (cid:48) ( t ) − εf ( t ) − ε δ m, f ( t )) ∂ c D − ( ρ (cid:48) ( t ) − c ( t ) + λ − εf ( t ) − ε δ m, f ( t )) D y + mε a (cid:48) ( ερ ) a ( ερ ) ( yQ m − c D ) y + O (( ε y Q m − c D ) y )+ ε ( f ( t ) + εδ m, f ( t )) ∂ c D − ε ( f ( t ) + εδ m, f ( t )) D y . We apply this identity to the functions w = εd ( t ) A c ( y ) (case m = 2 ,
4) and w = εd ( t ) A c ( y ) + ε B c ( t, y ) (case m = 3). We first deal with the cases m = 2 or 4. We have II ( w ) = εd (cid:48) ( t ) A c − εd ( t )( L A c ) (cid:48) + ( c (cid:48) ( t ) − εf ( t )) εd ( t ) ∂ c A c − ( ρ (cid:48) ( t ) − c ( t ) + λ − εf ( t )) εd ( t ) A (cid:48) c + ε d ( t ) f ( t ) ∂ c A c + O H ( R ) ( ε e − εγ | ρ ( t ) | ) . (Recall that A (cid:48) c ∈ Y .) This proves the first part of Lemma B.2.We treat now the cubic case, m = 3. Here we have f ( t ) ≡ A (cid:48) c ∈ Y and II ( w ) = εd (cid:48) ( t ) A c + ε ( B c ) t − ( L w ) y + ( c (cid:48) ( t ) − εf ( t ) − ε f ( t )) ∂ c w − ( ρ (cid:48) ( t ) − c ( t ) + λ − ε f ( t )) w y + 3 ε a (cid:48) ( ερ ) a ( ερ ) ( yQ c w ) y + O (( ε y Q c w ) y ) + ε ( f ( t ) + εf ( t )) ∂ c w − ε f ( t ) w y = − ( L w ) y − ( ρ (cid:48) ( t ) − c ( t ) + λ − ε f ( t )) w y + ( c (cid:48) ( t ) − εf ( t ) − ε f ( t )) ∂ c w + εd (cid:48) ( t ) A c + ε ( B c ) t + ε ( f ( t ) + εf ( t )) ∂ c ( d ( t ) A c + εB c ) − ε f ( t )( B c ) y + 3 ε d ( t ) a (cid:48) ( ερ ) a ( ερ ) ( yQ c A c ) y + O H ( R ) ( ε e − εγ | ρ ( t ) | ) . Therefore, we have II ( w ) = − ( L w ) y + ( c (cid:48) ( t ) − εf ( t ) − ε f ( t )) ∂ c w − ( ρ (cid:48) ( t ) − c ( t ) + λ − ε f ( t )) w y + ε (cid:2) ε d (cid:48) ( t ) A c + ( B c ) t + f ( t ) d ( t ) ∂ c A c + 3 d ( t ) a (cid:48) ( ερ ) a ( ερ ) ( yQ c A c ) y (cid:3) + ε (cid:2) d ( t ) f ( t ) ∂ c A c + f ( t ) ∂ c B c − f ( t )( B c ) y (cid:3) + ε f ( t ) ∂ c B c + O H ( R ) ( ε e − εγ | ρ ( t ) | ) . This concludes the proof. (cid:3)
Step 3. Nonlinear term.
Lemma B.3 (Decomposition of
III ) . Suppose that ( A c , B c ) satisfy (4.16)-(4.17). Then we have III = (cid:40) O H ( R ) ( ε e − εγ | ρ ( t ) | ) , m = 2 , ε a ( ερ ) d ( t )( Q c A c ) (cid:48) + 3 ε a ( εx )( d ( t ) A c + εB c ) B (cid:48) c + O H ( R ) ( ε e − εγ | ρ ( t ) | ) , m = 3 . (B.10) Proof.
First of all, define ˜III := a ( εx )[( R + w ) m − R m − mR m − w ]. We consider separate cases.Let us suppose m = 2 or 4. In these cases, we have w ( t ) = d ( t ) A c ( y ). Therefore, ˜III = (cid:40) ε d ( t ) a ( εx ) A c if m = 2; ε a ( εx ) d ( t ) A c [6 Q c + 4 εd ( t ) Q c A c + ε d ( t ) A c ] , in the case m = 4 . Thus taking space derivative we obtain
III = ε m +1 a (cid:48) ( εx ) d m ( t ) A mc + O H ( R ) ( ε e − εγ | ρ ( t ) | )= O H ( R ) ( ε m + e − εγ | ρ ( t ) | + ε e − εγ | ρ ( t ) | ) = O H ( R ) ( ε e − εγ | ρ ( t ) | ) . Note that ( A mc ) (cid:48) ∈ Y because A c satisfies (4.16).Suppose now m = 3. We have w ( t, x ) = εd ( t ) A c ( y ) + ε B c ( t, y ), and ˜III = a ( εx )[3 Q c w + w ] . From this identity we get
III = 3 ε a ( ερ ) d ( t )( Q c A c ) (cid:48) + εa (cid:48) ( εx ) w + 3 a ( εx ) w w x + O H ( R ) ( ε e − εγ | ρ ( t ) | )= 3 ε a ( ερ ) d ( t )( Q c A c ) (cid:48) + ε a (cid:48) ( εx )( d ( t ) A c + εB c ) + 3 ε a ( εx )( d ( t ) A c + εB c ) ( d ( t ) A (cid:48) c + εB (cid:48) c ) + O H ( R ) ( ε e − εγ | ρ ( t ) | ) . The first term above is of second order, so we keep it. The second term in the last identity is in H ( R ) and it can be estimated as follows: ε a (cid:48) ( εx )( d ( t ) A c + εB c ) == ε a (cid:48) ( εx )( d A c + 3 εd A c B c + 3 ε dA c B c + ε B c )= O H ( R ) ( ε / e − εγ | ρ ( t ) | ) + O L ∞ ( R ) ( ε a (cid:48) ( εx )( | y | + εy + ε | y | ) e − εγ | ρ ( t ) | )= O H ( R ) ( ε / e − εγ | ρ ( t ) | ) + O H ( R ) ( ε / ( | ρ ( t ) | + ε | ρ ( t ) | + ε | ρ ( t ) | ) e − εγ | ρ ( t ) | ) . Since we assume (4.10), we have | ρ ( t ) | ≤ KT ε inside the interval [ − T ε , ˜ T ε ], which gives ε a (cid:48) ( εx )( d ( t ) A c + εB c ) = O H ( R ) ( ε / − / e − εγ | ρ ( t ) | ) . Finally,3 ε a ( εx )( d ( t ) A c + εB c ) ( d ( t ) A (cid:48) c + εB (cid:48) c ) = 3 ε a ( εx )( d ( t ) A c + εB c ) B (cid:48) c + O H ( R ) ( ε e − εγ | ρ ( t ) | ) . Collecting all these estimates, we finally obtain (B.10). (cid:3)
Step 4. First conclusion.
Now we collect the estimates from Lemmas B.1, B.2 and B.3. Weobtain that, for all t ∈ [ − T ε , ˜ T ε ], S [˜ u ] = ( c (cid:48) ( t ) − εf ( t ) − ε δ m, f ( t )) ∂ c ˜ u + ( ρ (cid:48) ( t ) − c ( t ) + λ − εf ( t ) − ε δ m, f ( t )) ∂ ρ ˜ u + ˜ S [˜ u ] , (B.11)with ∂ ρ ˜ u := ∂ ρ R − w y ,˜ S [˜ u ] = ε [ F ( t, y ) − d ( t )( L A c ) y ] + O ( ε | ρ (cid:48) ( t ) − c ( t ) + λ − εf ( t ) | e − εγ | ρ ( t ) | | A c | )+ ε (cid:2) ( a (cid:48) ˜ a m ) (cid:48) ( ερ )( c − λ ) A c + f d ∂ c A c (cid:3) + ε O H ( R ) ( e − εγ | ρ ( t ) | + ε ) , (B.12)for the cases m = 2 and 4; and for the cubic case,˜ S [˜ u ] = ε [ F ( t, y ) − d ( t )( L A c ) y ] + ε [ F ( t, y ) − ( L B c ) y ]+ ε (cid:2) d ( t ) f ( t ) ∂ c A c + f ( t ) ∂ c B c − f ( t )( B c ) y (cid:3) (B.13)+ ε (cid:2) f ( t ) ∂ c B c + 3 a ( εx )( d ( t ) A c + εB c ) B (cid:48) c (cid:3) + ε O H ( R ) ( e − εγ | ρ ( t ) | + ε ) . (B.14) laudio Mu˜noz 43 In addition, f ( t ) , f ( t ) , f ( t ) and f ( t ) are given by (2.19), (4.18), (4.19) and (4.20) respectively, F := F I = f ( t )˜ a ( ερ ) Λ Q c + a (cid:48) ˜ a m (cid:2) ( yQ mc ) y − m − c − λ ) Q c (cid:3) − f ( t )˜ a ( ερ ) Q (cid:48) c , (B.15)(cf. (B.6).) Moreover, for any t ∈ [ − T ε , ˜ T ε ] one has (cid:90) R F ( t, y ) Q c ( y ) dy = 0 . (B.16)(See [34] for a proof of this identity.) On the other hand, F is given by F := ˜ F + O ( | ρ (cid:48) − c + λ − ε f | ( a (cid:48) a / ) (cid:48) ( ερ ) A c ) , with ˜ F := ( a (cid:48) a / ) (cid:48) ( ερ )( c − λ ) A c + f ( t ) a (cid:48) a / ∂ c A c + 3 a (cid:48) a / ( yQ c A c ) y + f ( t ) a / Λ Q c + a (cid:48)(cid:48) a / ( y Q c ) y − f ( t ) a / Q (cid:48) c + 3 a (cid:48) a / ( Q c A c ) y , (B.17)and | ˜ F ( t, y ) | ≤ Ke − εγ | ρ ( t ) | . Finally, one has, with the choice of f ( t ) in (4.19), (cid:90) R ˜ F ( t, y ) Q c ( y ) dy = 0 , (B.18)for all time t ∈ [ − T ε , ˜ T ε ]. (cf. (B.35) below for the proof.) Step 5. Resolution of the first linear problem.
The next step is the resolution of the linear differential equation involving the first order termsin ε . Indeed, from (B.12)-(B.13), we want to solve d ( t )( L A c ) y ( y ) = F ( t, y ) , for all y ∈ R , and t ∈ [ − T ε , ˜ T ε ] fixed; (B.19)with d ( t ) given by (4.13). We start with an important remark. Remark
B.1 (Simplified expression for F ) . Note that from (2.19), (4.18) and (B.15) one has F ( t ; y ) := a (cid:48) ˜ a m (cid:104) pc ( c − λλ )Λ Q c − m − c − λ ) Q c + ( yQ mc ) (cid:48) (cid:105) − f ( t )˜ a Q (cid:48) c , = a (cid:48) ˜ a m (cid:104) pc Λ Q c − cm − Q c + ( yQ mc ) (cid:48) − λ ξ m √ cQ (cid:48) c (cid:105) + λ a (cid:48) ˜ a m (cid:104) − c − m Λ Q c + 1 m − Q c + ξ m √ c Q (cid:48) c (cid:105) =: d ( t )( ˜ F ( t, y ) + λ ˆ F ( t, y )) . (B.20)Compared with the former term F described in [34], now F possesses an additional, oddcomponent given by − f ( t )˜ a Q (cid:48) c , which is orthogonal to Q c in L ( R ). The purpose of this term is toobtain a unique solution A c satisfying the additional orthogonality condition (cid:82) R A c Q c = 0 . Moreover, since f ≡ A c satisfies in this case,this condition for free.From the above remark, we are reduced to solve the following simple problem,( L A c ) y ( y ) = ˜ F ( t, y ) + λ ˆ F ( t, y ) , with ˜ F and ˆ F defined in (B.20), and from (B.16), (cid:90) R ( ˜ F ( t, y ) + λ ˆ F ( t, y )) Q c ( y ) = 0 . Now we introduce the following function, with the purpose of describing the effect of potential on the solution. Let c > ϕ ( x ) := − Q (cid:48) ( x ) Q ( x ) , ϕ c ( x ) := − Q (cid:48) c Q c = √ cϕ ( √ cx ) . (B.21)Note that ϕ is an odd function, and satisfies (see [27] for more details)lim x →±∞ ϕ ( x ) = ± ϕ ( k ) ∈ Y , k ≥ . (B.22)We recall the form of the solution A c that we are looking for. In addition to the simple structurerequired in [34], we seek for a bounded solution satisfying A c ( t ) ( y ) = β c ( t )( ϕ c ( y ) − (cid:112) c ( t )) + ˆ A c ( y ) + µ c ( t ) Q (cid:48) c ( y ) + δ c ( t )Λ Q c ( y ) , (B.23)for some β c ( t ) , µ c ( t ) , δ c ( t ) ∈ R , ϕ defined in (B.21), and ˆ A c ∈ Y . The parameters µ c , δ c will bechosen in order to find the unique solution A c satisfying some orthogonality conditions. This lastfact is one of the key new ingredients for the proof of our main result. Lemma B.4 (Solvability of system (B.19), improved version) . Suppose ≤ λ < , λ (cid:54) = ˜ λ , ( c, ρ ) given by (3.1), and f ( t ) , f ( t ) given by (2.19) and (4.18)respectively. There exists a solution A c = A c ( y ) of ( L A c ( t ) ) y ( y ) = ˜ F ( t, y ) + λ ˆ F ( t, y ) , (B.24) satisfying, for every t ∈ [ − T ε , ˜ T ε ] , A c ( y ) := β c ( ϕ c ( y ) − √ c ) + ˆ A c ( y ) + µ c Q (cid:48) c + δ c Λ Q c ( y ) , (B.25)lim −∞ A c = − √ cβ c ; | A c ( y ) | ≤ Ke − γy , as y → + ∞ , (B.26) with ˆ A c ∈ Y . In addition, we have β c ( t ) := 12 c / (cid:90) R ( ˜ F + λ ˆ F )( t ) (cid:54) = 0 . (B.27) Moreover, A c satisfies (cid:90) R A c Q c = (cid:90) R A c yQ c = 0 . (B.28) Proof.
First of all, note that from Remark B.1, we have used the explicit value of f ( t ) and f ( t )to obtain the simplified linear problem (B.24). Next, the existence of a solution A c ∈ L ∞ ( R ) ofthe form (B.25) for this equation was established in [34], provided (cid:90) R ( ˜ F ( t, y ) + λF ( t, y )) Q c = 0 , which is indeed the case (cf. (B.16) and Lemma 2.2). The novelty now is the inclusion of the termproportional to f ( t ) Q (cid:48) c in (B.15), which induces the new term δ c Λ Q c in (B.25) (Note that fromLemma 2.2 ( L Λ Q c ) (cid:48) = − Q (cid:48) c .) Furthermore, the limits in (B.26) are straightforward from (B.22).On the other hand, we choose the terms µ c and δ c in order to satisfy (B.28). Since we do notknow explicitly A c , we need another method to compute explicitly f ( t ) (and therefore, δ c ( t ).)Indeed, multiplying (B.24) by (cid:82) y −∞ Λ Q c and integrating, one has (cid:90) R ( L A c ) y (cid:90) y −∞ Λ Q c = (cid:90) R ( ˜ F + λF ) (cid:90) y −∞ Λ Q c . (B.29)Integrating by parts, we get( L A c ) (cid:90) y −∞ Λ Q c (cid:12)(cid:12)(cid:12) + ∞−∞ + (cid:90) R ( L A c ) y (cid:90) y −∞ Λ Q c = − (cid:90) R Λ Q c L A c = (cid:90) R Q c A c = 0 . laudio Mu˜noz 45 Using (4.16), we have ( L A c ) (cid:82) y −∞ Λ Q c (cid:12)(cid:12)(cid:12) + ∞−∞ = 0. Therefore, from (B.20), − f (cid:90) R Q c Λ Q c = a (cid:48) a (cid:90) R (cid:104) pc ( c − λλ )Λ Q c − m − c − λ ) Q c + ( yQ mc ) (cid:48) (cid:105) (cid:90) y −∞ Λ Q c . A simple computation, using Lemma C.1, gives us − θf c θ − (cid:90) R Q = a (cid:48) a (cid:104) pc ( c − λλ ) (cid:90) R Λ Q c − m − c − λ ) (cid:90) R Q c (cid:105) (cid:90) R Λ Q c = a (cid:48) ac (cid:104) p ( c − λλ )( θ −
14 ) − m − c − λ ) (cid:105) ( θ −
14 ) c θ − ( (cid:90) R Q ) . Using that p = m +3 , λ = − mm +3 and θ = m − − , we finally obtain f ( t ) = 3 − m − m ( 3 cm + 3 − λ − m ) a (cid:48) ( ερ ) √ ca ( ερ ) ( (cid:82) R Q ) (cid:82) R Q = 3 − m (5 − m ) (3 λ c − λ ) a (cid:48) ( ερ ) √ ca ( ερ ) ( (cid:82) R Q ) (cid:82) R Q , as desired (cf. (4.18).)Now, let us prove (B.27). Indeed, from (B.24), integrating over R and using (B.26), we get2 β c c √ c = cA c ( −∞ ) = L A c (+ ∞ ) − L A c ( −∞ ) = (cid:90) R ( ˜ F + λF ) , (B.30)which gives the value of β c .Let us now describe the dependence in c of the solution A c . From (B.20) (see also Lemma 4.5in [34]), one has ˜ F ( t, y ) + λ ˆ F ( t, y ) = c / ( m − ˜ F ( √ cy ) + λc / ( m − ˆ F ( √ cy ) , where ˜ F ( x ) := p Λ Q − m − Q + ( yQ m ) (cid:48) − λ ξ m Q (cid:48) , ˆ F ( x ) := − − m Λ Q + 1 m − Q + ξ m Q (cid:48) . Moreover, Claim 3 in [34] allows to conclude that A c satisfies the following decomposition: A c ( y ) = c / ( m − − / [ c ˜ A ( √ cy ) + λ ˆ A ( √ cy )] , (B.31)with ˜ A , ˆ A bounded solutions of ( L ˜ A ) (cid:48) = ˜ F and ( L ˆ A ) (cid:48) = ˆ F , respectively. Moreover, one has( ˜ A ) (cid:48) , ( ˆ A ) (cid:48) ∈ Y . Using this decomposition we have ∂ c A c = ( 1 m − −
32 ) 1 c A c + 12 c yA (cid:48) c + c / ( m − − / ˜ A ( √ cy ) . From this identity we see that ∂ c A c has the same behavior as A c : it is bounded, it is not L -integrable, and satisfies lim + ∞ ∂ c A c = 0, lim −∞ ∂ c A c (cid:54) = 0. The same result holds for ∂ c A c . (cid:3) Remark
B.2 (Cubic case) . In the special case m = 3, the algebra of functions involved in thelinear problem (B.24) is well understood, and it can be computed explicitly. Indeed, from (B.31)one has A c ( y ) = ˜ A ( √ cy ) + cλ ˆ A ( √ cy ) , with ˜ A ( s ) := 12 (1 − Q ) (cid:90) + ∞ s Q − y Q (cid:48) − yQ + Q (cid:48) ln Q + ˜ µ Q (cid:48) , and ˆ A ( s ) := −
12 (1 − Q ) (cid:90) + ∞ s Q + 14 y Q (cid:48) + 12 yQ + Q (cid:48) ln Q + ˆ µ Q (cid:48) . See Appendix C for the main ingredients of the proof of this result. In particular, we havelim −∞ A c = (1 − λc ) (cid:82) R Q , which is different from zero provided c ( t ) (cid:54) = λ . Finally, the constants˜ µ and ˆ µ are chosen such that (cid:90) R yQ c A c ( y ) = 0 . Step 6. Cubic case. resolution of a second linear system.
Since f ( t ) ≡ m = 3 (cf. (4.18)), we need to go beyond in our computations and solve a new linear system, inorder to find a formal defect in the solution. From (B.12), one has to consider a linear problemfor the unknown function B c ( t, · ), with fixed time t , and with source term non localized . Thenext result gives the existence of such a second order correction term. Lemma B.5 (Existence of a second order correction term) . Let f ( t ) , f ( t ) be given by (4.19)-(4.20), and consider ˜ F as in (B.17). For each fixed time t ∈ [ − T ε , ˜ T ε ] , there exists a unique solution B c ( t, · ) of ( L B c ) y = ˜ F ( t, y ) , (B.32) satisfying, for all t ∈ [ − T ε , ˜ T ε ] , (cid:90) R Q c B c = (cid:90) R yQ c B c = 0 . (B.33) In addition, one has, for some γ > independent of ε , (cid:40) | B c ( t, y ) | + | ∂ c B c ( t, y ) | ≤ Ke − γy e − εγ | ρ ( t ) | , as y → + ∞ , | B c ( t, y ) | + | ∂ c B c ( t, y ) | ≤ K | y | e − εγ | ρ ( t ) | , as y → −∞ . (B.34) Proof.
The proof is divided in several steps.1. Note that since A c ∈ L ∞ ( R ), one has from (B.17) that ˜ F ( t, · ) ∈ L ∞ ( R ). From Lemma 2.2 (seealso [34]), we get solvability in S (cid:48) ( R ) for (B.32) provided ˜ F satisfies the orthogonality condition (cid:90) R ˜ F Q c = 0 . (B.35)Let us prove this last identity. Indeed, we have (cid:90) R Q c ˜ F = (cid:90) R Q c ( f d∂ c A c + 3 d a (cid:48) a ( yQ c A c ) y + f ˜ a Λ Q c + 3 d a ( Q c A c ) y )= − f d (cid:90) R Λ Q c A c − d a (cid:48) a (cid:90) R yQ c Q (cid:48) c A c + f ˜ a (cid:90) R Q c Λ Q c − d a (cid:90) R Q c Q (cid:48) c A c = − a (cid:48) a / (cid:2)
13 ( c − λ ) (cid:90) R yQ (cid:48) c A c + 3 (cid:90) R yQ c Q (cid:48) c A c + 3 (cid:90) R Q c Q (cid:48) c A c (cid:3) + f a / (cid:90) R Q c Λ Q c . Let us define µ c := 13 ( c − λ ) (cid:90) R yQ (cid:48) c A c + 3 (cid:90) R yQ c Q (cid:48) c A c + 3 (cid:90) R Q c Q (cid:48) c A c . Our objective is to give a simple expression of this quantity. Indeed, first note that A (cid:48) c ∈ Y . Fromthe equation ( L A c ) (cid:48) = ˜ F + λ ˆ F , one has L A (cid:48) c = ˜ F + λ ˆ F + 6 Q c Q (cid:48) c A c . We multiply this identityby A c and integrate over R . We get (cid:90) R A c L A (cid:48) c = (cid:90) R A c ( ˜ F + λ ˆ F ) + 6 (cid:90) R Q c Q (cid:48) c A c . (B.36)On the other hand, after integration by parts, one has (cid:90) R A c L A (cid:48) c = (cid:90) R A (cid:48) c L A c = A c L A c | + ∞−∞ − (cid:90) R A c ( ˜ F + λ ˆ F )= − cA c ( −∞ ) − (cid:90) R A c ( ˜ F + λ ˆ F ) . For the sake of simplicity, we avoid the explicit dependence on time in this computation. Let us recall that m = 3. laudio Mu˜noz 47 From these two identities, we get3 (cid:90) R Q c Q (cid:48) c A c = − cA c ( −∞ ) − (cid:90) R A c ( ˜ F + λ ˆ F ) . We replace this identity above, in the definition of µ c , to obtain (recall that A c is orthogonal to Q c and Q (cid:48) c ) µ c = − cA c ( −∞ ) + 13 ( c − λ ) (cid:90) R yQ (cid:48) c A c + 3 (cid:90) R yQ c Q (cid:48) c A c − (cid:90) R A c ( 13 cyQ (cid:48) c + ( yQ c ) (cid:48) − λyQ (cid:48) c )= − cA c ( −∞ ) − (cid:90) R A c Q c . On the other hand, note that L ( − Q c ) = Q c . We have then µ c = − cA c ( −∞ ) + 12 (cid:90) R L A c Q c = − cA c ( −∞ ) − (cid:90) R Q c (cid:90) + ∞ y ( ˜ F + λ ˆ F )= − cA c ( −∞ ) − (cid:90) R Q c (cid:90) R ( ˜ F + λ ˆ F ) , (recall that f ≡ (cid:90) R ( ˜ F + λ ˆ F ) = −
12 ( c − λ ) (cid:90) R Q c , since from (B.30) and (B.20) one has A c ( −∞ ) = − c (cid:90) R ( ˜ F + λ ˆ F ) = 12 c ( c − λ ) (cid:90) R Q c , we finally get µ c = λ c ( c − λ )( (cid:90) R Q ) (cid:54) = 0 . From the definition of f ( t ) in (4.19), we get finally (B.35). In consequence, there exists at leastone solution B c ∈ S (cid:48) satisfying (B.32).2. Let us look for a solution B c with a special behavior. In fact, we will search for a solution withthe following structure: B c ( t, y ) = ˜ B c ( t, y ) + f ( t ) Q (cid:48) c ( y ) + f a / ( t )Λ Q c ( y ) , where ˜ B c has the following decomposition˜ B c ( t, y ) = α ( t ) (cid:90) + ∞ y A c + α ( t ) (cid:90) + ∞ y ∂ c A c + α ( t ) + ˆ B c ( t, y ) , ˆ B c ( t, · ) ∈ Y . (B.37)Here α ( t ) , α ( t ) , α ( t ) are real valued, exponentially decreasing, time-dependent functions, to befound. Note that this function satisfies (B.34), provided α ( t ) ≡
0, since A c ( y ) , ∂ c A c ( y ) , ∂ c A c ( y ) → y → + ∞ , at exponential rate. Moreover, we can choose unique f ( t ) , f ( t ) ∈ R such that(B.33) holds, respectively, for all time t ∈ [ − T ε , ˜ T ε ].Let us prove the existence of ˆ B c , with the desired properties. By replacing the form (B.37) in(B.32), we get( L ˆ B c ) y = − α (cid:2) L ( (cid:90) + ∞ y A c ) (cid:3) y − α (cid:2) L ( (cid:90) + ∞ y ∂ c A c ) (cid:3) y + 3 α ( Q c ) (cid:48) + ( a (cid:48) a / ) (cid:48) ( c − λ ) A c + f a (cid:48) a / ∂ c A c + 3 a (cid:48) a / ( yQ c A c ) y + f a / Λ Q c + a (cid:48)(cid:48) a / ( y Q c ) y + 3 a (cid:48) a / ( Q c A c ) y . On the other hand, one has (cid:2) L ( (cid:90) + ∞ y A c ) (cid:3) y = − cA c + A (cid:48)(cid:48) c − (cid:2) Q c (cid:90) + ∞ y A c (cid:3) y , and (cid:2) L ( (cid:90) + ∞ y ∂ c A c ) (cid:3) y = − c∂ c A c + ( ∂ c A c ) (cid:48)(cid:48) − (cid:2) Q c (cid:90) + ∞ y ∂A c (cid:3) y . Therefore, by defining α ( t ) := − ( a (cid:48) a / ) (cid:48) ( ερ ( t ))(1 − λc ( t ) ) , and α ( t ) := − c ( t ) f ( t ) a (cid:48) a / ( ερ ( t ));(note that both functions are exponentially decreasing in | ρ ( t ) | ), one has that ˆ B c ( t, y ) must be asolution of L ˆ B c = 3 a (cid:48) a / yQ c A c + f ca / yQ c + a (cid:48)(cid:48) a / y Q c + 3 a (cid:48) a / Q c A c + α (cid:2) A (cid:48) c − Q c (cid:90) + ∞ y A c (cid:3) + α (cid:2) ( ∂ c A c ) (cid:48) − Q c (cid:90) + ∞ y ∂ c A c (cid:3) + 3 α Q c . Note that the right hand side above is in Y and it is orthogonal to Q (cid:48) c , since there exists asolution B c of (B.32). Therefore, from Lemma 2.2, we have ˆ B c ( t, · ) ∈ Y , with (cid:107) ˆ B c ( t, · ) (cid:107) L ∞ ( R ) ≤ Ke − εγ | ρ ( t ) | + K | α ( t ) | . Let us adjust the value of α ( t ). Indeed, first note that( L ( c − Q c )) y = 0 . (cf. Lemma C.2 in Appendix C below.)Therefore, by substracting a suitable ponderation of the term c − Q c in the form of ˜ B c above (see(B.37)), we may suppose α ( t ) ≡
0, still having ˆ B c ∈ Y . This proves the existence of B c withthe required behavior.3. Finally, let us prove that f ( t ) has the form (4.20). Indeed, note that (cid:90) R B c Q c = − (cid:90) R B c L Λ Q c = − (cid:90) R Λ Q c L B c ; (B.38)From one has for y < r , L B c ( y ) = L B c ( r ) − (cid:82) ry ˜ F therefore( B.
38) = − (cid:90) R Λ Q c ( L B c ( r ) − (cid:90) ry ˜ F ) = (cid:90) R Λ Q c (cid:90) ry ˜ F . From the definition of Λ Q c , we get ( B.
38) = 12 c (cid:90) R yQ c ˜ F . Now we use the definition of F and the orthogonality conditions on A c to get2 c ( B.
38) = −
13 ( c − λ ) a (cid:48) a / (cid:90) R y Q (cid:48) c A c − a (cid:48) a / (cid:90) R ( yQ c ) (cid:48) yQ c A c − a (cid:48)(cid:48) a / (cid:90) R y Q c + f ( t )2 a / (cid:90) R Q c − a (cid:48) a / (cid:90) R ( yQ c ) (cid:48) Q c A c . Now we use the scaling property (B.31) of the function A c , with m = 3, to obtain a betterdescription of f ( t ): we have2 c ( B.
38) = −
13 ( c − λ ) a (cid:48) √ ca / (cid:90) R y Q (cid:48) [ ˜ A + λc ˆ A ] − √ ca (cid:48) a / (cid:90) R ( yQ ) (cid:48) yQ [ ˜ A ( y ) + λc ˆ A ( y )] − a (cid:48)(cid:48) √ c a / (cid:90) R y Q + f ( t )2 a / √ c (cid:90) R Q − √ ca (cid:48) a / (cid:90) R ( yQ ) (cid:48) Q [ ˜ A ( y ) + λc ˆ A ( y )] . Therefore, one has f ( t ) as in (4.20), with f ( t ) := 18 M [ Q ] (cid:90) R y Q > , laudio Mu˜noz 49 and f ( t ) := 13 M [ Q ] (1 − λc ) (cid:90) R y Q (cid:48) [ ˜ A + λc ˆ A ] + 3 M [ Q ] (cid:90) R ( yQ ) (cid:48) yQ [ ˜ A + λc ˆ A ]+ 3 M [ Q ] (cid:90) R ( yQ ) (cid:48) Q [ ˜ A + λc ˆ A ] . In conclusion, we have the existence of a unique B c ( t, y ) satisfying (B.32)-(B.33). Estimates(B.34) are direct from (B.37). (cid:3) Step 7. Final conclusion.
Having solved one linear problem in the cases m = 2 and 4, and twolinear equations in the case m = 3, from (B.11) and (B.12) we have S [˜ u ]( t, x ) = ( c (cid:48) ( t ) − εf ( t ) − ε δ m, f ( t )) ∂ c ˜ u +( ρ (cid:48) ( t ) − c ( t ) + λ − εf ( t ) − ε δ m, f ( t )) ∂ ρ ˜ u + ˜ S [˜ u ]( t, x ) , with ∂ ρ ˜ u := ∂ ρ R − w y + O ( ε e − εγ | ρ ( t ) | | A c | ) , and ˜ S [˜ u ] = (cid:40) ( B. , for the cases m = 2 , B.
13) + ( B. , in the cubic case.This proves the first part of Proposition 4.2.In addition, from Lemmas B.4 and B.5 we have (4.16) and (4.17), respectively. This proves thesecond part of Proposition 4.2. In addition, from these lemmas, f ( t ), f ( t ), f ( t ) and f ( t ) arewell determined. This proves the third part of Proposition 4.2.Finally, we prove the last part of Proposition 4.2. Let us recall that (B.12) is a bounded, nonlocalized term, and (B.13) + (B.14) is a polynomially growing term. Indeed, from (4.13), (2.19),(4.19), (4.20), (B.25) and (B.34) we have | ( B. | ≤ Kε | y | e − εγ | ρ ( t ) | , as y → −∞ , | ( B. | → y → + ∞ , and (cid:40) | ( B. | ≤ Kε (1 + | y | + ε | y | ) e − εγ | ρ ( t ) | , as y → −∞ , | ( B. | → , as y → + ∞ . Moreover, note that (cid:107) ( B. (cid:107) H ( y ≥− ε ) ≤ Kε / e − εγ | ρ ( t ) | ;(cf. [34] for this bound) and (cid:107) ( B. (cid:107) H ( y ≥− ε ) ≤ Kε / e − εγ | ρ ( t ) | , (cid:107) ( B. (cid:107) H ( y ≥− ε ) ≤ Kε / e − εγ | ρ ( t ) | . Finally, (4.21) is direct from this last estimate. On the other hand, from (B.12) one has (4.22),and from (B.13)-(B.14) we finally obtain (4.23).The proof of Proposition 4.2 is now complete.
Appendix C. Some identities related to the soliton Q This section has been taken from Appendix C in [26].
Lemma C.1 (Identities for the soliton Q ) . Suppose m > and denote by Q c := c m − Q ( √ cx ) the scaled soliton. Then (1) Energy . E [ Q ] = 12 ( λ − λ ) (cid:90) R Q = ( λ − λ ) M [ Q ] , with λ = 5 − mm + 3 . (2) Integrals . Recall θ = m − − . Then (cid:90) R Q c = c θ − (cid:90) R Q, (cid:90) R Q c = c θ (cid:90) R Q , E [ Q c ] = c θ ( λ − λ c ) M [ Q ] , and finally (cid:90) R Q m +1 c = 2( m + 1) c θ +1 m + 3 (cid:90) R Q , (cid:90) R Λ Q c = ( θ −
14 ) c θ − (cid:90) R Q, (cid:90) R Λ Q c Q c = θc θ − (cid:90) R Q . Lemma C.2 (Inverse functions, case m = 3) . Let L be the fixed, linearized operator defined in (2.5) for m = 3 . Then one has L ( Q (cid:48) ) = 0 , L ( yQ ) = − yQ − Q (cid:48) , L ( y Q (cid:48) ) = − yQ + 4 yQ − Q (cid:48) ; L ( (cid:90) + ∞ y Q ) = (1 − Q ) (cid:90) + ∞ y Q + Q (cid:48) , L ( Q (cid:90) + ∞ y Q ) = − Q (cid:90) + ∞ y Q + 5 Q Q (cid:48) , and L ( Q ) = − Q , L ( Q (cid:48) ln Q ) = − Q (cid:48) + 52 Q Q (cid:48) . The proof of these result is a lengthy but direct computation.
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