Asymptotic stability of solitons for mKdV
aa r X i v : . [ m a t h . A P ] M a r ASYMPTOTIC STABILITY OF SOLITONS FOR MKDV
PIERRE GERMAIN, FABIO PUSATERI, AND FR´ED´ERIC ROUSSET
Abstract.
We prove a full asymptotic stability result for solitary wave solutions of themKdV equation. We consider small perturbations of solitary waves with polynomial decayat infinity and prove that solutions of the Cauchy problem evolving from such data tenduniformly, on the real line, to another solitary wave as time goes to infinity. We describeprecisely the asymptotics of the perturbation behind the solitary wave showing that itsatisfies a nonlinearly modified scattering behavior. This latter part of our result relies ona precise study of the asymptotic behavior of small solutions of the mKdV equation.
Contents
1. Introduction 12. Stability of the zero solution 83. Stability of solitons 26Appendix A. Auxiliary Lemmas 41References 431.
Introduction
This paper is concerned with the Cauchy problem for the focusing modified Korteweg-deVries (mKdV) equation ∂ t u + ∂ x u + ∂ x ( u ) = 0 u ( t = 0) = u (mKdV)for u = u ( t, x ) ∈ R and ( t, x ) ∈ R × R . This equation admits a family of solitary wavesolutions of the form u c ( t, x ) = Q c ( x − ct ) with Q c ( ξ ) = √ cQ ( √ c ξ ) , Q ( s ) := √ / cosh( s ) , c > . (1.1)Our aim in this paper is to revisit the proof of global existence and modified scatteringfor (mKdV) for small and localized initial data, and then extend it in order to obtain newasymptotic stability results for solitary wave solutions.Important conserved quantities are the mass M , energy H , and momentum PM = Z R u dx H = Z R | ∂ x u | − | u | dx, P = Z R u dx. (1.2) Key words and phrases. mKdV, modified scattering, asymptotic stability, solitons.P. G. is partially supported by NSF grant DMS-1101269, a start-up grant from the Courant Institute,and a Sloan fellowship. F. P. is partially supported by NSF grant DMS-1265875. As we will remark later, these are not needed to prove the small data result, which also applies to moregeneral versions of (mKdV).
Moreover, we note that solutions of (mKdV) enjoy the scaling symmetry u λu ( λ t, λx ) , which is generated by the vector field S = 1 + x∂ x + 3 t∂ t .1.1. Known results.
Global well-posedness and asymptotic behavior.
There is a vast body of literature dealingwith the mKdV equation, and in particular with the local and global well-posedness of theCauchy problem. Without trying to be exhaustive, we mention the early works on the localand global well-posedness by Kenig-Ponce-Vega [29] and Kato [27]. Global well-posednessin low regularity spaces, and in particular in the energy space H , was established in theseminal work of Kenig-Ponce-Vega [30]. In this latter paper the authors considered the widerclass of generalized KdV (gKdV) equations ∂ t u + ∂ x u + ∂ x u p = 0, p ≥
2, which includes(mKdV) and the KdV equation ( p = 2). Sharp, up to the end-point, global well-posedness in H s for s > / s ≥ / Besides global regularity, another fundamental question for dispersive PDEs concerns theasymptotic behavior for large times. The first proof of global existence with a completedescription of the asymptotic behavior of solutions of (mKdV) in the defocusing case, is dueto Deift and Zhou [9], who used a steepest descent approach to oscillatory Riemann-Hilbertproblems and the inverse scattering transform [52, 2]. In [9], thanks to the complete inte-grability of the defocusing mKdV equation, the authors were able to treat suitably localizedinitial data with arbitrary size. A proof of global existence and a (partial) derivation of theasymptotic behavior for small localized solutions, without making use of complete integra-bility, was later given by Hayashi and Naumkin [18, 19], following the ideas introduced inthe context of the 1d nonlinear Schr¨odinger (NLS) equation in [17]. Recently, an alternativeproof of the results in [19], with a precise derivation of asymptotics and a proof of asymptoticcompleteness, was given by Harrop-Griffiths [16], following the approach used for the 1d NLSequation in [21].Our proof of global existence and asymptotic behavior - Theorem 1.1 - relies on the intu-ition developed in [28], where a very natural stationary phase argument is used to understandthe large time behavior of small and localized solutions and derive asymptotic corrections.This approach was inspired by the space-time resonance method put forward in [13, 14, 15].See section 1.3 below for a short explanation of these ideas in the present context. A simi-lar approach was also successfully employed in the proofs of global regularity and modifiedscattering for 2d gravity [22, 23, 24] and capillary [25, 26] water waves, and in other higherdimensional dispersive models [28, 48].
Stability of solitons.
The study of the stability of solitons also has a long history, but herewe will only address results which are closer in spirit to the present paper. The asymptoticstability in front of the soliton was first obtained by Pego-Weinstein [47] for initial per-turbations of a soliton with exponential decay as x → + ∞ . This result was then refined For more on the local and global well-posedness and ill-posedness of KdV and generalized KdV equationswe refer to the books of Tao [49] and Linares-Ponce [34]. Similarly to how it is stated in Theorem 3.1.
SYMPTOTIC STABILITY OF SOLITONS FOR MKDV 3 by Mizumachi [43], who treated perturbations belonging to polynomially weighted spacesof sufficiently high order. For perturbations in the energy space H , definitive asymptoticstability results in front of the solitary wave have been obtained for the whole class of sub-critical gKdV equations in a series of papers by Martel-Merle [35, 36, 37]. We also mention[41] on the L stability of KdV solitons, [5] on the H s s ≥ − N -soliton solutions of subcritical gKdV equations, and [46, 44] for a different ap-proach. For more on the asymptotic stability of solitons and multi-solitons for subcriticalgKdV equations we refer the reader to the survey articles [51, 38] and references therein.In [43] the author also obtained a full stability result for gKdV equations with a nonlin-earity of degree p ∈ (3 , p = 4), there are also scattering and asymptoticstability results in critical spaces rather than polynomially weighted ones, see [50, 33].The results we present extend the above mentioned works byi) proving the (modified) scattering result for the radiation in the (critical) case of (mKdV),ii) allowing a wider class of small perturbations belonging to weighted Sobolev spaces withweak polynomial decay at infinity.Because of the critical dispersive nature of the equation, in the case of (mKdV) the radiationdoes not behave linearly, but requires a nonlinear correction. See Theorem 1.5 and Remark1.6 for more details. The proof that we give below combines the virial approach of Martel-Merle [36] and the weighted estimates of Pego-Weinstein [47], in the spirit of the recent workof Mizumachi-Tzvetkov [46] on the L stability of solitons for KdV.1.2. Main results.
Our first main result concerns the stability of the zero solution undersmall perturbations.
Theorem 1.1 (Global Existence and Asymptotic Behavior) . Let an initial data u be givensuch that kh x i u k H ( R ) ≤ ε . (1.3) There exists ε > , such that for all ε ∈ (0 , ε ] the Cauchy problem (mKdV) admits aunique global solution u ∈ C ( R , H ( R )) . This solution satisfies the decay estimates | u ( t, x ) | . ε t − / h x/t / i − / , | ∂ x u ( t, x ) | . ε t − / h x/t / i / . (1.4) Moreover, for t ≥ the solution u has the following asymptotics: • In the region x ≥ t / we have the improved decay | u ( t, x ) | . ε t / ( x/t / ) / ; (1.5) • In the region | x | ≤ t / γ , for some γ > sufficiently small, the solution is approximatelyself-similar: (cid:12)(cid:12) u ( t, x ) − t / ϕ (cid:0) xt / (cid:1)(cid:12)(cid:12) . ε t / γ/ , (1.6) PIERRE GERMAIN, FABIO PUSATERI, AND FR´ED´ERIC ROUSSET where ϕ is a bounded solution of the Painlev´e II equation ϕ ′′ − ξϕ + ϕ = 0 , Z R ϕ ( x ) dx = Z R u ( x ) dx. • In the region x ≤ − t / γ , the solution has a nonlinearly modified asymptotic behavior:there exists f ∞ ∈ L ∞ ξ such that (cid:12)(cid:12)(cid:12) u ( t, x ) − √ tξ ℜ exp (cid:16) − itξ + iπ i | f ∞ ( ξ ) | log t (cid:17) f ∞ ( ξ ) (cid:12)(cid:12)(cid:12) ≤ ε t / ( − x/t / ) / , (1.7) where ξ := p − x/ (3 t ) , and ℜ denotes the real part. Remark 1.2.
In the proof of Theorem 1.1 above, the Hamiltonian structure of the equation,as well as the conservation of mass and energy, do not play any crucial role. For conveniencewe will work with (mKdV), but it will be clear that all our results also apply to the defocusingmKdV equation ∂ t u + ∂ x u − ∂ x ( u ) = 0, and to other (not necessarily Hamiltonian) versionsof the equation, such as ∂ t u + ∂ x u = a ( t ) ∂ x ( u ) , (1.8)where | a ( t ) | ≤ , | a ′ ( t ) | ≤ h t i − / . Remark 1.3.
In Theorem 1.1 we have decided to state the global existence and scatteringresult for initial data satisfying (1.3), that is xu ∈ H . However, in the proof we onlymake use of the assumption xu ∈ H α , for some α close to, but less than, 1 /
2. We cantherefore treat a larger class of initial data with respect to [19, 16]. Nevertheless, we havedecided to state Theorem 1.1 assuming the stronger initial condition (1.3), in order to makeits application in the proof of Theorem 1.5 below more convenient.
Remark 1.4.
We chose to characterize the modified asymptotic behavior of u (1.7) in L ∞ ,but statements analogous to (1.5)-(1.7) can be obtained for L -type norms.Our second main result is a strong asymptotic stability result for soliton solutions, undersmall perturbations belonging to a weak algebraically weighted space. Theorem 1.5 (Full Asymptotic Stability of Solitons) . Assume that u ( x ) = Q c ( x ) + v ( x ) , (1.9) for some c > , with kh x i v k H ( R ) + kh x + i m v k H ( R ) ≤ ε , (1.10) for some m > / . Then, for ε sufficiently small, there exists a unique solution u ∈ C ( R , H ( R )) of (mKdV) and a continuous function C ( · ) with C (0) = 0 , such that for some c + > and x + with | c + − c | + | x + | . C ( ε ) , (1.11) The smallness of R ϕ ( x ) dx guarantees the existence and uniqueness of a bounded solution to the Painlev´eII equation. Its asymptotics are as follows: ϕ ( ξ ) ∼ / Ai(3 / ξ ) ∼ / √ πξ / e − √ ξ / as ξ → ∞ , while ϕ ( ξ ) ∼ / √ π | ξ | / d cos (cid:16) − √ | ξ | / + π + d π log | y | / + θ (cid:17) as ξ → −∞ , where d and θ are constants depending on R ϕ ( x ) dx . We refer to [20] and [10] for this and much more on Painlev´e II. Note that our proof will actuallyprovide the existence of a bounded solution of the Painlev´e equation. SYMPTOTIC STABILITY OF SOLITONS FOR MKDV 5 we have | u ( t, x ) − Q c + ( x − c + t − x + ) − R ( t, x ) | . C ( ε ) h t i − (1.12) where: • The radiation R verifies the decay estimates k R ( t ) k L ∞ ( R ) . C ( ε ) h t i − / . (1.13) • R has the same asymptotics as a small solution to (mKdV) , and in particular possessesa modified scattering behavior as t → ∞ as in (1.7) . Note that we prove a full asymptotic stability result by describing the behavior of theperturbation behind the solitary wave and that, because of the critical dispersive decay ofthe mKdV equation, the radiation has nonlinear asymptotic oscillation.
Remark 1.6.
Note that the spatial decay (1.10) that we require in front of the solitaryis only slightly more than x / u in L . This is at the same scale as the decay propertywhich is used in the inverse scattering theory, where one requires xu in L . Spatial decayconditions on the data are not explicitly stated in the work of Deift-Zhou on the defocusingmKdV equation [9], but the condition above is used in the application of direct and inversescattering in [1, 40]. We also refer to [32] for a recent survey.1.3. Ideas of the proof.
We now briefly explain the main ideas and the intuition behindour results.
Global existence and modified scattering.
In what follows we let f ( t ) := e t∂ x u ( t ) (1.14)so that ∂ t f = − e t∂ x ∂ x ( u ). Then we can write (mKdV) as ∂ t b f ( t, ξ ) = − π Z Z e − itφ ( ξ,η,σ ) iξ b f ( t, ξ − η − σ ) b f ( t, η ) b f ( t, σ ) dη dσ. (1.15)with φ ( ξ, η, σ ) = ξ − ( ξ − η − σ ) − η − σ = 3( η + σ )( ξ − η )( ξ − σ ) . We follow the approach of the space-time resonances method [12] which is to view the aboveintegral as an oscillatory integral, whose large-time behavior will thus be dictated by thestationary points (in η , and in t after time integration) of the phase φ . As observed in [28],this will give a very simple means of computing the large-time correction to scattering, dueto the long-range effects of the critically dispersing nonlinear term.Before explaining this argument, we need to describe precisely the stationary points of φ .A small computation gives that ∂ η φ ( ξ, η, σ ) = 3( ξ − σ )( ξ − η − σ ) ∂ σ φ ( ξ, η, σ ) = 3( ξ − η )( ξ − η − σ ) (1.16)and det Hess η,σ φ = − η + σ + ησ − ξη − ξσ ) . (1.17) PIERRE GERMAIN, FABIO PUSATERI, AND FR´ED´ERIC ROUSSET
Notice that ∂ η φ = ∂ σ φ = 0 ⇐⇒ ( η, σ ) = ( η i , σ i ) , ≤ i ≤ ( η , σ ) = ( ξ, ξ )( η , σ ) = ( ξ, − ξ )( η , σ ) = ( − ξ, ξ )( η , σ ) = (cid:0) ξ , ξ (cid:1) , (1.19)and that, for i ∈ { , , } , φ ( ξ, η i , σ i ) = 0 φ ( ξ, η , σ ) = (8 / ξ det Hess η,σ φ ( ξ, η i , σ i ) = − ξ det Hess η,σ φ ( ξ, η , σ ) = 12 ξ sign Hess η,σ φ ( ξ, η i , σ i ) = 0sign Hess η,σ φ ( ξ, η , σ ) = 1 − sign ξ, (1.20)where sign M is the signature of the matrix M .The above computations are the basis to derive - heuristically for the moment - the largetime behavior of f . By the stationary phase lemma, assuming that b f is sufficiently smooth,(1.15) implies that ∂ t b f ( t, ξ ) = i sign ξ t | b f ( ξ ) | b f ( ξ ) + ict e − it ξ b f (cid:18) ξ (cid:19) + { integrable terms } . (1.21)The second summand on the right-hand side should not be asymptotically relevant, due tothe time-oscillating term. Thus the above reduces to ∂ t b f ( t, ξ ) ∼ i sign ξ t | b f ( ξ ) | b f ( ξ ) + { integrable terms } , from which we infer that | b f | converges as t → ∞ to an asymptotic profile F , while b f ( t, ξ ) ∼ F ( ξ ) exp (cid:16) i sign ξ t | F ( ξ ) | log t (cid:17) , as t → ∞ . Asymptotic stability of solitons.
The key idea when proving asymptotic stability for thesoliton will be the following decomposition u ( t, x ) = Q c ( t ) ( y ) + v ( t, y ) + v ( t, y ) | {z } v ( t,y ) . We now explain precisely how the new coordinate y , the soliton parameter c , and the radi-ation part v = v + v are determined. First, the new coordinates y = x − Z t c ( s ) ds + h ( t )correspond to adopting as a reference the moving frame of the soliton; the modulationparameters c and h will be chosen below to ensure a certain cancellation. The radiation part SYMPTOTIC STABILITY OF SOLITONS FOR MKDV 7 v satisfies the perturbed equation ∂ t v + ∂ y ( − c + ∂ y + 3 Q c ) | {z } L c v = ∂ y (( Q c + v ) − Q c − Q c v ) + { less important terms } v ( t = 0) = v . The asymptotic stability of solitons follows from the decay of v ; this in turn is given by decayestimates for the linear group e t L c . However, the functions in the generalized kernel of L c (of dimension 2) do not decay under this semi group. Thus one needs to make sure that, inthe spectral decomposition associated to L c , the component of v in the generalized kernel of L c is zero: this condition completely determines c and h .Following the work of Mizumachi [42], see also [45, 46], the radiation part v is then splitinto v = v + v , where v simply solves (mKdV) in the y coordinates, with data v , (cid:26) ∂ t v − ( c + ˙ h ) ∂ y v + ∂ y ( v ) + ∂ y v = 0 v ( t = 0) = v , while v , the remainder term with zero initial data, solves (cid:26) ∂ t v − L c v = − ∂ y (( Q c + v + v ) − Q c − v − Q c v ) + { less important terms } v ( t = 0) = 0 . The advantages of this decomposition become clear when one tries to obtain decay for thepart of the wave which is to the right of the soliton, that is the region { y > } . In moretechnical terms, we want to obtain decay for kh y + i m v k L + k e ay v k L . • The decay of kh y + i m v k L is obtained by a virial-type argument, which one can apply sincethe equation for v does not “see” the soliton, and the data are such that kh y + i m v k L < ∞ . • The decay of k e ay v k L is obtained via decay estimates in exponentially weighted spaces(with norm of the type k e ay · k L ) for the linear group e t L c . This requires the data, as wellas the right-hand side of the v equation, to belong to an exponentially weighted space.This is easily checked for the v equation: the data is zero, and expanding the right-handside, it appears that the slowly decaying v factors are always coupled to v or Q c , thusensuring exponential decay. Organization of the paper.
Section 2 contains the proof of Theorem 1.1 about the stabilityof the trivial solution. We begin by establishing some linear and simple multilinear decayestimates in 2.1. In 2.2 we prove energy estimates involving the scaling vector field. Sections2.3 and 2.4 contain the heart of the proof of Theorem 1.1, that is the justification of theasymptotic expansion (1.21) and the control of the remainder terms. In section 2.5 we derivethe complete asymptotic description of small solutions of (mKdV) relying on a refined linearestimate and the global bounds established before. Section 3 is devoted to the proof ofTheorem 1.5 about the stability of soliton solutions. We first prove asymptotic stability `ala Pego-Weinstein in section 3.1, and then give the proof of scattering for the radiation insection 3.2.1.4.
Notations.
For x ∈ R , we set h x i = √ x . We denote C for a constant whose value may change from one line to another. Given twoquantities X and Y , we write PIERRE GERMAIN, FABIO PUSATERI, AND FR´ED´ERIC ROUSSET • X . Y if X ≤ CY for a constant C . • X ∼ Y if X . Y and Y . X . • X ≪ Y if X ≤ cY for a small constant c .We define the Fourier transform by F g ( ξ ) = b g ( ξ ) := 1 √ π Z R e − ixξ g ( x ) dx = ⇒ g ( x ) = 1 √ π Z R e ixξ b g ( ξ ) dξ . The Fourier multiplier m ( ∂ x ) with symbol m is given by F [ m ( ∂ x ) f ]( ξ ) = m ( iξ ) b f ( ξ ) . and the pseudoproduct operator T m with symbol m ( ξ, η ) by F [ T m ( f, g )]( ξ ) = Z m ( ξ, η ) b f ( ξ − η ) g ( η ) dη. Let ψ be smooth, supported on [ − , − ] ∪ [ , X j ∈ Z ψ (cid:18) ξ j (cid:19) = 1 , for ξ = 0 . Define χ = X j< ψ (cid:18) ξ j (cid:19) and the Littlewood-Paley operators P j = ψ (cid:18) ∂ x i j (cid:19) , P
We assume that the following X -norm is a priori small: k u k X = sup t ≥ (cid:16) t − δ k u k H + t − / (cid:13)(cid:13) xf (cid:13)(cid:13) H + t α/ − / (cid:13)(cid:13) | ∂ x | α xf (cid:13)(cid:13) L + (cid:13)(cid:13) b f ( ξ ) (cid:13)(cid:13) L ∞ (cid:17) ≤ ε . (2.1)We will then show k u k X ≤ ε + Cε (2.2)for some absolute constant C . This a priori estimate, combined with a bootstrap argument,and the choice ε = ε / , gives global existence for sufficiently small ε . For simplicity, andwithout loss of generality, we only consider t ≥
1, assuming that a local solutions has beenalready constructed on the time interval [0 ,
1] by standard methods, such as those in [29].Using also time reversibility we obtain a solution for all times.
SYMPTOTIC STABILITY OF SOLITONS FOR MKDV 9
Linear and multilinear estimates.
We begin by proving a refined linear estimatewhich also gives useful L p bounds. Lemma 2.1 (Linear Estimates) . For any t ≥ , x ∈ R , and u ( t, x ) = e − t∂ x f ( t, x ) with f such that k b f ( t ) k L ∞ + t − / k xf ( t ) k L ≤ , (2.3) the following estimate holds true: (cid:12)(cid:12) | ∂ x | β u ( x, t ) (cid:12)(cid:12) . t − / − β/ (cid:0) | x/t / | (cid:1) − / β/ , ≤ β ≤ , (2.4) (cid:12)(cid:12) ∂ x u ( x, t ) (cid:12)(cid:12) . t − / (cid:0) | x/t / | (cid:1) / . (2.5) In particular, whenever u = e − t∂ x f satisfies the a priori bounds (2.1) , one has for any β ∈ [0 , ) , and all p with p (1 / − β/ > , k| ∂ x | β u ( t ) k L p . t − / − β/ / (3 p ) . (2.6) Remark 2.2.
The refined linear estimate (2.4) in the case β = 0, and the estimate (2.5)coincide with the estimates obtained in [19]; see also [8] for related work. The improvementfor β > Proof.
Denote Λ( ξ ) = ξ , and write u ( t, x ) = e − t∂ x f ( t, x ) = 1 √ π Z R e itφ ( ξ ) b f ( ξ ) dξ, φ ( ξ ) := ξ ( x/t ) + Λ( ξ ) . For x ≤
0, let ξ ± := ± p − x/ (3 t ) (2.7)denote the stationary points of the phase φ ( ξ ), φ ′ ( ξ ± ) = 0. In the case x > x ≤
0. Up to taking complex conjugates, we notice thatin order to obtain (2.4)-(2.5) it suffices to show that (cid:12)(cid:12)(cid:12) Z ∞ e itφ ( ξ ) | ξ | β b f ( ξ, t ) dξ (cid:12)(cid:12)(cid:12) . t − / − β/ (cid:0) | x/t / | (cid:1) − / β/ , for all β ∈ (0 , ξ := p − x/ (3 t ) the only stationary point in the aboveintegral. We see that since | x/t | / = 3( ξ t / ) , it is then enough to show (cid:12)(cid:12)(cid:12) Z ∞ e itφ ( ξ ) | ξ | β b f ( ξ, t ) dξ (cid:12)(cid:12)(cid:12) . t − / − β/ max (cid:0) t / ξ , (cid:1) − / β , (2.8)for any t ≥ x ≤
0, and any function f satisfying (2.3). We distinguish two cases dependingon the size of ξ . Case 1: ξ ≤ t − / . In this case we only need to obtain a bound of t − / − β/ for the term in(2.8). We split the integral in (2.8) as follows: Z ∞ e itφ ( ξ ) | ξ | β b f ( ξ ) dξ = A + B,A = Z ∞ e itφ ( ξ ) | ξ | β b f ( ξ ) χ (2 − t / ξ ) dξ,B = Z ∞ e itφ ( ξ ) | ξ | β b f ( ξ )(1 − χ (2 − t / ξ )) dξ. The first term can be very easily estimated using the first bound in (2.3). For the second termwe notice that | ξ | ≫ ξ on the support of the integral, so that | ∂ ξ φ | = 3 | ξ − ξ | & | ξ | & t − / .An integration by parts then gives: | B | . B + B ,B = 1 t Z ∞ (cid:12)(cid:12)(cid:12) ∂ ξ (cid:16) ∂ ξ φ | ξ | β (1 − χ (2 − t / ξ ) (cid:17)(cid:12)(cid:12)(cid:12) | b f ( ξ ) | dξ,B = 1 t Z ∞ (cid:12)(cid:12)(cid:12) ∂ ξ φ | ξ | β (1 − χ (2 − t / ξ ) (cid:12)(cid:12)(cid:12) | ∂ ξ b f ( ξ ) | dξ. We can then estimate B . t k b f k L ∞ Z ∞ | ξ | − β (1 − χ (2 − t / ξ )) + 1 | ξ | − β | χ ′ (2 − t / ξ ) | t / dξ . t − / − β/ . Similarly, we can use the second bound provided by (2.3) to obtain: B . t k ∂ ξ b f k L (cid:16) Z ∞ | ξ | − β (1 − χ (2 − t / ξ )) dξ (cid:17) / . t − / − β/ . Case 2: ξ ≥ t − / . In this case we aim to prove a bound of t − / ξ − / β for the left-handside of (2.8). To separate the non-stationary and stationary cases we split the integral asfollows (see 1.4): Z ∞ e itφ ( ξ ) | ξ | β b f ( ξ ) dξ = C + D,C = Z ∞ e itφ ( ξ ) | ξ | β b f ( ξ )(1 − ψ ( ξ/ξ )) dξ,D = Z ∞ e itφ ( ξ ) | ξ | β b f ( ξ ) ψ ( ξ/ξ ) dξ. (2.9)Integrating by parts we get | C | . C + C ,C = 1 t Z ∞ (cid:12)(cid:12)(cid:12) ∂ ξ (cid:16) ∂ ξ φ | ξ | β (1 − ψ ( ξ/ξ )) (cid:17)(cid:12)(cid:12)(cid:12) | b f ( ξ ) | dξ,C = 1 t Z ∞ | ∂ ξ φ | | ξ | β (1 − ψ ( ξ/ξ )) | ∂ ξ b f ( ξ ) | dξ. SYMPTOTIC STABILITY OF SOLITONS FOR MKDV 11
Using the fact that on the support of the above integrals | ∂ ξ φ | & max( ξ, ξ ) we obtain C . t k b f k L ∞ Z ∞ (cid:18) | ξ | β − max( ξ, ξ ) (1 − ψ ( ξ/ξ )) + 1 | ξ | − β | ψ ′ ( ξ/ξ ) | ξ − (cid:19) dξ . t − ξ − β , which is stronger than the desired bound since ξ ≥ t − / . Similarly C . t k ∂ ξ b f k L (cid:16) Z ∞ ξ, ξ ) − β (1 − ψ ( ξ/ξ )) dξ (cid:17) / . t − / ξ − / β which suffices since ξ ≥ t − / .To estimate the resonant contributions ξ ≈ ξ we let l be the smallest integer such that2 l ≥ / √ tξ and bound the term D in (2.9) as follows: | D | ≤ log ξ +10 X l = l | D l | ,D l := Z R e itφ ( ξ ) | ξ | β b f ( ξ ) ψ ( ξ/ξ ) χ (cid:0) − l ( ξ − ξ ) (cid:1) dξ,D l := Z R e itφ ( ξ ) | ξ | β b f ( ξ ) ψ ( ξ/ξ ) ψ (2 − l ( ξ − ξ )) dξ , l ≥ l + 1 . The choice of l and the first bound in (2.3) immediately give us | D l | . ξ β l . t − / ξ − / β .To control the terms D l we integrate by parts and see that: | D l | . D l, + D l, ,D l, = 1 t Z ∞ (cid:12)(cid:12)(cid:12) ∂ ξ (cid:16) ∂ ξ φ | ξ | β ψ ( ξ/ξ ) ψ (2 − l ( ξ − ξ )) (cid:17)(cid:12)(cid:12)(cid:12) | b f ( ξ ) | dξ,D l, = 1 t Z ∞ | ∂ ξ φ | | ξ | β ψ ( ξ/ξ ) ψ (2 − l ( ξ − ξ )) | ∂ ξ b f ( ξ ) | dξ. Using the fact that on the support of the integrals | ∂ ξ φ | = 3 | ξ − ξ | ≈ l ξ , we can estimate D l, . t − k b f k L ∞ − l ξ − β , which gives the desired bound upon summation in l . Similarly, we have D l, . t − k ∂ ξ b f k L − l/ ξ − β . Using the second bound in (2.3) and summing in l we obtain a bound of t − / ξ − β − l / ,which is better than our desired bound. This concludes the proof of (2.4). The estimate(2.6) follows by integrating in L p the inequality (2.4). (cid:3) Lemma 2.3 (Multilinear Estimates) . Let u = e t∂ x f be a function satisfying the a prioriassumptions sup t ≥ (cid:0) t − / (cid:13)(cid:13) xf ( t ) (cid:13)(cid:13) L + (cid:13)(cid:13) b f ( t ) (cid:13)(cid:13) L ∞ (cid:1) ≤ ε . (2.10) Then the following bilinear estimate holds: sup t ≥ t k u ( t ) u x ( t ) k L ∞ x . ε . (2.11) Moreover, for all ≤ α < we have k| ∂ x | α u ( t ) k L . ε | t | − / − α/ . (2.12) Proof.
To obtain (2.11) it suffices to multiply the bounds provided by (2.4) in the case β = 0and (2.6).To show (2.12) we start by choosing β ∈ ( α, / p, q, p , p ∈ (2 , ∞ ), such that1 /p + 1 /q = 1 / , /p = θ/p + (1 − θ ) /p with θ = α/β, (2.13)and p (1 / − β/ > , p > β ∈ ( α, / p and p ; one checks that theabove inequalities are satisfied if p, p → ∞ ). We use the fractional Leibniz rule, followedby the Gagliardo-Nirenberg inequality (Theorem 2.44 in[3]) to obtain k| ∂ x | α u k L . k| ∂ x | α u k L p k u k L q . k| ∂ x | β u k θL p k u k − θL p k u k L q Using the linear estimate (2.6) we have k| ∂ x | β u k L p . t − / − β/ / (3 p ) , k u k L p . t − / / (3 p ) , k u k L q . t − / / (6 q ) . Using these three inequalities we see that k| ∂ x | α u k L . t γ where, using (2.13), we have γ = ( − / − β/ / (3 p )) θ + ( − / / (3 p ))(1 − θ ) + 2( − / / (6 q ))= − / (cid:0) θ ( − β + 1 /p ) + (1 − θ ) /p ) + 1 / (3 q ) = − / − α/ (cid:3) Energy estimates.
We now prove energy and weighted energy estimates.
Lemma 2.4.
Let u ∈ C ([0 , T ); H ) be a solution of (mKdV) satisfying the apriori bounds (2.1) . Then k u ( t ) k H ≤ ε + Cε . Moreover, if u = e − t∂ x f , k xf ( t ) k H ≤ C ( ε + ε ) h t i / . (2.14) Finally, for ≤ α < , k| ∂ x | α xf ( t ) k L ≤ C ( ε + ε ) h t i / − α/ . (2.15) Proof.
The first estimate follows from the conservation of Mass and Energy (1.2). Theestimates (2.14) and (2.15) will be obtained by energy estimates performed on the (mKdV)equation itself, or on the equation obtained after commuting the scaling vector field S :=1 + x∂ x + 3 t∂ t , i.e. ∂ t Su + ∂ x Su + 3 ∂ x ( u Su ) = 0 . (2.16) SYMPTOTIC STABILITY OF SOLITONS FOR MKDV 13
Proof of (2.14). In the following, we denote, given a function a , Ia for the antiderivative of a vanishing at −∞ : [ Ia ]( x ) = R x −∞ a . Applying I to (2.16) gives ∂ t ISu + ∂ x Su + 3 u Su = 0 . Multiplying by
ISu and integrating in space yields12 ddt (cid:13)(cid:13)
ISu (cid:13)(cid:13) L + Z ∂ x Su ISu dx + 3 Z u Su ISu dx = 0 , or, after integrating by parts, and taking (2.11) into account,12 ddt (cid:13)(cid:13) ISu (cid:13)(cid:13) L = 32 Z ∂ x ( u ) (cid:0) ISu (cid:1) dx . ε t (cid:13)(cid:13) ISu (cid:13)(cid:13) L . By Gronwall’s lemma (cid:13)(cid:13)
ISu (cid:13)(cid:13) L . ε t Cε . (2.17)Observe now that xf = ISf − t∂ t If = e t∂ x ISu + 3 e t∂ x tu , from which the above inequality,combined with (2.12), gives the desired result: k xf k L ≤ k ISu k L + 3 t k u k L . ε t Cε + ε t / . A similar argument applies to give a bound for k ∂ x xf k L and completes the proof of (2.14). Proof of (2.15). As explained in Remark 1.3, for the proof here, we do not really need toassume that xf is in H . We shall give here a proof of (2.15) that do not require higherregularity. By using the H regularity, a shorter proof is possible ( we shall use this argumentin the proof of Lemma 3.10 below to handle the solitary wave stability). Starting from (2.16),a simple energy estimate leads to12 ddt k| ∂ x | α − Su k L = − Z | ∂ x | α − ∂ x ( u Su ) | ∂ x | α − Su dx = − Z | ∂ x | α − ∂ x ( wv ) | ∂ x | α − v dx where w = u and v = Su .
Recalling that w j = P j w and v j = P j v , a paraproduct decomposition of the above right-handside gives ddt k| ∂ x | α − v k L = Z | ∂ x | α − ∂ x (cid:16) X j ≫ k w j v k + X j ∼ k w j v k + X k ≫ j w j v k (cid:17) | ∂ x | α − v dx =: I + II + III.
To estimate I , we use the dispersive estimate (2.11) and the standard properties of theLittlewood-Paley decomposition (see for example [3]) to obtain I . (cid:13)(cid:13)(cid:13) | ∂ x | α − X k ≪ j ∂ x ( w j v k ) (cid:13)(cid:13)(cid:13) L k| ∂ x | α − v k L . "X j (cid:16) αj (cid:13)(cid:13)(cid:13) X k ≪ j w j v k (cid:13)(cid:13)(cid:13) L (cid:17) / k| ∂ x | α − v k L . "X j (cid:16) X k ≪ j ( α − j j k w j k L ∞ k (1 − α ) k| ∂ x | α − v k k L (cid:17) / k| ∂ x | α − v k L . ε h t i X j X k ≪ j ( j − k )( α − k| ∂ x | α − v k k L ! / k| ∂ x | α − v k L . ε h t i k| ∂ x | α − v k L . The estimate of II also relies on (2.11): II . (cid:13)(cid:13)(cid:13) | ∂ x | α − ∂ x X k ∼ j ( w j v k ) (cid:13)(cid:13)(cid:13) L k| ∂ x | α − v k L . "X ℓ (cid:16) αℓ (cid:13)(cid:13)(cid:13) P ℓ (cid:16) X k ∼ j w j v k (cid:17)(cid:13)(cid:13)(cid:13) L (cid:17) / k| ∂ x | α − v k L . X ℓ (cid:16) X k ∼ j & ℓ αℓ − αk j k w j k L ∞ k ( α − k v k k L (cid:17) / k| ∂ x | α − v k L . ε h t i X ℓ (cid:16) X k & ℓ α ( ℓ − k ) k| ∂ x | α − v k k L (cid:17) / k| ∂ x | α − v k L . ε h t i k| ∂ x | α − v k L . To estimate
III , we need the classical commutator estimate (cid:13)(cid:13)(cid:2) | ∂ x | α − ∂ x , P ≪ j w (cid:3) P j f (cid:13)(cid:13) L . ( α − j k ∂ x w k L ∞ k P j f k L , (2.18)proved in the appendix in Lemma A.3. Commuting | ∂ x | α − ∂ x with w ≪ k in III gives
III = Z X k (cid:2) | ∂ x | α − ∂ x , w ≪ k (cid:3) v k | ∂ x | α − v dx + Z X k w ≪ k ∂ x | ∂ x | α − v k | ∂ x | α − v dx =: III a + III b It follows easily from the commutator estimate (2.18) and (2.11) that | III a | . k ∂ x w k L ∞ k| ∂ x | α − v k L . ε h t i k| ∂ x | α − v k L . SYMPTOTIC STABILITY OF SOLITONS FOR MKDV 15
To estimate
III b , we integrate by parts to obtain III b = Z X j ∼ k w ≪ j ∂ x | ∂ x | α − v j | ∂ x | α − v k dx = Z X j ∼ k ∂ x w ≪ j | ∂ x | α − v j | ∂ x | α − v k dx + { remainder } . The remainder is easy to treat, thus we skip it, and the main term is not much harder: | III b | . k ∂ x w k L ∞ k| ∂ x | α − v k L . ε t k| ∂ x | α − v k L . Gathering the estimates on I , II and III , we obtain ddt k| ∂ x | α − v k L . ε t k| ∂ x | α − v k L , which implies, by Gronwall’s lemma, k| ∂ x | α − Su k L = k| ∂ x | α − v k L . ε t Cε . Finally, observe that k| ∂ x | α xf k L = k| ∂ x | α − ( x∂ x + 1) f k L ≤ k| ∂ x | α − Sf k L + 3 t k| ∂ x | α − ∂ t f k L = k| ∂ x | α − Su k L + 3 t k| ∂ x | α u k L . The desired estimate follows by combining the bound on k| ∂ x | α − Su k L and (2.12). (cid:3) Control of sup t k b f ( t ) k ∞ . This section is dedicated to proving the following key propo-sition:
Proposition 2.5.
Under the a priori assumptions (2.1) , the following estimate holds for asolution u of (mKdV) : sup t (cid:13)(cid:13) b f ( t ) (cid:13)(cid:13) ∞ ≤ (cid:13)(cid:13) b u (cid:13)(cid:13) ∞ + Cε . (2.19) Proof.
We will show the following key identity: for t > ∂ t b f ( t, ξ ) = i sign ξ t | b f ( t, ξ ) | b f ( t, ξ ) + ct e it ξ h | ξ | >t − / b f ( t, ξ/ i + R ( t, ξ ) , (2.20)where c ∈ C is a constant, and R satisfies the bound Z ∞−∞ | R ( t, ξ ) | dt . ε . (2.21)The proofs of (2.20) and (2.21) above will be given in Section 2.4 below; let us first show how(2.20)-(2.21) imply the desired conclusion (2.19). Define the modified profile b w as follows: b w ( t, ξ ) := e − iB ( t,ξ ) b f ( t, ξ ) B ( t, ξ ) := 16 sign ξ Z t (cid:12)(cid:12) b f ( s, ξ ) (cid:12)(cid:12) dss . (2.22)Then we have ∂ t b w ( t, ξ ) = e − iB ( t,ξ ) h ∂ t b f ( t, ξ ) − i∂ t B ( t, ξ ) b f ( t, ξ ) i = e − iB ( t,ξ ) (cid:26) ct e it ξ h | ξ | >t − / b f ( t, ξ/ i + R ( t, ξ ) (cid:27) Integrating in time the above identity, using the fact that B is real, | b w ( t, ξ ) | = | b f ( t, ξ ) | , andthe remainder estimate (2.21), we obtain (cid:12)(cid:12) b f ( t, ξ ) (cid:12)(cid:12) ≤ | b u ( ξ ) | + c (cid:12)(cid:12)(cid:12)(cid:12)Z t | ξ | − e is ξ e − iB ( s,ξ ) | ξ | >s − / b f ( s, ξ/ dss (cid:12)(cid:12)(cid:12)(cid:12) + ε . The desired conclusion will then follow once we show that, for t > | ξ | − , (cid:12)(cid:12)(cid:12) Z t | ξ | − e is ξ e − iB ( s,ξ ) b f ( s, ξ/ dss (cid:12)(cid:12)(cid:12) . ε . (2.23) Proof of (2.23) . Integrating by parts in s using the identity e is ξ = iξ ∂ s e is ξ , we see that (cid:12)(cid:12)(cid:12) Z t | ξ | − e is ξ e − iB ( s,ξ ) b f ( s, ξ/ dss (cid:12)(cid:12)(cid:12) . J + K + L + MJ = 1 | ξ | | b f ( s, ξ/ | s (cid:12)(cid:12)(cid:12) s = ts = | ξ | − ,K = Z t | ξ | − | ξ | | ∂ s b f ( s, ξ/ || b f ( s, ξ/ | dss ,L = Z t | ξ | − | ξ | | ∂ s B ( s, ξ ) || b f ( s, ξ/ | dss ,M = Z t | ξ | − | ξ | | b f ( s, ξ/ | dss . (2.24)Since t ≥ | ξ | − , the a priori assumption k b f ( t ) k L ∞ ≤ ε gives immediately that J . ε . Usingagain k b f ( t ) k L ∞ ≤ ε , and (2.20)-(2.21) we can estimate K . Z t | ξ | − | ξ | h ε s + R ( s, ξ ) i ε dss . ε . From the definition of B in (2.22) we see that L . Z t | ξ | − | ξ | ε s ε dss . ε . The last term, M , is easily bounded by | ξ | − ǫ R t | ξ | − dss . ǫ , which completes the proof of(2.23). (cid:3) Proof of (2.20) - (2.21) . Recall that we assume t > SYMPTOTIC STABILITY OF SOLITONS FOR MKDV 17
Some elementary estimates on f . Recall that f j = P j f ; we start by stating a few estimateson f j that follow immediately from the a priori assumption (2.1): k b f j k L ∞ . ε , k xf j k L . (cid:2) − j + min (cid:0) t , − αj t − α (cid:1)(cid:3) ε , k f j k L . k f j k / L k xf j k / L . (cid:0) j t (cid:1) ε k f j k L . (cid:0) j ( − α ) t − α (cid:1) ε , k| x | ρ f j k L . k f j k − ρL k xf j k +2 ρL . (cid:16) − ρj + 2 j ( − ρ ) t + ρ (cid:17) ε , for 0 ≤ ρ < . (2.25)Moreover, if 2 j ≥ t − , then f
1. Applying Lemma A.1, in light of (1.18)–(1.20), we get, for | ξ | ≥ t − / , I = i sign ξ t | b f ( ξ ) | b f ( ξ ) + ict e − it ξ b f (cid:18) ξ (cid:19) + 2 j O kh ( x, y ) i ρ b F k L (2 j t ) ρ ! , (2.28)where c is a constant whose exact value will not matter. Now observe that b F ( x, y ) = 2 − j π Z e − izξ f . j (2 − j ( z − x )) f . j (2 − j z ) f . j (2 − j ( y − z )) dz so that k b F k L . k f . j k L and k| ( x, y ) | ρ b F k L . ρj k f . j k L k| x | ρ f . j k L . (2.29)Combining (2.28) and (2.29) above, and using (2.26) gives (cid:12)(cid:12)(cid:12) I − i sign ξ t | b f ( ξ ) | b f ( ξ ) − ict e − it ξ b f (cid:18) ξ (cid:19) | ξ | >t − / (cid:12)(cid:12)(cid:12) . − ρj t − − ρ ( k f . j k L + 2 ρj k f . j k L k| x | ρ f . j k L ) . j ( − ρ − α ) t − − ρ − α ε . (2.30)Recall that α is close to, but less than, . Choosing ρ close to, but less than, , we get that − ρ − α =: − κ <
0, and − − ρ − α = − − κ . It follows that Z ∞ − j (cid:12)(cid:12)(cid:12) I − i sign ξ t | b f ( ξ ) | b f ( ξ ) − ict e − it ξ b f (cid:16) ξ (cid:17) | ξ | >t − / (cid:12)(cid:12)(cid:12) ds . ε Z ∞ − j − κj s − − κ ds . ε . Contribution of II . We essentially follow the same approach as for I , and keep in particularthe same value for ρ and α . A change of variables gives II = X k ∼ ℓ ≫ j k >t − / k i π Z Z e − it k φ ((2 − k ξ,η,σ ) ξ c f . k ( ξ − k ( η + σ )) c f . k (2 k η ) c f . k (2 k σ ) ψ ( η ) ψ ( σ ) dη dσ. SYMPTOTIC STABILITY OF SOLITONS FOR MKDV 19
Due to the absence of stationary points, Lemma A.1 implies | II | . X k ≫ j k >t − / j k kh ( x, y ) i ρ c F k k L (2 k t ) ρ where F k ( η, σ ) = c f . k ( ξ − k ( η + σ )) c f . k (2 k η ) c f . k (2 k σ )and, as above, k| ( x, y ) | ρ c F k k L . ρk k f . k k L k| x | ρ f . k k L . As before, using (2.26), this leads to | II | . X k ≫ j k >t − / j k k t ) ρ (2 ρk k f . k k L k| x | ρ f . k k L + k f . k k L ) . X k ≫ j k >t − / j k ( − − ρ − α ) t − − ρ − α ε , (2.31)and thus since + ρ + α > Z ∞ | II | ds . ε Z ∞ j s − − ρ − α max(2 j , s − / ) − − ρ − α ds . ε . Contribution of
III . For the summands in
III , 2 k ≫ ℓ , j , thus | η | is the largest variableand we can write III = P k A k , with A k ( ξ ) = i π Z Z e − itφ ( ξ,η,σ ) ξ d f ∼ k ( ξ − η − σ ) d f ∼ k ( η ) d f ≪ k ( σ ) ψ (cid:16) η k (cid:17) χ (cid:16) σ k (cid:17) dη dσ . On the support of the integrand, | ∂ σ φ | ∼ k and (cid:12)(cid:12)(cid:12)(cid:12) ∂ m η ∂ m σ ∂ σ φ ( ξ, η, σ ) (cid:12)(cid:12)(cid:12)(cid:12) . − k − ( m + m ) k (2.32)for all integers m , m ∈ { , . . . , } . We then integrate by parts in σ to get III = X k ≫ j ,t − / III (1) k + III (2) k + III (3) k III (1) k := i π Z Z e − itφ ( ξ,η,σ ) ξit∂ σ φ ∂ σ d f ∼ k ( ξ − η − σ ) d f ∼ k ( η ) d f ≪ k ( σ ) ψ (cid:16) η k (cid:17) χ (cid:16) σ k (cid:17) dη dσ ,III (2) k := i π Z Z e − itφ ( ξ,η,σ ) ξit∂ σ φ d f ∼ k ( ξ − η − σ ) d f ∼ k ( η ) ∂ σ d f ≪ k ( σ ) ψ (cid:16) η k (cid:17) χ (cid:16) σ k (cid:17) dη dσ ,III (3) k := i π Z Z e − itφ ( ξ,η,σ ) ξ ∂ σ (cid:20) it∂ σ φ ψ (cid:16) η k (cid:17) χ (cid:16) σ k (cid:17)(cid:21) d f ∼ k ( ξ − η − σ ) d f ∼ k ( η ) d f ≪ k ( σ ) dη dσ . From (2.32) and Lemma A.2 it follows that (cid:12)(cid:12)(cid:12)
III (1) kℓ (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) III (2) kℓ (cid:12)(cid:12)(cid:12) . j t k h k ∂ d f ∼ k k L k d f ≪ k k L + k d f ∼ k k L k ∂ d f ≪ k k L i k u ∼ k k L ∞ . j t / k/ ǫ . (2.33) This gives the desired estimate after summing and integrating in time: Z ∞ X k ≫ j k >t − / | III (1) k | + | III (2) k | ds . ε Z ∞ − k j s / max(2 j , s − / ) / ds . ε . The remaining term can be estimated similarly using again (2.32) and Lemma A.2: (cid:12)(cid:12)(cid:12)
III (3) k (cid:12)(cid:12)(cid:12) . j t k k d f ∼ k k L k d f ≪ k k L k u ∼ k k L ∞ . j t / k ε , (2.34)which gives the desired bound upon summation and time integration since Z ∞ − k j s / max(2 j , s − / ) ds . . Contribution of IV . Using simply k b f k ∞ ≤ ε , the term IV can be estimated by | IV | . X j , k , ℓ In this section we derive the asymptotic behavior of solutions of (mKdV)as time goes to infinity. We are going to show the following: Proposition 2.6 (Asymptotics for small solutions) . Let u be a solution of (mKdV) satis-fying the global bounds (2.1) - (2.2) . Then, for any t ≥ , the following holds: • In the region x ≥ t / we have the decay estimate | u ( t, x ) | . ε t / ( x/t / ) / ; (2.35) • In the region | x | ≤ t / γ , with γ = 1 / / − Cε ) , the solution is approximatelyself-similar: (cid:12)(cid:12) u ( t, x ) − t / ϕ (cid:0) xt / (cid:1)(cid:12)(cid:12) . ε t / γ/ , (2.36) where ϕ is a bounded solution of the Painlev´e II equation ϕ ′′ − ξϕ + ϕ = 0 , p . v . Z R ϕ ( x ) dx = Z R u ( x ) dx. (2.37) • In the region x ≤ − t / γ , the solution has a nonlinearly modified asymptotic behavior:there exists f ∞ ∈ L ∞ ξ such that (cid:12)(cid:12)(cid:12) u ( t, x ) − √ tξ ℜ exp (cid:16) − itξ + iπ i | f ∞ ( ξ ) | log t (cid:17) f ∞ ( ξ ) (cid:12)(cid:12)(cid:12) ≤ ε t / ( − x/t / ) / , (2.38) where ξ := p − x/ (3 t ) , and ℜ denotes the real part. The proof of Proposition 2.6 is given in the remaining of this section. SYMPTOTIC STABILITY OF SOLITONS FOR MKDV 21 Decaying region: Proof of (2.35) . The proof of (2.35) follows from similar argument to thoseused in the proof of Lemma 2.1. As before we denote Λ( ξ ) = ξ and write u ( t, x ) = e − t∂ x f ( t, x ) = 1 √ π Z R e itφ ( ξ ) b f ( ξ ) dξ, φ ( ξ ) := ξ ( x/t ) + Λ( ξ ) . Since for any x > ∂ ξ φ = x/t + 3 ξ ≥ max( x/t, ξ ), we integrate by parts in theabove formula and bound: | u ( t, x ) | . I + II,I = Z R (cid:12)(cid:12)(cid:12)(cid:12) t∂ ξ φ ( ξ ) ∂ ξ b f ( ξ ) (cid:12)(cid:12)(cid:12)(cid:12) dξ,II = Z R (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t [ ∂ ξ φ ( ξ )] ∂ ξ φ ( ξ ) b f ( ξ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dξ. Using the weighted L bound in (2.1)-(2.2) we can estimate | I | . t h Z R ( x/t + 3 ξ ) − dξ i / k xf k L . t ( x/t ) − / ε t / , which is the desired bound. Similarly, we can use the bound on b f to obtain | II | . t Z R ( x/t + 3 ξ ) − | ξ | dξ k b f k L ∞ . t ( x/t ) − ε , which is a stronger bound than what we need since x ≥ t / . Self-similar region: Proof of (2.36) . We now look at the self-similar region | x | ≤ t / γ .Define v through the identity u ( t, x ) = 1 t / v (cid:0) t, xt / (cid:1) , v ( t, x ) = t / u ( t, t / x ) . (2.39)Recall the definition of the scaling vectorfield S = 1 + x∂ x + 3 t∂ t . A simple computationshows that ∂ t v ( t, x ) = 13 t / (cid:0) Su (cid:1) ( t, t / x ) . (2.40)Moreover, since u is a solution of (mKdV), one can verify that ∂ t v ( t, x ) = 1 t ∂ x (cid:16) xv − v xx − v (cid:17) ( t, x ) . (2.41)Our aim is to show that v ( t, x ) is a Cauchy sequence in time with values in L ∞ x . For thiswe first show that, for all | x | ≤ t γ , one has (cid:12)(cid:12) P ≥ t γ v ( t, x ) (cid:12)(cid:12) ≤ ε t − γ/ , (2.42) (cid:12)(cid:12) ∂ t P ≤ t γ v ( t, x ) (cid:12)(cid:12) ≤ ε t − / γ/ Cε . (2.43)For (2.42), we recall that f = e t∂ x u , and write P ≥ t γ v ( t, x ) = t / Z R e iφ ( ξ ; x,t ) χ ( ξt / − γ − ) b f ( t, ξ ) dξ, φ ( ξ ; x, t ) := xξt / + tξ . Since for any | x | ≤ t γ , we have | ∂ ξ φ | & ξ t & t / γ on the support of the above integral,an integration by parts argument similar to those in the proof of Lemma 2.1, shows the validity of (2.42). Notice that a similar bound also holds for P ∼ t γ v ( t, x ). Because of this,in order to obtain (2.43), it suffices to prove the estimate for P ≤ t γ ∂ t v ( t, x ). Observe thatfrom (2.40) one has ∂ − x ∂ t v = 1 / (3 t )( ISu )( t, t / x ). Therefore, using Bernstein’s inequality,and the bound (2.17), we get (cid:12)(cid:12) P ≤ t γ ∂ t v ( t, x ) (cid:12)(cid:12) . t γ/ k ∂ − x ∂ t v k L . t γ/ − k ( ISu ) k L t − / . ε t − / γ/ Cε , as desired.We then write v ( t, v ) = v ( t, x )[1 − ψ ( x/t γ )] + P ≥ t γ v ( t, x ) ψ ( x/t γ ) + P ≤ t γ v ( t, x ) ψ ( x/t γ ) . Combining the decay estimate (2.4) which gives | v ( t, x )[1 − ψ ( x/t γ )] | . ε t − γ/ , with (2.42)-(2.43), we see that there exists ϕ := lim t →∞ v ( t ) with | v ( t ) − ϕ | . ε t − γ/ . It also followsthat, uniformly for | x | ≤ t γ , | v ( t, x ) − ϕ ( x ) | . ε t − γ/ + ε Z ∞ t t − / γ/ C ε . ε t − γ/ where we recall our choice of γ = 1 / / − Cε ).To verify that ϕ satisfies the first identity in (2.37) it suffices to notice that from (2.40)and (2.41) one has k xv − v xx − v k L = k ISu k L t − / . ε t − / cε . To prove the second identity in (2.37) we let 0 < a < γ/ | v ( t ) − ϕ | . t − γ/ to write Z ϕ ( x ) ψ ( x/t a ) dx = Z v ( t, x ) ψ ( x/t a ) dx + O ( t a − γ/ ) . Using Plancherel, and the moment conservation for u , we have Z v ( t, x ) ψ ( x/t a ) dx = Z ( b u ( t, ξ/t / ) − b u ( t, b ψ ( ξt a ) t a dx + Z u ( x ) dx. Using the bounds (2.1)-(2.2) we see that for all | ξ | ≤ | b u ( t, ξ/t / ) − b u ( t, | ≤ | b f ( t, ξ/t / )( e iξ − | + | b f ( t, ξ/t / ) − b f ( t, | . | ξ | / . This shows that (cid:12)(cid:12)(cid:12) Z ϕ ( x ) ψ ( x/t a ) dx − Z u ( x ) dx (cid:12)(cid:12)(cid:12) . t a − γ/ + t − a/ , which implies (2.37). Modified scattering: Proof of (2.38) . The next Lemma gives a refined version of the linearestimate (2.4). Lemma 2.7 (Refined linear estimate) . Let u = e − t∂ x f , for f ∈ L satisfying sup t ≥ (cid:0) t − / kh x i f ( t ) k L + k b f ( t ) k L ∞ ξ (cid:1) ≤ . (2.44) Then, for all t ≥ and x ≤ − t / , (cid:12)(cid:12)(cid:12) u ( t, x ) − √ tξ ℜ (cid:0) e − itξ + i π b f ( ξ ) (cid:1)(cid:12)(cid:12)(cid:12) . t / | x/t / | / , (2.45) where ξ := p − x/ (3 t ) , and ℜ denotes the real part. SYMPTOTIC STABILITY OF SOLITONS FOR MKDV 23 This result can be proven by similar arguments to those in the proof of Lemma 2.1, andthose of Lemma 3.2 in [24]. For completeness we give the main ideas of proof below. Proof of Lemma 2.7. We write u ( t, x ) = r π ℜ Z ∞ e itφ ( ξ ) b f ( ξ ) dξ, φ ( ξ ) := xt ξ + ξ . (2.46)As before we let ξ := p − x/ t ≈ t − / ( − x/t / ) / & t − / be the only stationary point ofthe phase φ in (2.46).We first look at the frequency region with | ξ − ξ | ≥ ξ / 2. There we have | ∂ ξ φ ( ξ ) | & max( ξ , ξ ). Then, an integration by parts like the one in the proof of Lemma 2.1 (cfr. theterms C and C there) gives us a bound of the form t − ξ − + t − / ξ − / . t − / ( − x/t / ) − / ,which is smaller than the right-hand side of (2.45).We then analyze the case with | ξ − ξ | ≤ ξ / 2. If | ξ − ξ | ≈ ℓ , for ℓ ≥ ℓ with2 ℓ ≈ t − / ( − x/t / ) − / , we integrate by parts in frequency. Using | ∂ ξ φ ( ξ ) | & ℓ ξ , we bound these contributions by t − (cid:16) k ∂ ξ b f k L ξ − − ℓ/ + k b f k L ∞ ξ − − ℓ (cid:17) . Using (2.44), and the definitions of ξ and ℓ , we see that the contribution from the region | ξ − ξ | ≥ ℓ is of the order of t − / ( − x/t / ) − / , which is an acceptable remainder.We are then left with estimating the contribution to the integral (2.46) coming from theregion | ξ − ξ | ≤ ℓ . We write this contribution as r π ℜ Z ∞ e itφ ( ξ ) χ (( ξ − ξ )2 − ℓ ) b f ( ξ ) dξ = A + B + CA = r π ℜ (cid:0) e itφ ( ξ ) b f ( ξ ) Z ∞ e it ξ ξ / χ ( ξ/ ℓ ) dξ (cid:17) B = r π ℜ Z ∞ (cid:16) e itφ ( ξ ) − e itφ ( ξ )+ itφ ′′ ( ξ )( ξ − ξ ) / (cid:17) χ (( ξ − ξ )2 − ℓ ) b f ( ξ ) dξC = r π ℜ e itφ ( ξ ) Z ∞ e itφ ′′ ( ξ )( ξ − ξ ) / χ (( ξ − ξ )2 − ℓ ) (cid:0) b f ( ξ ) − b f ( ξ ) (cid:1) dξ. Using the hypotheses we immediately see that | B | . t ℓ . t − / ( − x/t / ) − / , | C | . t / ℓ / . t − / ( − x/t / ) − / , so that these terms are acceptable remainders.Using the formula Z R e − ax dx = r πa , a ∈ C , ℜ a > , we see that Z ∞ e it ξ ξ / e − ξ / ℓ dξ = 12 r π − i tξ + O (cid:0) ℓ + 2 − ℓ ( tξ ) − / (cid:1) . Finally, it follows that A = ℜ s i tξ e itφ ( ξ ) b f ( ξ ) + O (cid:0) t − / | x/t / | − / (cid:1) , and this completes the proof of the Lemma. (cid:3) Notice that in the region x ≤ − t / γ one has ξ = p x/ ( − t ) & t − / γ ≫ t − / . Ournext goal is then to identify an asymptotic profile for b f ( ξ ), where f = e t∂ x u and u solves(mKdV), whenever | ξ | ≫ t − / γ . This will then determine the leading order asymptoticterm for u in this region via (2.45). Lemma 2.8. Let f = e t∂ x u with u satisfying the bounds (2.1) - (2.2) , and let us define themodified profile as in (2.22) : b w ( t, ξ ) := e − iB ( t,ξ ) b f ( t, ξ ) , B ( t, ξ ) := 16 sign ξ Z t (cid:12)(cid:12) b f ( s, ξ ) (cid:12)(cid:12) dss . (2.47) Then there exists w ∞ ∈ L ∞ such that, for all t ≥ , and | ξ | ≥ t − / γ | b w ( t, ξ ) − w ∞ ( ξ ) | . ε ( | ξ | t / ) − κ , (2.48) for any κ ∈ (0 , / . Moreover, there exists f ∞ ∈ L ∞ such that, for | ξ | ≥ t − / γ , we have (cid:12)(cid:12)(cid:12) b f ( t, ξ ) − exp (cid:16) i ξ | f ∞ ( ξ ) | log t (cid:17) f ∞ ( ξ ) (cid:12)(cid:12)(cid:12) . ε ( | ξ | t / ) − κ . (2.49) Proof. To prove (2.48) it suffices to show that for all times t ≥ t ≥ 2, one has | b w ( t , ξ ) − b w ( t , ξ ) | ≤ ε (cid:0) j t / (cid:1) − κ . (2.50)for every | ξ | ≈ j , with j ∈ Z and 2 j ≥ t − / γ . The starting point to prove (2.50) is theformula (2.27) which, for | ξ | ≥ t − / γ ≫ t − / , reads ∂ t b f ( t, ξ ) = I + II + III, where all the terms on the right-hand side are defined in (2.27). From (2.30) and thedefinition of the modified profile b w in (2.47), we see that, for t ≤ t ≤ t , (cid:12)(cid:12)(cid:12) ∂ t b w ( t, ξ ) − e − iB ( t,ξ ) ict e it ξ b f ( ξ/ (cid:12)(cid:12)(cid:12) . − jκ t − − κ/ ε + | II ( t, ξ ) | + | III ( t, ξ ) | , (2.51)where we recall that we have previously defined κ = − + 2 ρ + α , and we can choose0 < α < and 0 < ρ < so that κ = 1 / − β , for any small β > 0. To prove (2.48) it willthen suffice to show (cid:12)(cid:12)(cid:12) Z t t e − iB ( t,ξ ) ict e − it ξ b f ( ξ/ dt (cid:12)(cid:12)(cid:12) . ε (cid:0) j t / (cid:1) − κ , (2.52)for all | ξ | ≥ t − / γ , and | II ( t, ξ ) | + | III ( t, ξ ) | . ε t − − κ/ − κj , (2.53)for t ≤ t ≤ t , and | ξ | ≥ t − / γ . Here we have used the fact that the first term on the right-hand side of (2.51) matches the right-hand side of (2.53), which, upon integration between t and t , gives the desired bound. SYMPTOTIC STABILITY OF SOLITONS FOR MKDV 25 To prove (2.52) we use an integration by parts argument similar to the one that gave us(2.23). Proceeding as in (2.24), we see that (cid:12)(cid:12)(cid:12) Z t t e it ξ e − iB ( t,ξ ) b f ( t, ξ/ dtt (cid:12)(cid:12)(cid:12) . J ′ + K ′ + L ′ + M ′ J ′ = 1 | ξ | | b f ( t, ξ/ | t (cid:12)(cid:12)(cid:12) t t = t ,K ′ = Z t t | ξ | | ∂ t b f ( t, ξ/ || b f ( t, ξ/ | dtt ,L ′ = Z t t | ξ | | ∂ t B ( t, ξ ) || b f ( t, ξ/ | dtt ,M ′ = Z t t | ξ | | b f ( t, ξ/ | dtt . Using k b f ( t ) k L ∞ ≤ ε we immediately see that J ′ . ε − j t − , which is more than sufficient,since 2 j t / ≫ 1. Using (2.20) and (2.21) we see that K ′ . Z t t | ξ | h ε t + R ( t, ξ ) i ε dtt . ε − j t − h ε + Z t t R ( t, ξ ) dt i . ε − j t − .L ′ and M ′ can be bounded similarly, using also | ∂ t B ( t, ξ ) | ≤ ε t − .We now prove (2.53). To bound II we look back at the estimate (2.31), recall that κ = − + 2 ρ + α , and see that | II ( t, ξ ) | . ε j X k ≫ j ( − − κ ) k t − − κ/ . ε − κj t − − κ/ . To estimate III we recall (2.33) and (2.34), and, in the case 2 j ≥ t − / γ , deduce thefollowing: | III ( t, ξ ) | . ε j X k ≫ ℓ & j (cid:0) t − / − k/ + t − / − k (cid:1) . ε (2 − j/ t − / + 2 − j t − / ) . ε t − ( t / j ) − / . This completes the proof of (2.53) and gives us (2.50). We also deduce that b w ( t ) is a Cauchysequence and obtain the existence of a limit profile w ∞ as in (2.48).To prove (2.49) we begin by observing that (2.48) implies that for t ≥ (cid:12)(cid:12) | b f ( t, ξ ) | − | w ∞ ( ξ ) | (cid:12)(cid:12) . ε ( | ξ | t / ) − κ . (2.54)Next, for B as in (2.47), we define A ( t, ξ ) := B ( t, ξ ) − 16 sign ξ | b f ( t, ξ ) | log t. (2.55)Omitting the variable ξ , we calculate for 2 ≤ t ≤ t A ( t ) − A ( t ) = 16 sign ξ Z t t (cid:0) | b f ( s ) | − | b f ( t ) | (cid:1) dss + 16 sign ξ (cid:0) | b f ( t ) | − | b f ( t ) | (cid:1) log t . From this and (2.54) we deduce that A ( t, ξ ) is a Cauchy sequence in time, and there exists A ∞ ∈ L ∞ ξ such that | A ( t, ξ ) − A ∞ ( ξ ) | . ε ( | ξ | t / ) − κ log t. Thanks to (2.54) and (2.55) we see that (cid:12)(cid:12) B ( t, ξ ) − (cid:0) A ∞ ( ξ ) + 16 sign ξ | w ∞ ( ξ ) | log t (cid:1)(cid:12)(cid:12) . ε ( | ξ | t / ) − κ log t, and, in view of (2.47) and (2.48), we obtain (cid:12)(cid:12) b f ( t, ξ ) − w ∞ ( ξ ) exp (cid:0) iA ∞ ( ξ ) + i ξ | w ∞ ( ξ ) | log t (cid:1)(cid:12)(cid:12) . ε ( | ξ | t / ) − κ log t. The desired conclusion (2.49) follows by defining f ∞ ( ξ ) := w ∞ ( ξ ) exp( iA ∞ ( ξ )). (cid:3) Finally, we observe that in the space-time region x/t / ≤ − t γ we have ξ = p − x/ t ≈ ( − x/t / ) / t − / ≥ t − / γ , and we can then combine the refined linear estimate (2.45) inLemma 2.7, and the modified asymptotic estimate (2.49) in Lemma 2.8 to obtain: (cid:12)(cid:12)(cid:12) u ( t, x ) − √ tξ ℜ n exp (cid:16) − itξ + i π i | f ∞ ( ξ ) | log t (cid:17) f ∞ ( ξ ) o(cid:12)(cid:12)(cid:12) . ε ( tξ ) − / ( t / ξ ) − κ log t + ε t − / | x/t / | − / for f ∞ ∈ L ∞ , and whenever x/t / ≤ − t γ . Since κ can be chosen arbitrarily close to 1 / (cid:3) Stability of solitons In this section, shall study the the asymptotic stability of the solitons Q c ( x − ct ) = √ cQ ( √ c ( x − ct )) , Q ( s ) := √ / cosh( s ) , c > , for the focusing mKdV equation ∂ t u + ∂ x u + ∂ x u = 0 . (3.1)The aim is to prove Theorem 1.5. This will be obtained by combining the modified scatteringresult of the previous section and an asymptotic stability result in a weighted space for thesoliton (Theorem 3.1).For a smooth non-negative weight w , we shall use the following notation for the weightednorms: k u k L w = k w u k L , k u k H sw = X k ≤ s k w ∂ kx u k L . In the following, we shall use as weights, w ( x ) = (1 + tanh( δx )) / , with δ sufficiently small,and w ′ . We shall first prove: Theorem 3.1. For every ǫ > there exists ǫ such that the following holds true: if v satisfies k v k H + kh x + i m v k H ≤ ǫ (3.2) SYMPTOTIC STABILITY OF SOLITONS FOR MKDV 27 for some fixed m > / , then there exists a shift h ( t ) and a modulation speed c ( t ) such thatthe solution of (3.1) with u ( t = 0) = Q c + v satisfies u ( t, x ) = Q c ( t ) ( y ) + v ( t, y ) , y = y ( x, t ) = x − Z t c ( s ) ds + h ( t ) , (3.3) with kh y + i m v ( t ) k H + h t i m k v ( t ) k H w + h t i m ( | c ′ ( t ) | + | h ′ ( t ) | ) . ǫ , ∀ t ≥ . (3.4) Moreover, one has the bound Z ∞ k v k H w ′ dt . ǫ . (3.5)Note that this Theorem gives in particular that perturbations of a solitary wave decay toits right. This kind of result was already obtained in [47, 43, 35]. Nevertheless, we establishhere a form of the result which is appropriate for the proof of Theorem 1.5. In particular, weprove rates of decay that will be useful in order to describe the radiation behind the solitarywave, following the approach of the previous section in a second step.3.1. Proof of Theorem 3.1. We shall split the proof in several steps. Step 1: Linear estimates in exponentially weighted spaces. In this first step, we shall recallthe properties of the equation (3.1) linearized about the solitary wave Q c . By changingvariables from x to y = x − ct , we obtain the linearized equation ∂ t v − c∂ y v + ∂ y v + 3 ∂ y (cid:0) Q c v (cid:1) = 0 . (3.6)Let us denote by S c ( t ) the linear group associated to this linear equation, so that the solutionof (3.6) with initial value v can be written as v ( t ) = S c ( t ) v . We shall recall the decay resultsfor S c obtained by Pego-Weinstein [47] by using the weighted norms k f k L a := k e ay f k L , k f k H a := k e ay f k L + k e ay ∂ y f k L where a is chosen so that 0 < a < p c/ . (3.7)Let us define L c := ∂ y (cid:0) − cv + ∂ y v + 3 Q c v ) . (3.8)and ξ c ( y ) = ∂ y Q c , ξ c ( y ) = ∂ c Q c , that describe the generalized kernel of L c : L c ξ c = 0 , L c ξ c = ξ c . To define a projection on this generalized kernel, we use the generalized kernel of theadjoint (for the L scalar product) L ∗ c . Let us set ζ c ( y ) = − α (cid:18)Z y −∞ ∂ c Q c (cid:19) + α Q c ( y ) , ζ c ( y ) = α Q c , (3.9)where the normalization factors α and α are chosen so that Z ξ ic ζ jc = δ ij , ≤ i, j ≤ . Note that these integrals are well defined thanks to the fast decay of the ξ ic . Note that L ∗ c ζ = ζ , L ∗ c ζ = 0 . Define the projections P c v = ( v, ζ c ) L ξ c + ( v, ζ c ) L ξ c , Q c = I − P c (3.10)Note that these projections are well defined on L a and commute with L c as well as S c ( t ) forall t . From the linear stability of the solitary wave, one has: Theorem 3.2 (Pego-Weinstein [47], Theorem 4.2) . We have the following decay and smooth-ing estimates: k S c ( t ) Q c v k L a . e − bt k v k L a , k S c ( t ) Q c v k H a + k S c ( t ) Q c ∂ y v k L a . e − bt max (cid:0) , t − / (cid:1) k v k L a . for some b > . By induction, we can deduce from the above estimates and the Duhamel formula that k S c ( t ) Q c v k H ka . e − bt k v k H ka , k S c ( t ) Q c v k H k +1 a + k S c ( t ) Q c ∂ y v k H ka . e − bt max (cid:0) , t − / (cid:1) k v k H ka for every k ≥ Step 2: Decomposition of the perturbation. The perturbation of the solitary wave v ( t, y )defined in (3.3) evolves according to ∂ t v − ˜ c∂ y v + 3 ∂ y ( Q c ( t ) v ) + ∂ y v = ∂ y F ( v ) + e Q , v /t =0 = v ( x ) (3.11)where ˜ c ( t ) = c ( t ) − ˙ h ( t ) e Q ( t, y ) = ˙ c∂ c Q c ( t ) ( y ) + ˙ h∂ y Q c ( t ) ( y ) = ˙ cξ c ( t ) ( y ) + ˙ hξ c ( t ) ( y ) (3.12) F ( v ) = − (cid:0) ( Q c + v ) − Q c − Q c v (cid:1) . The modulation parameters ( h ( t ) , c ( t )) will be chosen to ensure the constraint( v, ζ c ) L = ( v, ζ c ) L = 0 . (3.13)Note that these constraints are always well defined (even the first one) when v is such that h y + i m v ∈ L for m > / v ( t, y ) defined in (3.3)into v ( t, y ) = v ( t, y ) + v ( t, y ) (3.14)where v will be estimated in H w , and v in H a . We choose v ( t, y ) as the solution of thefree nonlinear equation ∂ t v − ˜ c∂ y v + ∂ y v + ∂ y v = 0 , v (0) = v , (3.15)and v as the solution of ∂ t v − ˜ c∂ y v + 3 ∂ y ( Q c ( t ) v ) + ∂ y v = ∂ y N ( v ) + e Q , v (0) = 0 , (3.16)with N ( v ) = − ( Q c ( t ) + v + v ) + Q c ( t ) + v + 3 Q c ( t ) v . (3.17) SYMPTOTIC STABILITY OF SOLITONS FOR MKDV 29 We shall solve this equation for v in the weighted space H a by using estimates for thelinear semigroup S c . Note that the choice of the equation for v is made in order to ensurethat the source term N ( v ) that involves v lies in the weighted space L a .Let us define the norm: N ( t ) := h t i m ( k v ( t ) k H w + k v ( t ) k H a ) + kh y + i m v ( t ) k H + k v k H + | c ( t ) − c | + | h ( t ) − h | , (3.18)with the parameters δ in the definition of w , and a in the exponential weights, chosen sothat the following relations hold: Q / c ± ǫ e κ | x | . w + w ′ . e ax , ∀ x ∈ R , (3.19)for a small constant κ > N ( t ) . ˜ ǫ , ∀ t ∈ [0 , T ] (3.20)and we will prove that, for all t ∈ [0 , T ] N ( t ) . ǫ . It will be convenient to use also the quantity M ( t ) = sup s ∈ [0 ,t ] (cid:16) h s i m ( k v ( s ) k H w + k v ( s ) k H a (cid:17) . Note that by the bootstrap assumption, we also have that M ( t ) ≤ ˜ ǫ on [0 , T ]. Step 3: H estimate. In this step we shall prove that Proposition 3.3. For t ∈ [0 , T ] we have the estimate k v ( t ) k H . ǫ , k v ( t ) k H . ǫ + (1 + ˜ ǫ )( k v ( t ) k H w + k v ( t ) k H a + | c ( t ) − c | ) . Note that the last estimate does not seem appropriate for the bootstrap. Nevertheless, weshall prove below that the estimates for k v ( t ) k H w , k v ( t ) k H a and | c ( t ) − c | are much betterbehaved in the sense that these quantities can be estimated in terms of ǫ if ˜ ǫ is sufficientlysmall. We could use the orbital stability of the solitary wave to get better estimates at thisstage. Proof of Proposition 3.3. For the KdV type equation (3.15) we have the conservation of thequantities Z R | v | dx, Z R (cid:18) | ∂ x v | − v (cid:19) dx. Using these and Sobolev inequalities we easily get k v ( t ) k H . ǫ , ∀ t ∈ [0 , T ] . (3.21)To estimate v we use the conserved quantities for (3.1). The mass conservation Z R | u ( t, x ) | dx = Z R | Q c ( x ) + v ( x ) | dx implies, after expanding u as in (3.3) and (3.14), that Z R | v + v + Q c ( t ) | dx = Z Q c dx + O ( ǫ ) , and thus Z R | v | dy = Z R ( Q c − Q c ( t ) ) dy − Z R v dy − Z R Q c ( t ) v dy − Z R Q c ( t ) v dy − Z R v v dy + O ( ǫ ) . This yields k v ( t ) k L . ǫ + | c ( t ) − c | + k v ( t ) k H w + k v k H a , if ˜ ǫ is chosen small enough.To estimate k ∂ x v ( t ) k L one can proceed in a similar way, by using the conservation of theHamiltonian for (3.1). (cid:3) Step 4: Estimates of the modulation parameters. The existence of the modulation parametersis based on the following: Lemma 3.4. Let c > , h ≥ . There exists δ > such that for every w satisfying w ( t ) − Q c ( · − c t + h ) ∈ C ([0 , T ] , H h x i m + ) for some m > / , with sup [0 ,T ] kh ( · + h ) m + i ( w ( t ) − Q c ( · − c t + h )) k H < δ, there exists ( h ( t ) , c ( t )) ∈ C ([0 , T ]) such that Z R (cid:0) w ( t, x ) − Q c ( t ) ( y ) (cid:1) ζ kc ( t ) ( y ) dx = 0 , k = 1 , where y = x − R t c ( s ) ds + h ( t ) . The proof of this lemma is now very classical and relies on the use of the implicit functionTheorem. We refer to [47, Proposition 5.1] or [43, Proposition 3.1] for the proof.By using Lemma 3.4 for w = u , we get the existence of c ( t ) and h ( t ) such that thedecompositions (3.3), and (3.14) with (3.13) hold. Proposition 3.5. On [0 , T ] we have the following estimates for the modulation parameters: | ˙ h ( t ) | + | ˙ c ( t ) | . h t i − m M ( t ) . Note that by integrating in time the above estimate, we get that | c ( t ) − c | + | h ( t ) − h | ≤ M ( t ) , ∀ t ∈ [0 , T ] . (3.22) Proof of Proposition 3.5. By using the equation (3.11), we get by taking the time derivativesof the constraints (3.13) that the vector Γ( t ) = ( h ( t ) , c ( t )) t verifies the ODE A ( t ) ˙Γ( t ) = − (cid:18) ( F ( v ) , ∂ y ζ c )( F ( v ) , ∂ y ζ c ) (cid:19) , (3.23)(using once again ( v, ζ c ) = 0) with A ( t ) = Id − (cid:18) ( v, ∂ y ζ c ) ( v, ∂ c ζ c )( v, ∂ y ζ c ) ( v, ∂ c ζ c ) (cid:19) := Id − B ( t ) . Since | B ( t ) | . k v ( t ) k L w , we have that A ( t ) is invertible for ˜ ǫ sufficiently small, with thenorm of its inverse smaller than 2. Moreover, we can estimate the right hand side of (3.23)by using the localization provided by ∂ y ζ ic . In particular, we obtain that | ( F ( v ) , ∂ y ζ ic ) | . (1 + k v k H ) (cid:0) k v k L w + k v k L a ) . h t i − m M ( t ) , SYMPTOTIC STABILITY OF SOLITONS FOR MKDV 31 for t ∈ [0 , T ], which gives the desired estimate. (cid:3) Step 5: Estimates of v . We shall now use localized virial type estimates in order to estimatethe weighted norms of v . Proposition 3.6. For every t ∈ [0 , T ] , we have the estimates: k v ( t ) k H w . ǫ h t i − m , kh y + i m v k H . ǫ . (3.24) Moreover, we also have Z t k v ( s ) k H w ′ ds . ǫ . (3.25) Proof. We first notice that on [0 , T ], we have by assumption that | c ( t ) − c | ≤ ˜ ǫ and also byusing Proposition 3.5 that | ˙ h | . ˜ ǫ , consequently, by assuming that ˜ ǫ is sufficiently small,we can always ensure that c / ≤ ˜ c ( t ) ≤ c , ∀ t ∈ [0 , T ] . (3.26)We shall use weights φ k ( t, y ) := χ k,δ ( y + σt + x ) (3.27)with σ , 0 ≤ σ < c / x ∈ R , δ sufficiently small, and χ k,δ is given by χ k,δ ( y ) = (cid:0) A k + ( δy ) (cid:1) k (cid:0) δy ) (cid:1) We choose A k sufficiently big, so that the following inequalities hold: χ k,δ ∼ w h y i k , χ ′ k,δ ≥ , χ ′′ k,δ . δχ ′ k,δ , χ ′′′ k,δ . δ χ ′ k,δ . From (3.15), we first obtain ddt Z R φ k | v | + 12 (˜ c − σ ) Z φ ′ k | v | + 32 Z φ ′ k | ∂ y v | = 12 Z φ ′′′ k | v | + 14 Z φ ′ k | v | . Next, we observe that | φ ′′′ k | . δ φ ′ k and that k v k L ∞ . k v k H . ǫ thanks to Proposition3.3. We thus obtain that ddt Z R φ k | v | + ( 12 (˜ c − σ ) − Cǫ − Cδ ) Z φ ′ k | v | + 32 Z φ ′ k | ∂ y v | ≤ . (3.28)By setting e = 12 | ∂ y v | − | v | , d = ∂ y v + v , we also get from (3.15) that ∂ t e − ˜ c∂ y e − d ∂ y d = − ∂ y ( ∂ y d ∂ y v ) . Note that this is the infinitesimal conservation law corresponding to the conservation ofthe Hamiltonian. By integrating this identity against the weight φ k , we obtain after someintegration by parts that ddt Z φ k e + (˜ c − σ ) Z φ ′ k e + 12 Z φ ′ k | d | = Z φ ′ k ∂ y d ∂ y v . To control the last integral we can integrate by parts and use Proposition 3.3 and | φ ′′ k | . δφ ′ k to get Z φ ′ k ∂ y d ∂ y v + Z φ ′ k | d | . δ Z φ ′ k ( | d | + | ∂ y v | ) + ǫ Z φ ′ k ( | d | + | v | ) . We thus get that ddt Z φ k e + (˜ c − σ ) Z φ ′ k e + ( 32 − Cδ − Cǫ ) Z φ ′ k | d | . δ Z φ ′ k | ∂ y v | + ǫ Z φ ′ k | v | . By combining the last identity and (3.28), we thus obtain that ddt Z φ k ( e + 12 | v | ) + (˜ c − σ − Cǫ − Cδ ) Z φ ′ k ( e + 12 | v | )+ ( 32 − Cδ − Cǫ ) Z φ ′ k ( | d | + | ∂ y v | ) ≤ . (3.29)Note that for ǫ sufficiently small, e + | v | and | d | + | ∂ y v | + | v | are positive quantitiesthat control pointwise | ∂ y v | + | v | and | ∂ y v | + | ∂ y v | + | v | respectively.By using this identity with k = 0, σ = 0, x = 0, we obtain after integration in time that Z t Z R ( w ′ ) ( | ∂ yy v | + | ∂ y v | + | v | ) dydt . k v k H . (3.30)By taking k = 0, σ > 0, small and x = − στ , we also get by integrating between 0 and τ that for every τ > k v ( τ ) k H w . Z R φ ( y − στ )( | v | + | ∂ y v | ) dy. Since φ ( y − στ ) / h y + i m . / h τ i m , we also obtain that k v ( τ ) k H w . h τ i m kh y + i m v k H , ∀ τ ∈ [0 , T ] . (3.31)Finally, by using (3.29) with σ = 0 and x = 0 but for k = m , we get that Z R φ m (cid:0) | ∂ y v | + | v | (cid:1) ( t ) . kh y + i m v k H , ∀ t ∈ [0 , T ] . Since φ m behaves like y m for y ≥ 0, we get, using also Proposition 3.3, that kh y + i m v ( t ) k H . ǫ . This ends the proof of the proposition. (cid:3) Step 6: Estimate of v . We now estimate v mainly using the semi-group estimates of The-orem 3.2. Proposition 3.7. For all t ∈ [0 , T ] we have the estimates h t i m k v ( t ) k H a . ǫ , Z t k v ( τ ) k H a . ǫ . (3.32) Proof. We can first write the equation (3.16) for v as ∂ t v + L c v = ∂ y N ( v ) + e Q + ∂ y e Q , where e Q and N are defined in (3.12), (3.17) and e Q is given by e Q = − Q c ( t ) − Q c ) v + (˜ c − c ) v . (3.33)By using the semi-group S c we get that v is given by the following Duhamel formula v ( t ) = Z t S c ( t − τ ) ( ∂ y N ( v ) + e Q + ∂ y e Q ) ( τ ) dτ. (3.34) SYMPTOTIC STABILITY OF SOLITONS FOR MKDV 33 We shall first estimate P c v ( t ). By using the definition of P c and the fact that v satisfiesthe constraint (3.13), we get that kP c v ( t ) k H a . k v k L w + | c ( t ) − c |k v k L a . Since kP c v ( t ) k H a ≤ kP c v ( t ) k H a + kP c v ( t ) k H a . k v k L w + kP c v ( t ) k H a , by using Proposition 3.6 and (3.22), we get h t i m kP c v ( t ) k H a . ǫ + M ( t ) . (3.35)This also yields (since m > / (cid:18)Z t kP c v ( t ) k H a (cid:19) . ǫ + M ( t ) . (3.36)Next, we apply Q c to (3.34): Q c v ( t ) = Z t S c ( t − τ ) Q c ( ∂ y N ( v ) + e Q + ∂ y e Q ) ( τ ) dτ. Thanks to Theorem 3.2, we obtain kQ c v ( t ) k H sa . Z t e − b ( t − τ ) max (cid:0) , ( t − τ ) − / (cid:1) ( k e Q k H a + k e Q k H sa + kN ( v ) k H sa ) dτ, s = 1 , . (3.37)To estimate the right hand side above, we recall the definition (3.12) and observe that k e Q ( t ) k H a . | ˙ c | + | ˙ h | . Therefore, using Proposition 3.5, we obtain that on [0 , T ] k e Q ( t ) k H a . h t i − m M ( t ) . (3.38)Next, we observe that k e Q ( t ) k H a . ( | c ( t ) − c | + | ˙ h ( t ) | ) k v ( t ) k H a . h t i − m M ( t ) , (3.39)and in a similar way, we obtain Z t k e Q ( t ) k H a . ˜ ǫ Z t k v k H a . (3.40)To estimate N ( v ), we recall its definition in (3.17), and write N ( v ) = − (cid:0) Q c ( t ) v + 3 Q c ( t ) v + 3 Q c ( t ) v + 6 Q c ( t ) v v + 3 v v + 3 v v + v (cid:1) . (3.41)Using the localization provided by Q c we get that, on [0 , T ], kN ( v )( t ) k H a . (1 + k v k H ) k v k H w + ( k v k H + k v k H )(1 + k v k H + k v k H ) k v k H a . Therefore, by using Proposition 3.6 and Proposition 3.3, we get that kN ( v )( t ) k H a . ǫ h t i − m + ˜ ǫ h t i − m M ( t ) . (3.42)In a similar way, we obtain Z t kN ( v ) k H a . (1 + ˜ ǫ ) Z t k v k H w + ˜ ǫ Z t k v k H a . (3.43) Putting together (3.35), (3.37), (3.38), (3.39) and (3.42), we see that for all t ∈ [0 , T ] h t i m k v ( t ) k H a . ǫ + ˜ ǫ M ( t ) + ( ǫ + ˜ ǫ M ( t )) Z t h t i m e − b ( t − τ ) max (cid:0) , ( t − τ ) − / (cid:1) h τ i m dτ. Since sup t ≥ Z t h t i m e − b ( t − τ ) max (cid:0) , ( t − τ ) − / (cid:1) h τ i m dτ < + ∞ , we obtain, by using again Proposition 3.3, that h t i m k v ( t ) k H a . ǫ + ˜ ǫ sup [0 ,t ] ( h s i m k v ( s ) k H a ) . Taking ˜ ǫ sufficiently small, we get h t i m k v ( t ) k H a . ǫ , ∀ t ∈ [0 , T ] . (3.44)Consequently, the first part of (3.32) is proven.To get the second part, we use Young’s inequality and (3.36), (3.37) for s = 2, (3.38),(3.39), (3.43) to obtain (cid:18)Z t k v ( τ ) k H a dτ (cid:19) / . ǫ + M ( t ) + (cid:16) Z t k v k H w dτ (cid:17) + ˜ ǫ (cid:18)Z t k v ( τ ) k H a dτ (cid:19) / . By using Proposition 3.6 and (3.44), this yields (cid:18)Z t k v ( τ ) k H a dτ (cid:19) / . ǫ + ˜ ǫ (cid:18)Z t k v ( τ ) k H a dτ (cid:19) / , and we conclude again the proof by choosing ˜ ǫ sufficiently small. (cid:3) Step 7: Conclusion. By combining Propositions 3.6 and 3.7, we have already proven that M ( t ) . ǫ on [0 , T ]. From (3.22), we also obtain that | c ( t ) − c | + | h ( t ) − h | ≤ ǫ (actuallywe even have ǫ ). Finally from Proposition 3.3, we get k v ( t ) k H . ǫ on [0 , T ]. Since h t i m k v ( t ) k H w + kh y + i m v ( t ) k H . h t i m k v k H w + h t i m k v k H a + kh y + i m v k H + k v k H a , we obtain, by using again Proposition 3.6, that N ( t ) . ǫ , ∀ t ∈ [0 , T ] . By taking ǫ ≫ ǫ / and sufficiently small, we see, by a standard bootstrap argument, thatthe estimate (3.20) holds true for all times.Moreover, from Proposition 3.5, we have that | ˙ h | + | ˙ c | . ˜ ǫ h t i − m , and, since m > / 2, we deduce that there exists c + and h + such that | c ( t ) − c + | + | h ( t ) − h + | . ˜ ǫ h t i − (2 m − . This gives (3.4). Finally, note that the estimate (3.5) follows from Proposition 3.6 andProposition 3.7. (cid:3) SYMPTOTIC STABILITY OF SOLITONS FOR MKDV 35 Proof of Theorem 1.5. We now show how to obtain Theorem 1.5 from Theorem 3.1and the first part of the paper. We start again from the decompositions (3.3) and (3.14),so that v ( t, y ), v ( t, y ), c and h satisfy the estimates in the proof of Theorem 3.1. We thuswrite u ( t, x ) = Q c ( t ) ( y ) + ˜ v ( t, x ) , ˜ v ( t, x ) = ˜ v ( t, x ) + ˜ v ( t, x ) , ˜ v i ( t, x ) = v i ( t, y ) (3.45)where again y = y ( t, x ) = x − R t c + h ( t ). By using the estimates in Theorem 3.1, we alreadyhave that kh x + i m v ( t ) k H + h t i m ( k v ( t ) k H w + k v ( t ) k H a ) + h t i m ( | ˙ c ( t ) | + | ˙ h ( t ) | )+ h t i m − ( | c ( t ) − c + | + | h ( t ) − h + | ) . ˜ ǫ , ∀ t ≥ . (3.46)We now use the approach of the first part of the paper to estimate ˜ v = ˜ v + ˜ v behind thesolitary wave. Step 1: Estimates for ˜ v . By definition, since v ( t, y ) solves (3.15), we get that ˜ v ( t, x ) solvesthe mKdV equation ∂ t ˜ v + ∂ x ˜ v + ∂ x ˜ v = 0 , (˜ v ) /t =0 = v ( x ) . Consequently, ˜ v verifies the estimates of Theorem 1.1. In particular if we denote f := e t∂ x ˜ v we have k ˜ v k X := sup t (cid:0) h t i − δ k xf k H + h t i α − k| ∂ x | α xf k L + k b f k L ∞ (cid:1) . ǫ , (3.47)see the definition of the X -norm in (2.1). In particular, this also gives the linear estimate(2.4) and (2.6), the bilinear estimates (2.11) and the trilinear estimate (2.12) for ˜ v : h t i / β/ − / (3 p ) k| ∂ x | β ˜ v k L p . ǫ , for 0 ≤ β < / , p (1 / − β/ > , h t ik ˜ v ∂ x ˜ v k L ∞ . ǫ , h t i / α/ k| ∂ x | α ˜ v k L . ǫ , for 0 ≤ α < / . (3.48)Thanks to (2.17) we also have k IS ˜ v k L + k S ˜ v k L . ǫ t Cǫ . (3.49)For later use, we improve these latter estimates in front of the solitary wave: Lemma 3.8. Let us set H ( t, y ) = ( IS ˜ v )( t, x ) , again with y = x − R t c ( s ) ds + h ( t ) . Thenwe have the estimates k H ( t ) k L w . ǫ h t i m − − Cǫ , Z t k H ( τ ) k H w ′ dτ . ǫ . Proof. We observe that since v solves the free equation (3.15), and S commutes with theequation as in (2.16), then ∂ t H − ˜ c∂ y H + ∂ y H + 3 v ∂ y H = 0 . Note that the second part of this estimate, was not explicitly written down, but it is a direct consequenceof (2.16) and Gronwall’s inequality. Then we can use a virial type computation similar to (3.28), with the same φ k defined in(3.27), to find ddt Z R φ k H dy + 12 (˜ c − σ − Cδ − Cǫ ) Z R φ ′ k H dy + 32 Z R φ ′ k | ∂ y H | dy = 3 Z R | v ∂ y v | φ k H dy . ǫ h t i Z R φ k H dy, (3.50)where we have used (3.48) to obtain the above inequality. Note that for ǫ sufficiently small,we can ensure that ˜ c − σ − Cδ − Cǫ ≥ c . Consequently, integrating in time we get Z R φ k ( t, y ) | H ( t, y ) | dy . h t i Cǫ Z R φ k (0) | H (0 , y ) | dy. Moreover, by observing that H (0 , y ) = yv and taking the parameters in the weight so that σ > x = − στ , and k ≤ m − 1, we get in particular that for every τ ≥ kh x + i k H ( τ ) k L w . ǫ h τ i Cǫ h τ i m − . This proves the first part of the estimates by taking k = 0.Next, by using (3.50) with σ = 0, x = 0 and k = 0 in the weight, and integrating in timewe get, using that w ′ . χ ,δ , Z t k H k H w ′ ds . ǫ + ǫ Z t h s i m − − Cǫ ds . ǫ + ǫ , for ǫ sufficiently small,since we assumed m > / (cid:3) Step 2: Weighted estimates for IS ˜ v . In this section we shall use in a crucial way that ∂ c Q c = 12 c ∂ y ( yQ c ) . (3.51) Lemma 3.9. Set H ( t, y ) = ( IS ˜ v )( t, x ) again with y = x − R t c + h. Then k H k L a + Z ∞ k H k H a dτ . ˜ ǫ . Proof. We shall first estimate ∂ y H ( t, y ) = ( S ˜ v )( t, y ). Commuting the vector field S withthe equation ∂ t v + ∂ x v = F + ∂ x G gives ∂ t Sv + ∂ x Sv = ( S + 3) F + ∂ x ( S + 2) G . Applyingthis identity to (3.16), we get that ∂ y H solves ∂ t ∂ y H + L c + ∂ y H = ∂ y (cid:0) S ( ˜ N ( v ))( t, y ) + N ( v )( t, y ) + S ˜ e Q ( t, y ) + 3 e Q ( t, y ) (cid:1) − ∂ y (cid:0) ( S ˜ Q c ) v (cid:1) − ∂ y ( Q c v ) + ∂ y e Q ( t, y ) := F ( t, y ) , (3.52)where we have set˜ Q c ( t, x ) = Q c ( t, y ) , ˜ e Q ( t, x ) = e Q ( t, y ) , ˜ N ( v )( t, x ) = (cid:0) − ( ˜ Q c ( t ) + ˜ v + ˜ v ) + ˜ Q c ( t ) + ˜ v + 3 ˜ Q c ( t ) ˜ v (cid:1) ( t, x ) , (3.53)and e Q ( t, y ) = (˜ c − c + ) ∂ y H − Q c − Q c + ) ∂ y H ) . SYMPTOTIC STABILITY OF SOLITONS FOR MKDV 37 Let us first estimate P c + ∂ y H = P c + ( S ˜ v )( t, y ). By using the equation (3.16) to compute ∂ t ˜ v , and by putting the space derivatives on the functions ζ ic + , ξ ic + , we obtain that kP c + ∂ y H k H a . h t i (cid:16) k v k L a + k v k L w + | ˙ c | + | ˙ h | (cid:17) and hence by using (3.46), we obtain kP c + ∂ y H k H a . ˜ ǫ h t i m − , kP c + ∂ y H k L t H a . ˜ ǫ . (3.54)Note that the second estimate comes from the fact that m > / Q c + ∂ y H . Applying Q c + to the integral formulation of the equation ∂ y H ( t ) = R t S c + ( t − τ ) F ( τ, y ) dτ gives Q c + ∂ y H ( t ) = Z t S c + ( t − τ ) Q c + F ( τ, y ) dτ. Using the smoothing estimates in Theorem 3.2, the estimates (3.46), and Proposition 3.5,we get kQ c + ∂ y H ( t ) k L a . Z t (cid:16) e − b ( t − τ ) max (cid:0) , ( t − τ ) − / (cid:1)(cid:0) k S ˜ N ( v ) k L a + kN ( v ) k L a + ˜ ǫ h τ i m − k ∂ y H k L a + ˜ ǫ h τ i m (cid:1) + e − b ( t − τ ) h τ i ( | ¨ c ( τ ) | + | ¨ h ( τ ) | ) (cid:17) dτ. (3.55)Next, we claim that from the definitions of the nonlinearities in (3.41) and (3.53), andusing (3.46) and (3.49), one has kN ( v ) k L a . ˜ ǫ h t i − m , (3.56) k S ˜ N ( v ) k L a . ˜ ǫ h t i − ( m − + k ∂ y H k L w ′ + ˜ ǫ k ∂ y H k L a . (3.57)The first estimate is a direct consequence of (3.42). Most of the estimates involved in proving(3.57) are straightforward, so we only give details about one of the most complicated terms: k v ( S ˜ v )( t, y ) v k L a . k v k L ∞ k S ˜ v k L k v k H a . ˜ ǫ h t i m − Cǫ . We also have to estimate ¨ c and ¨ h . By differentiating in time the equation (3.23), using theequations for v and v to express ∂ t v and ∂ t v and always putting the space derivatives on ζ ic in the scalar products using integration by parts, we obtain by using (3.46) that | ¨ h ( t ) | + | ¨ c ( t ) | . ˜ ǫ h t i m + ˜ ǫ h t i m k ∂ yy v k L w ′ . Note that the last term comes from the estimate of cubic nonlinear term that yields, afterintegration by parts, terms of the form R R v∂ v v∂ yy v∂ x ζ ic dx , for i = 1 , 2. Consequently, byusing (3.5), we obtain that (cid:16) Z t h τ i ( | ¨ h ( τ ) | + | ¨ c ( τ ) | ) dτ (cid:17) . ˜ ǫ . (3.58) By plugging the above estimates into (3.55), we thus get that kQ c + ∂ y H ( t ) k L a . Z t e − b ( t − τ ) h τ i ( | ¨ c ( τ ) | + | ¨ h ( τ ) | ) dτ + Z t (cid:16) e − b ( t − τ ) max(1 , ( t − τ ) − / ) (cid:16) ˜ ǫ h τ i m − + k ∂ y H k L w ′ + ˜ ǫ k ∂ y H k L a (cid:17) dτ. By using Young’s inequality, (3.54) and Lemma 3.8, we thus get that k ∂ y H k L t L a . ˜ ǫ + ǫ + ˜ ǫ k ∂ y H k L t L a and hence for ˜ ǫ sufficiently small that k ∂ y H k L t L a . ˜ ǫ . (3.59)It remains to estimate k H k L a . Since H = R y −∞ ∂ y H , integrating in y (3.52), we get ∂ t H − ˜ c∂ y H + Q c ∂ y H + ∂ y H = S ( ˜ N ( v ))( t, y ) + 2 N ( v )( t, y ) − Z y −∞ [ e Q ( t, y ′ ) + 3( S ˜ e Q )( t, y ′ )] dy − (cid:0) S ˜ Q c v (cid:1) − Q c v = G ( t, y ) . By a weighted energy estimate, we get that for some b > 0, we have ddt Z R e ay | H | dy + b k H k H a . k G k L a k H k L a + k ∂ y H k L a k H k L a . Next, we use (3.51) to write e Q = ˙ c c ∂ y ( yQ c ) + ˙ h∂ y Q c , and see that Z y −∞ (cid:0) e Q ( t, y ′ ) + ( S ˜ e Q )( t, y ′ ) (cid:1) dy ′ is an exponentially decreasing function at ±∞ . Therefore, thanks to (3.46) and (3.58), weobtain (cid:13)(cid:13)(cid:13)(cid:13)Z y −∞ (cid:0) e Q ( t, y ′ ) + ( S ˜ e Q )( t, y ′ ) (cid:1) dy ′ (cid:13)(cid:13)(cid:13)(cid:13) L a . ˜ ǫ h t i m − . Consequently, by using (3.56), (3.57), we obtain ddt Z R e ay | H | dy + b k H k H a . (cid:16) ˜ ǫ h t i m − + k ∂ y H k L w ′ + k ∂ y H k L a (cid:17) k H k L a . By integrating in time and using Young’s inequality, we obtain k H ( t ) k L a + b Z t k H k H a . ˜ ǫ + Z t (cid:0) k ∂ y H k L w ′ + k ∂ y H k L a (cid:1) dτ. By using (3.59) and Lemma 3.8, we finally get that Z t k H k H a . ˜ ǫ . This completes the proof of the Lemma. (cid:3) SYMPTOTIC STABILITY OF SOLITONS FOR MKDV 39 Step 3: Estimates of e v . Let us recall that ˜ v ( t, x ) is defined in (3.45). We can write theequation for ˜ v under the form ∂ t ˜ v + ∂ x ˜ v + ∂ x (˜ v ) = − ∂ x (cid:0) ( ˜ Q c + ˜ v ) − ˜ Q c − ˜ v (cid:1) + ˙ h∂ x ˜ Q c + ˙ c∂ c ˜ Q c = ∂ x K. (3.60)where K := − ( ˜ Q c + ˜ v ) + ˜ Q c + ˜ v + ˙ h ˜ Q c + ˙ c c ^ ( yQ c )( t, x ) . (3.61)Above we have used the notation ^ ( yQ c )( t, x ) = ( x − R t c + h ) Q c ( x − R t c + h ). Note that inorder to write the right hand side of (3.60) as the derivative of a localized function, we haveused (3.51).We now study the profile f of ˜ v defined by f = e t∂ x ˜ v . By taking ǫ sufficiently small butsuch that ǫ ≪ ˜ ǫ ≪ ǫ , we will prove the following: Lemma 3.10. We have the estimate k ˜ v k X := sup t (cid:0) k ˜ v k H + h t i − (cid:13)(cid:13) xf (cid:13)(cid:13) H + h t i α − (cid:13)(cid:13) | ∂ x | α xf (cid:13)(cid:13) L + (cid:13)(cid:13) b f ( ξ ) (cid:13)(cid:13) L ∞ (cid:1) ≤ ǫ . Proof. We follow the same steps as in the first part of the paper. Note that the norm k ˜ v k H is already estimated in view of (3.46). Moreover, by combining Lemma 3.8 and Lemma 3.9we have that Z t k ( IS ˜ v )( τ, y ) k H w ′ dτ . ˜ ǫ . (3.62)Still following the steps of the first part of the paper, we shall first estimate k xf k H , usingagain IS ˜ v . We note that IS ˜ v solves ∂ t IS ˜ v + ∂ x IS ˜ v + 3˜ v ∂ x IS ˜ v = ( S + 2) K An energy estimate yields ddt k IS ˜ v k L . ǫ t k IS ˜ v k L + Z R ( | SK | + | K | ) | IS ˜ v | dx. (3.63)By using (3.46), we obtain Z R ( | SK | + | K | ) | IS ˜ v | . k IS ˜ v ( t, y ) k L w ′ (cid:18) ˜ ǫ h t i m − + h t i ( | ¨ c ( t ) | + | ¨ h ( t ) | ) + k S ˜ v ( t, y ) k L w ′ (cid:19) and thus by (3.62) and (3.58), we obtain Z t Z R ( | SK | + | K | ) | IS ˜ v ( t, y ) | dy dt . ˜ ǫ . Consequently, by integrating (3.63) in time, we obtain k IS ˜ v ( t ) k L . h t i Cǫ (cid:0) ǫ + ˜ ǫ (cid:1) . (3.64)Next, we observe that xf = e t∂ x IS ˜ v − tI∂ t f = e t∂ x IS ˜ v − te t∂ x (cid:0) − ˜ v + K (cid:1) . Using (2.12) we get t k ˜ v k L . ǫ h t i , and thanks to (3.46) we obtain t k K k L . ˜ ǫ h t i m − . Combining these estimates gives k xf k L . ˜ ǫ + ǫ t . Finally, we can estimate S ˜ v in L : first, we note that S ˜ v solves ∂ t S ˜ v + ∂ x S ˜ v + 3 ∂ x (˜ v S ˜ v ) = ∂ x ( S + 2) K. (3.65)From an energy estimate, we find after integrating by parts ddt k S ˜ v k L . ǫ t k S ˜ v k L + k S ˜ v k L w ′ (cid:18) ˜ ǫ h t i m − + h t i ( | ¨ c ( t ) | + | ¨ h ( t ) | ) + k S ˜ v ( t, y ) k L w ′ (cid:19) , and hence (3.62) and (3.58) yield k S ˜ v ( t ) k L . h t i Cǫ (cid:0) ǫ + ˜ ǫ (cid:1) . (3.66)By using | ∂ x | α ( xf ) = e t∂ x | ∂ x | α ( IS ˜ v ) − te t∂ x (cid:0) − | ∂ αx | ˜ v + | ∂ αx | K (cid:1) , we get that k| ∂ x | α ( xf ) k L . k IS ˜ v k H + t k| ∂ αx | ˜ v k L + t k K k H . Then, by using (3.66), (3.64), (3.48) and k K k H . ˜ ǫ h t i m + (1 + ˜ ǫ ) k v k H w . ˜ ǫ h t i m , we finally get k| ∂ x | α ( xf ) k L . ˜ ǫ + ǫ t − α . Note that another way to get this estimate (that avoids using the H regularity of the initialdata in this step) would be to start from (3.65) and to follow the same steps as in the proofof (2.15) in Lemma 2.4.It remains to estimate k ˆ f k L ∞ . We can follow the proof of Proposition 2.5. The equationfor ˆ f is now ∂ t b f ( t, ξ ) = i π Z Z e − itφ ( ξ,η,σ ) ξ b f ( ξ − η − σ ) b f ( η ) b f ( σ ) dη dσ + G ( t, ξ ) , where G ( t, ξ ) = F (cid:0) e t∂ x ∂ x K (cid:1) ( ξ ) . We can estimate k G ( t ) k L ∞ ≤ kF ( ∂ x K ) k L ∞ . k ∂ x K k L . | ˙ c | + | ˙ h | + (1 + ˜ ǫ ) k v ( t, y ) k H w by using the exponential decay provided by Q c , and hence deduce k G ( t ) k L ∞ . ˜ ǫ h t i m . This term is thus integrable in time for m > (cid:3) SYMPTOTIC STABILITY OF SOLITONS FOR MKDV 41 Appendix A. Auxiliary Lemmas In this appendix we gather several lemmas that are used throughout the paper. First, alemma about stationary phase. Lemma A.1 (Stationary phase in dimension 2) . Consider χ ∈ C ∞ such that χ = 0 in B (0 , c , and |∇ χ | + |∇ χ | . ; and ψ ∈ C ∞ such that | det Hess ψ | ≥ , and |∇ ψ | + |∇ ψ | + |∇ ψ | . . Let I = Z Z e iλψ ( η,σ ) F ( η, σ ) χ ( η, σ ) dη dσ. Then, for any α ∈ [0 , , (i) If ∇ ψ only vanishes at ( η , σ ) , I = 2 πe i π s √ ∆ e iλψ ( η ,σ ) λ F ( η , σ ) χ ( η , σ ) + O (cid:13)(cid:13)(cid:13) h ( x, y ) i α b F (cid:13)(cid:13)(cid:13) L λ α , where s = sign Hess ψ and ∆ = | det Hess ψ | . (ii) If |∇ ψ | ≥ , I = O (cid:13)(cid:13)(cid:13) h ( x, y ) i α b F (cid:13)(cid:13)(cid:13) L λ α . Proof. ( i ) We assume for simplicity that η = σ = ψ ( η , σ ) = 0. If necessary, it is possibleto restrict the support of χ to an arbitrarily small neighborhood of 0, since the remaindercan be treated by ( ii ). By Plancherel’s theorem, I = 12 π Z Z (cid:20)Z Z e i [ λψ ( η,σ ) − xη − yσ ] χ ( η, σ ) dη dσ (cid:21)| {z } K ( x,y ) b F ( x, y ) dx dy. (A.1)The function K can be written K ( x, y ) = Z Z e iλ Φ X,Y ( η,σ ) χ ( η, σ ) dη dσ with Φ X,Y ( η, σ ) = ψ ( η, σ ) − Xη − Y σ and X = x/λ , Y = y/λ . If X or Y is larger than2 max Supp χ |∇ ψ | , it is easy to bound K by integrating by parts in the above integral. Thuswe can assume that X , Y are less than 2 max Supp χ |∇ ψ | . Since Supp χ can be chosen to bea small neighborhood of 0, we can assume that X and Y are small.By assumption, Hess Φ is non-degenerate, thus by the implicit function theorem thereexists (¯ η ( X, Y ) , ¯ σ ( X, Y )) such that ∇ η,σ Φ X,Y (¯ η ( X, Y ) , ¯ σ ( X, Y )) = 0and furthermore | ¯ η ( X, Y ) | + | ¯ σ ( X, Y ) | . | X | + | Y | and | Φ X,Y (¯ η ( X, Y ) , ¯ η ( X, Y )) | . | X | + | Y | . (A.2)By the stationary phase lemma, K ( x, y ) = 2 πe i π s √ ¯∆ e iλ Φ(¯ η, ¯ σ ) λ χ (¯ η, ¯ σ ) + O (cid:18) λ (cid:19) , where ¯∆ = | det Hess ψ | (¯ η, ¯ σ ) Coming back to (A.1), I = 2 πe i π s λ Z Z b F ( x, y ) e iλ Φ(¯ η, ¯ σ ) χ (¯ η, ¯ σ ) dx dy + O k b F k L λ ! = 2 πe i π s √ ∆ 1 λ F (0 , χ (0 , e i π s √ ∆ 1 λ Z Z b F ( x, y ) (cid:20) e iλ Φ(¯ η, ¯ σ ) χ (¯ η, ¯ σ )¯∆ − χ (0 , (cid:21) dx dy + O k b F k L λ ! Therefore, using (A.2), we find for any α ∈ [0 , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) I − πe i π s √ ∆ 1 λ F (0 , 0) + O k b F k L λ !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . λ Z Z | b F ( x, y ) | (cid:2) | e iλ Φ(¯ η, ¯ σ ) − | + | χ (¯ η, ¯ σ ) − χ (0 , | (cid:3) dxdy . λ Z Z | b F ( x, y ) | (cid:20) | ( x, y ) | α λ α + | ( x, y ) | α λ α (cid:21) dxdy . λ α kh ( x, y ) i α b F k L , which is the desired result.( ii ) A direct stationary phase estimate as above, using only the non-degeneracy of Hess ψ ,gives the bound | I | . λ k b F k L . Furthermore, since we are now assuming that ψ does not have stationary points, it is possibleto integrate by parts in I before applying this stationary phase estimate. This gives | I | . λ kh ( x, y ) i b F k L . Interpolating between these two inequalities gives the desired estimate. (cid:3) The following lemma gives some bounds on pseudo-product operators satisfying certainstrong integrability conditions: Lemma A.2 (Bounds on pseudo-product operators) . Assume that m ∈ L ( R × R ) satisfies (cid:13)(cid:13)(cid:13) Z R × R m ( η, σ ) e ixη e iyσ dηdσ (cid:13)(cid:13)(cid:13) L x,y ≤ A , (A.3) for some A > . Then, for all p, q, r ∈ [1 , ∞ ] such that /p + 1 /q = 1 /r one has k T m ( f, g ) k L r . A k f k L p k g k L q . (A.4) Moreover, if /p + 1 /q + 1 /r = 1 (cid:12)(cid:12)(cid:12) Z R × R b f ( η ) b g ( σ ) b h ( − η − σ ) m ( η, σ ) dηdσ (cid:12)(cid:12)(cid:12) . A k f k L p k g k L q k h k L r . (A.5) SYMPTOTIC STABILITY OF SOLITONS FOR MKDV 43 Proof. We rewrite (cid:12)(cid:12)(cid:12) Z R × R b f ( η ) b g ( σ ) b h ( − η − σ ) m ( η, σ ) dηdσ (cid:12)(cid:12)(cid:12) = C (cid:12)(cid:12)(cid:12) Z R f ( x ) g ( y ) h ( z ) K ( z − x, z − y ) dxdydz (cid:12)(cid:12)(cid:12) , . Z R | f ( z − x ) g ( z − y ) h ( z ) | | K ( x, y ) | dxdydz, where K ( x, y ) := Z R × R m ( η, σ ) e ixη e iyσ dηdσ. The desired bound (A.5) follows easily from (A.3). The bilinear estimate (A.4) can be provensimilarly using a duality argument. (cid:3) Finally, for the energy estimates, we need the following lemma. Lemma A.3. The following commutator estimate holds: k (cid:2) | ∂ x | α − ∂ x , P ≪ j w (cid:3) P j f k L . ( α − j k ∂ x w k L ∞ k P j f k L . Proof. Denoting e P j for the Fourier multiplier with symbol e χ (cid:18) ξ j (cid:19) = ξ | ξ | α − αj (cid:20) χ (cid:18) ξ j +10 (cid:19) − χ (cid:18) ξ j − (cid:19)(cid:21) , observe that (cid:2) | ∂ x | α − ∂ x , P ≪ j w (cid:3) P j f ( x ) = 2 αj (cid:2) e P j , P ≪ j w (cid:3) P j f = 2 αj (cid:2) j be χ (2 j · ) ∗ , P ≪ j w (cid:3) P j f = 2 αj Z j be χ (2 j ( x − y )) (cid:2) P ≪ j w ( x ) − P ≪ j w ( y ) (cid:3) P j f ( y ) dy. Thus (cid:12)(cid:12)(cid:2) | ∂ x | α − ∂ x , P ≪ j w (cid:3) P j f ( y ) (cid:12)(cid:12) . αj k ∂ x w k ∞ Z j | be χ (2 j ( x − y )) || x − y || P j f ( y ) | dy, and the desired result follows by Young’s inequality. (cid:3) References [1] Ablowitz, M. J. Nonlinear dispersive waves: Asymptotic Analysis and solitons . Cambridge text inapplied mathematics. Cambridge University Press, New York, 2011. xiv+348 pp.[2] Ablowitz, M., Kaup, D., Newell, A. and Segur, H. The inverse scattering transform-Fourier analysisfor nonlinear problems. Studies in Appl. Math. 53 (1974), no. 4, 249-315.[3] Bahouri, H., J.Y. Chemin, J.Y. and Danchin, R. Fourier analysis and nonlinear partial differentialequations . Grundlehren der Mathematischen Wissenschaften 343. Springer, Heidelberg, 2011.[4] Barab, J. E. Non-existence of asymptotically free solutions for nonlinear Schr¨odinger equation. Journalof Math. Phys. , 25 (1984), no. 11, 3270-3273.[5] Buckmaster, T and Koch, H. The Korteweg de Vries equation at H − regularity. Preprint arXiv:1112.4657 .[6] Christ, M., Colliander, J. and Tao, T. Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocusing equations. Amer. J. Math. 125 (2003), no. 6, 1235-1293.[7] Colliander, J., Keel, M., Staffilani, G., Takaoka, H. and Tao T. Sharp global well-posedness for KdVand modified KdV on R and T . J. Amer. Math. Soc. 16 (2003), 705-749.[8] Cˆote, R. and Vega, L. Scaling-sharp dispersive estimates for the Korteweg-de Vries group. C. R. Math.Acad. Sci. Paris 346 (2008), no. 15-16, 845-848.[9] Deift, P. and Zhou, X. A steepest descent method for oscillatory Riemann-Hilbert problems. Asymp-totics for the MKdV equation. Ann. of Math. 137 (1993), no. 2, 295-368. [10] Deift, P. and Zhou, X. Asymptotics for the Painlev´e II equation. Comm. Pure Appl. Math. 48 (1995),no. 3, 277-337.[11] Delort, J.M. Existence globale et comportement asymptotique pour l’ ´equation de Klein-Gordon quasi-lin´eaire `a donn´ees petites en dimension 1. Ann. Sci. ´Ecole Norm. Sup. 34 (2001) 1-61.[12] Germain, P. Space-time resonances. Proceedings of the Journ´ees EDP 2010 , Exp. No 8.[13] Germain, P., Masmoudi, N. and Shatah, J. Global solutions for quadratic Schr¨odinger equations indimension 3. Int. Math. Res. Not. (2009), no. 3, 414-432.[14] Germain P., Masmoudi, N. and Shatah, J. Global solutions for 2D quadratic Schrdinger equations. J.Math. Pures Appl. (9) 97 (2012), no. 5, 505-543.[15] Germain P., Masmoudi, N. and Shatah, J. Global solutions for the gravity surface water waves equationin dimension 3. Ann. of Math. (2) 175 (2012), no. 2, 691-754.[16] Harrop-Griffiths, B. Long time behaviour of solutions to the mKdV. Preprint arXiv:1407.1406 .[17] Hayashi, N. and Naumkin, P. Asymptotics for large time of solutions to the nonlinear Schr¨odinger andHartree equations. Amer. J. Math. 120 (1998), 369-389.[18] Hayashi, N. and Naumkin, P. Large time behavior of solutions for the modified Korteweg-de Vriesequation. Int. Math. Res. Not. (1999), no. 8, 395-418.[19] Hayashi, N. and Naumkin, P. On the Modified Korteweg-De Vries Equation Mathematical Physics,Analysis and Geometry (2001), no. 4, 197-227.[20] Hastings, S. P. and McLeod J. B. A boundary value problem associated with the second Painlev´etranscendent and the Korteweg-de Vries equation. Arch. Rat. Mech. Anal. 73 (1980), 31-51.[21] Ifrim, M. and Tataru, D. Global bounds for the cubic nonlinear Schr¨odinger equation (NLS) in onespace dimension. Preprint arXiv:1404.7581 .[22] Ionescu, A. and Pusateri, F. Nonlinear fractional Schr¨odinger equations in one dimension. J. Funct.Anal. 266 (2014), 139-176.[23] Ionescu, A. and Pusateri, F. Global solutions for the gravity water waves system in 2D. Invent. Math. 199 (2015), no. 3, 653-804.[24] Ionescu, A. and Pusateri, F. A note on the asymptotic behavior of gravity water waves in two dimen-sions. https://web.math.princeton.edu/ fabiop/2dWWasym-f.pdf.[25] Ionescu, A. and Pusateri, F. Global analysis of a model for capillary water waves in 2D. arXiv:1406.6042 .[26] Ionescu, A. and Pusateri, F. Global regularity for 2D water waves with surface tension. arXiv:1408.4428 .[27] Kato, T. On the Cauchy problem for the (generalized) Korteweg-de Vries equation. Studies in appliedmathematics Diff. Int. Equations 24 (2011), no. 9-10, 923-940.[29] Kenig C. E., Ponce, G. and Vega, L. On the (generalized) Korteweg-de Vries equation. Duke Math. J. 59 (1989), no. 3, 585-610.[30] Kenig C. E., Ponce, G. and Vega, L. Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle. Comm. Pure Appl. Math. 46 (1993), no. 4, 527-620.[31] Kenig C. E., Ponce, G. and Vega, L. On the ill-posedness of some canonical dispersive equations, DukeMath. J. 106 (2001), 617-633.[32] Klein, C. and Saut J.-C. IST versus PDE, a comparative study Preprint arXiv:1409.2020 .[33] Koch, H. and Marzuola, J. Small data scattering and soliton stability in ˙ H − / for the quartic KdVequation. Anal. PDE Arch. Ration. Mech. Anal. 157 (2001), no. 3, 219-254.[36] Martel, Y. and Merle, F. Asymptotic stability of solitons of the subcritical gKdV equations revisited. Nonlinearity 18 (2005), no. 1, 55-80.[37] Martel, Y. and Merle, F. Asymptotic stability of solitons of the gKdV equations with general nonlin-earity. Math. Ann. 341 (2008), no. 2, 391-427. SYMPTOTIC STABILITY OF SOLITONS FOR MKDV 45 [38] Martel, Y. and Merle, F. Review of long time asymptotics and collision of solitons for the quarticgeneralized Korteweg-de Vries equation. Proc. Roy. Soc. Edinburgh Sect. A 141 (2011), no. 2, 287-317.[39] Martel, Y., Merle, F. and Tsai, T. Stability and asymptotic stability in the energy space of the sum of N solitons for subcritical gKdV equations. Comm. Math. Phys. 231 (2002), no. 2, 347-373.[40] Melin, A. Operator methods for inverse scattering on the real line. Comm. Partial Differential Equa-tions 10 (1985), no. 7, 677-766.[41] Merle, F. and Vega, L. L stability of solitons for KdV equation. Int. Math. Res. Not. Comm. Math. Phys. SIAM J. Math. Anal. 32 (2001), no. 5, 1050-1080.[44] Mizumachi, T. Weak interaction between solitary waves of the generalized KdV equations. SIAM J.Math. Anal. 35 (2003), no. 4, 1042-1080.[45] Mizumachi, T., Pego, R. and Quintero, J. Asymptotic stability of solitary waves in the Benney-Lukemodel of water waves. Differential Integral Equations 26 (2013), no. 3-4, 253-301.[46] Mizumachi, T. and Tzvetkov, N. L -stability of solitary waves for the KdV equation via Pego andWeinstein’s method. Preprint, arXiv:1403.5321 .[47] Pego, R. and Weinstein, M. I. Asymptotic stability of solitary waves. Comm. Math. Phys. 164 (1994),no. 2, 305-349.[48] Pusateri, F. Modified scattering for the Boson Star equation. Comm. Math. Phys. 332 (2014), no. 3,1203-1234.[49] Tao, T. Nonlinear dispersive equations. Local and global analysis. CBMS Regional Conference Seriesin Mathematics, 106. American Mathematical Society, 2006. xvi+373 pp. ISBN: 0-8218-4143-2.[50] Tao, T. Scattering for the quartic generalised Korteweg-de Vries equation, J. Diff. Equations Bull. Amer. Math. Soc. (N.S.) 46 (2009), no. 1, 1-33.[52] Zakharov, V. E. and Manakov S.V. Asymptotic behavior of nonlinear wave systems integrated by theinverse scattering method. Soviet Physics JETP 44 (1976), 106-112. Pierre Germain, Courant Institute of Mathematical Sciences, 251 Mercer Street, NewYork 10012-1185 NY, USA E-mail address : [email protected] Fabio Pusateri, Department of Mathematics, Princeton University, Washington Road,Princeton 08540 NJ, USA E-mail address : [email protected] Fr´ed´eric Rousset, Laboratoire de Math´ematiques d’Orsay (UMR 8628), Universit´e Paris-Sud, 91405 Orsay Cedex France et Institut Universitaire de France E-mail address ::