Effective dynamics of double solitons for perturbed mKdV
EEFFECTIVE DYNAMICS OF DOUBLE SOLITONS FORPERTURBED MKDV
JUSTIN HOLMER, GALINA PERELMAN, AND MACIEJ ZWORSKI
Abstract.
We consider the perturbed mKdV equation ∂ t u = − ∂ x ( ∂ x u + u − b ( x, t ) u ) where the potential b ( x, t ) = b ( hx, ht ), 0 < h (cid:28)
1, is slowly varying witha double soliton initial data. On a dynamically interesting time scale the solutionis O ( h ) close in H to a double soliton whose position and scale parameters followan effective dynamics, a simple system of ordinary differential equations. Theseequations are formally obtained as Hamilton’s equations for the restriction of themKdV Hamiltonian to the submanifold of solitons. The interplay between algebraicaspects of complete integrability of the unperturbed equation and the analytic ideasrelated to soliton stability is central in the proof. Introduction
We consider 2-soliton solutions of the modified KdV equation with a slowly varyingexternal potential (1.1). The purpose of the paper is to find minimal exact effective dy-namics valid for a long time in the semiclassical sense and describing non-perturbative2-soliton interaction. In standard quantum mechanics the natural long time for whichthe semiclassical approximation is valid is the Ehrenfest time, log(1 /h ) /h – see forinstance [7]. The semiclassical parameter, h , quantifies the slowly varying nature ofthe potential.Unlike in the case of single-particle semiclassical dynamics, that is, for the linearSchr¨odinger equation with a slowly varying potential, the exact effective dynamicsvalid for such a long time requires h -size corrections † . Those corrections appeared asunspecified O ( h ) additions to Newton’s equations (which give the usual semiclassi-cal approximation) in the work of Fr¨ohlich-Gustafson-Jonsson-Sigal [13] on 1-solitonpropagation. That paper and its symplectic point of view were the starting point for[18, 19].Following the 1-soliton analysis of [18, 19] the semiclassical dynamics for 2-solitonsconsidered here is obtained by restricting the Hamiltonian to the symplectic manifoldof 2-solitons and considering the finite dimensional dynamics there. The numericalexperiments [17] show a remarkable agreement with the theorem below. However,they also reveal an interesting scenario not covered by our theorem: the velocities of † A compensation for that comes however at having the semiclassical propagation accurate forlarger values of h . a r X i v : . [ m a t h . A P ] J un JUSTIN HOLMER, GALINA PERELMAN, AND MACIEJ ZWORSKI
Figure 1.
A gallery of numerical experiments showing agreement withthe results of the main theorem (clockwise from the left hand corner)for the external fields listed in (1.15) with the indicated initial data.The continuous lines are the numerically computed solutions and thedotted lines follow the evolution given by (1.4). The main theorem doesnot apply to the bottom two figures on the whole interval of time dueto the crossing of c j ’s – see Fig.3. In the first figure in the second line,(1.4) still apply directy, but in the second one further modification isneeded to account for the signs.the solitons can almost cross within exponentially small width in h and the effectivedynamics remains valid. Any long time analysis involving multiple interactions ofsolitons has to explain this avoided crossing which perhaps could be replaced by adirect crossing in a different parametrization. This seems the most immediate openproblem of phenomenological interest. FFECTIVE DYNAMICS OF DOUBLE SOLITONS FOR PERTURBED MKDV 3
The effective dynamics follows a long tradition of the use of modulation parametersin soliton propagation – see for instance [8],[25],[26],[30],[32] and the numerous refer-ences given there. For non-linear dispersive equations with non-constant coefficientsone can consult, in addition to [13], [3],[14],[15],[22], and references given there.Here we avoid generality and, as described above, the aims are more modest: for thephysically relevant cubic non-linearity we benefit from the completely integrable struc-ture and using classical methods we can give a remarkably accurate and phenomeno-logically relevant description of 2-soliton interaction. The equation (1.1) shares manyfeatures with the dynamical Gross-Pitaevskii equation, i∂ t u = − ∂ x u − | u | u + V ( x ) u , but is easier to study, mathematically and numerically. In a recent numerical studyPotter [31] showed that the same effective dynamics applies very well to N -solitontrains in the case of perturbed mKdV and NLS. The soliton matter-wave trains cre-ated for Bose-Einstein condensates [33] were a good testing ground and our effectivedynamics gives an alternative explanation of the observed phenomena. At the mo-ment it is not clear how to obtain exact effective dynamics for the perturbed NLS.To state the exact result we recall the perturbed mKdV equation [10],[11]: ∂ t u = − ∂ x ( ∂ x u − b ( x, t ) u + 2 u ) ,b ( x, t ) = b ( hx, ht ) , < h (cid:28) , ∂ α b ∈ L ∞ ( R ) . (1.1)For b ≡ N -solitonsolutions, q N ( x, a, c ), a ∈ R N , c ∈ R N – see § § N = 2 we obtain Theorem.
Let δ > and ¯ a, ¯ c ∈ R n . Suppose that u ( x, t ) solves (1.1) with (1.2) u ( x,
0) = q ( x, ¯ a, ¯ c ) , | ¯ c ± ¯ c | > δ > , δ < | ¯ c j | < (2 δ ) − . Then, for t < T ( h ) /h , (1.3) (cid:107) u ( · , t ) − q ( · , a ( t ) , c ( t )) (cid:107) H ≤ Ch e Cht , C = C ( δ , b ) > , where a ( t ) and c ( t ) evolve according to the effective equations of motion, ˙ a j = c j − sgn( c j ) ∂ c j B ( a, c, t ) ˙ c j = sgn( c j ) ∂ a j B ( a, c, t ) B ( a, c, t ) def = 12 (cid:90) b ( x, t ) q ( x, a, c ) dx . (1.4) The upper bound T ( h ) /h for the validity of (1.3) is given in terms of (1.5) T ( h ) = min( δ log(1 /h ) , T ( h )) , δ = δ ( δ , b ) > where for t < T ( h ) /h , | c ( t ) ± c ( t ) | > δ > and δ < | c j ( t ) | < δ − . Under theassumption (1.2) on ¯ c , T ( h ) > δ , where δ = δ ( δ , b ) > is independent of h – see (1.12) . JUSTIN HOLMER, GALINA PERELMAN, AND MACIEJ ZWORSKI
Remarks. 1.
We expect the same result to be true for all N with H replaced by H N . For N = 1 it follows directly from the arguments of [19]. That case is alsoimplicit in this paper: single soliton dynamics describes the propagation away fromthe interaction region. The Ehrenfest time bound, T ( h ) ≤ δ log(1 /h ), is probably optimal if we insist onthe agreement with classical equations of motion (1.4). We expect that the solution isclose to a soliton profile q ( x, a, c ) for much longer times ( h −∞ ?) but with a modifiedevolution for the parameters. One difficulty is the lack of a good description of thelong time behaviour of time dependent linearized evolution with b present – see § O ( h ) in (1.3) is optimal. As shown by the top two plots in Fig.1 the agreement of the approximations givenby (1.4) and numerical solutions of (1.1) is remarkable. The codes are available at[17], see also § The condition that | c ( t ) ± c ( t ) | > δ , that is, that the perturbed effectivedynamics avoids the lines shown in Fig.2, could most likely be relaxed. Allowing thatprovides more interesting dynamics as then the solitons can interact multiple times.As discussed in § ± c j ( t )’s getwithin exp( − c/h ) of each other – see Fig.3. Examples of such evolution, and thecomparisons with effective dynamics, are shown in the lower two plots in Fig.1. Oncloser inspection the agreement between the solutions and solitons moving accordingto effective dynamics is not as dramatic as in the case when ± c j ’s stay away fromeach other but for smaller values of h the result should still hold. We concentratedon the simpler case at this early stage. The equation (1.1) is globally well-posed in H k , k ≥ b . This can be shown by modifying the techniques of Kenig-Ponce-Vega [21] – see Appendix A. Although for k ≥ H k ’s at once. Studies of single solitons for perturbed KdV, mKdV, and their generalizationswere conducted by Dejak-Jonsson [10] and Dejak-Sigal [11]. The perturbative terms, b ( x, t ), were assumed to be not only slow varying but also small in size. The mKdVresults of [10] are improved by following [19]. For KdV one does not expect the samebehaviour as for mKdV and the O ( h )-approximation similar to (1.3) is not valid –see [28, 16] for finer analysis of that case. The conditions that u ( x,
0) = q ( x, ¯ a, ¯ c ) can be relaxed by allowing a small per-turbation in H – see [9] for the adaptation of [19] to that case. Similar statementsare possible here but we prefer the simpler formulation both in the statement of thetheorem and in the proofs. FFECTIVE DYNAMICS OF DOUBLE SOLITONS FOR PERTURBED MKDV 5 ! ! ! ! ! ! ! ! ! ! ! ! ! ! Figure 2.
On the left we show R \ C and on the right examplesof double solitons corresponding to ( c , c ) indicated on the left (with a = a = 0 in the first figure and a = − a = 1, in the other two). Atthe coordinate axes the double soliton degenerates into a single soliton.As one approaches the lines c = ± c the solitons escape to infinitiesin opposite direction.In the remainder of the introduction we will explain the origins of the effectivedynamics (1.4), outline the proof, and comment on numerical experiments.1.1. Double solitons for mKdV.
The single soliton solutions to mKdV, (1.1) with b ≡
0, are described in terms of the profile η ( x, a, c ) as follows. Let η ( x ) = sech x sothat − η + η (cid:48)(cid:48) + 2 η = 0, and let η ( x, c, a ) = cη ( c ( x − a )) for a ∈ R , c ∈ R \
0. Then asingle soliton defined by u ( x, t ) = η ( x, a + c t, c )is easily verified to be an exact solution to mKdV. Such solitary wave solutions areavailable for many nonlinear evolution equations. However, mKdV has richer struc-ture – it is completely integrable and can be studied using the inverse scatteringmethod (Miura [27], Wadati [35]). One of the consequences is the availability oflarger families of explicit solutions. In the case of mKdV, we have N -solitons and breathers . In this paper we confine our attention to the 2-soliton (or double soli-ton), which is described by the profile q ( x, a, c ) defined in (3.2) below. The four realparameters, a ∈ R , and c ∈ R \ C , C def = { ( c , c ) : c = ± c } ∪ R × { } ∪ { } × R , describe the position ( a ) and scale ( c ) of the double soliton. At the diagonal lines theparametrization degenerates: for c = ± c , q ≡
0. At the coordinate axes in the c JUSTIN HOLMER, GALINA PERELMAN, AND MACIEJ ZWORSKI space, we recover single solitons: q ( x, a, ( c , − c η ( x, a , c ) , q ( x, a, (0 , c )) = c η ( x, a , c ) . Fig.2 shows a few examples.Solving mKdV with u ( x,
0) = q ( x, a, c ) gives the solution u ( x, t ) = q ( x, a + tc , a + tc , c ) , that is, the double soliton solution.If, say, 0 < c < c , then for | a − a | large, q ( x, a, c ) ≈ η ( x, a + α , c ) + η ( x, a + α , c )where α j are shifts defined in terms of c , see Lemma 3.2 for the precise statement.This means that for large positive and negative times the evolving double soliton iseffectively a sum of single solitons. The decomposition can be made exact preservingthe particle-like nature of single solitons even during the interaction – see (3.11) andFig.4.We consider the set of 2-solitons as a submanifold of H ( R ; R ) with 8 open com-ponents corresponding to the components of R \ C :(1.6) M = { q ( · , a, c ) | a = ( a , a ) ∈ R , c = ( c , c ) ∈ R \ C } . As in the case of single solitons this submanifold is symplectic with respect to thenatural structure recalled in the next subsection.1.2.
Dynamical structure and effective equations of motion.
The equation(1.1) is a Hamiltonian equation of evolution for(1.7) H b ( u ) = 12 (cid:90) ( u x − u + bu ) dx , on the Schwartz space, S ( R ; R ) equipped with the symplectic form(1.8) ω ( u, v ) = 12 (cid:90) + ∞−∞ (cid:90) x −∞ ( u ( x ) v ( y ) − u ( y ) v ( x )) dydx . In other words, (1.1) is equivalent to(1.9) u t = ∂ x H (cid:48) b ( u ) , (cid:104) H (cid:48) b ( u ) , ϕ (cid:105) def = dds H b ( u + sϕ ) | s =0 , and ∂ x H (cid:48) b ( u ) is the Hamilton vector field of H b , Ξ H b , with respect to ω : ω ( ϕ, Ξ H b ( u )) = (cid:104) H (cid:48) b ( u ) , ϕ (cid:105) . For b = 0, Ξ H is tangent to the manifold of solitons (1.6). Also, M is symplecticwith respect to ω , that is, ω is nondegenerate on T u M , u ∈ M . Using the stabilitytheory for 2-solitons based on the work of Maddocks-Sachs [24], and energy methods(enhanced and simplified using algebraic identities coming from complete integrability FFECTIVE DYNAMICS OF DOUBLE SOLITONS FOR PERTURBED MKDV 7 of mKdV) we will show that the solution to (1.1) with initial data on M stays closeto M for t ≤ log(1 /h ) /h .A basic intuition coming from symplectic geometry then indicates that u ( t ) staysclose to an integral curve on M of the Hamilton vector field (defined using ω | M ) of H b restricted to M : H eff ( a, c ) def = H b | M ( a, c ) = H | M ( a, c ) + 12 (cid:90) b ( x ) q ( x, a, c ) dx ,H | M ( a, c ) = −
13 ( | c | + | c | ) ,ω | M = da ∧ d | c | + da ∧ d | c | , Ξ H eff = (cid:88) j =1 sgn( c j )( ∂ a j H eff ∂ c j − ∂ c j H eff ∂ a j ) . (1.10)The effective equations of motion (1.4) follow. This simple but crucial observationwas made in [18],[19] and it did not seem to be present in earlier mathematical workon solitons in external fields [13].The condition made in the theorem, that | c ( t ) ± c ( t ) | and | c j ( t ) | are boundedaway from zero for t < T ( h ) /h (where T ( h ) could be ∞ ), follows from a conditioninvolving a simpler system of decoupled h -independent ODEs – see Appendix B. Herewe state a condition which gives an h -independent T appearing in (1.5).Suppose we are given b ( x, t ) = b ( hx, ht ) in (1.1) and the initial condition is givenby q ( x, ¯ a, ¯ c ), ¯ a = (¯ a , ¯ a ), ¯ c = (¯ c , ¯ c ), | ¯ c ± ¯ c | > δ , | ¯ c j | > δ , We consider an h -independent system of two decoupled differential equations for A ( T ) = ( A ( T ) , A ( T )) , C ( T ) = ( C ( T ) , C ( T )) , given by(1.11) (cid:40) ∂ T A j = C j − b ( A j , T ) ∂ T C j = C j ∂ x b ( A j , T ) , A (0) = ¯ ah , C (0) = ¯ c , j = 1 , . Then, for a given δ < δ , T ( h ) in (1.5) can be replaced by(1.12) T = sup { T : | C ( T ) ± C ( T ) | > δ , | C j ( T ) | > δ , j = 1 , } . Outline of the proof.
To obtain the effective dynamics we follow a long tra-dition (see [13] and references given there) and define the modulation parameters a ( t ) = ( a ( t ) , a ( t )) , c ( t ) = ( c ( t ) , c ( t )) , be demanding that v ( x, t ) = u ( x, t ) − q ( x, a ( t ) , c ( t )) , q = q , JUSTIN HOLMER, GALINA PERELMAN, AND MACIEJ ZWORSKI
Figure 3.
The plots of c and a for the external potential given bythe last b ( x, t ) in (1.15), and ¯ c = (6 , a = ( − , − c j ’s (see also Fig.6). The crossings are avoidedwith exp( − /Ch ) width and a = a at the crossings. These cases arenot yet covered by our theory. Of the five crossings of a j ’s in the bottomfigure, three do not involve crossings of c j ’s are hence the descriptionby effective dynamics there is covered by our theorem. However, in theabsence of avoided crossing of c j ’s the solitons can interact only once.satisfies symplectic orthogonality conditions: ω ( v, ∂ a q ) = 0 ω ( v, ∂ a q ) = 0 ω ( v, ∂ c q ) = 0 ω ( v, ∂ c q ) = 0 FFECTIVE DYNAMICS OF DOUBLE SOLITONS FOR PERTURBED MKDV 9
These can be arranged by the implicit function theorem thanks to the nondegeneracyof ω | M . This makes q the symplectic orthogonal projection of u onto the manifold ofsolitons M .Since u = q + v and u solves mKdV, we have(1.13) ∂ t v = ∂ x ( L c,a v − qv − v + bv ) − F , where L c,a = − ∂ x − q ( x, a, c ) v , and F results from the perturbation and ∂ t landing on the parameters: F = (cid:88) j =1 ( ˙ a j − c j ) ∂ a j q + (cid:88) j =1 ˙ c j ∂ c j q − ∂ x ( bq ) . We decompose F = F (cid:107) + F ⊥ , where F (cid:107) is symplectic projection of F onto T q M , and F ⊥ is the symplectic projection onto its symplectic orthogonal ( T q M ) ⊥ . As seen in(5.4), F (cid:107) ≡ c > c > q , we show that F ⊥ is O ( h ). In fact it is important toobtain a specific form for the O ( h ) term so that it is amenable to finding a certaincorrection term later – see § F (cid:107) are obtained using the symplectic orthogonality properties of v . For example, 0 = (cid:104) v, ∂ − x ∂ a j q (cid:105) implies0 = ∂ t (cid:104) v, ∂ − x ∂ a j q (cid:105) = (cid:104) ∂ t v (cid:124)(cid:123)(cid:122)(cid:125) ↑ substitute equation (1.13) , ∂ − x ∂ a j q (cid:105) + (cid:104) v, ∂ t ∂ − x ∂ a j q (cid:105) , which can be used to show that(1.14) | F (cid:107) | ≤ Ch (cid:107) v (cid:107) H + (cid:107) v (cid:107) H , see § v satisfying (1.13) with v (0) = O ( h ) (in the theorem v (0) = 0, but we need this relaxed assumption for the bootstrap argument). We wantto show that on a time interval of length h − , that v at most doubles. The Lyapunovfunctional E ( t ) that we use to achieve this comes from the variational characterizationof the double soliton (see [23, §
2] and Lemma 4.1 below): if H c ( u ) = I ( u ) + ( c + c ) I ( u ) + c c I ( u ) , then H (cid:48) c ( q ( · , a, c )) = 0 , ∀ a ∈ R , and H (cid:48)(cid:48) c ( q ( · , a, c )) = K c,a , where K c,a is a fourth order operator given in (4.11) below. Hence E ( t ) def = H c ( t ) ( q ( • , a ( t ) , c ( t )) + v ( t )) − H c ( t ) ( q ( • , a ( t ) , c ( t ))) , satisfies E ( t ) ≈ (cid:104)K c,a v, v (cid:105) , and, as in Maddocks-Sachs [24] for KdV, K c,a has a two dimensional kernel and onenegative eigenvalue. However, the symplectic orthogonality conditions on v imply thatwe project far enough away from these eigenspaces and hence we have the coercivity δ (cid:107) v (cid:107) H ≤ E ( t ) . To get the upper bound on E ( t ), we compute ddt E ( t ) = O ( h ) (cid:107) v ( t ) (cid:107) H + (cid:104)K c,a v, F (cid:107) (cid:105) + (cid:104)K c,a v, F ⊥ (cid:105) , see §
9. Using (1.14) we can estimate the second term on the right-hand side but | F ⊥ | = O ( h ) only. We improve this to h using a correction term to v – see §
8, andthe comment at the end of this section.All of this combined gives, on [0 , T ], (cid:107) v (cid:107) H (cid:46) (cid:107) v (0) (cid:107) H + T ( | F (cid:107) |(cid:107) v (cid:107) H + h (cid:107) v (cid:107) H + (cid:107) v (cid:107) H ) , | F (cid:107) | ≤ Ch (cid:107) v (cid:107) H + (cid:107) v (cid:107) H , which implies (cid:107) v (cid:107) H (cid:46) h , | F (cid:107) | (cid:46) h , on [0 , h − ] . Iterating the argument δ log(1 /h ) times gives a slightly weaker bound for longer times.The O ( h ) errors in the ODEs can be removed without affecting the bound on v ,proving the theorem.In the proofs various facts due to complete integrability (such as the miraculousLemma 2.1) simplify the arguments, in particular in the above energy estimate.We conclude with the remark about the correction term added to v in order toimprove the bound on (cid:107) F ⊥ (cid:107) from h to h . A similar correction term was used in [19]for NLS 1-solitons. Together with the symplectic projection interpretation, it was thekey to sharpening the results in earlier works. Implementing the same idea in thesetting of 2-solitons is more subtle. The 2-soliton is treated as if it were the sum oftwo decoupled 1-solitons, the corrections are introduced for each piece, and the resultis that F ⊥ is corrected so that (cid:107) F ⊥ (cid:107) H (cid:46) h + h e − γ | a − a | That is, when | a − a | = O (1), there is no improvement. However, this happens onlyon an O (1) time scale and hence does not spoil the long time estimate. FFECTIVE DYNAMICS OF DOUBLE SOLITONS FOR PERTURBED MKDV 11
Numerical experiments.
Unlike NLS, KdV is a very friendly equation fromthe numerical point of view and
MATLAB is sufficient for producing good results.We first describe the simple codes on which our experiments are based. Instead ofconsidering (1.1) on the line, we consider it on the circle identified with [ − π, π ). Tosolve it numerically we adapt the code given in [34, Chapter 10] which is based onthe Fast Fourier Transform in x , the method of integrating factor for the − u xxx (cid:55)→− ik ˆ u ( k ) term, and the fourth-order Runge-Kutta formula for the resulting ODE intime. Unless the amplitude of the solution gets large (which results in large terms inthe equation due to the u term) it suffices to take 2 N , N = 8, discretization pointsin x .For X ∈ [ − π, π ) we consider B ( X, T ) periodic in X , and compute U ( X, T ) satis-fying ∂ T U = − ∂ X ( ∂ X U + 2 U − B ( X, T ) U ) , U ( π, T ) = U ( − π, T ) . A simple rescaling, u ( x, t ) = αU ( αx, α t ) , b ( x, t ) = α B ( αx, α t ) , gives a solution of (1.1) on [ − π/α, π/α ] with periodic boundary conditions. When α is small this is a good approximation of the equation on the line. If we use U ( X, T )in our numerical calculations with the initial data q ( X, A, C ), A ∈ R , C ∈ R \ C ,the initial condition on for u ( x, t ) is given by u ( x,
0) = q ( x, A/α, αC ) . If we want ¯ c = αC to satisfy the assumptions (1.2), the effective small constant h becomes h = α and b in (1.1) becomes b ( x, t ) = h B ( x, h t ) . In principle we have three scales: size of B , size of ∂ x B , and size of ∂ t B , whichshould correspond to three small parameters h . For simplicity we just use one scale h in the Theorem.Figure 1 shows four examples of evolution and comparison with effective dynamicscomputed using the MATLAB codes available at [17]. The external potentials used aregiven by B ( x, t ) = 100 cos ( x − t ) −
50 sin(2 x + 10 t ) ,B ( x, t ) = 100 cos ( x − t ) + 50 sin(2 x + 10 t ) ,B ( x, t ) = 60 cos ( x + 1 − t ) + 40 sin(2 x + 2 + 10 t ) ,B ( x, t ) = 40 cos(2 x + 3 − t ) + 30 sin( x + 1 + 10 t ) . (1.15)The rescaling the fixed size potential used in the theorem, b ( x, t ) = h B ( x, h t ),means that our h satisfies h (cid:39) / x are different than the ones in t : the potential is not slowly varying in t if h (cid:39) /
10. The agreement with the main theorem is very good in all cases.However, the theorem in the current version does not apply to the two bottom figuressince the condition in (1.12) is not satisfied for the full time of the experiment. Seealso Fig. 3 and Appendix B.We have not exploited numerical experiments in a fully systematic way but thefollowing conclusions can be deduced: • For the case covered by our theorem the agreement with the numerical solutionis remarkably close; the same thing is true for times longer than T /h , with T defined by (1.12) despite the crossings of C j ’s (resulting in the avoidedcrossing of c j ’s) The agreement is weaker but the experiments involve onlyrelatively large value of h . • The soliton profile persists for long times but we see a deviation from theeffective dynamics. This suggest the optimality of the bound log(1 /h ) /h in(1.3). • The slow variation in t required in the theorem can probably be relaxed.For instance, in the top plots in Fig.1 max | ∂ t b | / max | ∂ x b | ∼
10, while theagreement with the effective dynamics is excellent. For longer times it doesbreak down as can be seen using the
Bmovie.m code presented in [17, § • When the decoupled equations (1.11) predict crossing of C j ’s, we observe anavoided crossing of c j ’s – see Fig.3 and Fig.6 – with exponentially small width,exp( − /Ch ). At such times we also see the crossing of a j ’s, though it reallycorresponds to solitons changing their scale constants – see Fig.7. To havemultiple interactions of a pair of solitons, this type of crossing has to occur,and it needs to be investigated further.1.5. Acknowledgments.
The authors gratefully acknowledge the following sourcesof funding: J.H. was supported in part by a Sloan fellowship and the NSF grantDMS-0901582, G.P’s visit to Berkeley in November of 2008 was supported in partby the France-Berkeley Fund, and M.Z. was supported in part by the NSF grantDMS-0654436.2.
Hamiltonian structure and conserved quantities
The symplectic form, at first defined on S ( R ; R ) is given by(2.1) ω ( u, v ) def = (cid:104) u, ∂ − x v (cid:105) , (cid:104) f, g (cid:105) = (cid:90) f g , where ∂ − f ( x ) def = 12 (cid:18)(cid:90) x −∞ − (cid:90) + ∞ x (cid:19) f ( y ) dy FFECTIVE DYNAMICS OF DOUBLE SOLITONS FOR PERTURBED MKDV 13
Then the mKdV (equation (1.1) with b ≡
0) is the Hamiltonian flow ∂ t u = ∂ x H (cid:48) ( u )and (1.1) is the Hamiltonian flow ∂ t u = ∂ x H (cid:48) b ( u ), where H = 12 (cid:90) ( u x − u ) H b = 12 (cid:90) ( u x − u + bu )Solutions to mKdV have infinitely many conserved integrals and the first four aregiven by I ( u ) = (cid:90) u dx ,I ( u ) = (cid:90) u dx ,I ( u ) = (cid:90) ( u x − u ) dx ,I ( u ) = (cid:90) ( u xx − u x u + 2 u ) dx , which are the mass, momentum, energy, and second energy, respectively. In thispaper we will only use these particular conserved quantities.We write I j ( u ) = (cid:82) A j ( u ), which means that A j ( u ) denotes the j -th Hamiltonian density .For future reference, we record the expressions appearing in the Taylor expansionsof these densities,(2.2) A j ( q + v ) = A j ( q ) + A (cid:48) j ( q )( v ) + 12 A (cid:48)(cid:48) ( q )( v, v ) + O ( v ) ,A (cid:48) ( q )( v ) = 2 qv ,A (cid:48) ( q )( v ) = 2 q x v x − q v ,A (cid:48) ( q )( v ) = 2 q xx v xx − q x q v x − q x qv + 12 q v , and A (cid:48)(cid:48) ( q )( v, v ) = 2 v ,A (cid:48)(cid:48) ( q )( v, v ) = 2 v x − q v ,A (cid:48)(cid:48) ( q )( v, v ) = 2 v xx − q v x − q x v − qq x vv x + 60 q v . The differentials, I (cid:48) j ( q ), are identified with functions by writing: (cid:104) I (cid:48) j ( q ) , v (cid:105) = (cid:90) A (cid:48) j ( q )( v ) . It is useful to record a formal expression for I (cid:48) j ( q )’s valid when A j ( q )’s are polynomialsin ∂ (cid:96)x q :(2.3) I (cid:48) j ( q ) = (cid:88) (cid:96) ≥ ( − ∂ x ) (cid:96) ∂A j ( q ) ∂q ( (cid:96) ) x , q ( (cid:96) ) x = ∂ (cid:96)x q . The Hessians, I (cid:48)(cid:48) j ( q ), are the (self-adjoint) operators given by (cid:104) I (cid:48)(cid:48) j ( q ) v, v (cid:105) = (cid:90) A (cid:48)(cid:48) j ( q )( v, v ) . One way to generate the mKdV energies is as follows (see Olver [29]). Let us putΛ( u ) = − ∂ x − u − u x ∂ − x u , and recall that Λ( u ) ∂ x is skew-adjoint:Λ( u ) ∂ x = − ∂ x − u ∂ x − u x ∂ − x u∂ x = − ∂ x − u ∂ x − u x u + 4 u x ∂ − x u x , where we used the formal integration by parts ∂ − x ( uf x ) = − ∂ − x ( u x f ) + uf .With this notation we have the fundamental recursive identity:(2.4) ∂ x I (cid:48) k +1 ( u ) = Λ( u ) ∂ x I (cid:48) k − ( u ) , which together with skew-adjointness of Λ( u ) ∂ x shows that (cid:104) I (cid:48) j ( u ) , ∂ x I (cid:48) k ( u ) (cid:105) = (cid:104) I (cid:48) j − ( u ) , ∂ x I (cid:48) k +2 ( u ) (cid:105) , for j and k odd (if we use (2.4) with m even the choice I m ( u ) = 0, for m > (cid:104) I (cid:48) j ( u ) , ∂ x I (cid:48) k ( u ) (cid:105) = 0 , ∀ j , k . In fact, since j and k are odd we can iterate all the way down to j = 1 and apply(2.3): (cid:104) I (cid:48) ( u ) , ∂ x I (cid:48) k + j − ( u ) (cid:105) = −(cid:104) ∂ x u ( (cid:96) ) x , (cid:88) (cid:96) ≥ ∂A j + k − ( u ) /∂u ( (cid:96) ) x (cid:105) = − (cid:90) ∂ x ( A j + k − ( u )) dx = 0 . If u solves mKdV, then ∂ t u = ∂ x I (cid:48) ( u ) and hence by (2.5) we obtain ∂ t I j ( u ) = (cid:104) I (cid:48) j ( u ) , ∂ t u (cid:105) = 12 (cid:104) I (cid:48) j ( u ) , ∂ x I (cid:48) ( u ) (cid:105) = 0 . The following identities related to the conservation laws will be needed in §
9. Re-calling the definition (2.2) of A j , we have: FFECTIVE DYNAMICS OF DOUBLE SOLITONS FOR PERTURBED MKDV 15
Lemma 2.1.
For any function u ∈ S , and for b ∈ C ∞ ∩ S (cid:48) , we have (cid:104) I (cid:48) ( u ) , ( bu ) x (cid:105) = (cid:104) b x , A ( u ) (cid:105)(cid:104) I (cid:48) ( u ) , ( bu ) x (cid:105) = 3 (cid:104) b x , A ( u ) (cid:105) − (cid:104) b xxx , A ( u ) (cid:105)(cid:104) I (cid:48) ( u ) , ( bu ) x (cid:105) = 5 (cid:104) b x , A ( u ) (cid:105) − (cid:104) b xxx , A ( u ) (cid:105) + (cid:104) b xxxxx , A ( u ) (cid:105) Proof.
By taking arbitrary b ∈ S , we see that the claimed formulae are equivalent to u∂ x I (cid:48) ( u ) = ∂ x A ( u ) ,u∂ x I (cid:48) ( u ) = 3 ∂ x A ( u ) − ∂ x A ( u ) ,u∂ x I (cid:48) ( u ) = 5 ∂ x A ( u ) − ∂ x A ( u ) + ∂ x A ( u ) , and these can be checked by direct computation. (cid:3) Lemma 2.2.
For any function u, q ∈ S , and for b ∈ C ∞ ∩ S (cid:48) , we have (cid:104) I (cid:48)(cid:48) ( q ) v, ( bq ) x (cid:105) − (cid:104) ∂ x I (cid:48) ( q ) , bv (cid:105) = (cid:104) b x , A (cid:48) ( q )( v ) (cid:105)(cid:104) I (cid:48)(cid:48) ( q ) v, ( bq ) x (cid:105) − (cid:104) ∂ x I (cid:48) ( q ) , bv (cid:105) = 3 (cid:104) b x , A (cid:48) ( q )( v ) (cid:105) − (cid:104) b xxx , A (cid:48) ( q )( v ) (cid:105)(cid:104) I (cid:48)(cid:48) ( q ) v, ( bq ) x (cid:105) − (cid:104) ∂ x I (cid:48) ( q ) , bv (cid:105) = 5 (cid:104) b x , A (cid:48) ( q )( v ) (cid:105) − (cid:104) b xxx , A (cid:48) ( q )( v ) (cid:105) + (cid:104) b xxxxx , A (cid:48) ( q )( v ) (cid:105) Proof.
Differentiate the formulæ in Lemma 2.1 with respect to u at q in the directionof v . (cid:3) Double soliton profile and properties
Here we record some properties of mKdV and its double soliton solutions. Theparametrization of the family of double solitons follows the presentation for NLS inFaddeev–Takhtajan [12].The double-soliton is defined in terms of the profile q ( x, a, c ), where a = ( a , a ) ∈ R , c = ( c , c ) ∈ R \ C , C def = { ( c , c ) : c = ± c } ∪ R × { } ∪ { } × R . (3.1)The profile q = q (from now on we drop the subscript 2) is defined by(3.2) q ( x, a, c ) = det M det M where M = [ M ij ] ≤ i,j ≤ , M ij = 1 + γ i γ j c i + c j , M = M γ γ and γ j = ( − j − exp( − c j ( x − a j )) , j = 1 , . For conveninece we will consider the 0 < c < c connected component of R \ C throughout the paper. Since q ( x, a , a , c , c ) = − q ( x, a , a , c , c ) ,q ( x, a , a , − c , − c ) = − q ( − x, − a , − a , c , c ) , the only other component to consider would be, say, 0 < − c < c (see Fig.2), andthe analysis is similar.We should however mention that in numerical experiments it is more useful tointroduce a phase parameter (cid:15) = ( (cid:15) , (cid:15) ), (cid:15) j = ±
1, and define ˜ q ( x, a, c, (cid:15) ) by (3.2) butwith γ j ’s replaced by˜ γ j = ( − j − (cid:15) j exp( − c j ( x − a j )) , j = 1 , . We can then check that ˜ q ( x, a, c, (cid:15) ) = q ( x, a, ( (cid:15) c , (cid:15) c )) , but ˜ q seems more stable in numerical calculations.The corresponding double-soliton (3.3) u ( x, t ) = q ( x, a + c t, a + c t, c , c )is an exact solution to mKdV. For the double soliton this can be checked by an explicitcalculation but it is a consequence of the inverse scattering method. This is the onlyplace in this paper where we appeal directly to the inverse scattering method. Fig. 4illustrates some aspects of this evolution.The scaling properties of mKdV imply that q ( x + t, a + ( t, t ) , c ) = q ( x, a, c ) ,q ( tx, ta, c/t ) = q ( x, a, c ) /t . (3.4)Both properties also follow from the formula for q , with the second one being slightlyless obvious: q ( tx, ta, c/t ) = 1det tM det tM γ γ = 1det tM det t t
000 0 1 M /t = q ( x, a, c ) /t . FFECTIVE DYNAMICS OF DOUBLE SOLITONS FOR PERTURBED MKDV 17 ! ! ! ! ! ! ! ! ! Figure 4.
A depiction of the double soliton solution given by (3.3).The top figure shows the evolution of a double soliton. The bottom twofigures show the evolution of its two components defined using (3.11).One possible “particle-like” interpretation of the two soliton interaction[4] is that the slower soliton, shown in the left bottom plot is hit bythe fast soliton shown in the right bottom plot. Just like billiard balls,the slower one picks up speed, and the fast one slows down. But unlikebilliard balls, the solitons simply switch velocities.Now we discuss in more detail the properties of the profile q . Recalling that wesuppose that c > c >
0, let(3.5) α = 1 c log (cid:18) c + c c − c (cid:19) , α = 1 c log (cid:18) c − c c + c (cid:19) , noting that for c > c > α > α <
0. Fix a smooth function, θ ∈ C ∞ ( R , [0 , θ ( s ) = (cid:26) s ≤ − , − s ≥ . ! ! ! ! ! ! ! ! ! ! ! ! ! ! a2=3,c1=3,c2=5Single soliton with c=5Single soliton with c=3Double soliton with a2= ! a1=3,c1=3,c2=5Single soliton with c=3Single soliton with c=5 8 10 12 14 16 180123456 Evolution of the above data at t = 0.75 Figure 5.
The top plots show show q ( x, , , ∓ , ± η ( x, ˆ a j , c j ) given by Lemma 3.2. The bottom plots show the post-interaction pictures at times ∓ .
75. Since the sign of a − a changesafter the interaction we see the shift compared to the evotion of η ( x, ˆ a j , c j )’s.Define the shifted positions as(3.7) ˆ a j def = a j + α j θ ( a − a )that is, ˆ a j = (cid:26) a j + α j , a (cid:28) a ,a j − α j , a (cid:29) a . see Fig. 5. We note that ˆ a j = ˆ a j ( a j , c , c ). FFECTIVE DYNAMICS OF DOUBLE SOLITONS FOR PERTURBED MKDV 19
Let S denote the Schwartz space. We will next introduce function classes S sol and S err , and then show that q ∈ S sol and give an approximate expression for q with errorin S err . Definition 3.1.
Let S err denote the class of functions, ϕ = ϕ ( x, a, c ) , x ∈ R , a ∈ R , < δ < c < c − δ < /δ (for any fixed δ ) satisfying (cid:12)(cid:12) ∂ (cid:96)x ∂ kc ∂ pa ϕ (cid:12)(cid:12) ≤ C exp( − ( | x − a | + | x − a | ) /C ) , where C j depend on δ , (cid:96) , k , and p only.Let S sol denote the class of functions of ( x, a, c ) of the form p ( c , c ) ϕ ( c ( x − ˆ a )) + p ( c , c ) ϕ ( c ( x − ˆ a )) + ϕ ( x, a, c ) where (1) | ∂ (cid:96)k ϕ j ( k ) | ≤ C (cid:96) exp( −| k | /C ) , for some C , (2) p j ∈ C ∞ ( R \ C ) . (3) ϕ ∈ S err . Some elementary properties of S sol and S err are given in the following. Lemma 3.1 (properties of S err ) . (1) ∂ x S err ⊂ S err , ∂ a j S err ⊂ S err , ∂ c j S err ⊂ S err . (2) ( x − a j ) S err ⊂ S err and ( x − ˆ a j ) S err ⊂ S err . (3) If f ∈ S err and (cid:82) + ∞−∞ f = 0 , then ∂ − x f ∈ S err . The class S err allows to formulate the following Lemma 3.2 (asymptotics for q ) . Suppose that < c < c < c /(cid:15) < /(cid:15) , for (cid:15) > .Then for | a − a | ≥ C / ( c + c ) , (3.8) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ (cid:96)x ∂ kc ∂ pa (cid:32) q ( x, a, c ) − (cid:88) j =1 η ( x, ˆ a j , c j ) (cid:33)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C exp( − ( | x − a | + | x − a | ) /C ) , where C depends on k, (cid:96), p and (cid:15) , and C , C on (cid:15) only. In other words, q ( x, a, c ) − (cid:88) j =1 η ( x, ˆ a j , c j ) ∈ S err . Corollary 3.3. ∂ − x ∂ a j q , ∂ − x ∂ c j q ∈ S sol .Proof. By Lemma 3.2, we have ∂ c j q = ∂ c j (cid:88) j =1 η ( · , ˆ a j , c j ) + f where f ∈ S err . By direct computation with the η terms, we find that (cid:90) + ∞−∞ ∂ c j (cid:88) j =1 η ( · , ˆ a j , c j ) = 0 . By the remark in Lemma 3.5, we have (cid:82) + ∞−∞ ∂ c j q = 0. Hence (cid:82) + ∞−∞ f = 0. By Lemma3.1(3), we have ∂ − x f ∈ S err . Hence ∂ − x ∂ c j q = ∂ − x ∂ c j (cid:88) j =1 η ( · , ˆ a j , c j ) + S err and the right side is clearly in S sol . (cid:3) Proof of Lemma 3.2.
We define(3.9) Q ( x, α, δ ) def = q ( x, − α, α, − δ, δ ) , so that, using (3.4), q ( x, a , a , c , c ) = c + c Q (cid:18)(cid:18) c + c (cid:19) (cid:18) x − a + a (cid:19) , α, δ (cid:19) ,α = (cid:18) c + c (cid:19) (cid:18) a − a (cid:19) , δ = c − c c + c . (3.10)Hence it is enough to study the more symmetric expression (3.9). We decompose itin the same spirit as the decomposition of double solitons for KdV was performed in[4]:(3.11) Q ( x, α, δ ) = τ ( x, α, δ ) + τ ( − x, − α, δ ) , where(3.12) τ ( x, α, δ ) = 12 (1 + δ ) exp((1 − δ )( x + α )) + (1 − δ ) exp((1 + δ )( x − α )) δ sech ( x − δα ) + δ − cosh ( δx − α ) . This follows from a straightforward but tedious calculation which we omit.Thus, to show (3.8) we have to show that | ∂ (cid:96)x ∂ pα ∂ kδ ( τ ( x, α, δ ) − η ( x − | α | − log(1 /δ ) / (1 ± δ ) , ± δ )) |≤ C exp( − ( | x | + | α | ) /C ) , ± α (cid:29) , (3.13)uniformly for 0 < δ ≤ − (cid:15) .To see this put γ = (1 − δ ) / (1 + δ ), and multiply the numerator and denominatorof (3.12) by e − (1+ δ )( x − α ) :(3.14) τ ( x, α, δ ) = 2(1 − δ ) (cid:0) γ − e α − δx (cid:1) δe (1 − δ )( x + α ) (1 − e − x +2 δα ) + δ − e − (1 − δ )( x + α ) (1 + e − δx +2 α ) . FFECTIVE DYNAMICS OF DOUBLE SOLITONS FOR PERTURBED MKDV 21
Similarly, the multiplication by e − (1+ δ )( x − α ) gives τ ( x, α, δ ) = 2(1 + δ ) (cid:0) γe − α +2 δx (cid:1) δe (1+ δ )( x − α ) (1 − e − x +2 δα ) + δ − e − (1+ δ )( x − α ) (1 + e − δx − α ) = 2(1 + δ ) (cid:0) γe − α +2 δx (cid:1) (1 + e − δx − α ) − δe (1+ δ )( x − α ) ((1 − e − x +2 δα ) / (1 + e − δx − α )) + δ − e − (1+ δ )( x − α ) . (3.15)This shows that for negative values of x , τ is negligible: multiplying the numeratorand denominator by δ and using (3.14) for α ≤ α ≥
0, gives(3.16) τ ( x, α, δ ) ≤ (cid:26) δ (1 + δ )(1 + e − | α | + δ | x | ) ) e − (1+ δ )( | x | + | α | ) , α ≥ ,δ (1 + δ )(1 + e δ | x |− | α | ) − e − (1 − δ )( | x | + | α | ) , α ≤ , and in fact this is valid uniformly for 0 ≤ δ ≤
1. Similar estimates hold also forderivatives.For x ≥
0, 0 ≤ δ ≤ − (cid:15) , and for α (cid:28) −
1, we use (3.14) to obtain, τ ( x, α, δ ) = (1 − δ ) sech (cid:18) (1 − δ ) (cid:18) x − | α | − − δ log 1 δ (cid:19)(cid:19) + (cid:15) − ( x, α, δ ) , and for α (cid:29)
1, (3.15): τ ( x, α, δ ) = (1 + δ ) sech (cid:18) (1 + δ ) (cid:18) x − | α | −
11 + δ log 1 δ (cid:19)(cid:19) + (cid:15) + ( x, α, δ ) , where | ∂ kx (cid:15) ± | ≤ C k exp( − ( | x | + | α | ) /c ) , c > , uniformly in δ , 0 < δ < − (cid:15) . Inserting the resulting decomposition into (3.10)completes the proof. (cid:3) Lemma 3.4 (fundamental identities for q ) . With q = q ( · , a, c ) , we have (3.17) ∂ x I (cid:48) ( q ) = 2 ∂ x ( − ∂ x q − q ) = 2 (cid:88) j =1 c j ∂ a j q , (3.18) ∂ x I (cid:48) ( q ) = 2 ∂ x q = − (cid:88) j =1 ∂ a j q , (3.19) q = (cid:88) j =1 ( x − a j ) ∂ a j q + (cid:88) j =1 c j ∂ c j q . These three identities are analogues of the following three identities for the single-soliton η = η ( · , a, c ), which are fairly easily verified by direct inspection. ∂ x I (cid:48) ( η ) = ∂ x η = − ∂ a η ∂ x I (cid:48) ( η ) = ∂ x ( − ∂ x η − η ) = c ∂ a ηη = ( x − a ) ∂ a η + c∂ c η Proof.
The first identity is just the statement that (3.3) solves mKdV and we take iton faith from the inverse scattering method (or verify it by a computation). To see(3.18) and (3.19) we differentiate (3.4) with respect to t . (cid:3) The value of I j ( q ) for all j is recorded in the next lemma. Lemma 3.5 (values of I j ( q )) . (3.20) I ( q ) = 2 π For j = 1 , , , we have (3.21) I j ( q ) = 2( − j − c j + c j j . Also, (3.22) (cid:90) xq ( x, a, c ) dx = 2 a c + 2 a c . Note that by (3.20) , (cid:90) + ∞−∞ ∂ a j q = 0 , (cid:90) + ∞−∞ ∂ c j q = 0 , j = 1 , . from which it follows that ∂ − x ( ∂ a j q ) and ∂ − x ( ∂ c j q ) are Schwartz class functions.Proof. We prove (3.21), (3.20) by reduction to the 1-soliton case. Let u ( t ) = q ( · , a + tc , a + tc , c , c ). Then by the asymptotics in Lemma 3.2, I j ( q ) = I j ( u (0)) = I j ( u ( t )) = (cid:88) k =1 I j ( η ( · , ( a k + c k t )ˆ , c k )) + ω ( t )where | ω ( t ) | (cid:46) (cid:104) c (( a + tc ) − ( a + tc )) (cid:105) − But note that by scaling, I j ( η ( · , ( a k + c k t )ˆ , c k )) = c jk I j ( η )By sending t → + ∞ , we find that I j ( q ) = ( c j + c j ) I j ( η )To compute I j ( η ), we let η c ( x ) = cη ( cx ). By scaling I j ( η c ) = c j I j ( η ). Hence jI j ( η ) = ∂ c (cid:12)(cid:12) c =1 I j ( η c ) = (cid:104) I (cid:48) j ( η ) , ∂ c (cid:12)(cid:12) c =1 η c (cid:105) = (cid:104) I (cid:48) j ( η ) , ( xη ) x (cid:105) = 2( − j − (cid:104) η, ( xη ) x (cid:105) = 2( − j − , FFECTIVE DYNAMICS OF DOUBLE SOLITONS FOR PERTURBED MKDV 23 where we have used the identity(3.23) I (cid:48) j ( η ) = 2( − j − η , which follows from the energy hierarchy. In fact, I (cid:48) ( η ) = 2 η is just the definition of I (cid:48) . Assuming that I (cid:48) j ( η ) = 2( − j − η , we compute ∂ x I (cid:48) j +2 ( η ) = Λ( η ) ∂ x I (cid:48) j ( η )= 2( − − j − ( ∂ x + 4 η + 4 η x ∂ − x η ) η x = 2( − j +12 ∂ x ( η xx + 2 η )= 2( − j +12 ∂ x η We now prove (3.22). By direct computation, if u ( t ) solves mKdV, then ∂ t (cid:82) xu = − I ( u ). Again let u ( t ) = q ( · , a + tc , a + tc , c , c ). By (3.21) with j = 3, we have (cid:90) xq ( x, a, c ) dx = (cid:90) xu (0 , x ) dx = (cid:90) xu ( t, x ) dx − c + c ) t By the asymptotics in Lemma 3.2, (cid:90) xu ( t, x ) = (cid:88) j =1 (cid:90) xη ( x, ( a j + tc j )ˆ , c j ) + ω ( t )where | ω ( t ) | ≤ ( a + tc ) (cid:104) c (( a + c t ) − ( a + tc )) (cid:105) − But (cid:90) xη ( x, ˆ a j , c j ) = 2 c j ˆ a j Combining, and using that c ˆ a + c ˆ a = c a + c a , we obtain (cid:90) xq ( x, a, c ) dx = 2( c a + c a ) + ω ( t )Send t → + ∞ to obtain the result. (cid:3) We define the four-dimensional manifold of 2-solitons M as M = { q ( · , a, c ) | a = ( a , a ) ∈ R , c = ( c , c ) ∈ ( R ) \ C } Lemma 3.6.
The symplectic form (2.1) restricted to the manifold of -olitons is givenby ω | M = (cid:88) j =1 da j ∧ dc j . In particular, it is nondegenerate and M is a symplectic manifold. Proof.
By (3.21) with j = 1 and (3.18),0 = 12 ∂ a I ( q ) = 12 (cid:104) I (cid:48) ( q ) , ∂ a q (cid:105) = (cid:104) ∂ a q, ∂ − x ∂ a q (cid:105) + (cid:104) ∂ a q, ∂ − x ∂ a q (cid:105) = (cid:104) ∂ a q, ∂ − x ∂ a q (cid:105) Again by (3.21) with j = 1 and (3.18),(3.24) 1 = 12 ∂ c I ( q ) = 12 (cid:104) I (cid:48) ( q ) , ∂ c q (cid:105) = (cid:104) ∂ a q, ∂ − x ∂ c q (cid:105) + (cid:104) ∂ a q, ∂ − x ∂ c q (cid:105) By (3.21) with j = 3 and (3.17),(3.25) − c = 12 ∂ c I ( q ) = 12 (cid:104) I (cid:48) ( q ) , ∂ c q (cid:105) = − c (cid:104) ∂ a q, ∂ − x ∂ c q (cid:105) − c (cid:104) ∂ a q, ∂ − x ∂ c q (cid:105) Solving (3.24) and (3.25), we obtain that (cid:104) ∂ a q, ∂ − x ∂ c q (cid:105) = 1 and (cid:104) ∂ a q, ∂ − x ∂ c q (cid:105) = 0.We similarly obtain that (cid:104) ∂ a q, ∂ − x ∂ c q (cid:105) = 1 and (cid:104) ∂ a q, ∂ − x ∂ c q (cid:105) = 0. It remains toshow that (cid:104) ∂ c q, ∂ − x ∂ c q (cid:105) = 0: (cid:104) ∂ c q, ∂ − x ∂ c q (cid:105) = 1 c (cid:104) (cid:88) j =1 c j ∂ c j q, ∂ − x ∂ c q (cid:105) = 1 c (cid:104) q − (cid:88) j =1 ( x − a j ) ∂ a j q, ∂ − x ∂ c q (cid:105) by (3.19)= 1 c (cid:104) q + xq x , ∂ − x ∂ c q (cid:105) + 1 c (cid:88) j =1 a j (cid:104) ∂ a j q, ∂ − x ∂ c q (cid:105) by (3.18)= − c ∂ c (cid:90) xq + a c = 0 by (3.22) (cid:3) Remark. If | a − a | (cid:29)
2, and c < c then, in the notation of (3.7), (cid:88) j =1 , da j ∧ dc j = (cid:88) j =1 , d ˆ a j ∧ dc j , that is the map ( a, c ) (cid:55)→ (ˆ a, c ) is symplectic.The nondegeneracy of the symplectic form (2.1) restricted to the manifold of 2-olitons, M shows that H functions close to M can be uniquely decomposed intoan element q , of M and a function symplectically orthogonal T q M . We recall thisstandard fact in the following Lemma 3.7 (Symplectic orthogonal decomposition) . Given ˜ c , there exist constants δ > , C > such that the following holds. If u = q ( · , ˜ a, ˜ c ) + ˜ v with (cid:107) ˜ v (cid:107) H ≤ δ , then FFECTIVE DYNAMICS OF DOUBLE SOLITONS FOR PERTURBED MKDV 25 there exist unique a , c such that | a − ˜ a | ≤ C (cid:107) ˜ v (cid:107) H , | c − ˜ c | ≤ C (cid:107) ˜ v (cid:107) H and v def = u − q ( · , a, c ) satisfies (3.26) (cid:104) v, ∂ − x ∂ a j q (cid:105) = 0 and (cid:104) v, ∂ − x ∂ c j q (cid:105) = 0 , j = 1 , . Proof.
Let ϕ : H × R × ( R + ) → R be defined by ϕ ( u, a, c ) = (cid:104) u − q ( · , a, c ) , ∂ − x ∂ a q (cid:105)(cid:104) u − q ( · , a, c ) , ∂ − x ∂ a q (cid:105)(cid:104) u − q ( · , a, c ) , ∂ − x ∂ c q (cid:105)(cid:104) u − q ( · , a, c ) , ∂ − x ∂ c q (cid:105) Using that ω (cid:12)(cid:12) M = da ∧ dc + da ∧ dc , we compute the Jacobian matrix of ϕ withrespect to ( a, c ) at ( q ( · , ˜ a, ˜ c ) , ˜ a, ˜ c ) to be D a,c ϕ ( q ( · , ˜ a, ˜ c ) , ˜ a, ˜ c ) = . By the implicit function theorem, the equation ϕ ( u, a, c ) = 0 can be solved for ( a, c )in terms of u in a neighbourhood of q ( · , ˜ a, ˜ c ). (cid:3) We also record the following lemma which will be useful in the next section:
Lemma 3.8.
Suppose v solves a linearized equation ∂ t v = 12 ∂ x I (cid:48)(cid:48) ( q ( t )) v = ∂ x ( − ∂ x − q ( t ) ) v , q ( x, t ) = q ( x, a j + tc j , c j ) . Then ∂ t (cid:104) v ( t ) , ∂ − x ( ∂ c j q )( t ) (cid:105) = ∂ t (cid:104) v ( t ) , ∂ − x ( ∂ a j q )( t ) (cid:105) = 0 , where ( ∂ c j q )( t ) = ( ∂ c j q )( x, a j + tc j , c j ) (and not ∂ c j ( q ( x, a j + tc j , c j )) ). In addition,for v (0) = ∂ a j q , v ( t ) = ( ∂ a j q )( t ) , and for v (0) = ∂ c j q , v ( t ) = ( ∂ c j q )( t ) + 2 c j t ( ∂ a j q )( t ) . Lyapunov functional and coercivity
In this section we introduce the function H c adapted from the KdV theory ofMaddocks-Sachs [24]. We will build our Lyapunov functional E from H c .Thus let H c ( u ) def = I ( u ) + ( c + c ) I ( u ) + c c I ( u ) . We give a direct proof that q ( · , a, c ) is a critical point of H c : Lemma 4.1 ( q is a critical point of H ) . We have (4.1) H (cid:48) c ( q ( · , a, c )) = 0 , that is I (cid:48) ( q ) + ( c + c ) I (cid:48) ( q ) + c c I (cid:48) ( q ) = 0 . Proof.
We follow Lax [23, § A = A ( q ) and B = B ( q ) such that H (cid:48) ( q ) def = I (cid:48) ( q ) + AI (cid:48) ( q ) + BI (cid:48) ( q ) = 0 , for all q = q ( x, a, c ) ∈ M . If we consider the mKdV evolution of q given by (3.3),then Lemma 3.2 shows that as t → ±∞ we can express H (cid:48) ( q ) asymptotically using H (cid:48) ( η c ) and H (cid:48) ( η c ). From (3.23) we see that H (cid:48) ( η c ) = I (cid:48) ( η c ) + AI (cid:48) ( η c ) + BI (cid:48) ( η c ) = 2( c − Ac + B ) η c . Two parameters c and c are roots of this equation if A = c + c and B = c c andthis choice gives H (cid:48) ( q ( t )) = r ( t ) , (cid:107) r ( t ) (cid:107) L ≤ C exp( −| t | /C ) ,q ( t ) def = q ( x, a + c t, a + c t, c , c ) , (4.2)where the exponential decay of r ( t ) comes from Lemma 3.2 and the fact that c (cid:54) = c .To prove (4.1) we need to show that r (0) ≡
0. For the reader’s convenience weprovide a direct proof of this widely accepted fact. Since it suffices to prove that (cid:104) r (0) , w (cid:105) = 0, for all w ∈ S , we consider the mKdV linearized equation at q ( t ),(4.3) v t = 12 ∂ x I (cid:48)(cid:48) ( q ( t )) v , v (0) = w ∈ S , and will show that(4.4) ∂ t (cid:104) r ( t ) , v ( t ) (cid:105) = ∂ t (cid:104) H (cid:48) ( q ( t )) , v ( t ) (cid:105) = 0 . The conclusion (cid:104) r (0) , w (cid:105) = 0 will the follow from showing that(4.5) (cid:104) r ( t ) , v ( t ) (cid:105) → , t → ∞ . We first claim that ∂ t (cid:104) I (cid:48) k ( q ) , v (cid:105) = 0 , ∀ k . In fact, from (2.5) we have (cid:104) I (cid:48) k ( ϕ ) , ∂ x I (cid:48) ( ϕ ) (cid:105) = 0 for all ϕ ∈ S . Differentiating withrespect to ϕ in the direction of v , we obtain (cid:104) I (cid:48)(cid:48) k ( ϕ ) v, ∂ x I (cid:48) ( ϕ ) (cid:105) = −(cid:104) I (cid:48) k ( ϕ ) , ∂ x I (cid:48)(cid:48) ( ϕ ) v (cid:105) . FFECTIVE DYNAMICS OF DOUBLE SOLITONS FOR PERTURBED MKDV 27
Applying this with v = v ( t ) and ϕ = q ( t ) we conclde that ∂ t (cid:104) I (cid:48) k ( q ) , v (cid:105) = (cid:104) I (cid:48)(cid:48) k ( q ) ∂ t q, v (cid:105) + 12 (cid:104) I (cid:48) k ( q ) , ∂ x I (cid:48)(cid:48) ( q ) v (cid:105) = 12 (cid:104) I (cid:48)(cid:48) k ( q ) ∂ x I (cid:48) ( q ) , v (cid:105) + 12 (cid:104) I (cid:48) k ( q ) , ∂ x I (cid:48)(cid:48) ( q ) v (cid:105) , = 0 . Since H is a linear combination of I k ’s, k = 1 , ,
5, this gives (4.4).We now want to use the exponential decay of (cid:107) r ( t ) (cid:107) L in (4.2), and (4.4) to show(4.5). Clearly, all we need is a subexponential estimate on v ( t ), that is(4.6) ∀ (cid:15) > ∃ t , (cid:107) v ( t ) (cid:107) L ≤ e (cid:15)t , t > t . Let ψ be a smooth function such that ψ ( x ) = 1 for all | x | ≤ ψ ( x ) ∼ e − | x | for | x | ≥
1. With the notation of Lemma 3.2 define ψ j ( x, t ) = ψ ( δ ( x − ( a j + c j t ) (cid:98) )) . for 0 < δ (cid:28) j = 1 ,
2. We now establish that(4.7) (cid:12)(cid:12)(cid:12)(cid:12) ∂ t (cid:18) (cid:107) v (cid:107) L + (cid:107) v x (cid:107) L + 6 (cid:90) q v (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:88) j =1 (cid:107) ψ j v (cid:107) L . To prove (4.7), apply ∂ − x to (4.3) and pair with v t to obtain0 = (cid:104) ∂ − x v t , v t (cid:105) + (cid:104) v xx , v t (cid:105) + (cid:104) q v, v t (cid:105) which implies(4.8) ∂ t (cid:18) (cid:107) v x (cid:107) L + 3 (cid:90) q v (cid:19) = 6 (cid:90) qq t v Next, pair (4.3) with v to obtain0 = (cid:104) v t , v (cid:105) + (cid:104) v xxx , v (cid:105) + 6 (cid:104) ∂ x ( q v ) , v (cid:105) which implies(4.9) ∂ t (cid:107) v (cid:107) L = − (cid:90) qq x v Summing (4.8) and (4.9) gives (4.7).The inequality (4.7) shows that we need to control is (cid:107) ψ j v ( t ) (cid:107) , j = 1 ,
2. For t large ψ j provides a localization to the region where q decomposes into an approximate sumof decoupled solitons (see Lemma 3.2). Hence we define L j = c j − ∂ x − η ( x, ( a j + tc j ) (cid:98) , c j )(see also § t ≥ T ( δ ) = ⇒ ∂ t (cid:104)L j ψ j v, ψ j v (cid:105) = O ( δ ) (cid:107) v (cid:107) H , where T ( δ ) is large enough to ensure that the supports of ψ j ’s are separated. Itsuffices to assume that v (0) = w satisfies (cid:104) w, ∂ − x ∂ a j q (cid:105) = 0 and (cid:104) w, ∂ − x ∂ c j q (cid:105) = 0, sinceLemma 3.8 already showed that the evolutions of ∂ a j q and ∂ c j q are linearly boundedin t . Under this assumption, we have by Lemma 3.8 that (cid:104) v ( t ) , ∂ − x ∂ a j q ( t ) (cid:105) = 0 and (cid:104) v ( t ) , ∂ − x ∂ c j q ( t ) (cid:105) = 0.We now want to invoke the well known coercivity estimates for operators L j – seefor instance [18, §
4] for a self contained presentation. For that we need to check that |(cid:104) ψ j v, ∂ − x ∂ a η (ˆ a j + tc j , c j ) (cid:105)| (cid:28) , |(cid:104) ψ j v, ∂ − x ( ∂ c η (ˆ a j + tc j , c j ) |(cid:105)| (cid:28) . This follows from the fact that v is symplectically orthogonal to ( ∂ c j q )( t ) and ∂ a j q ( t )(Lemma 3.8 again), the fact that q decouples into two solitons for t large, and fromthe remark after the proof of Lemma 3.6.Hence, (cid:104)L j ψ j v, ψ j v (cid:105) (cid:38) (cid:107) ψ j v (cid:107) H . We now sum (4.7) and (4.10) multiplied by δ − to obtain, for t suffieciently large(depending on δ ), F (cid:48) ( t ) ≤ Cδ F ( t ) ,F ( t ) def = (cid:107) v ( t ) (cid:107) H + 6 (cid:90) q ( t ) v ( t ) + δ − (cid:104)L j ( t ) ψ j ( t ) v ( t ) , ψ j ( t ) v ( t ) (cid:105) (where we added the additional (cid:82) q v term to the right hand side at no cost). Con-sequently, F ( t ) ≤ exp( C (cid:48) δ t ), for t > T ( δ ).We recall that this implies (4.6) and going back to (4.4) show that r (0) = 0, andhence H (cid:48) ( q ) = 0. (cid:3) We denote the Hessian of H c at q ( • , a, c ) by K c,a : K c,a = I (cid:48)(cid:48) ( q ) + ( c + c ) I (cid:48)(cid:48) ( q ) + c c I (cid:48)(cid:48) ( q )It is a fourth order self-adjoint operator on L ( R ) and a calculation shows that(4.11) 12 K c,a = ( − ∂ x + c )( − ∂ x + c )+ 10 ∂ x q ∂ x + 10( − q x + ( q ) xx + 3 q ) − c + c ) q Lemma 4.2 (mapping properties of K ) . The kernel of K c,a in L ( R ) is spanned by ∂ a j q : (4.12) K c,a ∂ a j q = 0 , and (4.13) K c,a ∂ c j q = 4( − j c j ( c − c ) ∂ − x ∂ a j q FFECTIVE DYNAMICS OF DOUBLE SOLITONS FOR PERTURBED MKDV 29
Proof.
Equations (4.12) follow from differentiation of (4.1) with respect to a j . As x → ∞ , the leading part of K c,a is given by ( − ∂ x + c )( − ∂ x + c ) and hence the kernelin L is at most two dimensional.To see (4.13) recall that I (cid:48) ( q ) = 2 q = − ∂ − x ( ∂ a q + ∂ a q ) I (cid:48) ( q ) = − q (cid:48)(cid:48) − q = 2 ∂ − x ( c ∂ a q + c ∂ a q ) , where we used Lemma 3.4. By differentiating H (cid:48) ( q ) = I (cid:48) ( q ) + ( c + c ) I (cid:48) ( q ) + c c I (cid:48) ( q ) = 0 with respect to c j , we obtain(4.14) K ( ∂ c q ) = − c ( I (cid:48) ( q ) + c I (cid:48) ( q )) , K ( ∂ c q ) = − c ( I (cid:48) ( q ) + c I (cid:48) ( q )) . Inserting the above formulæ for I (cid:48) ( q ) and I (cid:48) ( q ) gives (4.13). (cid:3) The main result of this section is the following coercivity result:
Proposition 4.3 (coercivity of K ) . There exists δ = δ ( c ) > such that for all v ∈ H satisfying the symplectic orthogonality conditions (cid:104) v, ∂ − x ∂ a j q (cid:105) = 0 and (cid:104) v, ∂ − x ∂ c j q (cid:105) = 0 , j = 1 , , we have (4.15) δ (cid:107) v (cid:107) H ≤ (cid:104)K c,a v, v (cid:105) . The proposition is proved in a few steps. In Lemma 4.2 we already described thekernel K c,a and now we investigate the negative eigenvalues: Proposition 4.4 (Spectrum of K ) . The operator K c,a has a single negative eigen-value, h ∈ L ( R ) : (4.16) K c,a h = − µh , µ > . In addition, for < δ < c < c − δ < /δ , there exists a constant, ρ , depending only on δ , such that (4.17) min { λ > λ ∈ σ ( K c,a ) } > ρ , a ∈ R , Proof.
As always we assume 0 < c < c . We know the continuous spectrum of K c,a , σ ac ( K c,a ) = [2 c c , + ∞ )and that for all a, c , there is a two-dimensional kernel given by span { ∂ a q, ∂ a q } . Theeigenvalues depend continuously on a , c , and hence the constant dimension of thekernel shows that the number of negative eigenvalues is constant (since the creationor annihilation of a negative eigenvalue would increase the dimension of ker K c,a .)Hence it suffices to determine the number of negative eigenvalues of K for anyconvenient values of a , c . To do that we use the following fact: Lemma 4.5 (Maddocks-Sachs [24, Lemma 2.2]) . Suppose that K is a self-adjoint, th order operator of the form K = 2( − ∂ x + c )( − ∂ x + c ) + p ( x ) − ∂ x p ( x ) ∂ x , where the coefficients p j ( x ) are smooth, real, and rapidly decaying as x → ±∞ . Let r ( x ) , r ( x ) be two linearly independent solutions of K r j = 0 such that r j → as x → −∞ .Then the number of negative eigenvalues of K is equal to (4.18) (cid:88) x ∈ R dim ker (cid:20) r ( x ) r (cid:48) ( x ) r ( x ) r (cid:48) ( x ) (cid:21) . We apply this lemma with K = K c,a , in which case p = 20 q , p = 40 q xx q + 20 q x + 60 q − c + c ) q , q = q ( • , a, c ) . Convenient values of a and c are provided by a = a = 0 and c = 0 . c = 1 .
5. Inthe notation of (3.9) we then have q ( x, a, c ) = Q ( x, , . ∂ x Q = − ∂ a q − ∂ a q , ∂ α Q = − ∂ a q + ∂ a q , we can take r = ∂ x Q and r = ∂ α Q . A computation based on (3.11) and (3.12)shows that Q ( x, . ,
0) = sech( x/ , ∂ x Q ( x, . ,
0) = − sinh( x/ ( x/ ,∂ α Q ( x, . ,
0) = sinh( x/ ( x/
2) (9 − ( x/ x/ ( x/
2) + ∂ x Q ( x, . , . (4.19)Since x (cid:55)→ y = sinh( x/
2) is invertible, we only need to check the dimension of thekernel the Wronskian matrix of˜ r ( y ) = y y , ˜ r ( y ) = y (1 + y ) , and that is equal to 1 at y = 0 and 0 on R \ { } . In view of (4.18) this completes theproof of (4.16)To prove (4.17) we first note that by rescaling (3.10) we only need to prove theestimate for K ( c, α ) def = K (( c, , ( − α,α )) , c ∈ [ δ, − δ ] , < δ < / . For that we introduce another operator(4.20) P ( c ) def = ( − ∂ x + 1)( − ∂ x + c ) + 10 ∂ x η ∂ x + 10(3 η − η ) − c ) η , where η = sech x , c ∈ R + \ { } . FFECTIVE DYNAMICS OF DOUBLE SOLITONS FOR PERTURBED MKDV 31
The operator P ( c ) is the Hessian of H ( c, at η , which is also a critical point for H ( c, .In particular, P ( c ) ∂ x η = 0 . Putting, U α f ( x ) def = f ( x + α + log((1 + c ) / (1 − c )))) , and P + ( c, α ) def = U ∗ α P ( c ) U α , we see that K ( c, α ) = 2 P + ( c, α ) + O ( e − ( α + | x | ) /C ) ∂ x + O ( e − ( α + | x | ) /C ) , x ≥ . Similarly, if T c f ( x ) def = √ cf ( cx ) , and P − ( c, α ) def = c U α T c P (1 /c ) T ∗ c U ∗ α , then K ( c, α ) = 2 P − ( c, α ) + O ( e − ( α + | x | ) /C ) ∂ x + O ( e − ( α + | x | ) /C ) , x ≤ . We reduce the estimate (4.17) to a spectral fact about the operators P ( c ) and P (1 /c ): Lemma 4.6.
Suppose that there exists α (cid:55)−→ λ ( c, α ) ∈ R \ { } such that λ ( c, α ) ∈ σ ( K ( c, α )) , λ ( c, α ) −→ , α −→ ∞ . Then we have (4.21) dim ker L P ( c ) + dim ker L P (1 /c ) > , where ker L means the kernel in L .Proof. The assumption that 0 (cid:54) = λ ( c, α ) → α → ∞ implies that there exists afamily of quasimodes f α , (cid:107) f α (cid:107) L = 1,(4.22) (cid:107) K ( c, α ) f α (cid:107) L = o (1) , α −→ ∞ , f α ⊥ ker L K ( c, α ) . Since we know that the kernel of K ( c, α ) is spanned by U ∗ α ∂ x η + O ( e − ( | x | + α ) /C ) and U α T c ∂ x η + O ( e − ( | x | + α ) /C ), we can modify f α and replace the orthogonality conditionby f α ⊥ span ( U ∗ α ∂ x η, U α T c ∂ x η ) . The estimate in (4.22), and (cid:107) f α (cid:107) L = O (1), imply that(4.23) (cid:107) f α (cid:107) H = O (1) , α −→ ∞ . We first claim that(4.24) (cid:90) − | f α ( x ) | dx = o (1) , α −→ ∞ . In fact, on [ − α/ , α/ K ( c, α ) = ( − ∂ x + c )( − ∂ x + 1) + O ( e − α/C ) ∂ x + O ( e − α/c ) , and hence, using (4.23),( − ∂ x + c )( − ∂ x + 1) f α = r α , (cid:107) r α (cid:107) L ([ − α/ ,α/ = o (1) . Putting e α def = [( − ∂ x + c )( − ∂ x + 1)] − (cid:0) r α [ − α/ ,α/ (cid:1) , (cid:107) e α (cid:107) H = o (1) , we see that f α = g α + e α where(4.25) ( − ∂ x + c )( − ∂ x + 1) g α ( x ) = 0 , | x | < α/ . Suppose now that (4.24) were not valid. Then the same would be true for g α , and therewould exist a constant c >
0, and a sequence α j → ∞ , for which (cid:107) g α j (cid:107) L ([ − , > c .In view of (4.25) this implies that g α j ( x ) = (cid:88) ± (cid:0) a ± j e ± x + b ± j e ± cx (cid:1) , | x | < α/ , | a ± j | , | b ± j | = O (1) , and for at least one choice of sign, | a ± j | + | b ± j | > c > . We can choose a subsequence so that this is true for a fixed sign, say, +, for all j . Inthat case, a simple calculation shows that for M j → ∞ , M j ≤ α j / (cid:90) M j | g α j ( x ) | dx ≥ | a + j | e M j + 12 c | b + j | e cM j − c + 1 | a + j || b + j | e ( c +1) M j − − c | a + j || b − j | e (1 − c ) M j − O (1) ≥ (cid:18) − c c (cid:19) (cid:18) | a + j | e M j + 1 c | b + j | e M j c (cid:19) − − c ) | a + j | e − c ) M j − O (1) , where we used the fact that 0 < δ < c < − δ . Hence (cid:107) f α j (cid:107) L ≥ (cid:90) M j | f α j ( x ) | dx ≥ (cid:90) M j | g α j ( x ) | dx − o (1) ≥ (cid:18) − c c (cid:19) c e M j c − O (1) −→ ∞ , j → ∞ . Since (cid:107) f α (cid:107) L = 1 we obtain a contradiction proving (4.24). FFECTIVE DYNAMICS OF DOUBLE SOLITONS FOR PERTURBED MKDV 33
Now let χ ± C ∞ ( R ) be supported in ± [ − , ∞ ), and satisfy χ + χ − = 1. Then (4.24)(and the corresponding estimates for derivatives obtained from (4.22)) shows that (cid:107) P ± ( c, α )( χ ± f α ) (cid:107) L = o (1) , α −→ ∞ . For at least one of the signs we must have (cid:107) χ ± f α (cid:107) L > / α is large enough),and hence we obtain a quasimode for P ± ( c, α ), orthogonal to the known element ofthe kernel of P ± ( c, α ). This means that P ± ( c, α ), for at least one of the signs has anadditional eigenvalue approaching 0 as α → ∞ . Since the spectrum of P ± ( c, α ) isindependent of α it follows that for at least one sign the kernel is two dimensional.This proves (4.21). (cid:3) The next lemma shows that (4.21) is impossible:
Lemma 4.7.
For c ∈ R + \ { } (4.26) ker L P ( c ) = C · ∂ x η . Proof.
Let L def = ( I (cid:48)(cid:48) ( η ) + I (cid:48)(cid:48) ( η )) / L v = − v xx − η v + v , η ( x ) = sech( x ) . We recall (see the comment after (4.20)) that P ( c ) = 12 H (cid:48)(cid:48) ( c, ( η ) = 12 (cid:0) I (cid:48)(cid:48) ( η ) + (1 + c ) I (cid:48)(cid:48) ( η ) + c I (cid:48)(cid:48) ( η ) (cid:1) . We already noted that L ( ∂ x η ) = P ( c ) ∂ x η = 0 , and proceeding as in (4.14) we also have(4.27) L ( ∂ x ( xη )) = − η , P ( c )( ∂ x ( xη )) = 2(1 − c ) η . We claim that(4.28) P ( c ) ∂ x L = L ∂ x P ( c )Since I (cid:48) j ( η + tv ) = tI (cid:48)(cid:48) j ( η ) v + O ( t ), v ∈ S , the equation (2.5) implies that (cid:104) I (cid:48)(cid:48) j ( η ) v, ∂ x I (cid:48)(cid:48) k ( η ) v (cid:105) = 0 , ∀ j, k , v ∈ S . From this we see that (cid:104) P ( c ) v, ∂ x L v (cid:105) = 0 , ∀ v ∈ S , and hence by polarization, (cid:104) P ( c ) v, ∂ x L w (cid:105) = −(cid:104) P ( c ) w, ∂ x L v (cid:105) = (cid:104) ∂ x P ( c ) w, L v (cid:105) . which implies (4.28).Suppose now that dim ker L P ( c ) = 2 for some c (cid:54) = 1, and let η x and ψ be the basisof this kernel. Since P ( c ) is symmetric with respect to the reflection x (cid:55)→ − x , ψ can be chosen to be either even or odd. Applying (4.28) to ψ we get P ( c ) ∂ x L ψ = 0 andhence ∂ x L ψ = αη x + βψ , for some α, β ∈ R .If ψ is odd then ∂ x L ψ is even, and therefore α = β = 0. But then ψ ∈ ker L L = C · η x , giving a contradiction.If ψ is even then ∂ x L ψ is odd, β = 0 and L ψ = αη . We have α (cid:54) = 0 since ψ isorthogonal to the kernel of L , spanned by ∂ x η . From (4.27) we obtain ψ = − α ∂ x ( xη ) . Applying the second equation in (4.27) we then obtain P ( c ) ψ = − α (1 − c ) η , contradicting ψ ∈ ker L P ( c ). (cid:3) With this lemma we complete the proof of Proposition 4.4. (cid:3)
To obtain the coercivity statement in Proposition 4.3 we first obtain coercivityunder a different orthogonality condition:
Lemma 4.8.
There exists a constant ρ > depending only on c , c , such that thefollowing holds: If (cid:104) u, ∂ − x ∂ a q (cid:105) = 0 , (cid:104) u, ∂ − x ∂ a q (cid:105) = 0 , (cid:104) u, ∂ a q (cid:105) = 0 , (cid:104) u, ∂ a q (cid:105) = 0 ,then (cid:104)K c,a u, u (cid:105) ≥ ρ (cid:107) u (cid:107) L .Proof. To simplify notation we put K = K c,a in the proof. Using (4.13) and theexpression for the symplectic form, ω (cid:12)(cid:12) M = da ∧ dc + da ∧ dc , we have (cid:104)K ∂ c q, ∂ c q (cid:105) = − c ( c − c ) (cid:104) ∂ − x ∂ a q, ∂ c q (cid:105) = 4 c ( c − c )and similarly(4.29) (cid:104)K ∂ c q, ∂ c q (cid:105) = − c ( c − c ) . Since we assumed that c < c , (cid:104)K ∂ c q, ∂ c q (cid:105) < (cid:103) ∂ c q be the orthogonal projection of ∂ c q on (ker K ) ⊥ . We first claim that thereexists a constant α such that u = ˜ u + α (cid:103) ∂ c q with (cid:104) ˜ u, h (cid:105) = 0, where µ and h are definedin Proposition 4.4.To prove this, decompose ∂ c q as ∂ c q = ξ + βh with (cid:104) ξ, h (cid:105) = 0. Then by (4.29)0 > (cid:104)K ∂ c q, ∂ c q (cid:105) = (cid:104)K ξ, ξ (cid:105) + 2 β (cid:104)K h, ξ (cid:105) + β (cid:104)K h, h (cid:105) = (cid:104)K ξ, ξ (cid:105) − µβ FFECTIVE DYNAMICS OF DOUBLE SOLITONS FOR PERTURBED MKDV 35
Since (cid:104)K ξ, ξ (cid:105) ≥
0, we must have that β (cid:54) = 0. Hence there exists u (cid:48) and α such that u = u (cid:48) + α∂ c q with (cid:104) u (cid:48) , h (cid:105) = 0. Now take ˜ u to be the projection of u (cid:48) away from thekernel of K . This completes the proof of the claim.We have that (cid:104) u, K ∂ c q (cid:105) = − c ( c − c ) (cid:104) u, ∂ − x ∂ a q (cid:105) = 0by (4.13) and hypothesis. Substituting u = ˜ u + α (cid:103) ∂ c q , we obtain(4.30) (cid:104) ˜ u, K ∂ c q (cid:105) = − α (cid:104) (cid:103) ∂ c q, K ∂ c q (cid:105) = − α (cid:104) ∂ c q, K ∂ c q (cid:105) Now let ˜ ρ denote the bottom of the positive spectrum of K . We have (cid:104)K u, u (cid:105) = (cid:104)K (˜ u + α (cid:103) ∂ c q ) , (˜ u + α (cid:103) ∂ c q ) (cid:105) = (cid:104)K ˜ u, ˜ u (cid:105) + 2 α (cid:104)K ˜ u, ∂ c q (cid:105) + α (cid:104)K ∂ c q, ∂ c q (cid:105) = (cid:104)K ˜ u, ˜ u (cid:105) − α (cid:104)K ∂ c q, ∂ c q (cid:105) by (4.30) ≥ ˜ ρ (cid:107) ˜ u (cid:107) L + 4 c ( c − c ) α ≥ ˜ C ( (cid:107) ˜ u (cid:107) L + α )where ˜ C depends on c , c and ˜ ρ . However, since u = ˜ u + α (cid:103) ∂ c q , we have (cid:107) u (cid:107) L ≤ C ( (cid:107) ˜ u (cid:107) L + α )where C depends on c , c which completes the proof. (cid:3) We now put E = E a,c = ker K = span { ∂ a q, ∂ a q } ,F = F a,c = span { ∂ − x ∂ c q, ∂ − x ∂ c q } ,G = G a,c = span { ∂ − x ∂ a q, ∂ − x ∂ a q } . (4.31)In this notation Lemma 4.8 states that u ⊥ ( E + G ) = ⇒ (cid:104)K u, u (cid:105) ≥ θ (cid:107) u (cid:107) L , while to establish Proposition 4.3 we need u ⊥ ( F + G ) = ⇒ (cid:104)K u, u (cid:105) ≥ ˜ θ (cid:107) u (cid:107) L . That is, we would like to replace orthogonality with the kernel E by orthogonalitywith a “nearby” subspace F . For this, we apply the following analysis with D = F ⊥ . Definition 4.1.
Suppose that D and E are two closed subspaces in a Hilbert space.Then α ( D, E ) , the angle between D and E , is α ( D, E ) def = cos − sup (cid:107) d (cid:107) =1 , d ∈ D (cid:107) e (cid:107) =1 , e ∈ E (cid:104) d, e (cid:105) It is clear that 0 ≤ α ( D, E ) ≤ π/ α ( D, E ) = α ( E, D ), and that α ( E, D ) = π/ E ⊥ D . We will need slightly more subtle properties stated in thefollowing Lemma 4.9.
Suppose that D and E are two closed subspaces in a Hilbert space. Then (4.32) α ( D, E ) = cos − sup (cid:107) d (cid:107) =1 ,d ∈ D (cid:107) P E d (cid:107) , α ( D, E ) = sin − inf (cid:107) d (cid:107) =1 ,d ∈ D (cid:107) P E ⊥ d (cid:107) . In addition if E is finite dimensional then (4.33) α ( D, E ) = 0 ⇐⇒ D ∩ E (cid:54) = { } . Proof.
To see (4.32) let d ∈ D , with (cid:107) d (cid:107) = 1. By the definition of the projectionoperator, 1 − (cid:107) P E d (cid:107) = (cid:107) d − P E d (cid:107) = inf e ∈ E (cid:107) d − e (cid:107) = inf e ∈ E (cid:107) e (cid:107) =1 inf α ∈ R (cid:107) d − αe (cid:107) = inf e ∈ E (cid:107) e (cid:107) =1 inf α ∈ R (1 − α (cid:104) d, e (cid:105) + α ) = inf e ∈ E (cid:107) e (cid:107) =1 (1 − (cid:104) d, e (cid:105) )= 1 − sup e ∈ E (cid:107) e (cid:107) =1 (cid:104) d, e (cid:105) and consequently, (cid:107) P E d (cid:107) = sup e ∈ E (cid:107) e (cid:107) =1 (cid:104) d, e (cid:105) , from which the first formula in (4.32) follows. The second one is a consequence of thefirst one as 1 = (cid:107) P E d (cid:107) + (cid:107) P E ⊥ d (cid:107) .The ⇐ implication in (4.33) is clear. To see the other implication, we observe thatif D ∩ E = { } and E is finite dimensional theninf y ∈ E (cid:107) y (cid:107) =1 d ( y, D ) > , where d ( y, D ) = inf z ∈ D (cid:107) y − z (cid:107) is the distance from y to D . This implies that0 < inf y ∈ E (cid:107) y (cid:107) =1 inf z ∈ D (cid:107) y − z (cid:107) = inf y ∈ E (cid:107) y (cid:107) =1 inf z ∈ D (1 − (cid:104) y, z (cid:105) + (cid:107) z (cid:107) ) ≤ inf y ∈ E (cid:107) y (cid:107) =1 inf z ∈ D (cid:107) z (cid:107) =1 (2 − (cid:104) y, z (cid:105) ) = 2(1 − sup y ∈ E (cid:107) y (cid:107) =1 sup z ∈ D (cid:107) z (cid:107) =1 (cid:104) y, z (cid:105) )= 2(1 − cos α ( D, E )) . Thus if D ∩ E = { } then α ( D, E ) >
0. But that is the ⇒ implication in (4.33). (cid:3) In the notation of (4.31), the translation symmetry gives α ( E a,c , F ⊥ a,c ) = F ( c , c , a − a ) , FFECTIVE DYNAMICS OF DOUBLE SOLITONS FOR PERTURBED MKDV 37 where F is a continuous fuction in C × R . We claim that(4.34) F ( c , c , α ) ≥ κ δ > δ ≤ c ≤ c + δ ≤ c ≤ δ − . Consider now the case | a − a | ≤ A (where A is chosen large below), and hence c , c , and a − a vary within a compact set. Thus it suffices to check that α ( E a,c , F ⊥ a,c )is nowhere zero and this amounts to checking E ∩ F ⊥ = { } .Suppose the contrary, that is that there exists u = z ∂ a q + z ∂ a q ∈ F ⊥ . Since ω (cid:12)(cid:12) M = da ∧ dc + da ∧ dc , z j = (cid:104) u, ∂ − x ∂ c j q (cid:105) = 0 . This proves (4.34). To complete the argument in the case | a − a | ≤ A , we need: Lemma 4.10.
Let E = ker K , and suppose that G is a subspace such that E ⊥ G and the following holds: u ⊥ ( E + G ) = ⇒ (cid:104)K u, u (cid:105) ≥ θ (cid:107) u (cid:107) L . Then, for any other subspace F we have u ⊥ ( F + G ) = ⇒ (cid:104) Ku, u (cid:105) ≥ θ sin α ( E, F ⊥ ) (cid:107) u (cid:107) L . Proof.
Suppose u ⊥ ( F + G ) and consider its orthogonal decomposition, u = P E u + ˜ u .Since E ⊥ G and u ⊥ G , we have ˜ u ⊥ ( E + G ). Hence, by the hypothesis we have (cid:104)K u, u (cid:105) = (cid:104)K ˜ u, ˜ u (cid:105) ≥ θ (cid:107) ˜ u (cid:107) L = θ (cid:107) P E ⊥ u (cid:107) L . An application of (4.32),sin α ( E, F ⊥ ) = inf (cid:107) d (cid:107) =1 d ∈ F ⊥ (cid:107) P E ⊥ d (cid:107) L ≤ (cid:107) P E ⊥ u (cid:107) L (cid:107) u (cid:107) L , concludes the proof. (cid:3) Set-up of the proof
Recall the definition of T (for given δ > a , ¯ c ) stated in the introduction.Recall B ( a, c, t ) def = (cid:90) b ( x, t ) q ( x, a, c ) dx . In the next several sections, we establish the key estimates required for the proof ofthe main theorem. Let us assume that on some time interval [0 , T ], there are C parameters a ( t ) ∈ R , c ( t ) ∈ R such that, if we set(5.1) v ( · , t ) def = u ( · , t ) − q ( · , a ( t ) , c ( t )) then the symplectic orthogonality conditions (3.26) hold. Since u solves (1.1), v ( t )satisfies(5.2) ∂ t v = ∂ x ( − ∂ x v − q v − qv − v + bv ) − F where F results from the perturbation and ∂ t landing on the parameters:(5.3) F = (cid:88) j =1 ( ˙ a j − c j ) ∂ a j q + (cid:88) j =1 ˙ c j ∂ c j q − ∂ x ( bq )Now decompose F = F (cid:107) + F ⊥ where F (cid:107) is symplectically parallel to M and F ⊥ is symplectically orthogonal to M .Explicitly,(5.4) F (cid:107) = (cid:88) j =1 ( ˙ a j − c j + ∂ c j B ) ∂ a j q + (cid:88) j =1 ( ˙ c j − ∂ a j B ) ∂ c j q (5.5) F ⊥ = − ∂ x ( bq ) + 12 (cid:88) j =1 [ − ( ∂ c j B ) ∂ a j q + ( ∂ a j B ) ∂ c j q ]All implicit constants will depend upon δ > L ∞ norms of b ( x, t ) and itsderivatives. We further assume that(5.6) δ ≤ c ( t ) ≤ c ( t ) − δ ≤ δ − holds on all of [0 , T ].In § F ⊥ using the properties of q recalled in §
3. We note that F (cid:107) ≡ F (cid:107) are related to the quality of our effective dynamics andthey are provided in §
7. In § v . Finally energy estimates in § K lead to the final bootstrap argument in § Estimates on F ⊥ Using the identities in Lemma 3.4, we will prove that F ⊥ is O ( h ); in fact, we obtainmore precise information. For notational convenience, we will drop the t dependencein b ( x, t ), and will write b (cid:48) , b (cid:48)(cid:48) , b (cid:48)(cid:48)(cid:48) , to represent x -derivatives.We will use the following consequences of Lemma 3.2:(6.1) ∂ a j q = − ∂ x η ( · , ˆ a j , c j ) + S err FFECTIVE DYNAMICS OF DOUBLE SOLITONS FOR PERTURBED MKDV 39 and(6.2) c j ∂ c j q = ∂ x [( x − a j ) η ( x, ˆ a j , c j )] + 2 c − j θ ( a − a )( c + c )( c − c ) ∂ x η ( x, ˆ a j , c j ) − c j θ ( a − a )( c + c )( c − c ) ∂ x η ( x, ˆ a − j , c − j ) + S err , where θ is given by (3.6).Importantly, as the last formula shows, ∂ c j q is not localized around ˆ a j due to the c j -dependence of ˆ a − j . Also note that it is ( x − a j ) and not ( x − ˆ a j ) in the first terminside the brackets. Definition 6.1.
Let A denote the class of functions of a, c that are of the form h ϕ ( a − a , a, c ) + q ( a, c ) h ,a = ( a , a ) ∈ R , < δ < c < c − δ < /δ , where (cid:12)(cid:12) ∂ (cid:96)α ∂ kc ∂ pa ϕ ( α, a, c ) (cid:12)(cid:12) ≤ C (cid:104) α (cid:105) − N , (cid:12)(cid:12) ∂ kc ∂ pa q ( a, c ) (cid:12)(cid:12) ≤ C , where C depends on δ , N , (cid:96) , k , and p only. We note that if f ∈ S err , then (cid:82) f ( x ) dx has the form ϕ ( a − a , a, c ), ϕ ∈ A . Themost important feature of the class A is that for f ∈ A , | ∂ ka j ∂ (cid:96)c j f | (cid:46) h (cid:104) a − a (cid:105) − N + h with implicit constant depending on c , c . Lemma 6.1.
We have ∂ a j B ( a, c, · ) = 2 c j b (cid:48) (ˆ a j ) + A (6.3) ∂ c j B ( a, c, · ) = 2 b (ˆ a j ) + 2 b (cid:48) (ˆ a j )( a j − ˆ a j ) − π b (cid:48)(cid:48) (ˆ a j ) c − j − − j c − j ( b (cid:48) (ˆ a ) − b (cid:48) (ˆ a )) θ ( c + c )( c − c ) + A (6.4) Proof.
First we compute ∂ a j B ( a, c, t ). We have that ∂ a j q is exponentially localizedaround ˆ a j . Substituting the Taylor expansion of b around ˆ a j , we obtain ∂ a j B ( a, c, t ) = b (ˆ a j ) (cid:90) ∂ a j q + b (cid:48) (ˆ a j ) (cid:90) ( x − ˆ a j ) ∂ a j q + 12 b (cid:48)(cid:48) (ˆ a j ) (cid:90) ( x − ˆ a j ) ∂ a j q + O ( h )= I + II + III + O ( h ) Terms I and II are straightforward. Using (3.21) and (3.22),I = b (ˆ a j ) ∂ a j (cid:90) q = 0II = b (cid:48) (ˆ a j ) (cid:18) ∂ a j (cid:90) xq − ˆ a j ∂ a j (cid:90) q (cid:19) = 2 c j b (cid:48) (ˆ a j )For III, we will substitute (6.1) and hence pick up O ( h ) (cid:104) a − a (cid:105) − N errors.III = − b (cid:48)(cid:48) (ˆ a j ) (cid:90) ( x − ˆ a j ) ∂ x η ( x, ˆ a j , c j ) dx + A = A Thus, we obtain (6.3). Next, we compute ∂ c j B ( a, c, t ). Note that ∂ c j q is not localizedaround ˆ a j . Begin by rewriting ∂ c j B as ∂ c j B = (cid:90) b (ˆ a j ) ∂ c j q + (cid:90) b (cid:48) (ˆ a j )( x − ˆ a j ) ∂ c j q + (cid:90) ˜ b j ∂ c j q where ˜ b j ( x ) def = b ( x ) − b (ˆ a j ) − b (cid:48) (ˆ a j )( x − ˆ a j ) . Now substitute (6.2) into the last term and note that the S err term in (6.2) producesan A term here. ∂ c j B = (cid:90) b (ˆ a j ) ∂ c j q + (cid:90) b (cid:48) (ˆ a j )( x − ˆ a j ) ∂ c j q + 2 c j (cid:90) ˜ b j ( x ) ∂ x [( x − a j ) η ( x, ˆ a j , c j )] η ( x, ˆ a j , c j )+ c − j θc j ( c + c )( c − c ) (cid:90) ˜ b j ( x ) ∂ x η ( x, ˆ a j , c j ) − θ ( c + c )( c − c ) (cid:90) ˜ b j ( x ) ∂ x η ( x, ˆ a − j , c − j ) + A = I + II + III + IV + V + A where terms I-V are studied separately below.I = b (ˆ a j ) ∂ c j (cid:90) q = 2 b (ˆ a j )II = b (cid:48) (ˆ a j ) (cid:18) ∂ c j (cid:90) xq − ˆ a j ∂ c j (cid:90) q (cid:19) = 2 b (cid:48) (ˆ a j )( a j − ˆ a j ) FFECTIVE DYNAMICS OF DOUBLE SOLITONS FOR PERTURBED MKDV 41
Term III is localized around ˆ a j , and thus we integrate by parts in x and Taylor expand˜ b j around ˆ a j to obtainIII = 1 c j (cid:90) (cid:16) − ˜ b (cid:48) j ( x )( x − a j ) + ˜ b j ( x ) (cid:17) η ( x, ˆ a j , c j )= − b (cid:48)(cid:48) (ˆ a j ) c j (cid:90) ( x − ˆ a j ) η ( x, ˆ a j , c j ) − b (cid:48)(cid:48) (ˆ a j )(ˆ a j − a j ) (cid:90) ( x − ˆ a j ) η ( x, ˆ a j , c j ) + O ( h )= − π b (cid:48)(cid:48) (ˆ a j ) c − j + O ( h )Term IV is localized around ˆ a j , and thus we integrate by parts in x and Taylor expand˜ b j around ˆ a j to obtain (cid:90) ˜ b j ( x ) ∂ x η ( x, ˆ a j , c j ) = − (cid:90) ( b (cid:48) ( x ) − b (ˆ a j )) η ( x, ˆ a j , c j )= − b (cid:48)(cid:48) (ˆ a j ) (cid:90) ( x − ˆ a j ) η ( x, ˆ a j , c j ) + O ( h )= O ( h )Term V is localized around ˆ a − j , and thus we integrate by parts in x and Taylorexpand ˜ b j around ˆ a − j . (cid:90) ˜ b j ( x ) ∂ x η ( x, ˆ a − j , c − j ) = − (cid:90) ( b (cid:48) ( x ) − b (ˆ a j )) η ( x, ˆ a − j , c − j )= − ( b (cid:48) (ˆ a − j ) − b (cid:48) (ˆ a j )) (cid:90) η ( x, ˆ a − j , c − j ) − b (cid:48)(cid:48) (ˆ a − j ) (cid:90) ( x − ˆ a − j ) η ( x, ˆ a − j , c − j ) + O ( h )= − c − j ( b (cid:48) (ˆ a − j ) − b (cid:48) (ˆ a j )) + O ( h ) (cid:3) Lemma 6.2 (estimates on F ⊥ ) . (6.5) ∂ − x ∂ a j F ⊥ = O ( h ) · S sol , ∂ − x ∂ c j F ⊥ = O ( h ) · S sol , j = 1 , F ⊥ = − (cid:88) j =1 b (cid:48)(cid:48) (ˆ a j ) c j ∂ x τ ( · , ˆ a j , c j ) + A · S sol where (6.7) τ def = (cid:18) π
12 + x (cid:19) η ( x ) , τ ( x, ˆ a j , c j ) def = c j τ ( c j ( x − ˆ a j )) . In light of the above lemma, we introduce the notation F ⊥ = ( F ⊥ ) + ˜ F ⊥ , where(6.8) ( F ⊥ ) = − (cid:88) j =1 b (cid:48)(cid:48) (ˆ a j ) c j ∂ x τ ( · , ˆ a j , c j )and ˜ F ⊥ ∈ A · S sol . We make use of (6.5) in § § Proof.
We begin by proving (6.6). By (3.18), (3.19),(6.9) ∂ x ( bq ) = ( ∂ x b ) q + b ( ∂ x q )= ( ∂ x b ) (cid:88) j =1 (( x − a j ) ∂ a j q + c j ∂ c j q ) − b (cid:88) j =1 ∂ a j q = (cid:88) j =1 ( − b + ( ∂ x b )( x − a j )) ∂ a j q + (cid:88) j =1 ( ∂ x b ) c j ∂ c j q + O ( h ) · S sol The ∂ a j q term is well localized around ˆ a j , and thus we can Taylor expand the coeffi-cients around ˆ a j . The ∂ c j q term we leave alone for the moment.We have ∂ x ( bq ) = (cid:88) j =1 (cid:16) − b (ˆ a j ) + b (cid:48) (ˆ a j )(ˆ a j − a j ) + b (cid:48)(cid:48) (ˆ a j )(ˆ a j − a j )( x − ˆ a j ) + 12 b (cid:48)(cid:48) (ˆ a j )( x − ˆ a j ) (cid:17) ∂ a j q + (cid:88) j =1 b (cid:48) ( x ) c j ∂ c j q + A · S sol
Substituting the above together with (6.3) and (6.4) into (5.5), we obtain F ⊥ = 12 (cid:88) j =1 b (cid:48)(cid:48) (ˆ a j ) (cid:16) π c − j − a j − a j )( x − ˆ a j ) − ( x − ˆ a j ) (cid:17) ∂ a j q + ( b (cid:48) (ˆ a ) − b (cid:48) (ˆ a )) θ ( c + c )( c − c ) (cid:88) j =1 ( − j c − j ∂ a j q − (cid:88) j =1 ( b (cid:48) ( x ) − b (cid:48) (ˆ a j )) c j ∂ c j q + A · S sol
We now substitute (6.1) and (6.2) recognizing that this will only generate errors oftype A times a Schwartz class function. We also Taylor expand around ˆ a j or ˆ a − j FFECTIVE DYNAMICS OF DOUBLE SOLITONS FOR PERTURBED MKDV 43 depending upon the localization. F ⊥ = 12 (cid:88) j =1 b (cid:48)(cid:48) (ˆ a j ) (cid:16) − π c − j + 2(ˆ a j − a j )( x − ˆ a j ) + ( x − ˆ a j ) (cid:17) ∂ x η ( x, ˆ a j , c j ) ← I − ( b (cid:48) (ˆ a ) − b (cid:48) (ˆ a )) θ ( c + c )( c − c ) (cid:88) j =1 ( − j c − j ∂ x η ( x, ˆ a j , c j ) ← II − (cid:88) j =1 b (cid:48)(cid:48) (ˆ a j )( x − ˆ a j ) ∂ x [( x − a j ) η ( x, ˆ a j , c j )] ← III − (cid:88) j =1 c − j θ ( c + c )( c − c ) b (cid:48)(cid:48) (ˆ a j )( x − ˆ a j ) ∂ x η ( x, ˆ a j , c j ) ← IV+ (cid:88) j =1 c j θ ( c + c )( c − c ) b (cid:48)(cid:48) (ˆ a − j )( x − ˆ a − j ) ∂ x η ( x, ˆ a − j , c − j ) ← V+ (cid:88) j =1 c j θ ( c + c )( c − c ) ( b (cid:48) (ˆ a − j ) − b (cid:48) ( a j )) ∂ x η ( x, ˆ a − j , c − j ) ← VI+
A · S sol
We have that IV + V = 0 and II + VI = 0. Hence F ⊥ = I + III + A · S sol = − (cid:88) j =1 b (cid:48)(cid:48) (ˆ a j ) ∂ x (cid:18)(cid:0) π c − j + ( x − ˆ a j ) (cid:1) η ( x, ˆ a j , c j ) (cid:19) This completes the proof of (6.6). To obtain (6.5), we note that a consequence of(6.6) is F ⊥ = O ( h ) f , where f ∈ S sol . By the definition (5.5) of F ⊥ and Corollary3.3, we have ∂ − x F ⊥ ∈ S sol , and hence f ∈ S sol . (cid:3) Estimates on the parameters
The equations of motion are recovered (in approximate form) using the symplecticorthogonality properties (3.26) of v and the equation (5.2) for v . For a function G ofthe form G = g ∂ a q + g ∂ a q + g ∂ c q + g ∂ c q with g j = g j ( a, c ), define coef( G ) = ( g , g , g , g ) . Lemma 7.1.
Suppose we are given δ > and b ( x, t ) , and parameters a ( t ) , c ( t ) such that v defined by (5.1) satisfies the symplectic orthogonality conditions (3.26) .Suppose, moreover, that the amplitude separation condition (5.6) holds. Then (with implicit constants depending upon δ > and L ∞ norms of b and its derivatives), if (cid:107) v (cid:107) H (cid:46) , then we have (7.1) | coef( F (cid:107) ) | (cid:46) h (cid:107) v (cid:107) H + (cid:107) v (cid:107) H . Proof.
Since (cid:104) v, ∂ − x ∂ a j q (cid:105) = 0, we have upon substituting (5.2)0 = ∂ t (cid:104) v, ∂ − x ∂ a j q (cid:105) = (cid:104) ∂ t v, ∂ − x ∂ a j q (cid:105) + (cid:104) v, ∂ t ∂ − x ∂ a j q (cid:105) = (cid:104) ( ∂ x v + 6 q ) v, ∂ a j q (cid:105) + (cid:104) (6 qv + 2 v ) , ∂ a j q (cid:105) ← I + II − (cid:104) bv, ∂ a j q (cid:105) − (cid:104) F (cid:107) , ∂ − x ∂ a j q (cid:105) − (cid:104) F ⊥ , ∂ − x ∂ a j q (cid:105) ← III + IV + V+ (cid:104) v, ∂ − x ∂ a j (cid:32) (cid:88) k =1 ∂ a k q ˙ a k + (cid:88) k =1 ∂ c k q ˙ c k (cid:33) (cid:105) ← VIWe have, by (3.17), I = (cid:104) v, ∂ a j ( ∂ x q + 2 q ) (cid:105) = − (cid:104) v, ∂ − x ∂ a j ∂ x I (cid:48) ( q ) (cid:105) = −(cid:104) v, ∂ − x ∂ a j (cid:88) k =1 c k ∂ a k q (cid:105) Also, by (5.5)III = −(cid:104) bv, ∂ a j q (cid:105) = −(cid:104) v, ∂ a j ( bq ) (cid:105) = −(cid:104) v, ∂ − x ∂ a j ∂ x ( bq ) (cid:105) = −(cid:104) v, ∂ − x ∂ a j (cid:0) − F ⊥ −
12 2 (cid:88) k =1 ( ∂ c k B ) ∂ a k q +
12 2 (cid:88) k =1 ( ∂ a k B ) ∂ c k q (cid:1) (cid:105) Thus | I + III + VI | = |(cid:104) v, ∂ − x ∂ a j F ⊥ (cid:105) + (cid:104) v, ∂ − x ∂ a j F (cid:107) (cid:105)|≤ (cid:107) v (cid:107) L ( (cid:107) ∂ − x ∂ a j F ⊥ (cid:107) L + (cid:107) ∂ − x ∂ a j F (cid:107) (cid:107) ) ≤ (cid:107) v (cid:107) L ( h + | coef( F (cid:107) ) | )Next, we note that by Cauchy-Schwarz, | II | (cid:46) (cid:107) v (cid:107) H . Next, observe from (5.4) and Lemma 3.6 thatIV = (cid:104) F (cid:107) , ∂ − x ∂ a j q (cid:105) = − ( ˙ c j − ∂ a j B ) . FFECTIVE DYNAMICS OF DOUBLE SOLITONS FOR PERTURBED MKDV 45
Of course, we have V = (cid:104) F ⊥ , ∂ − x ∂ a j q (cid:105) = 0. Combining, we obtain(7.2) (cid:12)(cid:12)(cid:12)(cid:12) ˙ c j − ∂ a j B (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:107) v (cid:107) H ( h + | coef( F (cid:107) ) | ) + (cid:107) v (cid:107) H . A similar calculation, applying ∂ t to the identity 0 = (cid:104) v, ∂ − x ∂ c j q (cid:105) , yields(7.3) (cid:12)(cid:12)(cid:12)(cid:12) ˙ a j − c j + 12 ∂ c j B (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:107) v (cid:107) H ( h + | coef( F (cid:107) ) | ) + (cid:107) v (cid:107) H . Combining (7.2) and (7.3) gives (7.1). (cid:3) Correction term
Recall the definition (6.7) of τ . Let ρ be the unique function solving(1 − ∂ x − η ) ρ = τ , see [19, Proposition 4.2] for the properties of this equation. The function ρ is smooth,exponentially decaying at ∞ , and satisfies the symplectic orthogonality conditions(8.1) (cid:104) ρ, η (cid:105) = 0 , (cid:104) ρ, xη (cid:105) = 0Set ρ ( x, ˆ a j , c j ) def = c − j ρ ( c j ( x − ˆ a j ))and note that ( c j − ∂ x − η ( · , ˆ a j , c j )) ρ ( · , ˆ a j , c j ) = τ ( · , ˆ a j , c j )Define the symplectic projection operator P f def = (cid:88) j =1 (cid:104) f, ∂ − x ∂ c j q (cid:105) ∂ a j q + (cid:88) j =1 (cid:104) f, ∂ − x ∂ a j q (cid:105) ∂ c j q . Define(8.2) w def = −
12 ( I − P ) (cid:88) j =1 b (cid:48)(cid:48) (ˆ a j ) c j ρ ( · , ˆ a j , c j )Note that w = O ( h ) and clearly now w satisfies(8.3) (cid:104) w, ∂ − x ∂ a j q (cid:105) = 0 , (cid:104) w, ∂ − x ∂ c j q (cid:105) = 0 . Recall the definition (6.8) of ( F ⊥ ) . Lemma 8.1. If ˙ c j = O ( h ) , and ˙ a j = c j − b (ˆ a j ) + O ( h ) , then (8.4) ∂ t w + ∂ x ( ∂ x w + 6 q w − bw ) = − ( F ⊥ ) − G + A · S sol . where G is an O ( h ) term that is symplectically parallel to M , i.e. G ∈ span { ∂ − x ∂ a q, ∂ − x ∂ a q, ∂ − x ∂ c q, ∂ − x ∂ c q } . Proof.
Let w j = b (cid:48)(cid:48) (ˆ a j ) c j ρ ( · , ˆ a j , c j )Then ∂ t w j = b (cid:48)(cid:48)(cid:48) (ˆ a j ) ˙ˆ a j c − j ρ ( · , ˆ a j , c j ) − b (cid:48)(cid:48) (ˆ a j ) c − j ˙ c j ρ ( · , ˆ a j , c j )+ b (cid:48)(cid:48) (ˆ a j ) c − j ˙ˆ a j ∂ a j ρ ( · , ˆ a j , c j ) + b (cid:48)(cid:48) (ˆ a j ) c − j ˙ˆ c j ∂ c j ρ ( · , ˆ a j , c j ) + ∂ t b (cid:48)(cid:48) (ˆ a j ) c − j ρ ( · , ˆ a j , c j )= − ˙ a j ∂ x w j + A · S sol
Also, we have ( ∂ x + 6 q ) w j = ( ∂ x + 6 η ( · , ˆ a j , c j )) w j + A · S sol = c j w j − b (cid:48)(cid:48) (ˆ a j ) c − j τ ( · , ˆ a j , c j ) + A · S sol
Also, bw j = b (ˆ a j ) w j + A · S sol
Combining, we obtain ∂ t w j + ∂ x ( ∂ x w j + 6 q w j − bw j )= − b (cid:48)(cid:48) (ˆ a j ) c − j ∂ x τ ( · , ˆ a j , c j ) + ( − ˙ a j + c j − b (ˆ a j )) ∂ x w j + A · S sol = − b (cid:48)(cid:48) (ˆ a j ) c − j ∂ x τ ( · , ˆ a j , c j ) + A · S sol
Now we discuss ∂ t P w j . ∂ t P w j = (cid:104) ∂ t w j , ∂ − x ∂ a q (cid:105) ∂ c q + (cid:104) w j , ∂ t ∂ − x ∂ a q (cid:105) ∂ c j q + similar+ (cid:104) w j , ∂ − x ∂ a q (cid:105) ∂ t ∂ c q + similarThe first line of terms is symplectically parallel to M . For the second line, note thatby (8.1), we have (cid:104) w j , ∂ − x ∂ a q (cid:105) = A . Consequently, ∂ t P w j = T q M + A · S sol (cid:3)
Define ˜ u and ˜ v by(8.5) u = ˜ u + w , v = ˜ v + w . Of course, it follows that ˜ u = q + ˜ v . Note that by (3.26) and (8.3), we have(8.6) (cid:104) ˜ v, ∂ − x ∂ a j q (cid:105) = 0 and (cid:104) ˜ v, ∂ − x ∂ c j q (cid:105) = 0 , j = 1 , . Note that ˜ u solves(8.7) ∂ t ˜ u = − ∂ x ( ∂ x ˜ u + 2˜ u − b ˜ u ) − ∂ t w − ∂ x ( ∂ x w + 6˜ u w − bw ) + O ( h )where the O ( h ) terms arise from w and w . Moreover, if we make the mild assump-tion that ˜ v = O ( h ), then ˜ u w = q w + O ( h ). By (8.7) and (8.4), we have(8.8) ∂ t ˜ u = − ∂ x ( ∂ x ˜ u + 2˜ u − b ˜ u ) + ( F ⊥ ) + G + A · S sol
FFECTIVE DYNAMICS OF DOUBLE SOLITONS FOR PERTURBED MKDV 47
Since ˜ u = q + ˜ v , we have (in analogy with (5.2))(8.9) ∂ t ˜ v = ∂ x ( − ∂ x ˜ v − q ˜ v + b ˜ v ) − F (cid:107) − ˜ F ⊥ + G + A · S sol + O ( h ) H where we have made the assumption that ˜ v = O ( h / ) in order to discard the ˜ v and˜ v terms. We thus see that, in comparison to v , the equation for ˜ v has a lower-orderinhomogeneity, but still satisfies the symplectic orthogonality conditions (8.6) and v = ˜ v + O ( h ). 9. Energy estimate
Since w = O ( h ), to obtain the desired bound on v it will suffice to obtain a boundfor ˜ v . This will be achieved by the “energy method.” Lemma 9.1.
Suppose we are given δ > and b ( x, t ) , and parameters a ( t ) , c ( t ) such that v defined by (5.1) satisfies the symplectic orthogonality conditions (3.26) on [0 , T ] . Suppose, moreover, that the amplitude separation condition (5.6) holds on [0 , T ] . Then (with implicit constants depending upon δ > and L ∞ norms of b andits derivatives), if (cid:107) v (cid:107) H (cid:46) and T (cid:28) h − , then (cid:107) v (cid:107) L ∞ [0 ,T ] H (cid:46) (cid:107) v (0) (cid:107) H + h (cid:18) (cid:90) T (cid:104) a − a (cid:105) − N dt (cid:19) . Proof.
Recall that we have defined H c ( u ) = I ( u ) + ( c + c ) I ( u ) + c c I ( u ) . With w given by (8.2) and ˜ u given by (8.5), let E ( t ) = H c (˜ u ) − H c ( q ) . Then ∂ t E = (cid:104) H (cid:48) c (˜ u ) , ∂ t ˜ u (cid:105) − (cid:104) H (cid:48) c ( q ) , ∂ t q (cid:105) + 2( c ˙ c + c ˙ c )( I (˜ u ) − I ( q ))+ 2 c c ( c ˙ c + ˙ c c )( I (˜ u ) − I ( q ))= I + II + III + IVNote that II = 0 since Lemma 4.1 showed that H (cid:48) c ( q ) = 0. For III, we have by (3.17)and the orthogonality conditions (8.6),III = 2( c ˙ c + c ˙ c )( (cid:104) I (cid:48) ( q ) , ˜ v (cid:105) + O ( (cid:107) ˜ v (cid:107) H ))= 4( c ˙ c + c ˙ c ) (cid:104) (cid:88) j =1 c j ∂ − x ∂ a j q, ˜ v (cid:105) + O (( | ˙ c | + | ˙ c | ) (cid:107) ˜ v (cid:107) H )= O (( | ˙ c | + | ˙ c | ) (cid:107) ˜ v (cid:107) H ) Term IV is bounded similarly. It remains to study Term I. Writing (8.8) as ∂ t ˜ u = ∂ x I (cid:48) (˜ u ) + ∂ x ( b ˜ u ) + ( F ⊥ ) + G + A · S sol and appealing to (2.5), we have by Lemma2.1 (with u replaced by ˜ u in that lemma) thatI = (cid:104) H (cid:48) c (˜ u ) , ∂ x ( b ˜ u ) (cid:105) + (cid:104) H (cid:48) c (˜ u ) , ( F ⊥ ) + A · S sol (cid:105) = 5 (cid:104) b x , A (˜ u ) (cid:105) − (cid:104) b xxx , A (˜ u ) (cid:105) + (cid:104) b xxxxx , A (˜ u ) (cid:105) + ( c + c )(3 (cid:104) b x , A (˜ u ) (cid:105) − (cid:104) b xxx , A (˜ u ) (cid:105) ) + c c (cid:104) b x , A (˜ u ) (cid:105) + (cid:104) H (cid:48) c (˜ u ) , ( F ⊥ ) + A · S sol (cid:105)
Expand A j (˜ u ) = A j ( q + ˜ v ) = A j ( q ) + A (cid:48) j ( q )(˜ v ) + O (˜ v ) and H (cid:48) c (˜ u ) = H (cid:48) c ( q ) + K c,a ˜ v + O (˜ v ) = K c,a ˜ v + O (˜ v ) to obtain I = IA + IB + IC, whereIA = 5 (cid:104) b x , A ( q ) (cid:105) − (cid:104) b xxx , A ( q ) (cid:105) + (cid:104) b xxxxx , A ( q ) (cid:105) + ( c + c )(3 (cid:104) b x , A ( q ) (cid:105) − (cid:104) b xxx , A ( q ) (cid:105) ) + c c (cid:104) b x , A ( q ) (cid:105) IB = 5 (cid:104) b x , A (cid:48) ( q )(˜ v ) (cid:105) − (cid:104) b xxx , A (cid:48) ( q )(˜ v ) (cid:105) + (cid:104) b xxxxx , A (cid:48) ( q )(˜ v ) (cid:105) + ( c + c )(3 (cid:104) b x , A (cid:48) ( q )(˜ v ) (cid:105) − (cid:104) b xxx , A (cid:48) ( q )(˜ v ) (cid:105) ) + c c (cid:104) b x , A (cid:48) ( q )(˜ v ) (cid:105) IC = (cid:104)K c,a ˜ v, ( F ⊥ ) (cid:105) + O ( h (cid:107) ˜ v (cid:107) H ) + O ( A · (cid:107) ˜ v (cid:107) H )Then reapply Lemma 2.1 (with u replaced by q in that lemma) to obtain that IA = −(cid:104) H (cid:48) c ( q ) , ∂ x ( bq ) (cid:105) = 0. Applying Lemma 2.2,IB = (cid:104)K c,a ˜ v, ( bq ) x (cid:105) − (cid:104) ∂ x H (cid:48) c ( q ) , b ˜ v (cid:105) = (cid:104)K c,a ˜ v, ( bq ) x (cid:105) In summary thus far, we have obtained that ∂ t E = (cid:104)K c,a ˜ v, ( bq ) x + ( F ⊥ ) (cid:105) + O ( h (cid:107) ˜ v (cid:107) H ) + O ( A(cid:107) ˜ v (cid:107) H )By (4.12), (4.13), and (8.6) (recalling the definition (5.3) of F ), we obtain (cid:104)K c,a ˜ v, ∂ x ( bq ) (cid:105) = −(cid:104)K c,a ˜ v, F (cid:105) = −(cid:104)K c,a ˜ v, F (cid:107) + F ⊥ (cid:105) Hence ∂ t E = −(cid:104)K c,a ˜ v, F (cid:107) + ˜ F ⊥ (cid:105) + O ( h (cid:107) ˜ v (cid:107) H ) + O ( A(cid:107) ˜ v (cid:107) H )It follows from Lemma 7.1 and ˜ F ⊥ ∈ A · S sol (see (6.6), (6.8)) that | ∂ t E | (cid:46) ( h (cid:104) a − a (cid:105) − N + h ) (cid:107) ˜ v (cid:107) H + h (cid:107) ˜ v (cid:107) H If T = δh − , E ( T ) = E (0) + h (cid:18) (cid:90) T (cid:104) a − a (cid:105) − N (cid:19) (cid:107) ˜ v (cid:107) L ∞ [0 ,T ] H x + h (cid:107) ˜ v (cid:107) L ∞ [0 ,T ] H . By Lemma 4.1, the definition of E and K c,a , and the fact that ˜ u = q + ˜ v , we have |E − (cid:104)K c,a ˜ v, ˜ v (cid:105)| (cid:46) (cid:107) ˜ v (cid:107) H . FFECTIVE DYNAMICS OF DOUBLE SOLITONS FOR PERTURBED MKDV 49
Applying this at time 0 and T , together with the coercivity of K (Proposition 4.3), (cid:107) ˜ v ( T ) (cid:107) H (cid:46) (cid:107) ˜ v (0) (cid:107) H + h (cid:18) (cid:90) T (cid:104) a − a (cid:105) − N (cid:19) (cid:107) ˜ v (cid:107) L ∞ [0 ,T ] H x + h (cid:107) ˜ v (cid:107) L ∞ [0 ,T ] H . Replacing T by T (cid:48) such that 0 ≤ T (cid:48) ≤ T , and taking the supremum in T (cid:48) over0 ≤ T (cid:48) ≤ T , we obtain (cid:107) ˜ v (cid:107) L ∞ [0 ,T ] H (cid:46) (cid:107) ˜ v (0) (cid:107) H + h (cid:18) (cid:90) T (cid:104) a − a (cid:105) − N (cid:19) (cid:107) ˜ v (cid:107) L ∞ [0 ,T ] H x + h (cid:107) ˜ v (cid:107) L ∞ [0 ,T ] H . By selecting δ small enough, we obtain (cid:107) ˜ v (cid:107) L ∞ [0 ,T ] H (cid:46) (cid:107) ˜ v (0) (cid:107) H + h (cid:18) (cid:90) T (cid:104) a − a (cid:105) − N dt (cid:19) Finally, using that (cid:107) w (cid:107) H ∼ h , and v = ˜ v + w , we obtained the claimed estimate. (cid:3) Proof of the main theorem
We start with the proposition which links the ODE analysis with the estimates onthe error term v : Proposition 10.1.
Suppose we are given b ∈ C ∞ b ( R ) and δ > . (Implicit con-stants below depend only on b and δ ). Suppose that we are further given ¯ a ∈ R , ¯ c ∈ R \C , κ ≥ , h > , and v satisfying (3.26) , such that < h (cid:46) κ − , (cid:107) v (cid:107) H x ≤ κh . Let u ( t ) be the solution to (1.1) with b ( x, t ) = b ( hx, ht ) and initial data η ( · , ¯ a, ¯ c ) + v .Then there exist a time T (cid:48) > and trajectories a ( t ) and c ( t ) defined on [0 , T (cid:48) ] suchthat a (0) = ¯ a , c (0) = ¯ c and the following holds, with v def = u − η ( · , a, c ) : (1) On [0 , T (cid:48) ] , the orthogonality conditions (3.26) hold. (2) Either c ( T (cid:48) ) = δ , c ( T (cid:48) ) = c ( T (cid:48) ) − δ , c ( T (cid:48) ) = δ − , or T (cid:48) = ωh − , where ω (cid:28) . (3) | ˙ a j − c j + b ( a j , t ) | (cid:46) h . (4) | ˙ c j − c j b (cid:48) ( a j ) | (cid:46) h . (5) (cid:107) v (cid:107) L ∞ [0 ,T (cid:48) ] H x ≤ ακh , where α (cid:29) .Here α and ω are constants depending only on b and δ (independent of κ , etc)Proof. Recall our convention that implicit constants depend only on b and δ . ByLemma 3.7 and the continuity of the flow u ( t ) in H , there exists some T (cid:48)(cid:48) > a ( t ), c ( t ) can be defined so that (3.26) hold. Now take T (cid:48)(cid:48) to be the maximaltime on which a ( t ), c ( t ) can be defined so that (3.26) holds. Let T (cid:48) be first time0 ≤ T (cid:48) ≤ T (cid:48)(cid:48) such that c ( T (cid:48) ) = δ , c ( T (cid:48) ) = c ( T (cid:48) ) − δ , c ( T (cid:48) ) = δ − , T (cid:48) = T (cid:48)(cid:48) , or ωh − (whichever comes first). Here, 0 < ω (cid:28) suitably small at the end of the proof (depending only upon implicit constants in theestimates, and hence only on b and δ ). Remark . We will show that on [0 , T (cid:48) ], we have (cid:107) v ( t ) (cid:107) H x (cid:46) κh , and henceby Lemma 3.7 and the continuity of the u ( t ) flow, it must be the case that either c ( T (cid:48) ) = δ , c ( T (cid:48) ) = c ( T (cid:48) ) − δ , c ( T (cid:48) ) = δ − , or ωh − (i.e. the case T (cid:48) = T (cid:48)(cid:48) doesnot arise).Let T , 0 < T ≤ T (cid:48) , be the maximal time such that(10.1) (cid:107) v (cid:107) L ∞ [0 ,T ] H x ≤ ακh , where α is suitably large constant related to the implicit constants in the estimates(and thus dependent only upon b and δ > Remark . We will show, assuming that (10.1) holds, that (cid:107) v (cid:107) L ∞ [0 ,T ] H x ≤ ακh / and thus by continuity we must have T = T (cid:48) .In the remainder of the proof, we work on the time interval [0 , T ], and we are ableto assume that the orthogonality conditions (3.26) hold, δ ≤ c ( t ) ≤ c ( t ) − δ ≤ δ − ,and that (10.1) holds. By Lemma 7.1 and Taylor expansion, we have (since κ h (cid:46) h )(10.2) (cid:40) ˙ a j = c j − b ( a j , t ) + O ( h )˙ c j = c j ∂ x b ( a j , t ) + O ( h ) , with initial data a j (0) = ¯ a j , c j (0) = ¯ c j . Let ξ ( t ) def = b ( a ( t ) , t ) − b ( a ( t ) , t ) a ( t ) − a ( t )and let Ξ( t ) denote an antiderivative. By the mean-value theorem | ξ | (cid:46) h , and since T ≤ ωh − , we have e Ξ ∼
1. We then have ddt (cid:0) e Ξ ( a − a ) (cid:1) = e Ξ ( c − c ) + O ( h ) . Since δ ≤ c − c , we see that e Ξ ( a − a ) is strictly increasing. Let 0 ≤ t ≤ T denote the unique time at which e Ξ ( a − a ) = 0 (if the quantity is always positive, take t = 0, and if the quantity is always negative, take t = T , and make straightforwardmodifications to the argument below). If t < t , integrating from t to t we obtain δ ( t − t ) (cid:46) − e Ξ( t ) ( a ( t ) − a ( t )) = e Ξ( t ) | a ( t ) − a ( t ) | If t > t , integrating from t to t we obtain δ ( t − t ) (cid:46) e Ξ( t ) ( a ( t ) − a ( t )) . Hence, (cid:90) T (cid:104) a ( t ) − a ( t ) (cid:105) − (cid:46) . FFECTIVE DYNAMICS OF DOUBLE SOLITONS FOR PERTURBED MKDV 51
By Lemma 9.1, we conclude that (cid:107) v (cid:107) L ∞ T H x ≤ α (cid:107) v (0) (cid:107) H + h ) ≤ α κh + h ) ≤ ακh . (cid:3) We can now complete
Proof of the main Theorem.
Suppose that (cid:107) v (cid:107) H ≤ h . Iterate Prop. 10.1, as longas the condition(10.3) δ ≤ c ≤ c − δ ≤ δ − remains true, as follows: for the k -th iterate, put κ = α k in Prop. 10.1 and advancefrom time t k = kωh − to time t k +1 = ( k + 1) ωh − . At time t k , we have (cid:107) v ( t k ) (cid:107) H ≤ α k h , and we find from Prop. 10.1 that (cid:107) v (cid:107) L ∞ [ tk,tk +1] H x ≤ α k +1 h . Provided (10.3)holds on all of [0 , t K ], we can continue until κ − ∼ h , i.e. K ∼ log h − .Recall (1.5), and A j ( T ), C j ( T ) defined by (1.11). Let ˆ a j ( t ) = h − A j ( ht ), ˆ c j ( t ) = C j ( ht ). Then ˆ a j , ˆ c j solve (cid:40) ˙ˆ a j = ˆ c j − b (ˆ a j , t )˙ˆ c j = ˆ c j ∂ x b (ˆ a j , t )with initial data ˆ a j (0) = ¯ a j , ˆ c j (0) = ¯ c j . We know that (10.3) holds for ˆ c j on [0 , h − T ].Let ˜ a j = a j − ˆ a j , ˜ c j = c j − ˆ c j denote the differences. Let γ ( t ) def = b ( a j , t ) − b (ˆ a j , t ) a j − ˆ a j σ ( t ) def = ∂ x b ( a j , t ) − ∂ x b (ˆ a j , t ) a j − ˆ a j . By the mean-value theorem, | γ ( t ) | (cid:46) h and | σ ( t ) | (cid:46) h . We have(10.4) (cid:40) ˙˜ a j = ˜ c j + 2ˆ c j ˜ c j − γ ˜ a j + O ( h )˙˜ c j = ˜ c j ( ∂ x b )( a j , t ) + ˆ c j σ ˜ a j + O ( h ) . We conclude that | ˜ a j | (cid:46) e Cht and | ˜ c j | (cid:46) he Cht . This is proved by Gronwall’s methodand a bootstrap argument. Since (10.3) holds for ˆ c j on [0 , h − T ], it holds for c j onthe same time scale if T < ∞ , and up to the maximum time allowable by the aboveiteration argument, (cid:15)h − log h − , if T = + ∞ . (cid:3) Appendix A. Local and global well-posedness
In this appendix, we will prove that (1.1) is globally well-posed in H k , k ≥ M ( T ) def = k +1 (cid:88) j =0 (cid:107) ∂ jx b ( x, t ) (cid:107) L ∞ [0 ,T ] L ∞ x < ∞ . for all T >
0. This is proved for k = 1 under the additional assumption that (cid:107) b (cid:107) L x L ∞ T < ∞ in the appendix of Dejak-Sigal [11]. ‡ The removal of the assump-tion (cid:107) b (cid:107) L x L ∞ T < ∞ is convenient since it allows for us to consider potentials thatasymptotically in x converge to a nonzero number, rather than decay. Moreover, ourargument is self-contained.Well-posedness for KdV (nonlinearity ∂ x u ) with b ≡ b (cid:54) = 0 and to mKdV (1.1), it applies only for k > due to the derivativein the nonlinearity. Kenig-Ponce-Vega [21, 20] reduced the regularity requirements(for b ≡
0) below k = 1 by introducing new local smoothing and maximal functionestimates and applying the contraction method. These estimates were obtained byFourier analysis (Plancherel’s theorem, van der Corput lemma). At the H level ofregularity (and above) for mKdV, the full strength of the maximal function estimatein [21, 20] is not needed. Here, we prove a local smoothing estimate and a (weak)maximal function estimate (see (A.2) and (A.3) in Lemma A.1 below) instead by theintegrating factor method, which easily accomodates the inclusion of a potential termsince integration by parts can be applied. The estimates proved by Kenig-Ponce-Vegawere directly applied by Dejak-Sigal, treating the potential term as a perturbation,which required introducing the norm (cid:107) b (cid:107) L x L ∞ T . Our argument does not apply directlyto KdV since we are lacking the (strong) maximal function estimate used by [21, 20].Let Q n = [ n − , n + ] so that R = ∪ Q n . Let ˜ Q n = [ n − , n + 1]. An example ofour notation is: (cid:107) u (cid:107) (cid:96) ∞ n L T L Qn = sup n (cid:107) u (cid:107) L ,T ) L Qn . We will use variants like (cid:96) n L ∞ T L Q n etc. Note that due to the finite incidence of overlap,we have (cid:107) u (cid:107) (cid:96) ∞ n L T L Qn ∼ (cid:107) u (cid:107) (cid:96) ∞ n L T L Qn ‡ It is further assumed in [11] that (cid:107) b (cid:107) L ∞ T L ∞ x is small, although this appears to be unnecessary intheir argument. FFECTIVE DYNAMICS OF DOUBLE SOLITONS FOR PERTURBED MKDV 53
Theorem A.1 (local well-posedness) . Take k ∈ Z , k ≥ . Suppose that M def = k +1 (cid:88) j =0 (cid:107) ∂ jx b ( x, t ) (cid:107) L ∞ [0 , L ∞ x < ∞ . For any R ≥ , take T (cid:46) min( M − , R − ) . (1) If (cid:107) u (cid:107) H k ≤ R , there exists a solution u ( t ) ∈ C ([0 , T ]; H kx ) to (1.1) on [0 , T ] with initial data u ( x ) satisfying (cid:107) u (cid:107) L ∞ T H kx + (cid:107) ∂ k +1 x u (cid:107) (cid:96) ∞ n L T L Qn (cid:46) R . (2)
This solution u ( t ) is unique among all solutions in C ([0 , T ]; H x ) . (3) The data-to-solution map u (cid:55)→ u ( t ) is continuous as a mapping H k → C ([0 , T ]; H kx ) . The main tool in the proof of Theorem A.1 is the local smoothing estimate (A.2)below.
Lemma A.1.
Suppose that v t + v xxx − ( bv ) x = f . We have, for T (cid:46) (1 + (cid:107) b x (cid:107) L ∞ T L ∞ x + (cid:107) b (cid:107) L ∞ T L ∞ x ) − , the energy and local smoothing estimates (A.2) (cid:107) v (cid:107) L ∞ T L x + (cid:107) v x (cid:107) (cid:96) ∞ n L T L Qn (cid:46) (cid:107) v (cid:107) L x + (cid:40) (cid:107) ∂ − x f (cid:107) (cid:96) n L T L Qn (cid:107) f (cid:107) L T L x and the maximal function estimate (A.3) (cid:107) v (cid:107) (cid:96) n L ∞ T L Qn (cid:46) (cid:107) v (cid:107) L x + T / (cid:107) v (cid:107) L T H x + T / (cid:107) f (cid:107) L T L x . The implicit constants are independent of b .Proof. Let ϕ ( x ) = − tan − ( x − n ), and set w ( x, t ) = e ϕ ( x ) v ( x, t ). Note that 0 < e − π ≤ e ϕ ( x ) ≤ e π < ∞ , so the inclusion of this factor is harmless in the estimates, althoughhas the benefit of generating the “local smoothing” term in (A.2). We have ∂ t w + w xxx − ϕ (cid:48) w xx +3( − ϕ (cid:48)(cid:48) +( ϕ (cid:48) ) ) w x +( − ϕ (cid:48)(cid:48)(cid:48) +3 ϕ (cid:48)(cid:48) ϕ (cid:48) − ( ϕ (cid:48) ) ) w − ( bw ) x + ϕ (cid:48) bw = e ϕ f . This equation and manipulations based on integration by parts show that ∂ t (cid:107) w (cid:107) L x = 6 (cid:104) ϕ (cid:48) , w x (cid:105) − (cid:104) ( − ϕ (cid:48)(cid:48) + ( ϕ (cid:48) ) ) (cid:48) , w (cid:105) + 2 (cid:104)− ϕ (cid:48)(cid:48)(cid:48) + 3 ϕ (cid:48)(cid:48) ϕ (cid:48) − ( ϕ (cid:48) ) , w (cid:105)− (cid:104) b x , w (cid:105) + 2 (cid:104) bϕ (cid:48) , w (cid:105) + 2 (cid:104) w, e ϕ f (cid:105) . We integrate the above identity over [0 , T ], move the smoothing term 6 (cid:82) T (cid:104) ϕ (cid:48) , w x (cid:105) x dt over to the left side, and estimate the remaining terms to obtain: (cid:107) w ( T ) (cid:107) L x + 6 (cid:107)(cid:104) x − n (cid:105) − w x (cid:107) L T L x ≤ (cid:107) w (cid:107) L x + CT (1 + (cid:107) b x (cid:107) L ∞ T L ∞ x + (cid:107) b (cid:107) L ∞ T L ∞ X ) (cid:107) w (cid:107) L ∞ T L x + C (cid:90) T (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) e ϕ f w dx (cid:12)(cid:12)(cid:12)(cid:12) dt . Replacing T by T (cid:48) , and taking the supremum over T (cid:48) ∈ [0 , T ], we obtain, for T (cid:46) (1 + (cid:107) b x (cid:107) L ∞ T L ∞ x + (cid:107) b (cid:107) L ∞ [0 ,T ] L ∞ x ) − , the estimate (cid:107) w (cid:107) L ∞ T L x + (cid:107)(cid:104) x − n (cid:105) − w x (cid:107) L T L x (cid:46) (cid:107) w (cid:107) L x + (cid:90) T (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) e ϕ f w dx (cid:12)(cid:12)(cid:12)(cid:12) dt Using that 0 < e − π/ ≤ e ϕ ≤ e π/ < ∞ , this estimate can be converted back to anestimate for v : (cid:107) v (cid:107) L ∞ T L x + (cid:107) v x (cid:107) L T L Qn (cid:46) (cid:107) v (cid:107) L x + (cid:90) T (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) e ϕ f v dx (cid:12)(cid:12)(cid:12)(cid:12) dt . Estimating as (cid:90) T (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) e ϕ f v dx (cid:12)(cid:12)(cid:12)(cid:12) dt (cid:46) (cid:107) f (cid:107) L T L x (cid:107) v (cid:107) L ∞ T L x , and then taking the supremum in n yields the second bound in (A.2). Estimatinginstead as: (cid:90) T (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) e ϕ f v dx (cid:12)(cid:12)(cid:12)(cid:12) dt = (cid:90) T (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) e ϕ ( ∂ x ∂ − x f ) v dx (cid:12)(cid:12)(cid:12)(cid:12) dt ≤ (cid:90) T (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) ( ∂ − x f ) ∂ x ( e ϕ v ) dx (cid:12)(cid:12)(cid:12)(cid:12) dt ≤ (cid:88) m (cid:107) ∂ − x f (cid:107) L T L Qm (cid:107)(cid:104) ∂ x (cid:105) v (cid:107) L T L Qm ≤ (cid:107) ∂ − x f (cid:107) (cid:96) m L T L Qm (cid:107)(cid:104) ∂ x (cid:105) v (cid:107) (cid:96) ∞ m L T L Qm and taking the supremum in n yields the second bound in (A.2).For the estimate (A.3), we take ψ ( x ) = 1 on [ n − , n + ] and 0 outside [ n − , n +1],set w = ψv , and compute, similarly to the above, (cid:107) v (cid:107) L ∞ T L Qn (cid:46) (cid:107) v (cid:107) L Qn + T (cid:107) v x (cid:107) L T L Qn + T (cid:107) f (cid:107) L T L Qn The proof is completed by summing in n . (cid:3) Proof of Theorem A.1.
We prove the existence by contraction in the space X , where X = { u | (cid:107) u (cid:107) C ([0 ,T ]; H kx ) + (cid:107) ∂ k +1 x u (cid:107) (cid:96) ∞ n L T L Qn + sup α ≤ k − (cid:107) ∂ αx u (cid:107) (cid:96) n L ∞ T L Qn ≤ CR } . FFECTIVE DYNAMICS OF DOUBLE SOLITONS FOR PERTURBED MKDV 55
Here C is just chosen large enough to exceed the implicit constant in (A.2). Given u ∈ X , let ϕ ( u ) denote the solution to(A.4) ∂ t ϕ ( u ) + ∂ x ϕ ( u ) − ∂ x ( bϕ ( u )) = − ∂ x ( u ) . with initial condition ϕ ( u )(0) = u . A fixed point ϕ ( u ) = u in X will solve (1.1). Weseparately treat the case k = 1 for clarity of exposition. Case k = 1. Applying ∂ x to (A.4) gives, with v = ϕ ( u ) x , v t + v xxx − ( bv ) x = − u ) xx + ( b x ϕ ( u )) x . Now, (A.2) gives (cid:107) ϕ ( u ) x (cid:107) L ∞ T L x + (cid:107) ϕ ( u ) xx (cid:107) (cid:96) ∞ n L T L Qn (cid:46) (cid:107) u (cid:107) H x + (cid:107) ( u ) x (cid:107) (cid:96) n L T L Qn + (cid:107) ( b x ϕ ( u )) x (cid:107) L T L x . (A.5)Using that (cid:107) u (cid:107) L ∞ Q (cid:46) ( (cid:107) u (cid:107) L Q + (cid:107) u x (cid:107) L Q ) (cid:107) u (cid:107) L Q , we also have (cid:107) ( u ) x (cid:107) L Q (cid:46) (cid:107) u x (cid:107) L Q (cid:107) u (cid:107) L ∞ Q (cid:46) (cid:107) u x (cid:107) L Q (cid:107) u (cid:107) L Q ( (cid:107) u (cid:107) L Q + (cid:107) u x (cid:107) L Q ) . Taking the L T norm and applying the H¨older inequality, we obtain (cid:107) ( u ) x (cid:107) L T L Q (cid:46) (cid:107) u x (cid:107) L ∞ T L Q (cid:107) u (cid:107) L ∞ T L Q ( (cid:107) u (cid:107) L T L Q + (cid:107) u x (cid:107) L T L Q ) . Taking the (cid:96) n norm and applying the H¨older inequality again yields (cid:107) ( u ) x (cid:107) (cid:96) L T L Qn (cid:46) (cid:107) u x (cid:107) (cid:96) ∞ n L ∞ T L Qn (cid:107) u (cid:107) (cid:96) n L ∞ T L Qn ( (cid:107) u (cid:107) (cid:96) n L T L Qn + (cid:107) u x (cid:107) (cid:96) n L T L Qn ) . Using the straightforward bounds (cid:107) u x (cid:107) (cid:96) ∞ n L ∞ T L Qn (cid:46) (cid:107) u x (cid:107) L ∞ T L x , (cid:107) u (cid:107) (cid:96) n L T L Qn (cid:46) (cid:107) u (cid:107) L T L x (cid:46) T / (cid:107) u (cid:107) L ∞ T L x and (cid:107) u x (cid:107) (cid:96) n L T L Qn (cid:46) (cid:107) u x (cid:107) L T L x (cid:46) T / (cid:107) u x (cid:107) L ∞ T L x , we obtain (cid:107) ( u ) x (cid:107) (cid:96) n L T L Qn (cid:46) T / (cid:107) u (cid:107) L ∞ T H x (cid:107) u (cid:107) (cid:96) n L ∞ T L Qn . Inserting these bounds into (A.5),(A.6) (cid:107) ϕ ( u ) x (cid:107) L ∞ T L x + (cid:107) ϕ ( u ) xx (cid:107) (cid:96) ∞ n L T L Qn (cid:46) (cid:107) u (cid:107) H x + T / (cid:107) u (cid:107) L ∞ T H x (cid:107) u (cid:107) (cid:96) n L ∞ T L Qn + T ( (cid:107) b x (cid:107) L ∞ x + (cid:107) b xx (cid:107) L ∞ x ) (cid:107) ϕ ( u ) (cid:107) H x . The local smoothing estimate (A.2) applied to v = ϕ ( u ) (not v = ϕ ( u ) x as above),and the estimate (cid:107) ( u ) x (cid:107) L T L x (cid:46) T (cid:107) u (cid:107) L ∞ T H x , provides the estimate(A.7) (cid:107) ϕ ( u ) (cid:107) L ∞ T L x (cid:46) T (cid:107) u (cid:107) L ∞ T H x The maximal function estimate (A.3) applied to v = ϕ ( u ) and the estimate (cid:107) ( u ) x (cid:107) L T L x (cid:46) T / (cid:107) u (cid:107) L ∞ T H x , give the estimate(A.8) (cid:107) ϕ ( u ) (cid:107) (cid:96) n L ∞ T L Qn (cid:46) (cid:107) u (cid:107) L x + T (cid:107) ϕ ( u ) (cid:107) L ∞ T H x + T (cid:107) u (cid:107) L ∞ T H x . Summing (A.6), (A.7), (A.8), we obtain that (cid:107) ϕ ( u ) (cid:107) X ≤ CR if (cid:107) u (cid:107) X ≤ CR provided T is as stated above. Thus ϕ : X → X . A similar argument establishes that ϕ is acontraction on X . Case k ≥
2. Differentiating (A.4) k times with respect to x we obtain, with v = ∂ kx ϕ ( u ), ∂ t v + ∂ x v − ∂ x ( bv ) = − ∂ k +1 x ( u ) − ∂ x (cid:88) α + β ≤ k +1 β ≤ k − ∂ αx b ∂ βx ϕ ( u ) . Using (A.2) gives (cid:107) ∂ kx ϕ ( u ) (cid:107) L ∞ T L x + (cid:107) ∂ k +1 x ϕ ( u ) (cid:107) (cid:96) ∞ n L T L Qn (cid:46) (cid:107) ∂ kx u (cid:107) (cid:96) n L T L Qn + sup α + β ≤ k +1 β ≤ k − (cid:107) ∂ x ( ∂ αx b ∂ βx ϕ ( u )) (cid:107) L T L x . Expanding, and applying Leibniz rule gives ∂ kx u = (cid:88) α + β + γ = kα ≤ β ≤ γ c αβγ ∂ αx u ∂ βx u ∂ γx u , which is then estimated as follows (cid:107) ∂ kx u (cid:107) (cid:96) n L T L Qn (cid:46) (cid:88) α + β + γ = kα ≤ β ≤ γ (cid:107) ∂ αx u (cid:107) (cid:96) n L ∞ T L ∞ Qn (cid:107) ∂ βx u (cid:107) (cid:96) n L T L ∞ Qn (cid:107) ∂ γx u (cid:107) (cid:96) ∞ n L ∞ T L Qn . By the Sobolev embedding theorem (as in the k = 1 case) we obtain (cid:107) ∂ kx u (cid:107) (cid:96) n L T L Qn (cid:46) (cid:88) α + β + γ = kα ≤ β ≤ γ (cid:18) sup σ ≤ α +1 (cid:107) ∂ σx u (cid:107) (cid:96) n L ∞ T L Qn (cid:19) (cid:18) sup σ ≤ β +1 (cid:107) ∂ σx u (cid:107) (cid:96) n L T L Qn (cid:19) (cid:107) ∂ γx u (cid:107) L ∞ T L x When k ≥
2, we have α ≤ [[ k ]] ≤ k − β ≤ [[ k ]] ≤ k −
1, and therefore (cid:107) ∂ kx u (cid:107) (cid:96) n L T L Qn (cid:46) T / (cid:18) sup α ≤ k − (cid:107) ∂ αx u (cid:107) (cid:96) n L ∞ T L Qn (cid:19) (cid:107) u (cid:107) L ∞ T H kx . Also, (cid:107) ∂ x ( ∂ αx b ∂ βx ϕ ( u )) (cid:107) L T L x ≤ T (cid:18) sup α ≤ k +1 (cid:107) ∂ αx b (cid:107) L ∞ T L ∞ x (cid:19) (cid:107) ϕ ( u ) (cid:107) L ∞ T H kx FFECTIVE DYNAMICS OF DOUBLE SOLITONS FOR PERTURBED MKDV 57
Combining these estimates, we obtain (cid:107) ∂ kx ϕ ( u ) (cid:107) L ∞ T L x + (cid:107) ∂ k +1 x ϕ ( u ) (cid:107) (cid:96) ∞ n L T L Qn (cid:46) (cid:107) u (cid:107) H kx + T / (cid:18) sup α ≤ k − (cid:107) ∂ αx u (cid:107) (cid:96) n L ∞ T L Qn (cid:19) (cid:107) u (cid:107) L ∞ T H kx + T (cid:18) sup α ≤ k +1 (cid:107) ∂ αx b (cid:107) L ∞ T L ∞ x (cid:19) (cid:107) ϕ ( u ) (cid:107) L ∞ T H kx (A.9)The local smoothing (cid:107) ( u ) x (cid:107) L T L x (cid:46) T (cid:107) u (cid:107) L ∞ T H x to obtain(A.10) (cid:107) ϕ ( u ) (cid:107) L ∞ T L x (cid:46) T (cid:107) u (cid:107) L ∞ T H x We apply the maximal function estimate (A.3) to v = ∂ αx ϕ ( u ) for α ≤ k − (cid:107) ∂ α +1 x u (cid:107) L T L x ≤ T (cid:107) u (cid:107) L ∞ T H kx and (cid:107) ∂ α +1 x ( bϕ ( u )) (cid:107) L T L x ≤ T (cid:18) sup β ≤ k (cid:107) ∂ βx b (cid:107) L ∞ T L ∞ x (cid:19) (cid:107) ϕ ( u ) (cid:107) L ∞ T H kx to obtain(A.11) (cid:107) ∂ αx ϕ ( u ) (cid:107) (cid:96) n L ∞ T L Qn (cid:46) (cid:107) u (cid:107) H k − x + T (cid:107) ϕ ( u ) (cid:107) L ∞ T H kx + T (cid:107) u (cid:107) L ∞ T H kx + T (cid:18) sup β ≤ k (cid:107) ∂ βx b (cid:107) L ∞ T L ∞ x (cid:19) (cid:107) ϕ ( u ) (cid:107) L ∞ T H kx Summing (A.9), (A.10), (A.11), we obtain that ϕ : X → X , and a similar argumentshows that ϕ is a contraction. This concludes the case k ≥ C ([0 , T ]; H x ), we argue as follows. Suppose u, v ∈ C ([0 , T ]; H x ) solve (1.1). By (A.3), (cid:107) v (cid:107) (cid:96) n L ∞ T L Qn (cid:46) (cid:107) v (cid:107) L + T (cid:107) v (cid:107) L ∞ T H x + T (cid:107) v (cid:107) L ∞ T H x . By taking T small enough in terms of (cid:107) v (cid:107) L ∞ T H x , we have that(A.12) (cid:107) v (cid:107) (cid:96) n L ∞ T L Qn (cid:46) (cid:107) v (cid:107) L ∞ T H x . Similarly,(A.13) (cid:107) u (cid:107) (cid:96) n L ∞ T L Qn (cid:46) (cid:107) u (cid:107) L ∞ T H x . Set w = u − v . Then, with g = ( u − v ) / ( u − v ) = u + uv + v , we have w t + w xxx − ( bw ) x ± ( gw ) x = 0 . Apply (A.2) to v = w x to obtain(A.14) (cid:107) w x (cid:107) L ∞ T L x + (cid:107) w xx (cid:107) (cid:96) ∞ n L T L Qn (cid:46) (cid:107) ( gw ) x (cid:107) (cid:96) n L T L Qn + (cid:107) ( b x w ) x (cid:107) L T L x The terms of (cid:107) ( gw ) x (cid:107) (cid:96) n L T L Qn are bounded following the method used above: (cid:107) u x vw (cid:107) (cid:96) n L T L Qn (cid:46) (cid:107) u x (cid:107) (cid:96) ∞ n L ∞ T L Qn (cid:107) vw (cid:107) (cid:96) n L T L ∞ Qn (cid:46) (cid:107) u x (cid:107) (cid:96) ∞ n L ∞ T L Qn ( (cid:107) vw (cid:107) (cid:96) n L T L Qn + (cid:107) ( vw ) x (cid:107) (cid:96) n L T L Qn ) The term in parentheses is bounded by (cid:107) v (cid:107) (cid:96) n L T L Qn (cid:107) w (cid:107) (cid:96) n L ∞ T L Qn + (cid:107) v x (cid:107) (cid:96) n L T L Qn (cid:107) w (cid:107) (cid:96) n L ∞ T L Qn + (cid:107) v (cid:107) (cid:96) n L ∞ T L Qn (cid:107) w x (cid:107) (cid:96) n L T L Qn which leads to the bound(A.15) (cid:107) u x vw (cid:107) (cid:96) n L T L Qn (cid:46) T / (cid:107) u (cid:107) L ∞ T H x ( (cid:107) v (cid:107) L ∞ T H x (cid:107) w (cid:107) (cid:96) n L ∞ T L Qn + (cid:107) v (cid:107) (cid:96) n L ∞ T L Qn (cid:107) w (cid:107) L ∞ T H x )We now allow implicit constants to depend upon (cid:107) u (cid:107) L ∞ T H x and (cid:107) v (cid:107) L ∞ T H x . Appealingto (A.14), (A.15) (and analogous estimates for other terms in gw ), (A.12), (A.13) toobtain (cid:107) w (cid:107) L ∞ T H x (cid:46) T / ( (cid:107) w (cid:107) (cid:96) n L ∞ T L Qn + (cid:107) w (cid:107) L ∞ T H x )Combining this estimate with the maximal function estimate (A.3) applied to w yields (cid:107) w (cid:107) (cid:96) n L ∞ T L Qn (cid:46) T / (cid:107) w (cid:107) L ∞ T H x + T (cid:107) g (cid:107) L ∞ T H x (cid:107) w (cid:107) L ∞ T H x . This gives w ≡ T sufficiently small. The continuity of the data-to-solution mapis proved using similar arguments. (cid:3) Next, we prove global well-posedness in H k by proving a priori bounds. TheoremA.1 shows that doing it suffices for global well-posedness Theorem A.2 (global well-posedness) . Fix k ≥ and suppose M ( T ) < ∞ for all T ≥ , where M ( T ) is defined in (A.1) . For u ∈ H k , there is a unique global solution u ∈ C loc ([0 , + ∞ ); H kx ) to (1.1) with (cid:107) u (cid:107) L ∞ T H kx controlled by (cid:107) u (cid:107) H k , T , and M ( T ) .Proof. Before beginning, we note that by the Gagliaro-Nirenberg inequality, (cid:107) u (cid:107) L (cid:46) (cid:107) u (cid:107) L (cid:107) u x (cid:107) L , we have (in the focusing case) (cid:107) u x (cid:107) L − (cid:107) u x (cid:107)(cid:107) u (cid:107) L ≤ I ( u ) ≤ (cid:107) u x (cid:107) L . With α = (cid:107) u x (cid:107) L / (cid:107) u (cid:107) L and β = I ( u ) / (cid:107) u (cid:107) L , this is α − α / ≤ β ≤ α , which impliesthat (cid:104) α (cid:105) ∼ (cid:104) β (cid:105) , i.e. (cid:107) u x (cid:107) L + (cid:107) u (cid:107) L ∼ I ( u ) + (cid:107) u (cid:107) L The same statement holds in the defocusing case.Another fact we need is based on the ddt I j ( u ) = (cid:104) I (cid:48) j ( u ) , ∂ t u (cid:105) = (cid:104) I (cid:48) j ( u ) , − u xxx − u ) x + ( bu ) x (cid:105) = (cid:104) I (cid:48) j ( u ) , ∂ x I (cid:48) ( u ) (cid:105) + (cid:104) I (cid:48) j ( u ) , ( bu ) x (cid:105) = (cid:104) I (cid:48) j ( u ) , ( bu ) x (cid:105) For u ( t ) ∈ L , we compute near conservation of momentum and energy from Lemma2.1: ddt I ( u ) = (cid:104) b x , A ( u ) (cid:105) FFECTIVE DYNAMICS OF DOUBLE SOLITONS FOR PERTURBED MKDV 59
Estimate |(cid:104) b x , A ( u ) (cid:105)| ≤ (cid:107) b x (cid:107) L ∞ I ( u ), and apply Gronwall to obtain a bound on (cid:107) u (cid:107) L ∞ T L x in terms of (cid:107) b x (cid:107) L ∞ T L ∞ and (cid:107) u (cid:107) L . For u ( t ) ∈ H , we compute near conser-vation of energy from Lemma 2.1: ddt I ( u ) = 3 (cid:104) b x , A ( u ) (cid:105) − (cid:104) b xxx , A ( u ) (cid:105) . We have |(cid:104) b x , A ( u ) (cid:105)| (cid:46) (cid:107) b x (cid:107) L ∞ ( (cid:107) u x (cid:107) L + (cid:107) u (cid:107) L ) (cid:46) (cid:107) b x (cid:107) L ∞ ( (cid:107) u x (cid:107) L + (cid:107) u x (cid:107) L (cid:107) u (cid:107) L ) (cid:46) (cid:107) b x (cid:107) L ∞ ( (cid:107) u x (cid:107) L + (cid:107) u (cid:107) L ) (cid:46) (cid:107) b x (cid:107) L ∞ ( I ( u ) + (cid:107) u (cid:107) L )and |(cid:104) b xxx , A ( u ) (cid:105)| (cid:46) (cid:107) b xxx (cid:107) L ∞ (cid:107) u (cid:107) L . Combining these gives (cid:12)(cid:12)(cid:12)(cid:12) ddt I ( u ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:107) b x (cid:107) L ∞ I ( u ) + (cid:107) b x (cid:107) L ∞ (cid:107) u (cid:107) L + (cid:107) b xxx (cid:107) L ∞ (cid:107) u (cid:107) L Gronwall’s inequality, combined with the previous bound on (cid:107) u (cid:107) L , gives the boundon I ( u ) and hence (cid:107) u (cid:107) H .For u ( t ) ∈ H , we apply Lemma 2.1 to obtain ddt I ( u ) = (cid:104) I (cid:48) ( u ) , ( bu ) x (cid:105) = 5 (cid:104) b x , A ( u ) (cid:105) − (cid:104) b xxx , A ( u ) (cid:105) + (cid:104) b xxxxx , A ( u ) (cid:105) We have |(cid:104) b x , A ( u ) (cid:105)| (cid:46) (cid:107) b x (cid:107) L ∞ ( (cid:107) u xx (cid:107) L + (cid:107) u (cid:107) H + (cid:107) u (cid:107) H ) (cid:46) (cid:107) b x (cid:107) L ∞ I ( u ) + (cid:107) b x (cid:107) L ∞ ( (cid:107) u (cid:107) H + (cid:107) u (cid:107) H )Also, |(cid:104) b xxx , A ( u ) (cid:105)| (cid:46) (cid:107) b xxx (cid:107) L ∞ ( (cid:107) u (cid:107) H + (cid:107) u (cid:107) H )and |(cid:104) b xxxxx , A ( u ) (cid:105)| (cid:46) (cid:107) b xxx (cid:107) L ∞ (cid:107) ( u ) xx (cid:107) L (cid:46) (cid:107) b xxx (cid:107) L ∞ (cid:107) u (cid:107) H (cid:107) u (cid:107) L Combining, applying Gronwall’s inequality, and appealing to the bound on (cid:107) u (cid:107) H obtained previously, we obtain the claimed a priori bound in the case k = 2.Bounds on H k for k ≥ k = 3, we donot need such refined information. By direct computation from (1.1), ddt (cid:107) ∂ kx u (cid:107) L = − (cid:90) ∂ k +1 x ( bu ) ∂ kx u + 2 (cid:90) ∂ k +1 x u ∂ kx u In the Leibniz expansion of ∂ k +1 x u , we isolate two cases: ∂ k +1 x u = 3 u ∂ k +1 x u + (cid:88) α + β + γ = k +1 α ≤ β ≤ γ ≤ k c αβγ ∂ αx u ∂ βx u ∂ γx u For the first term, (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) u ∂ k +1 x u ∂ kx u (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) ( u ) x ( ∂ kx u ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:107) u (cid:107) H (cid:107) u (cid:107) H k By the H¨older’s inequality and interpolation, if α + β + γ = k + 1 and γ ≤ k , (cid:107) ∂ αx u ∂ βx u ∂ γx u (cid:107) L (cid:46) (cid:107) u (cid:107) H (cid:107) u (cid:107) H k Thus we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) ∂ k +1 x u ∂ kx u (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:107) u (cid:107) H (cid:107) u (cid:107) H k Similarly, we can bound (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) ∂ k +1 x ( bu ) ∂ kx u (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) M ( t ) (cid:107) u (cid:107) H k by separately considering the term b ∂ k +1 x u ∂ kx u and integrating by parts. We obtain (cid:12)(cid:12)(cid:12)(cid:12) ddt (cid:107) ∂ kx u (cid:107) L (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) ( M + (cid:107) u (cid:107) H ) (cid:107) u (cid:107) H k and can apply the Gronwall inequality to obtain the desired a priori bound. (cid:3) Appendix B. Comments about the effective ODEs
Here we make some comments about the differential equations for the parameters a and c .B.1. Conditions on T . First we give a reason for replacing T ( h ) in the definitionof T ( h ) (1.5) by T defined by (1.12). In (10.2) we have seen that the a and c solvingthe system (1.4) give the following equations for (cid:101) A = ha , (cid:101) C = c , T = ht : (cid:40) ∂ T (cid:101) A j = (cid:101) C j − b ( (cid:101) A j , T ) + O ( h ) ∂ T (cid:101) C j = (cid:101) C j ∂ x b ( (cid:101) A j , T ) + O ( h ) , (cid:101) A (0) = ¯ ah , (cid:101) C (0) = ¯ c , j = 1 , . This can also be seen by analysing (B.6) using Lemma 3.2.As in (10.4) we can write the equations for (cid:101) A j − A j and (cid:101) C j − C j : ∂ T ( (cid:101) A j − A j ) = ( (cid:101) C j − C j ) + 2 C j ( (cid:101) C j − C j ) + γ ( (cid:101) A j − A j ) + O ( h ) ∂ T ( (cid:101) C j − C j ) = ( (cid:101) C j − C j )( ∂ x b )( A j , t ) + C j σ ( (cid:101) A j − A j ) + O ( h ) , (cid:101) A j (0) − A j (0) = 0 , (cid:101) C j (0) − C j (0) = 0 , FFECTIVE DYNAMICS OF DOUBLE SOLITONS FOR PERTURBED MKDV 61 where γ , σ = O (1). This implies that (cid:40) (cid:101) A j ( T ) − A j ( T ) = O ( h ) e CT , (cid:101) C j ( T ) − C j ( T ) = O ( h ) e CT . This means that for
T < δ log(1 /h ), we have C j ( T ) = (cid:101) C j ( T ) + O ( h − δC ). Hence, if δ is small enough, then for small h we have that T ( h ) defined in (1.5) and T in (1.12)can be interchanged.B.2. Examples with C j going to . In the decoupled equations (1.11) we can have C j ( T ) → , T → ∞ , which implies that T < ∞ in the definition (1.12). That prevents log(1 /h ) /h lifespanof the approximation (1.3).Let us put a = A j , c = C j , so that the system (1.11) becomes(B.1) a (cid:48) T = c ( T ) − b ( a, T ) , c (cid:48) T = c ∂ a b ( a, T ) . For simplicity we consider the case of b ( a, T ) = b ( a ). In that case the Hamiltonian E ( a, c ) = − c + cb ( a )is conserved in the evolution and we have(B.2) exp( T min ∂ a b ) ≤ | c ( T ) | ≤ exp( T max ∂ a b ) . In particular this means that c > δ >
T < T ( δ ).We cannot improve on (B.2), and in general we may have | c ( T ) | ≤ e − γT , T → ∞ , but this behaviour is rare. First we note that the conservation of E shows that if c ( T j ) → T j → ∞ , then E = 0. We can then solve for c , and theequation reduces to da/dT = 2 b ( a ), c = 3 b ( a ), that is to(B.3) 12 (cid:90) aa d ˜ ab (˜ a ) = T , b ( a (0)) > . If b ( a ) > a then(B.4) a ( T ) → ∞ , T → ∞ , and c ( T ) = (3 b ( a ( T ))) . If b ( a ) = 0 for some a > a (0) ( a (cid:48) T = 2 b > a , the smallestsuch a and assume that the order of vanishing of b there is (cid:96) . The analysis of (B.3)shows that a ( T ) = a + O (1) Ke − γT (cid:96) = 1 ,KT − / ( (cid:96) − (cid:96) > , which gives the rate of decay of c ( T ).Hence we have shown the following statement which is almost as long to state asto prove: Lemma B.1.
Suppose that in (B.1) b = b ( a ) . Then E (cid:54) = 0 , | c (0) | > δ > ⇒ ∃ δ > ∀ T > , | c ( T ) | > δ . If E = 0 , let a = min { a : a > a (0) , b ( a ) = 0 } , with a not defined if the set is empty (note that c (0) (cid:54) = 0 and E = 0 imply that b ( a (0)) > ). Now suppose that a exists, and that ∂ (cid:96) b ( a ) = 0 , (cid:96) < (cid:96) , ∂ (cid:96) b ( a ) (cid:54) = 0 . Then as T → ∞ , | c ( T ) | ≤ Ke − γT (cid:96) = 1 ,KT − (cid:96) / ( (cid:96) − (cid:96) > , for some constants γ and K , and a ( T ) → a .If a does not exist then c ( T ) = (3 b ( a ( T ))) , a ( T ) → ∞ , T → ∞ . We excluded the case of infinite order of vanishing since it is very special from ourpoint of view.The lemma suggests that c → t in (1.1) we would like to go up to time δ log(1 /h ) /h wecannot do it in some cases as then c ( t ) | t = δ log(1 /h ) /h ∼ h γδ/ (cid:96) = 1 , log − (cid:96) / ( (cid:96) − (1 /h ) (cid:96) > . B.3.
Avoided crossing for the effective equations of motion.
Here we makesome comments about the puzzling avoided crossing which needs further investigation.For the decoupled equations it is easy to find examples in which(B.5) c ( T ) = c ( T ) . One is shown in Fig.6. We take b independent of T and equal to cos x . If wechoose the initial conditions so that c j = 3 cos A j , A j = ha j as in (1.11), and FFECTIVE DYNAMICS OF DOUBLE SOLITONS FOR PERTURBED MKDV 63 ! ! Figure 6.
The plots of ( A j , c j ), j = 1 ,
2, solving (B.6) for for b ( x, t ) = cos x and initial data A (0) = − π/ A (0) = π/
6, and c (0) = √ π/ c (0) = √ π/ A j , c j ), j = 1 , − π/ < A < − A <
0, then when A ( T ) = − A ( T ) we have (B.5) (this alsoprovides an example of c ( T ) → T → ∞ ).The decoupled equations (1.11) should be compared the rescaled version of (1.4): ∂ T c j = ∂ x j B ( c, A, h ) , ∂ T A j = c j − ∂ c j B ( c, A, h ) ,B ( c, A, h ) def = 12 (cid:90) q ( x/h, c, A/h ) b ( x ) dx . (B.6)For the example above the comparison between the solutions of the decoupled h -independent equations and solutions to the equation (B.6) are shown in Fig.6 (thesolutions (1.11) are shown as a single curve which both solutions with these initialdata follow). ! ! ! ! ! ! ! ! ! Figure 7.
The plots of q ( x, c, A/h ) for ( A j , c j ), j = 1 ,
2, solving (B.6)for for b ( x, t ) = cos x and initial data A (0) = − π/ A (0) = π/ c (0) = √ π/ c (0) = √ π/ h = 0 . h = 0 . b in Fig.3) are not seen in the behaviour of q ( x, c, A/h ) which is theapproximation of the solution to (1.1) – see Fig.7. The masses of the right and leftsolitons are switched and that corresponds to the switch of positions of A and A .It is possible that a different parametrization of double solitons would resolve thisproblem. Another possibility is to study the decomposition (3.11) in the proof ofLemma 3.2 uniformly α → a − a → | A − A | > (cid:15) > h log (cid:18) c − c c + c (cid:19) , see Lemma 3.2. For this to affect the motion of trajectories on finite time scales in T we need(B.7) c − c (cid:39) exp (cid:16) − γh (cid:17) . This means that c j ’s have to get exponentially close to each other (but does notexplain avoided crossing).On the other hand if | a − a | > (cid:15) >
0, where a j ’s are the original variables in (1.4), A j (0) = ha j (0), then we can use the decomposition in Lemma 3.2 and variables ˆ a j defined by (3.7). The remark after the proof of Lemma 3.6 shows that the equationsof motion take essentially the same form written in terms of (cid:98) a j ’s and c j ’s and henceˆ a j has to stay bounded. And that means that c − c is bounded away from 0. Hence,when c − c → a − a → FFECTIVE DYNAMICS OF DOUBLE SOLITONS FOR PERTURBED MKDV 65
Appendix C. Alternative proof of Lemma 4.7 (with Bernd Sturmfels)
We note that the standard substition reduces the equation P ( c ) u = 0, where P ( c )is defined in (4.20), to an equation with rational coefficients: z = tanh x , ∂ x = (1 − z ) ∂ z , η = 1 − z . This means that P ( c ) u = 0 is equivalent to Q ( c ) v = 0, u ( x ) = v (tanh x ), where Q ( c ) = ( L + 1)( L + c ) − LR ( z ) L + 10(3 R ( z ) − R ( z ) ) − c ) R ( z ) , and L = 1 i (1 − z ) ∂ z , R ( z ) = 1 − z , − < z < . Lemma 4.7 will follow from finding a basis of solutions of Q ( c ) v = 0 and fromseeing that the only bounded solution is the one corresponding to ∂ x η , that is, to v ( z ) = z (1 − z ) . Remarkably, and no doubt because of some deeper underlying structure due to com-plete integrability, this can be achieved using
MAPLE package
DEtools .First, the operator Q ( c ) is brought to a convenient form Q = ( z − ( z + 1) d dz f ( z ) + 12 z ( z − ( z + 1) d dz f ( z )+ ( z − ( z + 1) (26 z − c + 1) d dz f ( z ) − z ( z − z + 1)(8 z −
11 + c ) ddz f ( z )+ (4 − z + 6 c z − c + 16 z ) f ( z )Applying the MAPLE command
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