Asymptotic stability manifolds for solitons in the generalized Good Boussinesq equation
aa r X i v : . [ m a t h . A P ] F e b ASYMPTOTIC STABILITY MANIFOLDS FOR SOLITONS IN THEGENERALIZED GOOD BOUSSINESQ EQUATION
CHRISTOPHER MAULÉN
Abstract.
We consider the generalized Good-Boussinesq model in one dimension, with powernonlinearity and data in the energy space H ˆ L . This model has solitary waves with speeds ´ ă c ă . When | c | approaches 1, Bona and Sachs showed orbital stability of such waves.It is well-known from a work of Liu that for small speeds solitary waves are unstable. In thispaper we consider in more detail the long time behavior of zero speed solitary waves, or standingwaves. By using virial identities, in the spirit of Kowalczyk, Martel and Muñoz, we constructand characterize a manifold of even-odd initial data around the standing wave for which thereis asymptotic stability in the energy space. Contents
1. Introduction 21.1. Setting 21.2. Standing waves 31.3. Main results 41.4. Idea of the proof 5Organization of this paper 7Acknowledgments 72. A virial identity for the ( g GB) system 72.1. Decomposition of the solution in a vicinity of the soliton 72.2. Notation for virial argument 82.3. Virial estimate 82.4. End of Proposition 2.1 123. Transformed problem and second virial estimates 123.1. The transformed problem 133.2. Virial functional for the transformed problem 133.3. Proof of Proposition 3.1: first computations 133.4. First technical estimates 153.5. Controlling error and nonlinear terms 183.6. End of proof of Proposition 3.1 194. Gain of derivatives via transfer estimates 204.1. A virial estimate related to M p t q N p t q Key words and phrases.
Generalized Boussinesq Boussinesq, decay, virial.Ch.M. was partially funded by Chilean research grants FONDECYT 1191412, and CONICYTPFCHA/DOCTORADO NACIONAL/2016-21160593 and CMM ANID PIA AFB170001. ´B x L ´B x L p u q “ B x z i and B x v i B x z i and B x v i Introduction
Setting.
In the 1870’s, J. Boussinesq [7] deduced a system of equations to describe two-dimensional irrotational and inviscid fluids in a uniform rectangular channel with flat bottom.He was the first to give a favorable explanation to the traveling-waves, solitons, or solitary wavessolutions discovered by Scott Rusell thirty years earlier [31], which remained in their form andtravelled with constant velocity.In a first order approximation, Boussinesq’s matrix model reduces to a scalar, fourth ordermodel B t φ ´ B x φ ´ B x φ ` B x p f p φ qq “ , (1.1)However, this model, known as the bad Boussinesq equation, is strongly linearly ill-posed. Con-sequently, in order to repair this problem, the following equation was proposed [37, 28]: B t φ ` B x φ ´ B x φ ` B x p f p φ qq “ . (1.2)Here the physical model considers the nonlinearity as quadratic, i.e. f p φ q “ φ and φ p t, x q is areal-valued function. This model is called good Boussinesq, and if formally u “ φ and v “ B ´ x B t φ ,this model has the following representation as ˆ system: p g GB) B t u “ B x v B t v “ B x p´B x u ` u ´ f p u qq . (1.3)This will be the exact model worked in this paper, which is Hamiltonian, and has the followingassociated conserved quantities: E r u, v s “ ż “ v ` u ` pB x u q ´ F p u q ‰ p Energy q ,P r u, v s “ ż uv p Momentum q . (1.4)(Here ş means ş R dx .) These laws define a standard energy space p u, v q P H ˆ L . As well asthe Korteweg-de Vries (KdV) equation, ( g GB) is considered as a canonical model of shallow waterwaves, see [36]. In addition, ( g GB) arises in the so-called "nonlinear string equation" describingsmall nonlinear oscillations in an elastic beam (see [11]).The study of the Boussinesq-type equations has increased recently, mainly due to the versatilityof these models when describing nonlinear phenomena. There are several authors that focus on thegood Boussinesq equation. The fundamental works Bona and Sachs [6], using abstract techniquesof Kato, proved that the Cauchy problem is locally and globally well-posed for small data, and
SYMPTOTICS FOR GOOD BOUSSINESQ SOLITONS 3 showed the existence of solitary waves for velocities c ă . Linares [22, 14], using Stricharzestimates, proved that the Cauchy problem is globally well-posed in the energy space in the caseof small data. Kishimoto [16], in the case of a quadratic nonlinearity, proved that the Cauchyproblem is globally well-posed in H s p R q , for s ď ´ { , and ill-posed for s ă ´ { . In [30], itwas proved that small solutions in the energy space must decay to zero as time tends to infinityin proper subsets of space. Recently, Charlier and Lenells [9] developed the inverse scatteringtransform and a Riemann-Hilbert approach for the quadratic ( g GB), which is integrable. Ingeneral, solitons (solitary waves in integrable equations) are stable objects. However, this is notthe case of good Boussinesq (similar to Klein-Gordon). Indeed, small perturbations of solitonsmay decay or form singularities in finite time, see [11, 23, 3, 37].In this paper, we are motivated by the long time behavior problem for solitary waves of thegGB (1.2) in the case where f p s q “ | s | p ´ s for p ą . A solitary wave is a solution to (1.2) of theform p u, v q “ p Q c , ´ cQ c qp x ´ ct ´ x q , | c | ă , x P R , with Q c solving p c ´ q Q c ` Q c ` f p Q c q “ in H p R q . This interesting question has attracted theattention of several authors before us, showing that the behavior of solitary waves in the standardenergy space H ˆ L is not an easy problem. Bona and Sachs [6], applying the theory developedby Grillakis, Shatath and Strauss (see [13]), proved that solitary waves are stable if the speed c obeys the condition p p ´ q{ ă c ă and p ą . Li, Ohta, Wu and Xue [21] proved the orbitalinstability in the degenerate case ă p ă and speed c “ p p ´ q{ . Additionally, Kalantarovand Ladyzhenkaya in [15] proved that solutions associated to initial data with nonpositive energymay blow up in some sense. Inspired by this work, Liu [23] showed that there are solutions withinitial data arbitrarily near the ground state ( c “ ) that blow up in finite time.1.2. Standing waves.
In the case that f is a pure power nonlinearity of the form f p s q “ | s | p ´ s for p ą , it is well-known that (up to shifts) standing solitary waves have the form u p t, x q “ Q p x q “ ˜ p `
12 cosh ` p ´ x ˘ ¸ {p p ´ q , v p t, x q “ . (1.5)Here, Q satisfies the equation Q p x q ´ Q p x q ` f p Q p x qq “ . (1.6)Let us consider a perturbation in (1.3) of Q of the form u p t, x q “ Q p x q ` w p t, x q , v p t, x q “ z p t, x q . Then one can see that this perturbation satisfies the following linear system at first order: B t w “ B x z B t z “ B x L w, (1.7)where L p w q “ ´B x w ` V p x q w, with V p x q “ ´ f p Q q . (1.8) L is the classical Schrödinger operator associated to the soliton Q . This operator has been exten-sively studied in [8] for instance.Therefore, from (1.7) one has B t w “ B x L w . Consequently, for the well-understanding of theproblem we require to study the fourth order operator ´B x L , much in the spirit of the fundamentalworks by Pego and Weinstein results [32, 33]. In Appendix A, we will prove the following: for any p ą , the linear operator ´ B x L p u q “ B x u ´ B x u ` B x p pQ p ´ u q , (1.9)has a unique eigenfunction φ p x q associated to a negative first eigenvalue ´ ν ă , satisfying xB ´ x φ , B ´ x φ y “ , ´B x L p φ q “ ´ ν φ , | φ p x q| À e ´ ´ | x | . (1.10) CHRISTOPHER MAULÉN
Note that we also have B ´ x φ well-defined, exponentially decreasing and part of L . Here x¨ , ¨y is the inner product in L p R q , and ´ is a number slightly below 1. The second eigenvalue of ´B x L is 0 but it is also a resonance in the classical sense (in L z L ), but the unique L eigenvalueis φ p x q “ c Q p x q . Therefore, by the Spectral Theorem, orthogonal to φ the operator ´B x L isnonnegative. See Appendix A for more details and full proofs of all the previous statements.Let Y ˘ “ ˆ φ ˘ ν B ´ x φ ˙ , Z ˘ “ ˆ B ´ x φ ˘ ν ´ B ´ x φ ˙ . (1.11)These are even-odd functions, i.e. the first coordinate is even and the second odd (see AppendixA.7). The functions u ˘ p t, x q “ e ˘ ν t Y ˘ p x q are solutions of the linearized problem (1.7), showingthe presence of exponentially stable and unstable linear manifolds relevant for the dynamics ofnonlinear solutions in a neighborhood of the soliton.In that follows, we refers to global solution of (1.3) to a function C pr , , H ˆ L q that satisfies(1.3) for all t ě .1.3. Main results.
It is not difficult to realize that (1.3) preserves the even-odd parity in itsvariables p u, v q . In this paper, we will prove that any even-odd small perturbation of the staticsoliton ( c “ ) in the energy space, under certain orthogonality condition, is orbitally stable andin fact, it is (locally) asymptotically stable. Furthermore, we will construct a manifold of initialdata such that the associated solutions are orbitally stable in H ˆ L , and locally asymptoticallystable in the space L X L . Our first result is: Theorem 1.1.
Let p ě . There exists δ ą such that if a global even-odd solution p φ, B t B ´ x φ q of (1.3) satisfies for all t ě , }p φ, B t B ´ x φ qp t q ´ p Q, q} H p R qˆ L p R q ă δ, (1.12) then, for any γ ą small enough and any compact interval I of R , lim t Ñ`8 ` } φ p t q ´ Q } L p I qX L p I q ` }p ´ γ B x q ´ B t φ p t q} L p I q ˘ “ . (1.13)This is, as far as we understand, the first description of the standing wave dynamics in the GoodBoussinesq model, which is unstable by nature. Clearly the data under which (1.12) is satisfiedis not empty, the soliton p Q, q being its most important representative. However, (1.12) cannotdefine an open set in the energy space as simple as in some stable, subcritical dynamics, such asKdV. Our second result will describe the manifold of initial data leading to (1.12), but first weneed to clarify some remarks. Remark 1.1.1 (On the lack of decay of derivatives) . Estimate (1.13) provides a clean and cleardescription of the local decay of φ p t q in the Lebesgue spaces L X L . However, no clear descriptionof the derivative B x φ p t q has been found, which remains an interesting open problem. Remark 1.1.2 (On the B t B ´ x φ term) . We have been unable to provide a clean description ofdecay for the second component of the Good Boussinesq system. This is due to some deep problemspresent at the level of the dynamics. However, (1.13) provides additional information on the decayof a suitable modification of the second variable. The constant γ depends on δ , but it can be takenarbitrarily small if needed. Remark 1.1.3 (About general data) . The construction performed in this paper uses in severalsteps the parity of the data. Extending our results to general data is a challenging problem, mainlybecause one needs to introduce shifts that may affect in a strong fashion the dynamics. We hopeto consider this problem in a forthcoming publication.
Remark 1.1.4 (About the condition p ě ) . The condition p ě is of technical type, and itis needed to ensure a control on the unstable direction, sufficiently good for our purposes. Webelieve that the situation for p close to 1 may be very complicated because of the weak decay of theamplitude associated to the unstable direction. SYMPTOTICS FOR GOOD BOUSSINESQ SOLITONS 5
The following result provides a description of the manifold of initial data leading to globalsolutions for which (1.12) holds.Let δ ą , and let A be the manifold given by A “ ǫ P H p R q ˆ L p R q| ǫ is even-odd , } ǫ } H ˆ L ă δ and x ǫ , Z ` y “ ( . (1.14) Theorem 1.2.
Let p ą . There exist C, δ ą and a Lipschitz function h : A Ñ R with h p q “ and | h p ǫ q| ď C } ǫ } { H ˆ L such that, denoting M “ tp Q, q ` ǫ ` h p ǫ q Y ` with ǫ P A u , (1.15) the following holds:(1) If φ P M then the solution of (1.3) with initial data φ is global and satisfies, for all t ě , } φ p t q ´ p Q, q} H p R qˆ L p R q ď C } φ ´ p Q, q} H p R qˆ L p R q . (1.16) (2) If a global even-odd solution φ of (1.3) satisfies, for all t ě , } φ p t q ´ p Q, q} H p R qˆ L p R q ď δ , (1.17) then for all t ě , φ p t q P M . Remark 1.2.1 (About blow-up) . Liu [23] showed that initial data p u , v q for which E p u , v q ă ,or E p u , v q ě and less than a particular function of Im ş B ´ x u v (which is zero in our case),lead to blow up solutions in finite time. In our case, we work with perturbation of the soliton p Q, q . One can easily check that E p Q, q “ p ´ p p ` q ş Q p ` ą , therefore we are not in the blow-upregime determined by Liu. Remark 1.2.2 (Extension to other models) . We believe that our results open the door to the un-derstanding of long time solitary wave dynamics in several other Boussinesq models. We mentionfor instance the asymptotic stability of abcd solitary waves, at least in the zero speed even data case [4, 5] , and the more involved case of the Improved Boussinesq solitary wave; see [29] for furtherdetails on this challenging problem.
Idea of the proof.
The proofs in this paper follow the lines of the ideas used recently byKowalczyk, Martel and Muñoz in [18] to understand the unstable soliton dynamics in the nonlinearKlein-Gordon equation, and by Kowalczyk, Martel, Muñoz and Van Den Bosch [19] to study thestability properties of kinks for (1+1)-dimensional nonlinear scalar field theories.More precisely, the proofs are based in a series of localized virial type arguments, similar to theones used in [1, 2, 18, 19, 17, 25, 27]. In our case, we will use a combination of virials to obtainthe integrability in time of the L ˆ L -norm of p φ p t q ´ Q, p ´ γ B x q ´ B t φ p t qq , for any γ ą smallenough, and in any compact interval I , i.e., ż ´ } φ p t q ´ Q } L p I q ` }p ´ γ B x q ´ B t φ p t q} L p I q ¯ dt ă 8 . However, some important issues, not present in the previously mentioned works [18, 19] will appearalong the proofs. The beginning of the proof is similar to [18]: The first step is to decompose thesolution close to the solitary waves in an adequate way. We will consider p u , u q P H ˆ L be aneven-odd perturbation of the solitary waves, which are in some sense orthogonal to Y ` and Y ´ ,and the flow on these directions: for a , a unique, u p t, x q “ Q p x q ` a p t q φ p x q ` u p t, x q ,v p t, x q “ a p t q ν B ´ x φ p x q ` u p t, x q . CHRISTOPHER MAULÉN
Then, we will focus on p u , u q P H ˆ L , which satisfy the linearized equation (1.7). Following[30], for an adequate weight function ϕ A placed at scale A large, we obtain the virial estimate ddt ż ϕ A p x q u u ď ´ ż “ w ` pB x w q ` ` ´ C A ´ ˘ w ‰ ` C a ` C ż sech p x q u , (1.18)where p w , w q is localized version of p u , u q at A scale, and C denotes a fixed constant. Thisvirial estimate has no good sign because of the term C ş sech p x q u . Then we require to transformthe system to a new one which has better virial estimates, in the spirit of Martel [24]. For any γ ą small enough, we define new variables p v , v q P H ˆ H by v “ p ´ γ B x q ´ L u ,v “ p ´ γ B x q ´ u . (see (3.1)). Note that p v , v q P H ˆ H , which is bad news because of the lack of a correctregularity order in the variables. This will cause problems later on. However, the new system for p v , v q (see (3.2)) satisfies, for an adequate weight function ψ A,B , B ! A , the virial estimate ddt ż ψ A,B v v ď ´ C ż “ z ` p V p x q ´ B ´ q z ` pB x z q ‰ ` B ´ ˆ } w } L ` } w } L ˙ ` | a | , (1.19)where p z , z q is a lozalized version of p v , v q , at the smaller scale B , V given by (1.8), and C denotes a fixed constant.Following [18], in order to combine estimates (1.18) and (1.19) we need an estimate for the lastterm in (1.18). However, unlike previous works, here we have the following coercivity estimate interms of the variables p w , w q and p z , z q : ż sech p x q u À B ´ { ` } w } L ` }B x w } L ˘ ` B { } z } L ` B ´ }B x z } L . (1.20)We can directly observe that the term B x z does not appears in (1.19), leading to the mainobstruction present in this paper. This problem is deeply related to the fact that p v , v q P H ˆ H ,i.e., the new variables are in opposed order of regularity.In order to overcome this problem, we introduce a series of modifications that will allow us toclose estimates (1.18) and (1.19) properly. First, we must gain derivates. In a new virial estimatefor the system of pB x v , B x v q (see (4.1)), we obtain the third virial estimate ddt ż ψ A,B B x v B x v ď ´ ż ` pB x z q ` ` V p x q ´ C B ´ ˘ pB x z q ` pB x z q ˘ ` C } z } L ` C B ´ } z } L ` C B ´ ` }B x w } L ` } w } L ` } w } L ˘ ` C | a | , (1.21)with C ą fixed. This new estimate give us local L control on B x z and B x z , which wasnot present before. Finally, our last contribution is a transfer virial estimate that exchangesinformation between B x z , B x z and B x z , in the form of ż pB x z q ď ddt ż ρ A,B B x v v ` ż “ pB x z q ` pB x z q ` z ` z ‰ ` C B ´ ˆ } w } L ` } w } L ˙ ` C | a | . (1.22)Here C ą is fixed and ρ A,B is a suitable weight function. Finally, we consider a functional H being a well-chosen linear combination of (1.18), (1.19), (1.21), (1.20) and (1.22). We get ddt H p t q ď ´ C B ´ ` } w } L ` }B x w } L ` } w } L ˘ ` C | a | , for all t ě . SYMPTOTICS FOR GOOD BOUSSINESQ SOLITONS 7
This final estimate allows us to close estimates, and prove local decay for u after some standardchange of variables from w j to u j . Organization of this paper.
This paper is organized as follows. Section 2 deals with a first virialestimate for a decomposition system, namely (2.1). In Section 3 we introduce the transformedproblem and prove first virial estimates on that system. In Section 4 we obtain virial estimates forhigher order derivatives of the transformed problem. Section 5 is devoted to a technical transferestimate dealing with higher order transformed variables. Finally, in Section 6 we prove Theorem1.1, and in Section 7 we prove Theorem 1.2.
Acknowledgments.
I deeply thank professors Didier Pilod (U. Bergen), Juan Soler (U. Granada),Francisco Gancedo (U. Sevilla) and Miguel A. Alejo (U. Córdoba) for the funding and their hos-pitality during the research stays where this work was completed.2.
A virial identity for the ( g GB) system
Recall the ( g GB) system (1.3). The first step in our proof is to consider a small even-odd perturbation of soliton p Q, q . In what follows we will describe this decomposition, introduce somenotation, and develop a virial estimate for the good Boussinesq system.2.1. Decomposition of the solution in a vicinity of the soliton.
Let p u, v q “ p φ, B t B ´ x φ q be a solution of (1.3) satisfying (1.12) for some small δ ą . Using Y ` as in (1.11), we decompose p u, v q as follows u p t, x q “ Q p x q ` a p t q φ p x q ` u p t, x q ,v p t, x q “ a p t q ν B ´ x φ p x q ` u p t, x q , (2.1)where (see (1.10)) a p t q “ x u p t q ´ Q, ν ´ L φ y “ x u p t q ´ Q, B ´ x φ y ,a p t q “ ν xB x v, ν ´ B x L φ y “ ν xB x v, B ´ x φ y , such that x u p t q , B ´ x φ y “ “ x u p t q , B ´ x φ y , (2.2)or equivalently, x u p t q , L φ y “ “ x u p t q , B x L φ y . (2.3)Orthogonalities (2.2) are nonstandard particular choices motivated by key cancelation properties.See Appendix A for a detailed construction of B ´ x φ and B ´ x φ . Setting b ` “ p a ` a q , b ´ “ p a ´ a q , (2.4)from (1.12), we have for all t P R ` } u p t q} H ` } v p t q} L ` | a p t q| ` | a p t q| ` | b ` p t q| ` | b ´ p t q| ď C δ. (2.5)Moreover, using (1.6), (1.10) and (2.2), p a , a q satisfies the following differential system $&% a “ ν a a “ ν a ` N ν , or equivalently $’’&’’% b ` “ ν b ` ` N ν b ´ “ ´ ν b ´ ´ N ν . (2.6)where N “ B x ` f p Q q ` f p Q qp a φ ` u q ´ f p Q ` a φ ` u q ˘ ,N K “ N ´ N B ´ x φ , and N “ x N, B ´ x φ y . (2.7)Then, p u , u q satisfies the system u “ B x u u “ B x L p u q ` N K , (2.8)with u even and u odd. CHRISTOPHER MAULÉN
Notation for virial argument.
We consider a smooth even function χ : R Ñ R satisfying χ “ on r´ , s , χ “ on p´8 , s Y r , , χ ď on r , . (2.9)For A ą , we define the functions ζ A and ϕ A as follows ζ A p x q “ exp ˆ ´ A p ´ χ p x qq| x | ˙ , ϕ A p x q “ ż x ζ A p y q dy, x P R . (2.10)For B ą , we also define ζ B p x q “ exp ˆ ´ B p ´ χ p x qq| x | ˙ , ϕ B p x q “ ż x ζ B p y q dy, x P R . (2.11)We consider the function ψ A,B defined as ψ A,B p x q “ χ A p x q ϕ B p x q where χ A p x q “ χ ´ xA ¯ , x P R . (2.12)These functions will be used in two distinct virial arguments with different scales ! B ! B ! A. (2.13)The following remark will be essential for the well-boundedness of some nonlinear terms in whatfollow. Remark 2.0.1.
One can see that for each function v ż χ A v ď ż | x |ď A v ď C ż | x |ď A e ´ | x |{ A v À ż v ζ A ď } ζ A v } L . This estimate will be useful later on (see Subsections 3.5.1 and 3.5.2).
Virial estimate.
Set I p t q “ ż R ϕ A p x q u u , (2.14)and w i “ ζ A u i , i “ , . (2.15)Here, p w , w q represents a localized version of p u , u q at scale A . The following virial argumenthas been used in [18, 19] in a similar context. Proposition 2.1.
There exist C ą and δ ą such that for any ă δ ď δ , the followingholds. Fix A “ δ ´ . Assume that for all t ě , (2.5) holds. Then for all t ě , ddt I p t q ď ´ ż “ w ` pB x w q ` ` ´ C A ´ ˘ w ‰ ` C a ` C ż sech ´ x ¯ w . (2.16)Some remarks are in order. Remark 2.1.1.
This virial has several similarities with the developed in [18] for nonlinear Klein-Gordon equation. In that paper, the main part of the virial is composed by the H -norm of w . Inour case, this main part is similar to the H ˆ L -norm of p w , w q , and the rest of the terms arethe same. Unlike [18] , we did not use a correction term since the momentum of the equation (1.4) works well in this case. This virial was already used in [30] in a different context (small solutionsaround zero). The proof of Proposition 2.1 follows after the next intermediate lemma.
Lemma 2.2.
Let p u , u q P H p R q ˆ L p R q a solution of (2.8) . Consider ϕ A “ ϕ A p x q a smoothbounded function to be chosen later. Then ddt I p t q “ ´ ż ϕ A ` u ` u ` pB x u q ˘ ` ż ϕ A u ` ż p ϕ A u ` ϕ A B x u q ` f p Q q ` f p Q q a φ ´ f p Q ` a φ ` u q ´ N B ´ x φ ˘ . (2.17) SYMPTOTICS FOR GOOD BOUSSINESQ SOLITONS 9
Proof.
Taking derivative in (2.14) and using (2.8), ddt I p t q “ ż ϕ A p u u ` u u q “ ż ϕ A pB x u u ` u pB x L p u q ` N K qq“ ´ ż ϕ A u ` ż ϕ A u B x L p u q ` ż ϕ A u N K . (2.18)For the second integral in the RHS of the above equation, we have ż ϕ A u B x L p u q “ ż ϕ A u p´B x u ` B x u ´ B x p f p Q q u qq“ ´ ż ϕ A u B x u ` ż ϕ A B x p u q ´ ż ϕ A u B x p f p Q q u q . Integrating by parts ż ϕ A u B x L p u q “ ´ ż ϕ A u ` ż p ϕ A u ` ϕ A B x u qB x u ´ ż ϕ A u B x p f p Q q u q“ ´ ż ϕ A “ u ` pB x u q ‰ ` ż ϕ A u B x u ´ ż ϕ A u B x p f p Q q u q . (2.19)Integrating by parts in the second integral in the RHS of the above equation, we get ż ϕ A u B x u “ ´ ż p ϕ A u ` ϕ A B x u qB x u “ ´ ż ˆ ϕ A B x p u q ` ϕ A pB x u q ˙ “ ´ ż ϕ A pB x u q ` ż ϕ A u . For the last integral in the RHS of (2.18), separating terms and integrating by parts we obtain ż ϕ A u N K “ ż ϕ A u ` B x ` f p Q q ` f p Q qp a φ ` u q ´ f p Q ` a φ ` u q ˘ ´ N B ´ x φ ˘ “ ´ ż p ϕ A u ` ϕ A B x u q ` f p Q q ` f p Q q a φ ´ f p Q ` a φ ` u q ˘ ` ż ϕ A u B x p f p Q q u q ´ N ż ϕ A u B ´ x φ . Cancelling terms, we finally obtain ddt I p t q “ ´ ż ϕ A ˆ u ` u ` pB x u q ` pB x u q ˙ ` ż ϕ A u ´ ż ϕ A u B x p f p Q q u q` ż p ϕ A u ` ϕ A B x u q ` f p Q q ` f p Q q a φ ´ f p Q ` a φ ` u q ˘ ` ż ϕ A u B x p f p Q q u q ´ N ż ϕ A u B ´ x φ “ ´ ż ϕ A ` u ` u ` pB x u q ˘ ` ż ϕ A u ´ N ż ϕ A u B ´ x φ . ` ż p ϕ A u ` ϕ A B x u q ` f p Q q ` f p Q q a φ ´ f p Q ` a φ ` u q ˘ . (2.20)This concludes the proof. (cid:3) Now we rewrite the main part of the virial identity using the new variables p w , w q . Lemma 2.3.
It holds ż ϕ A ` u ` u ` pB x u q ˘ ´ ż ϕ A u “ ż ˆ w ` pB x w q ` ˆ ` ζ A ζ A ´ p ζ A q ζ A ˙ w ˙ , with ˇˇˇˇ ζ A ζ A ´ p ζ A q ζ A ˇˇˇˇ À A . (2.21)
Proof.
Considering w i “ ζ A u i , i “ , , and ϕ A “ ζ A , we have ż ϕ A ` u ` u ˘ “ ż ζ A ` u ` u ˘ “ ż ` w ` w ˘ . (2.22)Also, ż ϕ A pB x u q “ ż pB x w q ` ż w ζ A ζ A . (2.23)In the case of the last terms we have, ż ϕ A u “ ż p ζ A q ζ A w “ ż ˆ ζ A ζ A ` p ζ A q ζ A ˙ w . By (2.10), we have ζ A ζ A “ ´ A “ ´ χ p x q| x | ` p ´ χ p x q sgn p x qq ‰ ,ζ A ζ A “ ˆ ζ A ζ A ˙ ` A “ χ p x q| x | ` χ p x q sgn p x q ‰ . (2.24)Then, substracting ζ A ζ A ´ ˆ ζ A ζ A ˙ “ ´ A “ ´ χ p x q| x | ` p ´ χ p x qq sgn p x q ‰ ` A “ χ p x q| x | ` χ p x q sgn p x q ‰ . For ď | x | ď , one can see that ˇˇˇˇˇ ζ A ζ A ´ ˆ ζ A ζ A ˙ ˇˇˇˇˇ À A ` A . For | x | ě , we have that ˇˇˇˇˇ ζ A ζ A ´ ˆ ζ A ζ A ˙ ˇˇˇˇˇ “ A . Then, ˇˇˇˇˇ ζ A ζ A ´ ˆ ζ A ζ A ˙ ˇˇˇˇˇ À ˆ A ` A ˙ t| x |ě u À A .
This ends the proof. (cid:3)
Next, we deal with the nonlinear terms.
Lemma 2.4. ˇˇˇˇż p ϕ A u ` ϕ A B x u q ` f p Q ` a φ ` u q ´ f p Q q ´ f p Q q a φ ˘ ´ N ż ϕ A u B ´ x φ ˇˇˇˇ À a ` ż sech ´ x ¯ w ` A } u } p ´ L ż |B x w | . (2.25) Proof.
First, we treat the term N ş ϕ A u B ´ x φ . Noticing that N “ x N, B ´ x φ y “ ´x f p Q q ` f p Q qp a φ ` u q ´ f p Q ` a φ ` u q , φ y , (2.26)and by Taylor’s expansion, one has | f p Q ` a φ ` u q ´ f p Q q ´ f p Q qp a φ ` u q| À a f p Q q φ ` f p Q q u ` | a | p φ p ` | u | p . (2.27)Thus, by exponential decay estimates on Q and φ (see Appendix A), and by (2.5), | a | À , } u } L ď } u } H À , it holds | N | À a ` ż sech p x { q u , (2.28) SYMPTOTICS FOR GOOD BOUSSINESQ SOLITONS 11 taking A ě , we have | N | À a ` ż sech ´ x ¯ w . (2.29)Noticing that for all x P R , | ϕ A | ď | x | , | ϕ A B ´ x φ | À | x sech p x { q| ď ˇˇˇˇ sech ˆ x ˙ˇˇˇˇ , and using Hölder inequality, we have ˇˇˇˇż u ϕ A B ´ x φ ˇˇˇˇ À ˇˇˇˇż w sech ´ x ¯ˇˇˇˇ À ˇˇˇˇż w sech ´ x ¯ˇˇˇˇ { ˇˇˇˇż sech ´ x ¯ˇˇˇˇ { À ˇˇˇˇż w sech ´ x ¯ˇˇˇˇ { . (2.30)We conclude using Cauchy-Schwarz inequality ˇˇˇˇ N ż ϕ A u B ´ x φ ˇˇˇˇ À | N | ` ˇˇˇˇż u ϕ A B ´ x φ ˇˇˇˇ À a ` ż sech ´ x ¯ w . For the remaining terms, we consider the following decomposition ż ` ϕ A B x u ` ϕ A u ˘ “ f p Q ` a φ ` u q ´ f p Q q ´ f p Q q a φ ‰ “ ż ϕ A B x “ F p Q ` a φ ` u q ´ F p Q ` a φ q ´ p f p Q q ` f p Q q a φ q u ‰ ´ ż ϕ A Q “ f p Q ` a φ ` u q ´ f p Q ` a φ q ´ p f p Q q ` f p Q q a φ q u ‰ ´ a ż ϕ A B x φ “ f p Q ` a φ ` u q ´ f p Q ` a φ q ´ f p Q q u ‰ ` ż ϕ A u r f p Q ` a φ ` u q ´ f p Q q ´ f p Q q a φ s“ : I ` I ` I ` I , and rewriting as I “ ´ ż ϕ A “ F p Q ` a φ ` u q ´ F p Q ` a φ q ´ F p Q ` a φ q u ´ F p u q ‰ ´ ż ϕ A “ f p Q ` a φ q ´ f p Q q ` f p Q q a φ q ‰ u ´ ż ϕ A F p u q ,I “ ´ ż ϕ A Q “ f p Q ` a φ ` u q ´ f p Q ` a φ q ´ f p Q ` a φ q u ‰ s´ ż ϕ A Q ` f p Q ` a φ q ´ f p Q q ` f p Q q a φ ˘ u ,I “ ´ a ż ϕ A B x φ “ f p Q ` a φ ` u q ´ f p Q ` a φ q ´ f p Q ` a φ q u ‰ ´ a ż ϕ A B x φ ` f p Q ` a φ q ´ f p Q q ˘ u , and I “ ż ϕ A u r f p Q ` a φ ` u q ´ f p Q ` a φ q ´ f p u qs` ż ϕ A u r f p Q ` a φ q ´ f p Q q ´ f p Q q a φ s ` ż ϕ A u f p u q . By Taylor expansion, p ě , | a | , } u } L À , we have | F p Q ` a φ ` u q ´ F p Q ` a φ q ´ F p Q ` a φ q u ´ F p u q| , À | Q ` a φ | p ´ u ` | Q ` a φ || u | p , À | Q ` a φ | p ´ u ` | Q ` a φ || u | À sech p x { q u À sech ´ x ¯ w . Similarly, using (2.10) and A ě , we find the following estimates | ϕ A Q “ f p Q ` a φ ` u q ´ f p Q ` a φ q ´ f p Q ` a φ q u ‰ | À sech ´ x ¯ w , | a ϕ A B x φ “ f p Q ` a φ ` u q ´ f p Q ` a φ q ´ f p Q ` a φ q u ‰ | À sech ´ x ¯ w , | ϕ A u r f p Q ` a φ ` u q ´ f p Q ` a φ q ´ f p u qs | À sech ´ x ¯ w . Furthermore, once again by Taylor expansion, we have | ϕ A “ f p Q ` a φ q ´ f p Q q ` f p Q q a φ q ‰ u |` | ϕ A Q “ f p Q ` a φ q ´ f p Q q ` f p Q q a φ ‰ u |` | a ϕ A B x φ “ f p Q ` a φ q ´ f p Q q ‰ u |` | ϕ A u “ f p Q ` a φ q ´ f p Q q ´ f p Q q a φ ‰ |À sech ´ x ¯ | a | | u | À sech ´ x ¯ | w | ` sech ´ x ¯ | a | . (2.31)For the last step, we need the following claim proved in [18]. Claim 2.5.
It holds ż ζ A | u | p ` “ ż ζ ´ p ` A | w | p ` À A } u } p ´ L ż |B x w | . Using this claim, we have ˇˇˇˇż ϕ A F p u q ˇˇˇˇ ` ˇˇˇˇż ϕ A u f p u q ˇˇˇˇ À ż ζ A | u | p ` À A } u } p ´ L ż |B x w | . Finally, we get ˇˇˇˇż pB x ϕ A u ` ϕ A B x u q ` f p Q ` a φ ` u q ´ f p Q q ´ f p Q q a φ ` N ν ´ L φ ˘ˇˇˇˇ À a ` ż sech ´ x ¯ w ` A } u } p ´ L ż |B x w | . (2.32)This ends the proof of Lemma 2.4. (cid:3) End of Proposition 2.1.
Applying Lemmas 2.3 and 2.4, and taking } u } L ď δ A , for δ A small enough, we have proved ddt I p t q ď ´ ż „ w ` pB x w q ` ˆ ´ ˆ A ` A ˙ t| x |ě u ˙ w ` C a ` C ż sech ´ x ¯ w ` A } u } p ´ L ż |B x w | ď ´ ż „ w ` pB x w q ` ˆ ´ C A ˙ w ` C a ` C ż sech ´ x ¯ w . (2.33)This concludes the proof.3. Transformed problem and second virial estimates
Following the idea of Martel [24], we will consider the function v “ L u instead u to obtain atransformed problem with better virial properties. However, we must be careful since our originalvariables p u , u q belong to H p R qˆ L p R q , and by using L , the new variables are not well-defined.Therefore, we need a regularization procedure, as in [18]. SYMPTOTICS FOR GOOD BOUSSINESQ SOLITONS 13
The transformed problem.
Let γ ą small, to be determined later, set v “ p ´ γ B x q ´ L u ,v “ p ´ γ B x q ´ u . (3.1)From the system (2.8), follows that p v , v q P H p R q ˆ H p R q , and satisfies the system v “ L pB x v q ` G p x q , v “ B x v ` H p x q , (3.2)where H p x q “ p ´ γ B x q ´ N K ,G p x q “ γ p ´ γ B x q ´ “ B x p f p Q qqB x v ` B x p f p Q qqB x v ‰ . (3.3)Now we compute a second virial estimate, this time on p v , v q .3.2. Virial functional for the transformed problem.
Set now J p t q “ ż ψ A,B v v , (3.4)with ψ A,B “ χ A ϕ B , z i “ χ A ζ B v i , i “ , . (3.5)Here, p z , z q represents a localized version of the variables p v , v q at the scale B . This scale isintermediate, and J p t q involves a cut-off at scale A , which is needed to bound some bad error andnonlinear terms; see [19] for a similar procedure. Proposition 3.1.
There exist C ą and δ ą such that for γ “ B ´ and for any ă δ ď δ ,the following holds. Fix B “ δ ´ { . Assume that for all t ě , (2.5) holds. Then for all t ě , ddt J p t q ď ´ ż “ z ` p V p x q ´ C B ´ q z ` pB x z q ‰ ` C B ´ ˆ } w } L ` } w } L ˙ ` C | a | , (3.6) where V p x q is given by (1.8) . The rest of this section is devoted to the proof of this proposition, which has been divided inseveral subsections.3.3.
Proof of Proposition 3.1: first computations.
We have from (3.4) and (3.2), ddt J p t q “ ż ψ A,B „ p L B x v q v ` B x p v q ` H p x q v ` G p x q v “ ż ψ A,B p L B x v q v ´ ż ψ A,B v ` ż ψ A,B r G p x q v ` H p x q v s . (3.7)In a similar way to the computation in (2.19), we have ż ψ A,B p L B x v q v “ ´ ż ψ A,B ` v ` pB x v q ˘ ` ż ψ A,B v ´ ż ψ A,B f p Q qB x p v q . We consider now the following decomposition ddt J p t q “ ´ ż ψ A,B ` v ` v ` pB x v q ˘ ` ż ψ A,B v ´ ż ψ A,B f p Q qB x p v q ` ż ψ A,B G p x q v ` ż ψ A,B H p x q v “ : p J ` J q ` p J ` J ` J q . (3.8) By definition of ψ A,B (see (3.5)), it follows that ψ A,B “ χ A ζ B ` p χ A q ϕ B ,ψ A,B “ χ A p ζ B q ` p χ A q ζ B ` p χ A q ϕ B ,ψ A,B “ χ A p ζ B q ` p χ A q p ζ B q ` p χ A q ζ B ` p χ A q ϕ B . (3.9)Also, by the definition of z i in (3.5), we have: ´ J “ ż ψ A,B ` v ` v ` pB x v q ˘ “ ż p χ A ζ B ` p χ A q ϕ B q ` v ` v ` pB x v q ˘ “ ż p z ` z q ` ż p χ A q ϕ B ` v ` v ` pB x v q ˘ ` ż χ A ζ B pB x v q . (3.10)Derivating z “ ζ B χ A v , replacing and integrating by parts, we obtain ż χ A ζ B pB x v q “ ż ζ B ζ B z ` ż pB x z q ` ż „ χ A ` χ A ζ B ζ B χ A ζ B v . (3.11)Then, for J we obtain ´ J “ ż p z ` z q ` ż ζ B ζ B z ` ż pB x z q ` ż „ χ A ` χ A ζ B ζ B χ A ζ B v ` ż p χ A q ϕ B ` v ` v ` pB x v q ˘ . (3.12)Now we turn into J . By (3.9), J satisfies the following decomposition J “ ż p χ A p ζ B q ` p χ A q p ζ B q ` p χ A q ζ B ` p χ A q ϕ B q v “ ż «ˆ ζ B ζ B ˙ ` ζ B ζ B ff z ` ż p p χ A q p ζ B q ` p χ A q ζ B ` p χ A q ϕ B q v . For J , integrating by parts and using the definition of z , we obtain ´ J “ ´ ż B x p ψ A,B f p Q qq v “ ´ ż “ p χ A ζ B ` p χ A q ϕ B q f p Q q ` χ A ϕ B B x p f p Q qq ‰ v “ ´ ż „ f p Q q ` B x p f p Q qq ϕ B ζ B z ´ ż p χ A q ϕ B f p Q q v . Finally, we obtain that the main part of the virial can be write as J ` J ` J “ ´ ż “ z ` V p x q z ` pB x z q ‰ ` ˜ J , where V p x q “ ` ζ B ζ B ´ p ζ B q ζ B ´ f p Q q ´ B x p f p Q qq ϕ B ζ B , and the error term is given by ˜ J “ ´ ż „ ` χ A χ A ´ p χ A q ˘ ζ B ´ p χ A q p ζ B q ` ` p χ A q ´ p χ A q ˘ ϕ B v ` ż p χ A q ϕ B f p Q q v ´ ż p χ A q ϕ B v ´ ż p χ A q ϕ B pB x v q . (3.13)To control the main part of the virial is necessary a lower bound for the potential V p x q . We havethe following result: Lemma 3.2.
There are C ą and B ą such that for all B ě B , one has V p x q ě V p x q ´ CB ´ , where V p x q “ ´ f p Q q . (3.14) SYMPTOTICS FOR GOOD BOUSSINESQ SOLITONS 15
Proof.
First, recalling (2.21) and changing the scale, we have ˇˇˇˇˇ ζ B ζ B ´ ˆ ζ B ζ B ˙ ˇˇˇˇˇ À B . (3.15)Using that for x P r ,
8q ÞÑ ζ B p x q is a non-increasing function, we have for x ą ,ϕ B ζ B “ ş x ζ B ζ B ą , and B x p f p Q qq ă for x ą . Then, V p x q ě ´ CB ´ ´ f p Q q ` |B x p f p Q qq x | ě ´ CB ´ ´ f p Q q “ V p x q ´ CB ´ . (3.16)The case x ď is similar. These estimates hold for any x P R . This concludes the proof. (cid:3) First conclusion . Using this lemma, and the above definition of ˜ J , we conclude ddt J p t q ď ´ ż “ z ` p V ´ CB ´ q z ` pB x z q ‰ ` ˜ J ` J ` J , (3.17)where J and J are related with the nonlinear term in (3.8). To control the terms ˜ J , J , J , andthe terms that will appear in the sections below, some technical estimates will be needed.3.4. First technical estimates.
For γ ą , let p ´ γ B x q ´ be the bounded operator from L to H defined by its Fourier transform as { pp ´ γ B x q ´ g qp ξ q “ p g p ξ q ` γξ , for any g P L . We start with a basic but essential result, in the spirit of [19].
Lemma 3.3.
Let f P L p R q and ă γ ă , we have the following estimates p i q }p ´ γ B x q ´ f } L p R q ď } f } L p R q , p ii q }p ´ γ B x q ´ B x f } L p R q ď γ ´ { } f } L p R q , p iii q }p ´ γ B x q ´ f } H p R q ď γ ´ } f } L p R q . We also enunciate the following result that appears in [19, 18]:
Lemma 3.4.
There exist γ ą and C ą such that for any γ P p , γ q , ă K ď and g P L ,the following estimates holds ›› sech p Kx q p ´ γ B x q ´ g ›› L ď C ›› p ´ γ B x q ´ r sech p Kx q g s ›› L , (3.18) and ›› cosh p Kx q p ´ γ B x q ´ r sech p Kx q g s ›› L ď C ›› p ´ γ B x q ´ g ›› L . From this lemma, we obtain the following result.
Corollary 3.5.
For any ă K ď and γ ą small enough, for any f P L , } sech p Kx qp ´ γ B x q ´ B x f } L À γ ´ { } sech p Kx q f } L , (3.19) where the implicit constant is independent of γ and K .Proof. Using (3.18) and rewriting, we have } sech p Kx qp ´ γ B x q ´ B x f } L À }p ´ γ B x q ´ r sech p Kx qB x f s } L À }p ´ γ B x q ´ B x r sech p Kx q f s } L ` }p ´ γ B x q ´ pB x sech p Kx qq f } L . The proof concludes applying Lemma 3.3. (cid:3)
Following the spirit of Lemma 3.4, we obtain
Lemma 3.6.
For any ă K ď and γ ą small enough, for any f P L , } sech p Kx qp ´ γ B x q ´ p ´ B x q f } L À γ ´ } sech p Kx q f } L , (3.20) where the implicit constant is independent of γ and K .Proof. Set h “ sech p Kx q p ´ γ B x q ´ p ´ B x q f and k “ sech p Kx q f . We have cosh p Kx q h “ p ´ γ B x q ´ p ´ B x q r cosh p Kx q k s . (3.21)Thus, we obtain cosh p Kx q h “ p ´ γ B x q ´ r cosh p Kx q k s ´ B x p ´ γ B x q ´ r cosh p Kx q k s“ p ´ γ B x q ´ r cosh p Kx q k s ` γ ´ p ´ γ B x ´ qp ´ γ B x q ´ r cosh p Kx q k s“ p ´ γ B x q ´ r cosh p Kx q k s ` γ ´ cosh p Kx q k ´ γ ´ p ´ γ B x q ´ r cosh p Kx q k s . Thus, γh “p γ ´ q sech p Kx q p ´ γ B x q ´ r cosh p Kx q k s ` k, using Lemma 3.4 and dividing by γ , we obtain } h } L À γ ´ } k } L . This concludes the proof. (cid:3)
We need some additional auxiliary estimates to related the several variables defined.
Lemma 3.7.
One has:(a) Estimates on v : } v } À γ ´ } u } L , }B x v } À γ ´ { } f p Q q u } L ` γ ´ }B x u } L . (3.22) (b) Estimates on v : } v } L À } u } L , }B x v } L À γ ´ { } u } L , }B x v } L À γ ´ } u } L . (3.23)The proof of the above results are a direct application of Lemma 3.3. Lemma 3.8.
Let ď K ď A fixed. Then(a) Estimates on v : } ζ K v } À γ ´ } w } L , } ζ K B x v } À γ ´ }B x w } L ` γ ´ } w } L . (3.24) (b) Estimates on v : } ζ K v } L À } w } L , } ζ K B x v } L À γ ´ { } w } L , } ζ K B x v } L À γ ´ } w } L . (3.25) Proof.
Proof of (3.24). p i q Applying the definition of v (3.1), we have } ζ K v } L À } ζ K p ´ γ B x q ´ “ p ´ B x qp u q ´ f p Q q u ‰ } L . Using Lemma 3.6 and Lemma 3.3, (2.10) and ď K ď A , we conclude } ζ K v } L À γ ´ } ζ K u } L ` } ζ K f p Q q u } L À γ ´ } ζ K ζ ´ A w } L ` } ζ K ζ ´ A f p Q q w } L À γ ´ } w } L . Proof of p ii q . First, by the definition of w (2.15), we get ζ K B x u “ ζ K ζ ´ A ˆ B x w ´ ζ A ζ A w ˙ . (3.26)Then, by definition of v in (3.1), } ζ K B x v } L À } ζ K p ´ γ B x q ´ p ´ B x qB x u } L ` } ζ K p ´ γ B x q ´ B x r f p Q q u s} L . SYMPTOTICS FOR GOOD BOUSSINESQ SOLITONS 17 and using Lemma 3.6, } ζ K B x v } L À γ ´ } ζ K B x u } L ` } ζ K f p Q q u } L À γ ´ p}B x w } L ` A ´ } w } L q ` } w } L À γ ´ p}B x w } L ` } w } L q . This ends the proof of (3.24). Following the preceding steps for v , the proof concludes. (cid:3) Now we perform some technical estimates on the variable z i . Corollary 3.9.
One has:(a) Estimates on z : } z } À γ ´ } w } L , }B x z } À γ ´ }B x w } L ` γ ´ } w } L . (3.27) (b) Estimates on z : } z } L À } w } L , }B x z } L À γ ´ { } w } L , }B x z } L À γ ´ } w } L . (3.28) Proof.
Proof of (3.27). For p i q , from definition of z “ χ A ζ B v , we have } z } L À } ζ B v } L , and using Lemma 3.8, we conclude } z } L À γ ´ } w } L . For p ii q , derivating z , we obtain B x z “ χ A ζ B v ` χ A ζ B v ` χ A ζ B B x v “ χ A ζ B v ` ζ B ζ B z ` χ A ζ B B x v . Then, by Lemma 3.8 we have }B x z } ď γ ´ ` } w } L ` }B x w } L ˘ . For z we use the same strategy, and we skip the details. This ends the proof. (cid:3) Lemma 3.10.
One has:(a) Estimates on u : ››› sech { p x q u ››› L À } w } L , ››› sech { p x qB x u ››› L À }B x w } L ` } w } L . (3.29) (b) Estimates on u : ››› sech { p x q u ››› L À } w } L . (3.30) Proof.
Proof of (3.29) . Recalling definition of w i “ ζ A u i for i “ , . . We have } sech { p x q u i } L À } sech { p x q ζ ´ A w i } L ď } w i } L . Furthermore, derivating w , we have ζ A B x u “ B x w ´ ζ A ζ A w . Then, } sech { p x qB x u } L “ ›››› sech { p x q ζ ´ A ˆ B x w ´ ζ A ζ A w ˙›››› ď }B x w } L ` A ´ } w } L . This concludes the proof. (cid:3)
Controlling error and nonlinear terms.
By the definition of ζ B and χ A in (2.11) and(2.12), it holds ζ B p x q ď e ´ | x | B , | ζ B p x q| À B e ´ | x | B , | ϕ B | À B, |p χ A q | À A ´ , |p χ A q | À A ´ , |p χ A q | À A ´ . (3.31)3.5.1. Control of ˜ J . Considering the following decomposition ˜ J : ˜ J “ ´ ż p χ A q ϕ B r v ` pB x v q s ` ż “ p χ A q f p Q q ` ` p χ A q ´ p χ A q ˘‰ ϕ B v ´ ż „` χ A χ A ´ p χ A q ˘ ´ p χ A q ζ B ζ B ζ B v “ : H ` H ` H . For H and H , using |p χ A q ϕ B | À A ´ B and Remark 2.0.1, we obtain | H | À A ´ B p} v } L p| x |ď A q ` }B x v } L p| x |ď A q q À A ´ B p} ζ A v } L ` } ζ A B x v } L q , and | H | À A ´ B } v } L p A ď| x |ď A q À A ´ B } v } L p| x |ď A q À A ´ B } ζ A v } L . For H , using (3.31), we have | H | ď ż ˇˇˇˇ` χ A χ A ´ p χ A q ˘ ´ p χ A q ζ B ζ B ˇˇˇˇ ζ B v À p AB q ´ } ζ B v } L p| x |ď A q À p AB q ´ } ζ B v } L . Finally, we get | ˜ J | À A ´ B ` } ζ A v } L ` } ζ A v } L ` } ζ A B x v } L ˘ , (3.32)since ζ B À ζ A .3.5.2. Control of J . Recall that J “ γ ż ψ A,B v p ´ γ B x q ´ “ B x pB x p f p Q qqB x v q ´ B x p f p Q qqB x v ‰ . Using Hölder’s inequality | J | À γ } ψ A,B v } L }p ´ γ B x q ´ r B x pB x p f p Q qqB x v q ´ B x p f p Q qqB x v s} L loooooooooooooooooooooooooooooooooooomoooooooooooooooooooooooooooooooooooon J . First we focus on J . Using (3.3), J À γ ´ { }pB x p f p Q qqB x v } L ` }B x p f p Q qqB x v } L . Recall that |B x p f p Q qq| , |B x p f p Q qq| „ Q p ´ | Q | À e ´p p ´ q| x | . Therefore, we are led to the estimateof } e ´p p ´ q| x | B x v } L . Differentiating z “ χ A ζ B v , we obtain χ A ζ B B x v “ B x z ´ ζ B ζ B z ´ χ A ζ B v , we get e ´ p p ´ q| x | pB x v q “ e ´ p p ´ q| x | χ A pB x v q ` e ´ p p ´ q| x | p ´ χ A qpB x v q À e ´p p ´ q| x | « pB x z q ` ˆ ζ B ζ B ˙ z ` p χ A ζ B v q ff ` e ´p p ´ q A e ´p p ´ q| x | pB x v q À e ´p p ´ q| x | „ pB x z q ` B z ` e ´p p ´ q| x | p χ A ζ B v q ` e ´p p ´ q A e ´p p ´ q| x | pB x v q . Hence, } e ´| x | B x v } L À }B x z } L ` } z } L ` e ´p p ´ q A p} ζ B v } L ` } ζ B B x v } L q . SYMPTOTICS FOR GOOD BOUSSINESQ SOLITONS 19
By the above inequality, we have J À γ ´ { ˆ }B x z } L ` } z } L ` e ´p p ´ q A p} ζ B v } L ` } ζ B B x v } L q ˙ . (3.33)Second, using ψ A,B “ χ A ϕ B in (3.5), and Remark 2.0.1, one can see that } ψ A,B v } L À B } χ A v } L À B } v } L p| x |ă A q À B } ζ A v } L . We conclude | J | À γ { B } ζ A v } L ˆ }B x z } L ` } z } L ` e ´p p ´ q A p} ζ B v } L ` } ζ B B x v } L q ˙ . (3.34)3.5.3. Control of J . Recalling that ψ A,B “ χ A ϕ B , using the Hölder inequality and Remark 2.0.1,we get | J | “ ˇˇˇˇż ψ A,B H p x q v ˇˇˇˇ À } χ A ϕ B v } L } χ A p ´ γ B x q ´ N K } L À } χ A ϕ B v } L } ζ A p ´ γ B x q ´ N K } L . By the definition of N K (see (2.7)), it follows that } ζ A p ´ γ B x q ´ p N K q} L ď } ζ A p ´ γ B x q ´ N } L ` | N |} ζ A p ´ γ B x q ´ B ´ x φ } L À } ζ A p ´ γ B x q ´ N } L ` | N | , since B ´ x φ P L and ď ζ A À .Furthermore, by definition of N in (2.7), and using Corollary 3.5, (2.27) and Lemma 3.3, we have } ζ A p ´ γ B x q ´ N } L ď γ ´ { } ζ A r f p Q q ` f p Q qp a φ ` u q ´ f p Q ` a φ ` u qs} L ď γ ´ { } ζ A r a f p Q q φ ` f p Q q u ` | a | p φ p ` | u | p s} L ď γ ´ { ` a } f p Q q ζ A φ } L ` } f p Q q ζ A w } L ` | a | p } ζ A φ p } L ` } ζ A | u | p ´ w } L ˘ À γ ´ { ´ a ` } u } L } f p Q q w } L ` | a | p ` } u } p ´ L } w } L ¯ À γ ´ { ` a ` } u } L } w } L ˘ . (3.35)Note that we have used that p ě . Since | χ A ϕ B | À B , we have } χ A ϕ B v } L À B } χ A v } L À B } v } L p| x |ă A q À B } ζ A v } L . (3.36)Finally, by (2.29), (3.35) and (3.36) , we conclude | J | À Bγ ´ { } ζ A v } L ` a ` } u } L } w } L ˘ . (3.37)3.6. End of proof of Proposition 3.1.
From (3.32), (3.34), (3.37), and choosing ă γ “ B ´ , (3.38)it follows | ˜ J ` J ` J | À A ´ B ` } ζ A v } L ` } ζ A v } L ` } ζ A B x v } L ˘ ` γ { B } ζ A v } L ˆ }B x z } L ` } z } L ` e ´p p ´ q A p} ζ B v } L ` } ζ B B x v } L q ˙ ` Bγ ´ { } ζ A v } L ˆ a ` } u } L } w } L ˙ . (3.39)Applying Lemma 3.8-(3.24) and (2.13), we obtain | ˜ J ` J ` J | À B ´ ˆ } w } L ` } w } L ` } z } L ` }B x z } L ˙ ` B ˆ a ` } u } L } w } L ˙ . (3.40) Choosing B ď δ ´ { , (3.41)(to be fixed later) and using (2.5), we arrive to B p a ` } u } L } w } L q À δ ´ p a ` } u } L } w } L q À | a | ` δ } w } L . Then, using the above estimates, we obtain that the error term and the associated to the nonlinearpart are bounded as follows: | ˜ J ` J ` J | À B ´ ` } w } L ` } w } L ` } z } L ` }B x z } L ˘ ` | a | . Finally, the virial estimate is concluded as follows: for some C ą independent of B large, ddt J p t q ď ´ ż “ z ` p V p x q ´ CB ´ q z ` pB x z q ‰ ` CB ´ ˆ } w } L ` } w } L ` } z } L ` }B x z } L ˙ ` C | a | . ď ´ ż “ z ` p V p x q ´ C B ´ q z ` pB x z q ‰ ` C B ´ ˆ } w } L ` } w } L ˙ ` C | a | . (3.42)This ends the proof of Proposition 3.1.4. Gain of derivatives via transfer estimates
We must note that in (2.16) the last term is a localized one, which in the language of estimate(3.42) will correspond to a term of type B x z , not appearing in this last estimate. However, thisnew term will be well-defined by the regularity of the original variables p u , u q . We think thatthis problem appears as a product of the lack of balance in the regularity of p v , v q (see Subsection3.1). Therefore, we need new estimates to control B x z .To solve this new problem, we will focus on a new virial obtained for a new system of equationsinvolving the variables ˜ v i “ B x v i , for i “ , . Formally taking derivatives in (3.2), we have ˜ v “ L pB x ˜ v q ´ B x p f p Q qq v ` ˜ G p x q , ˜ G p x q “ B x G p x q , ˜ v “ B x ˜ v ` ˜ H p x q , ˜ H p x q “ B x H p x q , (4.1)where G and H are given in (3.3).For this new system, we consider the virial M p t q “ ż φ A,B ˜ v ˜ v “ ż φ A,B B x v B x v . (4.2)Later we will choose φ A,B “ ψ A,B “ χ A ϕ B (see (3.5)).4.1. A virial estimate related to M p t q .Lemma 4.1. Let p v , v q P H p R q ˆ H p R q a solution of (3.2) . Consider φ A,B an odd smoothbounded function to be a choose later. Then ddt M p t q “ ´ ż φ A,B ` pB x v q ` pB x v q ` pB x v q ˘ ` ż φ A,B pB x v q ´ ż φ A,B f p Q qB x ppB x v q q ` ż φ A,B ˜ G p x qB x v ` ż φ A,B ˜ H p x qB x v . (4.3)The identity (4.3) is interesting because it has exactly the same structure that ddt J p t q in (3.8).This holds despite the new derivative terms appearing in (4.1). To obtain this we will benefit froma cancellation given by the parity of the data. SYMPTOTICS FOR GOOD BOUSSINESQ SOLITONS 21
Proof of Lemma 4.1.
From (3.2), (4.1) and (3.8), we have ddt M p t q “ ´ ż φ A,B ` ˜ v ` ˜ v ` pB x ˜ v q ˘ ` ż φ A,B ˜ v ´ ż φ A,B f p Q qB x p ˜ v q ` ż φ A,B “ ´ B x p f p Q qq v ` ˜ G p x q ‰ ˜ v ` ż φ A,B ˜ H p x q ˜ v . (4.4)Rewriting the above identity in term of the variables p v , v q , we have ddt M p t q “ ´ ż φ A,B ` pB x v q ` pB x v q ` pB x v q ˘ ` ż φ A,B pB x v q ´ ż φ A,B f p Q qB x ppB x v q q ´ ż φ A,B B x p f p Q qq v B x v ` ż φ A,B ˜ G p x qB x v ` ż φ A,B ˜ H p x qB x v . (4.5)Noticing that v B x v , B x p f p Q qq and φ A,B are odd functions (see (3.1) and (2.8)), we have ż φ A,B B x p f p Q qq v B x v “ . This ends the proof of Lemma 4.1. (cid:3)
The following proposition connects two virial identities in the variable p z , z q . Recall that from(3.38) and (3.41), γ “ B ´ , B ď δ ´ { . Proposition 4.2.
There exist C ą and δ ą such that for any ă δ ď δ , the followingholds. Fix B “ δ ´ { ď δ ´ { . Assume that for all t ě , (2.5) holds. Then for all t ě , ddt M p t q ď ´ ż ` pB x z q ` ` V p x q ´ C B ´ ˘ pB x z q ` pB x z q ˘ ` C } z } L ` C B ´ } z } L ` C B ´ ` }B x w } L ` } w } L ` } w } L ˘ ` C | a | . (4.6)The proof of the above result requires some technical estimates. We first state them, to thenprove Proposition 4.2 (Subsection 4.3).4.2. Second set of technical estimates.
Now, we recall the following technical estimates onthe variables ζ B and other related error terms. These estimates are similar to the ones obtainedin (2.21), therefore we only prove the new ones. Lemma 4.3.
Let ζ B and χ be defined by (2.11) and (2.9) , respectively. Then ζ B ζ B “ ´ B r´ χ p x q| x | ` p ´ χ p x qq sgn p x qs , ζ B ζ B “ ˆ ζ B ζ B ˙ ` B r χ p x q| x | ` χ p x q sgn p x qs , (4.7) and ζ B ζ B “ ζ B ζ B ζ B ζ B ´ ˆ ζ B ζ B ˙ ` B ´ “ χ p x q| x | ` χ p x q sgn p x q ‰ ,ζ p q B ζ B “ ζ B ζ B ζ B ζ B ` ˆ ζ B ζ B ˙ ´ ζ B ζ B ˆ ζ B ζ B ˙ ` ˆ ζ B ζ B ˙ ` B ” χ p q p x q| x | ` χ p x q sgn p x q ı . (4.8) Proof.
Direct. (cid:3)
Remark 4.3.1.
From the previous lemma we observe that ˇˇˇˇ ζ B ζ B ˇˇˇˇ À B ´ t| x |ą u p x q , ˇˇˇˇ ζ B ζ B ˇˇˇˇ À B ´ t| x |ą u p x q ` B ´ t ă| x |ă u p x q À B ´ ` B ´ sech p x q , ˇˇˇˇ ζ B ζ B ˇˇˇˇ À B ´ ` B ´ sech p x q , ˇˇˇˇˇ ζ p q B ζ B ˇˇˇˇˇ À B ´ ` B ´ sech p x q . In particular, for A large enough, the following estimates hold: ˇˇˇˇ ζ B ζ B ˇˇˇˇ À B ´ , ˇˇˇˇ t A ă| x |ă A u ζ B ζ B ˇˇˇˇ À B ´ , ˇˇˇˇ t A ă| x |ă A u ζ B ζ B ˇˇˇˇ À B ´ , ˇˇˇˇˇ t A ă| x |ă A u ζ p q B ζ B ˇˇˇˇˇ À B ´ . (4.9) Finally, ˇˇˇˇ ζ B ζ B ˇˇˇˇ ` ˇˇˇˇ ζ B ζ B ˇˇˇˇ ` ˇˇˇˇˇ ζ p q B ζ B ˇˇˇˇˇ À B ´ . (4.10)These estimates will be useful in Claim 4.5. Now we prove a formula for changing variables. Claim 4.4.
Let P P W , p R q , v i be as in (3.1) , and z i be as in (3.5) . Then ż P p x q χ A ζ B pB x v i q “ ż P p x qpB x z i q ` ż „ P p x q ζ B ζ B ` P p x q ζ B ζ B z i ` ż E p P p x q , x q ζ B v i , (4.11) where E p P p x q , x q “ P p x q „ χ A χ A ` p χ A q ζ B ζ B ` P p x qp χ A q , (4.12) and | E p P p x q , x q| À A ´ } P } L ` p AB q ´ } P } L . (4.13)For the proof of these results, see Appendix B.1. Remark 4.4.1.
For P “ , we get ż χ A ζ B pB x v i q “ ż pB x z i q ` ż ζ B ζ B z i ` ż E p , x q ζ B v i , (4.14) where E p , x q “ χ A χ A ` p χ A q ζ B ζ B . (4.15) Finally, one has the following estimate: } χ A ζ B B x v i } À }B x z i } L ` B ´ } z i } L ` p AB q ´ } w i } L . We need a second claim on the second derivative of v i . Claim 4.5.
Let R be a W , p R q function, v i be as in (3.1) , and z i be as in (3.5) . Then ż R p x q χ A ζ B pB x v i q “ ż R p x qpB x z i q ` ż ˜ R p x q z i ` ż P R p x qpB x z i q ` ż „ P R p x q ζ B ζ B ` P R p x q ζ B ζ B z i ` ż E p R p x q , x q ζ B v i ` ż E p P R p x q , x q ζ B v i ` ż E p R p x q , x q ζ B pB x v i q , where ˜ R p x q “ ´ R p x q « ζ p q B ζ B ` ζ B ζ B ζ B ζ B ff ´ R p x q ζ B ζ B ´ R p x q ζ B ζ B , (4.16) P R p x q “ R p x q „ ζ B ζ B ´ ˆ ζ B ζ B ˙ ` R p x q ζ B ζ B , (4.17) SYMPTOTICS FOR GOOD BOUSSINESQ SOLITONS 23 E is defined in (4.12) , E p R p x q , x q “ ´ R p x q ˆ χ p q A χ A ` χ A χ A ζ B ζ B ` χ A χ A ζ B ζ B ` p χ A q ζ B ζ B ˙ ´ R p x q ˆ χ A χ A ` χ A χ A ζ B ζ B ` χ A χ A ζ B ζ B ˙ ´ R p x q ˆ χ A χ A ` p χ A q ζ B ζ B ˙ , (4.18) and E p R p x q , x q “ R p x q „ χ A χ A ´ p χ A q ` ζ B ζ B p χ A q ` R p x qp χ A q . (4.19) Finally, P R , E and E satisfy the following inequalities | P R | À B ´ } R } L ` B ´ } R } L , | P R | À B ´ } R } L ` B ´ } R } L ` B ´ } R } L , | E | À p AB q ´ } R } L ` p AB q ´ } R } L ` p AB q ´ } R } L , | E | À A ´ } R } L ` p AB q ´ } R } L . (4.20)For the proof of these results, see Appendix B.2. Remark 4.5.1.
For R “ , we obtain ż χ A ζ B pB x v i q “ ż pB x z i q ` ż ˜ R p x q z i ` ż P p x qpB x z i q ` ż „ P p x q ζ B ζ B ` P p x q ζ B ζ B z i ` ż E p , x q ζ B v i ` ż E p P p x q , x q ζ B v i ` ż E p , x q ζ B pB x v i q , (4.21) where, ˜ R p x q “ ´ « ζ p q B ζ B ` ζ B ζ B ζ B ζ B ff , P p x q “ ζ B ζ B ´ ˆ ζ B ζ B ˙ , (4.22) E is defined in (4.12) , E p , x q “ ´ ˆ χ p q A χ A ` χ A χ A ζ B ζ B ` χ A χ A ζ B ζ B ` p χ A q ζ B ζ B ˙ , (4.23) and E p , x q “ χ A χ A ´ p χ A q ` ζ B ζ B p χ A q . (4.24) By Lemma 3.8, we obtain the estimate: } χ A ζ B B x v i } À }B x z i } L ` B ´ }B x z i } L ` B ´ } z i } L ` A ´ B } w i } L . Start of proof of Proposition 4.2.
The proof of this result is based in the followingcomputation:
Lemma 4.6.
Let p v , v q P H p R q ˆ H p R q a solution of (3.2) . Consider φ A,B “ ψ A,B “ χ A ϕ B .Then ddt M “ ´ ż ˆ pB x z q ` ˆ V p x q ´ ϕ B ζ B B x p f p Q qq ˙ pB x z q ` pB x z q ˙ ` ż ϕ B ζ B ζ B ζ B B x p f p Q qq z ` R z p t q ` R v p t q ` DR v p t q` ż φ A,B ˜ G p x qB x v ` ż φ A,B ˜ H p x qB x v , (4.25) where R z p t q , R v p t q and DR v p t q are error terms that satisfy the following bounds | R z p t q| ` | R v p t q| ` | DR v p t q| À B ´ ` } w } L ` }B x w } L ` } w } L ˘ ` B ´ ` } z } L ` } z } L ` }B x z } L ˘ , (4.26) valid for B sufficiently large.Proof. First, we recall that z i “ χ A ζ B v i , and by (3.9) and Claim 4.4 ż φ A,B “ pB x v q ` pB x v q ‰ “ ż pB x z q ` pB x z q ` ż ζ B ζ B ` z ` z ˘ ` ż E p , x q ζ B ` v ` v ˘ ` ż p χ A q ϕ B “ pB x v q ` pB x v q ‰ , (4.27)where E p , x q is given by (4.15). Now, using Remark 4.5.1 (4.21), we get ż φ A,B pB x v q “ ż pB x z q ` ż P p x qpB x z q ` ż „ ˜ R p x q ` P p x q ζ B ζ B ` P p x q ζ B ζ B z ` ż E p , x q ζ B v ` ż E p P p x q , x q ζ B v ` ż E p , x q ζ B pB x v q ` ż p χ A q ϕ B pB x v q , (4.28)where ˜ R p x q , P p x q , E p , x q , E p , x q are given by (4.22), (4.23), (4.24), and E is gyven by (4.12).Now, continuing with the second integral in the RHS of (4.3), we have ż φ A,B pB x v q “ ż p ζ B q ζ B χ A ζ B pB x v q ` ż „ p χ A q ζ B ζ B ` p χ A q ` p χ A q ϕ B ζ B ζ B pB x v q , and using Claim 4.4, ż φ A,B pB x v q “ ż p ζ B q ζ B pB x z q ` ż p ζ B q ζ B ζ B ζ B z ` ż ˆ p ζ B q ζ B ˙ ζ B ζ B z ` ż p ζ B q ζ B „ χ A χ A ` p χ A q ζ B ζ B ζ B v ` ż ˆ p ζ B q ζ B ˙ p χ A q ζ B v ` ż „ p χ A q ζ B ζ B ` p χ A q ` p χ A q ϕ B ζ B ζ B pB x v q . (4.29)For the third integral in the RHS of (4.3), integrating by parts ż φ A,B f p Q qB x ppB x v q q “ ´ ż ˆ f p Q q ` ϕ B ζ B B x p f p Q qq ˙ χ A ζ B pB x v q ´ ż p χ A q ϕ B f p Q qpB x v q . By the extended version of Claim 4.4 and expanding the derivates in terms of z , we have ż φ A,B f p Q qB x ppB x v q q“ ´ ż „ ˆ f p Q q ` ϕ B ζ B B x p f p Q qq ˙ ζ B ζ B ´ ˆ B x p f p Q qq ´ ϕ B p ζ B q ζ B B x p f p Q qq ` ϕ B ζ B B x p f p Q qq ˙ ζ B ζ B z ´ ż ˆ f p Q q ` ϕ B ζ B B x p f p Q qq ˙ pB x z q ´ ż ˆ f p Q q ` ϕ B ζ B B x p f p Q qq ˙ „ χ A χ A ` p χ A q ζ B ζ B ζ B v ´ ż B x ˆ f p Q q ` ϕ B ζ B B x p f p Q qq ˙ p χ A q ζ B v ´ ż p χ A q ϕ B f p Q qpB x v q . (4.30)Collecting (4.27), (4.28),(4.29) and (4.30), we obtain ddt M “ ´ ż pB x z q ` ˆ V p x q ´ ϕ B ζ B B x p f p Q qq ˙ pB x z q ` pB x z q ` ż „ ϕ B ζ B ζ B ζ B B x p f p Q qq ` ϕ B ζ B ζ B ζ B B x p f p Q qq z ` R z p t q ` R v p t q ` DR v p t q` ż φ A,B ˜ G p x qB x v ` ż φ A,B ˜ H p x qB x v , SYMPTOTICS FOR GOOD BOUSSINESQ SOLITONS 25 where the error terms are the following: associated to p z , z q is R z p t q “ ´ ż ζ B ζ B ` z ` z ˘ ´ ż „ ˜ R p x q ` P p x q ζ B ζ B ` P p x q ζ B ζ B z ` ż « p ζ B q ζ B ζ B ζ B ` ˆ p ζ B q ζ B ˙ ζ B ζ B ff z ` ż „ f p Q q ζ B ζ B ` ˆ B x p f p Q qq ´ ϕ B ζ B ζ B ζ B B x p f p Q qq ˙ ζ B ζ B z ` ż ˆ p ζ B q ζ B ´ P p x q ˙ pB x z q , (4.31)associated to p v , v q is R v p t q “ ´ ż E p , x q ζ B ` v ` v ˘ ´ ż „ E p , x q ` E p P p x q , x q ζ B v ` ż p ζ B q ζ B „ χ A χ A ` p χ A q ζ B ζ B ζ B v ` ż ˆ p ζ B q ζ B ˙ p χ A q ζ B v ` ż ˆ f p Q q ` ϕ B ζ B B x p f p Q qq ˙ „ χ A χ A ` p χ A q ζ B ζ B ζ B v ` ż B x ˆ f p Q q ` ϕ B ζ B B x p f p Q qq ˙ p χ A q ζ B v , (4.32)and associated to pB x v , B x v q is DR v p t q “ ´ ż p χ A q ϕ B “ pB x v q ` pB x v q ` pB x v q ‰ ´ ż E p , x q ζ B pB x v q ` ż “ p χ A q p ζ B q ` p χ A q ζ B ` p χ A q ϕ B ` p χ A q ϕ B f p Q q ‰ pB x v q . (4.33)We have obtained the identity (4.25). To conclude the proof of Lemma 4.6, we must estimate theerror terms.4.4. Controlling error terms.
We consider the following decomposition for R z p t q from (4.31), R z p t q “ R z p t q ` R z p t q ` R z p t q , where R z p t q “ ´ ż ζ B ζ B ` z ` z ˘ ` ż „ p ζ B q ζ B ζ B ζ B ` ˆ p ζ B q ζ B ˙ ζ B ζ B ´ R p x q z , R z p t q “ ´ ż „ P p x q ζ B ζ B ` P p x q ζ B ζ B z ` ż ˆ p ζ B q ζ B ´ P p x q ˙ pB x z q , R z p t q “ ż „ f p Q q ζ B ζ B ` ˆ B x p f p Q qq ´ ϕ B ζ B ζ B ζ B B x p f p Q qq ˙ ζ B ζ B z . For R z p t q , recalling estimate (4.10) and ˜ R (see (4.22)), and we obtain | R z p t q| ď B ´ } z } L ` B ´ } z } L . (4.34)For R z p t q , we recall the form of P (see (4.22)) and by (4.10), we conclude | R z p t q| À B ´ } z } L ` B ´ }B x z } L . (4.35)For R z p t q , first we note ˇˇˇˇ ϕ B ζ B B x p f p Q qq ˇˇˇˇ À B, and by (4.9), we obtain | R z p t q| À B ´ } z } L . (4.36) Collecting (4.34),(4.35) and (4.36), we have | R z p t q| ď B ´ ˆ } z } L ` } z } L ` }B x z } L ˙ . (4.37)For R v p t q , given by (4.32), we consider the following decomposition R v p t q “ ´ ż E p , x q ζ B ` v ` v ˘ ´ ż „ E p , x q ` E p P p x q , x q ζ B v ` ż p ζ B q ζ B „ χ A χ A ` p χ A q ζ B ζ B ζ B v ` ż ˆ p ζ B q ζ B ˙ p χ A q ζ B v , R v p t q “ ż B x ˆ f p Q q ` ϕ B ζ B B x p f p Q qq ˙ p χ A q ζ B v ` ż ˆ f p Q q ` ϕ B ζ B B x p f p Q qq ˙ „ χ A χ A ` p χ A q ζ B ζ B ζ B v . We note that the terms E p P p x q , x q and E p , x q (see (4.12), (4.22) and (4.23)), by (4.9), arebounded and satisfy the following estimates: | E p , x q| À p AB q ´ , and | E p P p x q , x q| À p AB q ´ , and for E p , x q in (4.15), | E p , x q| À p AB q ´ . Then, we have | R v p t q| À p AB q ´ ` } ζ B v } L ` } ζ B v } L ˘ . For R v p t q , expanding the derivative and using (3.31), we obtain ˇˇ R v p t q ˇˇ À A ´ ż ˇˇˇˇ ϕ B ζ B B x p f p Q qq ` ˆ ´ ζ B ζ B ϕ B ζ B ˙ B x p f p Q qq ˇˇˇˇ ζ B v ` A ´ ż ˇˇˇˇ f p Q q ` ϕ B ζ B B x p f p Q qq ˇˇˇˇ ζ B v À A ´ B } ζ B v } L . Then, | R v p t q| À A ´ B ` } ζ B v } L ` } ζ B v } L ˘ . (4.38)For DR v , given by (4.33), computing directly and using Remark 2.0.1, we have | DR v p t q| À BA ´ ` } ζ A B x v } L ` } ζ A B x v } L ` } ζ A B x v } L ˘ . (4.39)And, by (4.37), (4.38) and (4.39), we obtain | R z p t q| ` | R v p t q| ` | DR v p t q|À BA ´ ` } ζ A B x v } L ` } ζ A B x v } L ` } ζ A B x v } L ` } ζ B v } L ` } ζ B v } L ˘ ` B ´ ` } z } L ` } z } L ` }B x z } L ˘ . Using Lemma 3.8, we conclude | R z p t q| ` | R v p t q| ` | DR v p t q| À A ´ B ` } w } L ` }B x w } L ` } w } L ˘ ` B ´ ` } z } L ` } z } L ` }B x z } L ˘ . (4.40)This ends the proof of Lemma 4.6. (cid:3) Controlling nonlinear terms.
Recall (4.25). We set M “ ż φ A,B ˜ G p x qB x v , M “ ż φ A,B ˜ H p x qB x v . These are the two remaining terms in (4.25) to be controlled.
SYMPTOTICS FOR GOOD BOUSSINESQ SOLITONS 27
Control of M . Recalling that ˜ G p x q “ B x G p x q and G is given by (3.3), we have M “ ´ γ ż pp χ A q ζ B ` χ A p ζ B q qB x v p ´ γ B x q ´ “ B x p f p Q qqB x v ` B x p f p Q qqB x v ‰ ´ γ ż χ A ϕ B B x v p ´ γ B x q ´ “ B x p f p Q qqB x v ` B x p f p Q qqB x v ‰ “ : M ` M . First, we focus on M . Using Remark 4.4.1 and Lemma 3.8, we have }pp χ A q ζ B ` χ A p ζ B q qB x v } L À A ´ } ζ B B x v } L ` B ´ } χ A ζ B B x v } L À A ´ B } w } L ` B ´ ” }B x z } L ` B ´ } z } L ` p AB q ´ { } w } L ı À B ´ “ }B x z } L ` } z } L ‰ ` B ´ } w } L , and by (3.33), we conclude | M | À B ´ „ }B x z } L ` } z } L ` } w } L . (4.41)Secondly, for M . Set ρ p x q “ sech p x { q , making the following separation | M | À γ } χ A ϕ B ρ K p x qB x v } L ›››› p ρ K p x qq ´ p ´ γ B x q ´ „ B x p f p Q qqB x v ` B x p f p Q qqB x v ›››› L À γB } χ A ζ B B x v } L ›››› p ρ K p x qq ´ p ´ γ B x q ´ „ B x p f p Q qqB x v ` B x p f p Q qqB x v ›››› L looooooooooooooooooooooooooooooooooooooooomooooooooooooooooooooooooooooooooooooooooon M . Using Lemma 3.4 in M , we obtain M “ ›››› p ρ K p x qq ´ p ´ γ B x q ´ „ ρ K p x q B x p f p Q qqp ρ K p x qq ´ B x v ` B x p f p Q qqp ρ K p x qq ´ B x v ( ›››› L ď ›››› p ´ γ B x q ´ „ pB x p f p Q qqp ρ K p x qq ´ B x v ` B x p f p Q qqp ρ K p x qq ´ B x v q ›››› L À}pB x p f p Q qqp ρ K p x qq ´ B x v } L ` }B x p f p Q qqp ρ K p x qq ´ B x v q} L . (4.42)Since B x p f p Q qq ρ ´ K „ e ´ | x |{ and making the following decomposition, we have e ´ | x |{ pB x v q “ e ´ | x |{ ζ ´ B p χ A ζ B B x v q ` e ´ | x |{ ζ ´ A p ´ χ A qp ζ A B x v q . Since e ´ | x |{ ζ ´ B ď and e ´ A { ζ ´ A p A q „ e ´ A { , using Remark 4.4.1 and Lemma 3.8, weconclude } e ´ | x |{ B x v } L À “ }B x z } L ` B ´ } z } L ` A ´ } ζ B v } L ‰ ` e ´ A { } ζ A B x v } L À }B x z } L ` B ´ } z } L ` A ´ } w } L . And, for the second term in the RHS of (4.42). We note B x p f p Q qq ρ ´ „ e ´ | x |{ and repeatingthe decomposition, we have e ´ | x |{ pB x v q “ e ´ | x |{ ζ ´ B χ A ζ B pB x v q ` e ´ | x |{ ζ ´ A p ´ χ A q ζ A pB x v q . By a similar argument as before, e ´ | x |{ ζ ´ B ď and e ´ A { ζ ´ A p A q „ e ´ A { , applying Remark4.5.1 and Lemma 3.8, we have } e ´ | x |{ B x v } L À } χ A ζ B B x v } L ` e ´ A { } ζ A B x v } L À }B x z } L ` B ´ }B x z } L ` B ´ } z } L ` p A ´ B q { } w } L . Finally, for M we have | M | À B ´ ` }B x z } L ` }B x z } L ` B ´ } z } L ` A ´ B } w } L ˘ . (4.43) Collecting (4.41) and (4.43), we conclude M À B ´ „ }B x z } L ` }B x z } L ` } z } L ` } w } L . (4.44)4.5.2. Control of M . Recalling that ˜ H “ B x H , H is given by (3.3) and using Lemma 3.3, weobtain | M | À } χ A ϕ B B x v } L } χ A p ´ B x q ´ B x N K } L À } χ A ϕ B B x v } L } ζ A p ´ B x q ´ B x N K } L . Now, by a similar computation that (3.35), we have } ζ A p ´ γ B x q ´ B x p N K q} L ď γ ´ p a ` } u } L } w } L q . (4.45)Then, by (3.36), Lemma 3.8, (2.29) and the above estimates, we conclude | M | À γ ´ B } w } L p a ` } u } L } w } L q À B ´ } w } L ` γ ´ B p a ` } u } L } w } L q . (4.46)4.6. End of proof Proposition 4.2.
Using a similar computation that Lemma 3.2, we are ableto estimates ddt M . Set B “ δ ´ { , and considering (4.40), (4.46), and (4.44), we obtain ddt M ď ´ ż ˆ pB x z q ` ˆ V p x q ´ ϕ B ζ B B x p f p Q qq ˙ pB x z q ` pB x z q ˙ ` ż ˆ ϕ B ζ B ζ B ζ B B x p f p Q qq ` ϕ B ζ B ζ B ζ B B x p f p Q qq ˙ z ` C max t B A ´ , B ´ , δ u ` }B x w } L ` } w } L ` } w } L ˘ ` CB ´ „ }B x z } L ` } z } L ` } z } L ` C | a | . Since ϕ B ζ B B x p f p Q qq ă , and ˇˇˇˇ ϕ B ζ B ζ B ζ B B x p f p Q qq ` ϕ B ζ B ζ B ζ B B x p f p Q qq ˇˇˇˇ À , we conclude ddt M ď ´ ż pB x z q ` ` V p x q ´ CB ´ ˘ pB x z q ` pB x z q ` C } z } L ` CB ´ } z } L ` C max t B A ´ , B ´ , δ u ` }B x w } L ` } w } L ` } w } L ˘ ` C | a | . Calling C “ C , and using (2.13), the proof is completed.5. A second transfer estimate
The variation of the virial M p t q involve the terms B x z and B x z , these terms do not appear inthe variation of the virial related to the dual problem. Hence, we need to find a way to transferinformation between the terms B x z to B x z . The virial N , defined as N “ ż ρ A,B ˜ v v “ ż ρ A,B B x v v , (5.1)where ρ A,B is a well-chosen localized weight depending on A and B , its variation will give us thatrelation. A similar quantity was considered in [19]. Note that the virial N considers the dynamicsin (3.2) and (4.1). SYMPTOTICS FOR GOOD BOUSSINESQ SOLITONS 29
A virial identity for N p t q .Lemma 5.1. Let p v , v q P H p R q ˆ H p R q a solution of (3.2) . Consider ρ A,B an even smoothbounded function to be a choose later. Then ddt N “ ż ρ A,B pB x v q ´ ż ρ A,B “ pB x v q ` V p x qpB x v q ‰ ` ż ρ A,B pB x v q ` ż B x r ρ A,B V p x qs v ´ ż ρ p q A,B v ` ż ρ A,B v ˜ G p x q ` ż ρ A,B B x v H p x q . (5.2) Proof.
Computing the variation of N , using (3.2) and (4.1), we obtain ddt N “ ż ρ A,B v L B x ˜ v ` ż ρ A,B ˜ v ` ż ρ A,B v p ˜ G p x q ´ B x p f p Q qq v q ` ż ρ A,B ˜ v H p x q . Integrating by parts the first integral of the RHS N , we have ż ρ A,B v L B x ˜ v “ ż ρ A,B v p´B x ˜ v ` V p x qB x ˜ v q“ ´ ż B x p ρ A,B v qB x ˜ v ´ ż ` B x r ρ A,B V p x qs v ˜ v ` ρ A,B V p x q ˜ v ˘ “ ż ρ A,B v ˜ v ` ż ρ A,B ˜ v ´ ż ρ A,B ` pB x ˜ v q ` V p x q ˜ v ˘ ´ ż B x r ρ A,B V p x qs v ˜ v . Then, we get ddt N “ ´ ż ρ A,B ` pB x ˜ v q ` V p x q ˜ v ˘ ` ż ρ A,B ˜ v ` ż ρ A,B ˜ v ´ ż B x r ρ A,B V p x qs v ˜ v ` ż ρ A,B v ˜ v ` ż ρ A,B v p ˜ G p x q ´ B x p f p Q qq v q ` ż ρ A,B ˜ v H p x q . Rewriting the last expression in terms of p v , v q , we obtain ddt N “ ´ ż ρ A,B ` pB x v q ` V p x qpB x v q ˘ ` ż ρ A,B pB x v q ` ż ρ A,B pB x v q ´ ż ρ p q A,B v ` ż B x r ρ A,B V p x qs v ` ż ρ A,B v p ˜ G p x q ´ B x p f p Q qq v q ` ż ρ A,B B x v H p x q . The proof concludes from the fact that ρ A,B v is even and B x p f p Q qq is an odd function. (cid:3) Now we choose the weight function ρ A,B . As in [19], let ρ A,B p x q “ χ A ζ B , (5.3)with χ A and ζ B introduced in (2.12) and (2.11).We will make the connection between (5.2) and the variables p z , z q , through the followingresult. Recall that from (3.38) and Proposition 4.2, γ “ B ´ , B “ δ ´ { . Proposition 5.2.
Under (5.3) , the following holds. There exist C and δ ą such that for γ “ B ´ and for any ă δ ď δ , the following holds. Fix B “ δ ´ { holds. Assume that for all t ě , (2.5) holds. Then, for all t ě , ddt N p t q ě ż pB x z q ´ C ż “ pB x z q ` pB x z q ` z ` z ‰ ´ C δ { ˆ } w } L ` } w } L ` | a | ˙ . (5.4) Start of proof of Proposition 5.2.
The proof of this result is based in the following result,which relates Lemma 5.1 and the variables z i . Lemma 5.3.
Let p v , v q P H p R q ˆ H p R q a solution of (3.2) . Consider ρ A,B “ χ A ζ B , then ddt N “ ´ ż „ pB x z q ` ` ´ f p Q q ˘ pB x z q ` B x p f p Q qq z ` ż pB x z q ` RZ p t q ` RV p t q ` RDV p t q ` ż ρ A,B v ˜ G p x q ` ż ρ A,B B x v H p x q , (5.5) where RZ p t q , RV p t q and RDV p t q are error term that satisfy the following estimates | RZ p t q| À B ´ } z } L ` B ´ } z } L ` B ´ }B x z } L , | RV p t q| À p AB q ´ γ ´ } w } L ` A ´ } w } L , | RDV p t q| À p AB q ´ γ ´ } w } L . Proof.
First, we consider the following decomposition from (5.2): ddt N “ ż ρ A,B pB x v q ´ ż ρ A,B pB x v q ´ ż ρ A,B V p x qpB x v q ` ż ρ A,B pB x v q ` ż B x r ρ A,B V p x qs v ´ ż ρ p q A,B v ` ż ρ A,B B x v H p x q ` ż ρ A,B v ˜ G p x q“ : p N ` N ` N ` N q ` p N ` N q ` p N ` N q . (5.6)Secondly, from the definition of ρ A,B (5.3) ρ A,B “p χ A q ζ B ` χ A p ζ B q ,ρ A,B “ p χ A q ζ B ` p χ A q p ζ B q ` χ A p ζ B q ,ρ A,B “ p χ A q ζ B ` p χ A q p ζ B q ` p χ A q p ζ B q ` χ A p ζ B q ,ρ p q A,B “ p χ A q p q ζ B ` p χ A q p ζ B q ` p χ A q p ζ B q ` p χ A q p ζ B q ` χ A p ζ B q p q . (5.7)For N , applying Claim 4.4 with i “ and P p x q “ V p x q , we have ´ N “ ż V p x qpB x z q ` ż „ V p x q ζ B ζ B ` B x p V p x qq ζ B ζ B z ` ż E p V p x q , x q ζ B v , (5.8)where E is given by (4.12) For N , by (5.7), we have N “ ż rp χ A q ζ B ` p χ A q p ζ B q spB x v q ` ż p ζ B q ζ B χ A ζ B pB x v q , (5.9)and using Claim 4.4, with i “ and P p x q “ p ζ B q { ζ B , we get N “ ż p ζ B q ζ B pB x z q ` ż «ˆ p ζ B q ζ B ˙ ζ B ζ B ` p ζ B q ζ B ζ B ζ B ff z ` ż E ˆ p ζ B q ζ B , x ˙ ζ B v ` ż rp χ A q ` p χ A q ζ B ζ B s ζ B pB x v q . (5.10)Now, for N , expanding the derivative, replacing (5.7) and using definition of z , we have N “ ´ ż „ ´B x p V p x qq ´ p ζ B q ζ B V p x q ´ p ζ B q ζ B B x p V p x qq z ` ż ˆ „ p χ A q ζ B ζ B ` p χ A q V p x q ` p χ A q B x p V p x qq ˙ ζ B v . (5.11)Finally, for N , reeplacing (5.7), we have N “ ż p ζ B q p q ζ B z ` ż „ p χ A q p ζ B q ζ B ` p χ A q p ζ B q ζ B ` p χ A q ζ B ζ B ` p χ A q p q ζ B v . (5.12) SYMPTOTICS FOR GOOD BOUSSINESQ SOLITONS 31
Therefore, collecting (4.21), (4.14), (5.8), (5.10), (5.11) and (5.12) (and also for N and N we usethe relations in Remarks 4.4.1 and 4.5.1), we conclude ddt N “ ´ ż „ pB x z q ` V p x qpB x z q ` B x p f p Q qq z ` ż pB x z q ` RZ p t q ` RV p t q ` RDV p t q ` ż ρ A,B v ˜ G p x q ` ż ρ A,B B x v H p x q , where the error term related to z “ p z , z q is RZ p t q “ ż ζ B ζ B z ´ ż „ V p x q ζ B ζ B ` B x p V p x qq ζ B ζ B z ´ ż „ ´ p ζ B q ζ B V p x q ´ p ζ B q ζ B B x p V p x qq ` p ζ B q p q ζ B z ´ ż „ ˜ R p x q ` P p x q ζ B ζ B ` P p x q ζ B ζ B z ` ż «ˆ p ζ B q ζ B ˙ ζ B ζ B ` p ζ B q ζ B ζ B ζ B ff z ` ż „ p ζ B q ζ B ´ P p x q pB x z q , the related to p v , v q is RV p t q “ ż E p , x q ζ B v ´ ż „ E p V p x q , x q ` E p , x q ` E p P p x q , x q ´ E ˆ p ζ B q ζ B , x ˙ ζ B v ` ż „ ˆ p χ A q ζ B ζ B ` p χ A q ˙ V p x q ` p χ A q B x p V p x qq ζ B v ´ ż „ p χ A q p ζ B q ζ B ` p χ A q p ζ B q ζ B ` p χ A q ζ B ζ B ` p χ A q p q ζ B v , and the related to B x v is RDV p t q “ ż „ p χ A q ` p χ A q ζ B ζ B ´ E p , x q ζ B pB x v q . It is clear, from (4.10), that the error terms satisfies the following estimates | RZ p t q| À B ´ ˆ } z } L ` } z } L ` }B x z } L ˙ , (5.13)and | RV p t q| À p AB q ´ } ζ B v } L ` A ´ } ζ B v } L , | RDV p t q| À p AB q ´ } ζ B B x v } L . Recalling that γ “ B ´ and applying Lemma 3.8 , we conclude | RV p t q| À A ´ B } w } L ` A ´ } w } L , | RDV p t q| À A ´ B } w } L . (5.14)This concludes the proof of the Lemma 5.3. (cid:3) Control of nonlinear terms.
The nonlinear terms in (5.5) are denoted N “ ż ρ A,B v ˜ G p x q , N “ ż ρ A,B B x v H p x q . Control of N . Recalling that ˜ G “ B x G and G is given by (3.3), using definition of z and (5.3),we have | N | “ ˇˇˇˇż pp χ A ζ B q z ` χ A ζ B B x z q G p x q ˇˇˇˇ À γ p} z } L ` }B x z } L q} G } L . By Cauchy-Schwarz inequality, (3.33) and Lemma 3.8, we conclude | N | À γ { p} z } L ` }B x z } L q ˆ }B x z } L ` } z } L ` e ´p p ´ q A p} ζ B v } L ` } ζ B B x v } L q ˙ À γ { ˆ }B x z } L ` } z } L ` e ´ p p ´ q A p} ζ B v } L ` } ζ B B x v } L q ˙ À γ { ˆ }B x z } L ` } z } L ` e ´ p p ´ q A γ ´ } w } L ˙ . (5.15) Control of N . We observe that p χ A ζ B q B x v “ χ A ζ B B x z ´ p χ A ζ B q z , then, we have N “ ż r χ A ζ B B x z ´ p χ A ζ B q z s H p x q . Recalling that H p x q is given by (3.3). Moreover, using (3.35), (2.29) and (2.5) we have | N | À B ´ ` }B x z } L ` } z } L ˘ ` B ` a ` } u } L } w } L ˘ À B ´ ` }B x z } L ` } z } L ˘ ` B δ ` | a | ` } w } L ˘ . (5.16)5.4. End of proof of Proposition 5.2.
Since γ “ B ´ and B “ δ ´ { , collecting (5.13), (5.14),(5.15) and (5.16), we obtain for some C ą fixed in (5.5) ddt N p t q ě ż pB x z q ´ ż „ pB x z q ` ` ´ f p Q q ˘ pB x z q ` B x p f p Q qq z ´ | RZ p t q| ´ | RV p t q| ´ | RDV p t q| ´ | N | ´ | N |ě ż pB x z q ´ C ż “ pB x z q ` pB x z q ` z ` z ‰ ´ C max t A ´ B , δ { u ˆ } w } L ` } w } L ` | a | ˙ . Using (2.13), A ´ B ! B ´ ! δ { . This ends the proof.6. Proof of Theorem 1.1
Before starting the proof of Theorem 1.1, we need a coercivity result to deal with the term ż sech p x q w that appears in the virial estimates of I p t q (see (2.16)). We will decompose this term in terms ofthe variables p w , w q and p z , z q . The last ones involve the variables p v , v q ; then we should beable to reconstruct the operator L from our computations.6.1. Coercivity.
We shall prove a coercivity result adapted to the orthogonality conditions x u, Q y “ x u, L p φ qy “ in (2.2), where φ was introduced in (1.10). The idea is to followthe strategy used in [35] and [10]. Recently, in [18] the operator L was appeared in a similarsetting. It has a unique negative single eigenvalue τ “ ´p p ` qp p ` q{ , associated to an L eigenfunction denoted Y .Our first result is a coercivity property for L whenever the first eigenfunction Y is changed by L p φ q . Lemma 6.1 (Coercivity lemma) . Consider the bilinear form H p u, v q “ x L p u q , v y “ ż pB x u B x v ` uv ´ f p Q q uv q . Then, there exists λ ą such that H p v, v q ě λ } v } H , (6.1) for all v P H p R q satisfying x v, Q y “ x v, L p φ qy “ . SYMPTOTICS FOR GOOD BOUSSINESQ SOLITONS 33
Proof.
See Appendix C. (cid:3)
We will need a weighted version of the previous result. See e.g. Côte-Muñoz-Pilod-Simpson[10] for a very similar proof of this result.
Lemma 6.2 (Coercivity with weight function) . Consider the bilinear form H φ ℓ p u, v q “ x a φ ℓ L p u q , a φ ℓ v y “ ż φ ℓ pB x u B x v ` uv ´ f p Q q uv q . for φ ℓ smooth and bounded and such that ă φ ℓ ď Cℓφ ℓ , where C is independent from ℓ . Then,there exists λ ą independent of ℓ small such that H φ ℓ p v, v q ě λ ż φ ℓ ppB x v q ` v q , for all v P H p R q satisfying x v, Q y “ x v, L p φ qy “ , and provided ℓ is taken small enough. The key element of the proof of Theorem 1.1 is the following transfer estimate.
Lemma 6.3.
Let u be even and satisfying (2.3) , p w , w q be as in (2.15) , and p z , z q as in (3.5) .Then, for any B large enough, it holds ż sech p x q u À B ´ { ` } w } L ` }B x w } L ˘ ` B { } z } L ` γ }B x z } L . (6.2) Proof.
Set B ă ℓ ă min t , ? λ u ď . We note that ż sech p x q u À ż sech p ℓx q u . Now, we focus on the term on the RHS of the last equation. Applying Lemma 6.2 for φ “ sech p ℓx q ,since | φ | ď Cℓφ . We obtain ż sech p ℓx q u ď ż sech p ℓx q “ u ` pB x u q ‰ ď λ ż sech p ℓx q “ u ` pB x u q ´ f p Q q u ‰ . Now, integrating by parts ż sech p ℓx qpB x u q “ ´ ż sech p ℓx q u B x u ` ż p sech p ℓx qq u , and by |p sech p ℓx qq | ď ℓ sech p ℓx q . Choosing ℓ small enough ( ă ℓ ď ? λ ), we obtain ż sech p ℓx q u À ż sech p ℓx q L p u q u . Now, using definition of v , we obtain ż sech p ℓx q L p u q u À ż sech p ℓx q u v ´ γ ż sech p ℓx q u B x v . (6.3) For the first integral in RHS of (6.3), using definition of z and w , one can see that ż sech p ℓx q u v “ ż χ A sech p ℓx q u v ` ż p ´ χ A q sech p ℓx q u v “ ż χ A sech p ℓx q p ζ A ζ B q ´ w z ` ż p ´ χ A q sech p ℓx q ζ ´ A w p ζ A v qÀ max | x |ă A t sech p ℓx q p ζ A ζ B q ´ u} w } L } z } L ` max | x |ą A t sech p ℓx q ζ ´ A u} w } L } ζ A v } L À max | x |ă A t sech p ℓx q p ζ A ζ B q ´ u} w } L } z } L ` γ ´ max | x |ą A t sech p ℓx q ζ ´ A u} w } L À ǫ } w } L ` ǫ ´ } z } L ` γ ´ e ´ A B } w } L . (6.4)Note that the last inequality holds if B ´ ă ℓ .Now, for the second integral on the RHS of (6.3), integrating by parts we obtain the followingexpression ż B x “ sech p ℓx q u ‰ B x v “ ż “ p sech p ℓx qq u ` sech p ℓx q B x u ‰ B x v “ ż p sech p ℓx qq χ A u B x v ` ż p ´ χ A qp sech p ℓx qq u B x v ` ż sech p ℓx q χ A B x u B x v ` ż p ´ χ A q sech p ℓx q B x u B x v . (6.5)Using the following decomposition and by Hölder inequality, we get ˇˇˇˇż p sech p ℓx qq u B x v ˇˇˇˇ À ˇˇˇˇż p sech p ℓx qq χ A u B x v ˇˇˇˇ ` ˇˇˇˇż p sech p ℓx qq p ´ χ A q u B x v ˇˇˇˇ À ˇˇˇˇż p sech p ℓx qq χ A u B x v ˇˇˇˇ ` ˇˇˇˇż p sech p ℓx qq ζ ´ A p ´ χ A q w p ζ A B x v q ˇˇˇˇ À ℓ } χ A u } L } ζ B χ A B x v } L ` ℓ max | x |ą A t sech p ℓx q ζ ´ A u} w } L } ζ A B x v } L . Furthermore, by the definition of z , we can check χ A ζ B B x v “ χ A B x z ´ χ A ζ B ζ B z ´ χ A z ; (6.6)and by Lemma 3.8 and Remark 2.0.1, we obtain ˇˇˇˇż p sech p ℓx qq u B x v ˇˇˇˇ À ℓ } w } L p}B x z } L ` B ´ } z } L q` ℓγ ´ max | x |ą A t sech p ℓx q ζ ´ A up}B x w } L ` } w } L q . (6.7)In similar way, we obtain ˇˇˇˇż sech p ℓx q B x u B x v ˇˇˇˇ À ˇˇˇˇż sech p ℓx q χ A B x u B x v ˇˇˇˇ ` ˇˇˇˇż sech p ℓx q p ´ χ A qB x u B x v ˇˇˇˇ À } χ A B x u } L } ζ B χ A B x v } L ` max | x |ą A t sech p ℓx q ζ ´ A u} ζ A B x u } L } ζ A B x v } L . By (6.6), Lemma 3.8 and Remark 2.0.1, we get ˇˇˇˇż sech p ℓx q B x u B x v ˇˇˇˇ À } ζ A B x u } L p}B x z } L ` } z } L q ` γ ´ max | x |ą A t sech p ℓx q ζ ´ A u} ζ A B x u } L . SYMPTOTICS FOR GOOD BOUSSINESQ SOLITONS 35
We conclude using (3.26) with K “ A , we have ˇˇˇˇż sech p ℓx q B x u B x v ˇˇˇˇ À p} w } L ` }B x w } L qp}B x z } L ` } z } L q` γ ´ max | x |ą A t sech p ℓx q ζ ´ A up} w } L ` }B x w } L q . (6.8)Collecting (6.4), (6.7), (6.8) and by Cauchy-Schwarz inequality, we obtain ż sech p x q u À ǫ } w } L ` ǫ } z } L ` γ ` } w } L ` }B x w } L ˘ ` γ ` }B x z } L ` B ´ } z } L ˘ À max t ǫ, A ´ , γ u} w } L ` γ }B x w } L ` max t ǫ ´ , B ´ u} z } L ` γ }B x z } L . Finally, choosing ǫ “ B ´ { , we conclude ż sech p x q u À B ´ { ` } w } L ` }B x w } L ˘ ` B { } z } L ` γ }B x z } L . This ends the proof of Lemma 6.3. (cid:3)
We will need a third coercivity estimate, related to the function z in (3.5). Lemma 6.4.
Recall L “ ´B x ` V p x q , with V defined in (1.8) . Assume that ş Qφ ‰ . Thenthere exists m ą fixed such that x L p u q , u y ě m } u } H ´ m ˇˇ x u, B ´ x φ y ˇˇ , for any u P H p R q odd.Proof. Since u is odd, one clearly has x L p u q , u y ě . Since ker L “ span t Q u , we only need tocheck that x L p u q , u y ě m } u } L , (6.9)for any u P H p R q odd, and provided x u, B ´ x φ y “ . First of all, it is not difficult to check thatfor some m ą , x L p u q , u y ě m ` } u } L ´ } Q } ´ L x u, Q y ˘ . Assume that } u } L “ . The term on the right hand side is zero only if u is parallel to Q , whichis not possible since ş Qφ ‰ . Therefore, after rescaling, (6.9) is proved. (cid:3) Remark 6.4.1.
Lemma 6.4 will be used in the following way: from (2.2) we have x u , B ´ x φ y “ ,and from (3.1) , we have x v , p ´ γ B x qB ´ x φ y “ . Using (3.5) and (2.15) , and the exponential decay of B ´ x φ we obtain |x z , B ´ x φ y| ď |x z , p ´ γ B x qB ´ x φ y| ` γ |x z , B x φ y|À |x v χ A ζ B , p ´ γ B x qB ´ x φ y| ` γ } z } L À |x v , p ´ χ A ζ B qp ´ γ B x qB ´ x φ y| ` γ } z } L À |x u ζ A , ζ ´ A p ´ γ B x q ´ p ´ χ A ζ B qp ´ γ B x qB ´ x φ y| ` γ } z } L À e ´ B } w } L ` γ } z } L . Finally, we prove that
Lemma 6.5. ş Qφ ‰ . Proof. If ş Qφ “ , from (1.10) one has xB x L Q, Q y ď . However ě xB x L Q, Q y “ ´p p ´ qx Q p , Q y “ ´p p ´ qx Q p , Q ´ Q p y “ ´p p ´ q ż R p Q p ` ´ Q p q . Finally, from the equation Q “ Q ´ Q p and multiplying by Q p and integrating by parts, we get ´ p ż Q p ´ Q “ ż Q p ` ´ ż Q p . Finally, using that Q “ Q ´ p ` Q p ` , we get ş Q p ` “ p ` p p ` q ş Q p , and replacing, ě xB x L Q, Q y “ ´p p ´ q ż R p Q p ` ´ Q p q “ p p p ´ q p p ` q ż R Q p ą , a contradiction. (cid:3) Now we are ready to conclude the proof of Theorem 1.1.6.2.
Proof of Theorem 1.1.
Recalling that the constants γ , C i and δ i ą for i “ , . . . , weredefined in Propositions 2.1 3.1, 4.2, 5.2. Proposition 6.6.
There exist C and ă δ ď min t δ , δ , δ , δ u such that for any ă δ ď δ ,the following holds. Fix A “ δ ´ , B “ δ ´ { and γ “ B ´ . Assume that for all t ě , (2.5) holds. Let H “ J ` C B ´ I ` B ´ M ´ B ´ C C N . Then, for all t ě , ddt H p t q ď ´ C B ´ ` } w } L ` }B x w } L ` } w } L ˘ ` C | a | . (6.10) Proof.
First, from (2.16) and (6.2) we obtain for some C ą fixed, ddt I p t q ď ´ „ } w } L ` }B x w } L ` } w } L ` C a ` C B ´ { ` } w } L ` }B x w } L ˘ ` C B { } z } L ` C γ }B x z } L . Using (5.4) and γ “ B ´ , we get ddt I p t q ď ´ ˆ } w } L ` }B x w } L ` } w } L ˙ ` C | a | ` B ´ C ddt N p t q` B { C } z } L ` B ´ C “ }B x z } L ` }B x z } L ` } z } L ` } z } L ‰ . Secondly, for ddt J , using (3.6), Lemma 6.4, and Remark 6.4.1, ddt J p t q ď ´ m ˆ } z } L ` }B x z } L ` } z } L ˙ ` C B ´ ˆ } w } L ` } w } L ˙ ` C | a | . We conclude that ddt J p t q ` C B ´ ddt I p t q ď ´ m ˆ } z } L ` }B x z } L ` } z } L ˙ ´ C B ´ ˆ } w } L ` }B x w } L ` } w } L ˙ ` B ´ C C }B x z } L ` C | a | ` B ´ C C ddt N p t q . Thirdly, using (4.6) for ddt M , ddt M p t q À ´ ` }B x z } L ` }B x z } L ˘ ` C }B x z } L ` C } z } L ` C B ´ } z } L ` C B ´ ` }B x w } L ` } w } L ` } w } L ˘ ` C | a | . SYMPTOTICS FOR GOOD BOUSSINESQ SOLITONS 37
Therefore, ddt ` J p t q ` C B ´ I p t q ` B ´ M p t q ´ B ´ C C N p t q ˘ ď ´ m ˆ } z } L ` }B x z } L ` } z } L ˙ ´ B ´ ` }B x z } L ` }B x z } L ˘ ´ C B ´ ` } w } L ` }B x w } L ` } w } L ˘ ` C | a | . Setting H “ J ` C B ´ I ` B ´ M ´ B ´ C C N , and C “ C , we obtain the desiredproperty. (cid:3) We define now B “ b ` ´ b ´ , where b ` , b ´ are given in (2.4). Lemma 6.7.
There exist C and δ ą , such that for any ă δ ď δ , the following holds.Assume that for all t ě (2.5) holds. Then, for all t ě , | b ` ´ ν b ` | ` | b ´ ` ν b ´ | ď C ˆ b ` ` b ´ ` ››› sech { p x { q w ››› L ˙ , (6.11) and ˇˇˇˇ ddt b ` ´ ν b ` ˇˇˇˇ ` ˇˇˇˇ ddt b ´ ` ν b ´ ˇˇˇˇ ď C ˆ b ` ` b ´ ` ››› sech { p x { q w ››› L ˙ { . (6.12) In particular, ddt B ě ν p a ` a q ´ C ››› sech { p x { q w ››› L . (6.13) Proof.
From (2.29) and (2.4), it holds | N | À b ` ` b ´ ` ż sech ´ x ¯ w . From (2.6) we conclude the estimates (6.11) and (6.12). Finally, (6.13) follows directly from (6.12)and taking δ ą small enough. (cid:3) Combining (6.10) and (6.13), it holds ddt ˆ B ´ B C C H ˙ ě ν p a ` a q ` C ` } w } L ` }B x w } L ` } w } L ˘ . (6.14)By the choice of A “ δ ´ , the bound | ϕ A | À A , (2.14) and (2.5), we have for all t ě | I p t q| À A } u } L } u } L À δ. Analogously, using Lemma 3.7, we have | J p t q| À B } v } L } v } L À Bγ ´ } u } L } u } L À B } u } L } u } L À B δ À δ, | M p t q| À B }B x v } L }B x v } L À Bγ ´ { } u } L } u } L À B } u } L } u } L À B δ À δ, and | N p t q| À }B x v } L } v } L À γ ´ } u } H } u } L À B } u } H } u } L À B δ À δ. Then, we have | H | ď δ. Estimate | B | ď δ is also clear from (2.5). Therefore, integrating estimates (6.14) on r , t s andpassing the limit as t Ñ 8 , we have ż “ a ` a ` } w } L ` }B x w } L ` } w } L ‰ dt À δ. By Lemma 3.10 one can see ż ˆ a ` a ` ż p u ` pB x u q ` u q sech p x q ˙ dt ď δ. (6.15)Using the above equation, we will conclude the proof of Theorem 1.1.Let K p t q “ ż sech p x q u ` ż sech p x qpp ´ γ B x q ´ B x u q “ : K p t q ` K p t q . For K , using (2.8) and integrating by parts, we have d K dt “ ż sech p x qp u u q “ ż sech p x qp u B x u q “ ´ ż p sech p x q u ` sech p x qB x u q u . Then, ˇˇˇˇ ddt K p t q ˇˇˇˇ ď ż sech p x qp u ` pB x u q ` u q . For K , passing to the variables p v , v q (see (3.1)) K “ ż sech p x qpB x v q , and using (3.2), we get ddt K “ ż sech p x qB x v B x v ` ż sech p x qB x v B x H p x q “ : K ` K . Integrating by parts in K , we have K “ ´ ż p sech p x qB x v ` sech p x qB x v qB x v , besides using Cauchy-Schwarz inequality and Lemma 3.7, we obtain | K | À ż sech p x qppB x v q ` pB x v q ` pB x v q qÀ γ ´ ż sech p x qp u ` pB x u q q . For K , we use Cauchy-Schwartz inequality, Corollary 3.5 and a similar computation of (4.45),then | K | À ż sech p x qrpp ´ γ B x q ´ B x u q ` pp ´ γ B x q ´ B x N K q sÀ ż sech p x qr γ ´ u ` pp ´ γ B x q ´ B x N K q s À γ a ` ż sech p x qr u ` u s . Then, we conclude ˇˇˇˇ ddt K p t q ˇˇˇˇ À γ a ` ż sech p x qp u ` pB x u q ` u q . By (6.15), there exists and increasing sequence t n Ñ 8 such that lim n Ñ8 “ a p t n q ` a p t n q ` K p t n q ` K p t n q ‰ “ . For t ě , integrating on r t, t n s , and passing to the limit as n Ñ 8 , we obtain K p t q À ż t „ a ` ż sech p x qp u ` pB x u q ` u q dt. By (6.15), we deduce lim t Ñ8 K p t q “ . Finally, by (2.6) and (2.29), we get ˇˇˇˇ ddt p a q ˇˇˇˇ ` ˇˇˇˇ ddt p a q ˇˇˇˇ À a ` a ` ż sech p x q u . SYMPTOTICS FOR GOOD BOUSSINESQ SOLITONS 39
In a similar way as above, integrating on r t, t n s and taking n Ñ 8 , we conclude a p t q ` a p t q À ż t „ a ` a ` ż u sech p x q dt, which proves lim t Ñ8 | a p t q| ` | a p t q| “ . By the decomposition of solution (2.1) this implies(1.13). This ends the proof of Theorem 1.1. Remark 6.7.1.
We have not being able to describe the asymptotic behavior of pB x u q and u , dueto the fact that we are working in the energy space, and any variation of the virial that involvesthese terms is not well-defined. In fact, the regularity considered for the variation of K and K issharp, in the sense that we do not have a gap where to include terms with higher-order derivatives.For example, for K “ ż sech p x q u , its variation is ddt K “ ż sech p x q u pB x L u ` N K q . One can see that L u P H ´ and u P L . Then, the last estimate may not be well-defined. Proof of Theorem 1.2
Now we construct initial data for which Theorem 1.1 remains valid. We follow the ideas in [18],with some particular differences in some estimates.7.1.
Conservation of Energy.
Using (1.4), (2.1), (1.8), and by the orthogonality condition (2.2),we have “ E p u, v q ´ E p Q, q ‰ “ ż “ v ` u ` pB x u q ´ F p u q ‰ ´ E p Q, q“ a ż ν pB ´ x φ q ` a ż ppB x φ q ` V p x q φ q ` ż f p Q qp a φ ` u q ` ż ppB x u q ` V p x q u ` u q ´ ż ` F p u q ´ F p Q q ´ f p Q qp a φ ` u q ˘ “ a ν }B ´ x φ } L ` a x L p φ q , φ y ` x L p u q , u y ` } u } L ´ ż ´ F p u q ´ F p Q q ´ f p Q qp a φ ` u q ´ f p Q q p a φ ` u q ¯ . Using (1.10), we get x L p φ q , φ y “ x ν B ´ x φ , φ y “ ´ ν xB ´ x φ , B ´ x φ y “ ´ ν , and, by (2.4), we obtain the identity “ E p u, v q ´ E p Q, q ‰ “ ´ ν b ` b ´ ` x L p u q , u y ` } u } L ´ ż ˆ F p u q ´ F p Q q ´ f p Q qp a φ ` u q ´ f p Q q p a φ ` u q ˙ . (7.1)Let δ be defined by δ “ b ` p q ` b ´ p q ` } u p q} H ` } u p q} L . Considering (7.1) at t “ follows | “ E p u, v q ´ E p Q, q ‰ | À δ . Besides, by the conservation ofenergy, estimate (7.1) at some t ą gives | ´ ν b ` b ´ ` x L p u q , u y ` } u } L ´ O p| b ` | ` | b ´ | ` } u } H q| À δ . Considering the orthogonality condition x u , Q y “ x u , L p φ qy “ , the parity of u , and usingthe Lemma 6.1, it follows that for some λ P p , q , x L p u q , u y ě λ } u } H . Due to } u } H ` } u } L ` | b ` | ` | b ´ | À δ , the following estimate holds } u } H ` } u } L À | b ` | ` | b ´ | ` δ . (7.2) Construction of the graph.
We will construct initial data that directs to global solutionsclose to the ground state Q . To accomplish this objective, we use the energy estimate (7.2),Lemma 6.7 and a standard contradiction argument.Let ǫ “ p ǫ , ǫ q P A . Let Z ` be as in (1.11). Then, the condition x ǫ , Z ` y “ rewrites x ǫ , B ´ x φ y ` x ǫ , ν ´ B ´ x φ y “ . Define b ´ p q and p u p q , u p qq such that b ´ p q “ ´x ǫ , B ´ x φ y “ x ǫ , ν ´ B ´ x φ y , and ǫ “ b ´ p q φ ` u p q , ǫ “ ´ b ´ p q ν B ´ x φ ` u p q . Then, it holds x u p q , B ´ x φ y “ x u p q , B ´ x φ y “ . From (1.15) and (1.14), we observe that the initial condition in Theorem 1.2 holds the followingdecomposition: φ “ φ p q “ p Q, q ` p u , u qp q ` b ´ p q Y ´ ` h p ǫ q Y ` . We will prove that there is a function h p ǫ q such that the corresponding solution φ is global andsatisfies (1.16). We show that at least h p ǫ q “ b ` p q satisfies this statement.Let δ ą small enough and K ą large enough to be chosen. Following the scheme of [18],we introduce the following bootstrap estimates } u } H ď K δ and } u } L ď K δ , (7.3) | b ´ | ď Kδ , (7.4) | b ` | ď K δ . (7.5)Given any p u p q , u p qq and b ´ p q such that } u p q} H ď δ , } u p q} L ď δ , | b ´ p q| ď δ (7.6)and b ` p q satisfying | b ` p q| ď K δ . Let T “ sup t t ě such that (7.3) , (7.4) , (7.5) hold on r , t su . Since K ą follow that T is well-defined in r , `8s . Our aim is to prove that there exists at leasta value of b ` p q P r´ K δ , K δ s such that T “ 8 . To prove this we argue by contradiction: weassume that for all values of b ` p q P r´ K δ , K δ s , one has T ă 8 .The first step is improve the estimates (7.3). By (7.3), we have } u } H ` } u } L ď K δ (7.7)Otherwise, using the energy estimates (7.2) it holds } u } H ` } u } L ď C p K δ ` K δ ` δ q , for some constant C ą . Thus, using the smallness of δ and largeness of K , it holds C ď K , C K δ ď K , ď K , (7.8)and we obtain } u } H ` } u } L ď K δ , that it is a clear improve of the inequality (7.7).The second step is control b ´ . Using (6.12), (7.3), (7.4) and (7.5), we have ˇˇˇˇ ddt p e ν t b ´ q ˇˇˇˇ ď C p K δ ` K δ q e ν t , for some constant C ą . Therefore, by integration on r , t s and using (7.6), we obtain b ´ ď C ν p K δ ` K δ q ` δ . SYMPTOTICS FOR GOOD BOUSSINESQ SOLITONS 41
Under the constraints C ν K δ ď K , C ν K δ ď K , ď K , (7.9)we get b ´ ď K δ , that is an improvement of (7.4).By the improved estimates (7.3) and (7.4), and a continuity argument, we observe that if T ă `8 , then | b ` p T q| “ K δ .The third step is to analyze the growth of b ` . If t P r , T s is such that | b ` p t q| “ K δ , thenfollows from (6.11) that ddt b ` ě ν b ` ´ C | b ` |p b ` ` b ´ ` } u } H qě ν b ` ´ C | b ` |p b ` ` K δ ` K δ qě ν K δ ´ C p K δ ` K δ q , for some constant C ą . Under the constraints C K δ ď ν K , C K ď ν K , (7.10)the following inequality holds ddt b ` ě ν K δ ą . By standard arguments, such transversality condition implies that T is the first time for which | b ` p t q| “ K δ and moreover that T is continuous in the variable b ` p q . The image of thecontinuous map b ` p q P r´ K δ , K δ s ÞÝÑ b ` p T q P t´ K δ , K δ u is exactly t´ K δ , K δ u which is a contradiction. We conclude that there exists at least onevalue of b ` p q P p´ K δ , K δ q such that T “ 8 , when constraints in (7.8), (7.9), (7.10) arefulfilled. Finally, to satisfy the conditions (7.8), (7.9), (7.10) it is sufficient first to fix K ą largeenough, depending only on C , C , C , and then to choose δ ą small enough.7.3. Uniqueness and Lipschitz regularity.
To finish the proof of Theorem 2, we will provethe following proposition that implies the uniqueness of the choice of h p ǫ q “ b ` p q , for a given ǫ P A , as well the Lipschitz regularity of the graph M (see (1.15)) Proposition 7.1.
There exist
C, δ ą such if φ and r φ are two even-odd solution of (1.3) satisfying for all t ě , } φ p t q ´ p Q, q} H ˆ L ă δ, } r φ p t q ´ p Q, q} H ˆ L ă δ (7.11) then, decomposing φ p q “ p Q, q ` ǫ ` b ` p q Y ` , r φ p q “ p Q, q ` ˜ ǫ ` ˜ b ` p q Y ` (7.12) with x ǫ , Z ` y “ x ˜ ǫ , Z ` y “ , it holds | b ` p q ´ ˜ b ` p q| ď Cδ { } ǫ ´ ˜ ǫ } H ˆ L . (7.13) Proof.
Let φ and ˜ φ solutions of (1.3) likes in the Subsection 2.1, i.e., satisfies the decomposition(2.1) and the smallest condition (2.5). Then, } u } H ` } ˜ u } H ` } u } L ` } ˜ u } L ` | b ˘ | ` | ˜ b ˘ | ď C δ. (7.14)Let q a “ a ´ r a , q a “ a ´ r a , q b ` “ b ` ´ r b ` , q b ´ “ b ´ ´ r b ´ , q u “ u ´ r u , q u “ u ´ r u , q N “ N ´ r N, q N K “ N K ´ r N K , q N “ N ´ r N . (7.15) Then, by (2.6) and (2.8), p ˇ u , ˇ u q and p ˇ b ` , ˇ b ´ q satisfy the following equations: q u “ B x q u q u “ B x L p q u q ` q N K and $’’’&’’’% q b ` “ ν q b ` ` q N ν q b ´ “ ´ ν q b ´ ´ q N ν . (7.16)Furthermore, let β ` “ q b ` , β ´ “ q b ´ , β c “ x L q u , q u y ` x q u , q u y . Computing the variation of β c , we obtain β c “ x q N K , q u y . Now, recalling (2.7) and (2.29), we get ˇ N “ p Q ´ ˜ a B x φ ´ B x ˜ u q „ f p Q q ` f p Q qp a φ ` u q ´ f p Q ` a φ ` u q´ ´ f p Q q ` f p Q qp ˜ a φ ` ˜ u q ´ f p Q ` ˜ a φ ` ˜ u q ¯ ` p q a B x φ ` B x q u qp f p Q q ´ f p Q ` a φ ` u qq` p q a φ ` q u q f p Q qp ˜ a B x φ ` B x ˜ u q . By Taylor expansion, for any v, ˜ v , it holds ˇˇ f p Q ` v q ´ f p Q q ´ f p Q q v ´ p f p Q ` ˜ v q ´ f p Q q ´ f p Q q ˜ v q ˇˇ À | v ´ ˜ v |p| v | ` | ˜ v |qp Q p ´ ` | v | p ´ ` | ˜ v | p ´ q À | v ´ ˜ v |p| v | ` | ˜ v |q Then, | ˇ N | À | q a φ ` q u || Q ´ ˜ a B x φ ´ B x ˜ u | ` | ˜ a φ ` ˜ u | ` | a φ ` u | ˘ ` f p Q q| q a B x φ ` B x q u || a φ ` u | ` f p Q q| q a φ ` q u || ˜ a B x φ ` B x ˜ u | . Then, using Sobolev embbeding, L - norm of q N is bounded by } ˇ N } L À } q a φ ` q u } L } Q ´ ˜ a B x φ ´ B x ˜ u } L ` } ˜ a φ ` ˜ u } L ` } a φ ` u } L ˘ ` } q a B x φ ` B x q u } L } a φ ` u } L ` } q a φ ` q u } L } ˜ a B x φ ` B x ˜ u } L À p| q a | ` } q u } H q} Q ´ ˜ a B x φ ´ B x ˜ u } L ` | ˜ a | ` } ˜ u } H ` | a | ` } u } H ˘ ` p| q a | ` } q u } H qp| a | ` } u } H q ` p| q a | ` } q u } H qp| ˜ a | ` } ˜ u } H qÀ p| q a | ` } q u } H q “ | a | ` | ˜ a | ` } u } H ` } ˜ u } H ‰ . (7.17)Then, by (7.16), (7.17), and using | q N | À } q N } L }B ´ x φ } L , we get | β c | ` | β ` ´ ν β ` | ` | β ´ ` ν β ´ | ď Kδ p β c ` β ` ` β ´ q for some K ą . (7.18)In order to obtain a contradiction, assume that the following holds ă Kδ p β c p q ` β ` p q ` β ´ p qq ă ν β ` p q . (7.19)Now, we consider the following bootstrap estimate Kδ p β c ` β ` ` β ´ q ď ν β ` . (7.20)and let T “ sup t t ą such that (7.20) holds u ą . From (7.18) and (7.20), it holds ν β ` ď ν β ` ´ Kδ p β c ` β ` ` β ´ q ď β ` , for t P r , T s . (7.21)Then, β ` is positive and increasing function on r , T s .Now, by (7.18) and (7.20), we get β c ď ν β ` ď β ` SYMPTOTICS FOR GOOD BOUSSINESQ SOLITONS 43 integrating and using that β ` p q ą , we obtain β c p t q ď β c p q ` β ` p t q ´ β ` p q ď β c p q ` β ` p t q . Furtheremore, by (7.19) and for δ small enough, we get Kδβ c p t q ď Kδ p β c p q ` β ` p t qq ď ν β ` p q ` Kδβ ` p t q ď ν β ` p t q . For β ´ , using (7.18) and (7.20), we get β ´ ď ´ ν β ´ ` ν β ` , integrating and using (7.19), we have β ´ p t q ď e ´ ν t β ´ p q ` ν β ` e ´ ν t ż t e ν s ds ď β ´ p q ` β ` p t q . For δ small enough, we get Kδβ ´ p t q ď Kδ p β ´ p q ` β ` p t qq ď ν β ` p q ` Kδβ ` p t q ď ν β ` p t q . For β ` , it is clear that holds Kδ ď ν for δ small enough.We have proved that, for all t P r , T s , Kδ p β c p t q ` β ` p t q ` β ´ p t qq ď ν β ` p t q . By a continuity argument, we get that T “ 8 . However, by the exponential growth (7.21) and β ` p q ą , we obtain a contradiction with (7.14) on | b ` | .Since it holds ǫ “ u p q ` b ´ p q Y ´ , ˜ ǫ “ ˜ u p q ` ˜ b ´ p q Y ´ , with x u p q , Y ´ y “ x ˜ u p q , Y ´ y “ , and estimates (7.19) is contradicted, we have proved (7.13). (cid:3) Appendix A. Linear spectral theory for ´B x L In this section we describe the spectral properties of the operator ´B x L , where L is introducedin (1.8). Notice that this last operator has been widely studied (see [25, 26]). For the study of theoperator ´B x L we shall start with the following result. Lemma A.1.
Let p ą . The operator L defined in (1.8) satisfies the following properties.(1) The continuum spectrum of L is r , .(2) The kernel of L is only spanned by the function Q .(3) The generalized kernel of L is given by span " Q p x q , Q p x q ż xǫ p Q p r qq ´ dr * , for any x ě ǫ ą or x ď ǫ ă . In what follows, and with a slight abuse of notation, we will write ż x p Q p r qq ´ dr instead of ş xǫ p Q p r qq ´ dr ; but it is understood that the zero limit of integration corresponds to any ε sufficiently close to zero.An important remark is the following: Remark A.1.1.
Note that L p f g q “ g L p f q ´ f g ´ f g . (A.1) This property will be useful in the following computations.
Now, we study the properties of the operator ´B x L . Remark A.1.2.
A direct analysis shows that the null space of B x L “ B x ´ B x is spanned byfunctions of the type e x , e ´ x , , x, as x Ñ 8 . Note that this set is linearly independent and among these four functions there is only one L integrable in the semi-infinite line r , . Therefore, since B x L is a compact perturbation of thescalar operator B x L , the null space of B x L | H p R q is spanned by at most one L -function. Lemma A.2.
Let p ą . The operators ´B x L satisfy the following properties.(1) The continuum spectrum of ´B x L is r , .(2) The generalized kernel of ´B x L is spanned byspan "„ ´ p ´ ` ´ p ` p ´ Q p x q ż x Q ´ p r q dr ,Q p x q ż x ´ sQ p s q ´ ş s ´8 Q p y q dy ¯ p Q q p s q ds, Q p x q , Q p x q ż x p Q p r qq ´ dr ,.- . (A.2) Proof.
The proof of (1) follows directly from the form of the operator.Proof of (2). Clearly u p x q : “ Q p x q , u p x q : “ Q p x q ż x p Q p r qq ´ dr, are solutions to ´B x L p u q “ . Notice that if ´B x L p u q “ is equivalent to L p u q “ ax ` b with a, b P R . Then we should solve this equation. First, we consider the case a “ . Without loss ofgenerality, we consider b “ . One has L p q “ ´ pQ p ´ . Computing, L ˆ Q ż x Q n ˙ “ L p Q q ż x Q n ´ Q n p Q q ´ nQ n ´ p Q q “ ´ Q n p Q ´ Q p q ´ nQ n ´ ˆ Q ´ p ` Q p ` ˙ “ Q n ` p´ ´ n q ` ˆ n ` p ` p ` ˙ Q p ` n . If n “ ´ , we have L ˆ Q ż x Q ´ ˙ “ ´ ` pp ` Q p ´ . Set u p x q “ ´ p ´ „ ` p ` Q p x q ż x Q ´ p r q dr . We observe that L p u p x qq “ . Therefore, upto the generalized kernel of L , u solves the equation L p u q “ .Now, without loss of generality, we consider a “ and b “ , then we must solve L p u q “ x .Using the method of reduction of order with an unknown function ψ , consider u “ Q ψ . Using(A.1), we have L p Q ψ q “ ´ Q ψ ´ Q ψ “ x. We obtain that the solution of this equation is u p x q “ Q p x q ż x p Q q ´ p s q ˆż s Q p y q ydy ˙ ds ´ ˆż ´8 Q p y q dy ˙ Q p x q ż x p Q q ´ p s q ds “ Q p x q ż x p Q q ´ p s q ˆż s yQ p y q dy ´ ż ´8 Q p y q dy ˙ ds “ Q p x q ż x p Q q ´ p s q ˆ sQ p s q ´ ż s ´8 Q p y q dy ˙ ds. We finally conclude that the fundamental set of solutions for B x L p u q “ is given by t u p x q , u p x q , u p x q , u p x qu . SYMPTOTICS FOR GOOD BOUSSINESQ SOLITONS 45
This ends the proof. (cid:3)
Corollary A.3.
There is, up to constant, only one solution of ´B x L p u q “ in L p R q . Now, we focus on describing the eigenfunctions and negative eigenvalues of operator ´B x L .This analysis will be the main ingredient to describe the stability of the soliton. Our first resultestablishes the parity of eigenfunctions associated to nonzero eigenvalues. Lemma A.4. If φ P H p R q is an eigenfunction associated to an eigenvalue λ ‰ of the operator ´B x L , then B ´ x φ P H and and B ´ x φ P H , i.e., are well-defined. Furthermore, if φ is an evenfunction then B ´ x φ is an odd function and ż φ p y q dy “ . Proof.
We have ´B x L φ “ λ φ , with λ ‰ , this is equivalent to B x φ ´ B x φ ` B x p pQ p ´ φ q “ λ φ . (A.3)Applying Fourier transform, we have ξ x φ ` ξ x φ ´ ξ p p { Q p ´ φ q “ λ x φ . From this identity, and the fact that φ P H p R q , we observe that lim ξ Ñ x φ p ξ q “ λ ´ lim ξ Ñ ´ ξ x φ ` ξ x φ ´ ξ { pQ p ´ φ ¯ “ . Also lim ξ Ñ ξ ´ x φ p ξ q “ λ ´ lim ξ Ñ ´ ξ x φ ` ξ x φ ´ ξ { pQ p ´ φ ¯ “ , lim ξ Ñ ξ ´ x φ p ξ q “ λ ´ lim ξ Ñ ´ ξ x φ ` x φ ´ { pQ p ´ φ ¯ “ ´ pλ ´ { Q p ´ φ p q . Then, we obtain that ż φ p x q “ , ż ż x ´8 φ p s q ds “ . Also, we know that { Q p ´ φ is well defined (the Fourier transform is an homeomorphism from L into L ). Then B ´ x φ and B ´ x φ are well-defined, and exponentially decreasing, provided φ andits derivatives are also exponentially decreasing.Now, suppose that φ is an even function. Integrating between 0 and x in (A.3), we obtain pB x φ ´ B x φ ` B x p pQ p ´ φ qqp x q ´ rB x φ ´ B x φ ` B x p pQ p ´ φ qs| x “ “ λ ż x φ . Since Q p ´ is an even function and B x φ , B x φ and B x p Q p ´ φ q are odd functions, satisfying B x φ p q “ B x φ p q “ B x p Q p ´ φ qp q “ , we conclude B x L p φ qp x q “ pB x φ ´ B x φ ` B x p pQ p ´ φ qqp x q “ λ ż x φ p y q dy. Now, given that φ P H p R q , one has B x φ p x q , B x φ p x q , B x p Q p ´ φ qp x q Ñ as x Ñ ˘8 . Weconclude ż x φ p y q dy “ ´ ż ´ x φ p y q dy and ż φ p y q dy “ . This proves the oddness of B ´ x φ and concludes the proof. (cid:3) We observe that ´B x L is not a self-adjoint operator. In fact, if ϕ, ψ P H p R q , @ ´B x L ϕ, ψ D “ x ϕ, ´B x L p ψ qy ` x ϕ, f p Q qB x ψ ´ B x p f p Q q ψ qy , since the operators B x and L do not commute. For this reason, we need to consider this operatorin an appropriate sense. A way to face this problem is to consider the following result. Lemma A.5.
The operator ´B x L has only real eigenvalues.Proof. Given ϕ P H p R q eigenfunction of the operator ´B x L with eigenvalue λ P C , we consider ϕ “ B x ψ or ψ “ B ´ x ϕ . We know that this function is well defined by Lemma A.4. Now, wehave ´B x L pB x ψ q “ ´B x L p ϕ q “ λ ϕ “ λ B x ψ . Integrating, we obtain ´B x L pB x ψ q “ λ ψ . We can easily check that the operator ´B x L B x is self-adjoint with eigenvalue λ and eigenfunction ψ . We conclude that λ is real, hence the eigenvalues of ´B x L are real. (cid:3) Therefore, the operator ´B x L has a similar structure of a self-adjoint operator. This fact al-lows to follow the strategy of Greenberg and Maddocks-Sachs [12, 20] for counting the negativeseigenvalues of this operator.The most important property about ´B x L is that it possesses only one negative eigenvalue. Theorem A.6.
The operator ´B x L has a unique negative eigenvalue ´ ν ă of multiplicity one.The associated eigenfunction φ satisfies the exponential decay in (1.10) , along with its derivatives. This is just a consequence of the fact that the only solution of ´B x L p u q “ converging to zeroat ´8 is Q p x q , see Claim A.8. This function has a unique zero. The exponential decay is justconsequence of Remark A.1.2. Corollary A.7.
Given φ eigenfunction associated to the unique negative eigenvalue ´ ν , then φ is an even function and B ´ x φ is an odd function.Proof. Consider the function ψ p x q “ φ p´ x q , we have ´B x L p ψ q “ B x p ψ q ´ B x ψ ` B x p pQ p ´ ψ q . Notice that Q p ´ , B x p Q p ´ q are even functions and B x p Q p ´ q is an odd function, also B x p pQ p ´ ψ qp x q “ B x p pQ p ´ p x q φ p´ x qq “ B x p pQ p ´ φ qp´ x q . Then, we observe that ´B x L p ψ q “ ´B x L p φ qp´ x q “ ´ ν φ p´ x q “ ´ ν ψ p x q . Finally, since λ is the unique negative eigenvalue of multiplicity one, we conclude that φ p x q “ ψ p x q “ φ p´ x q , i.e., φ is an even function. Finally, by Lemma A.4 we know B ´ x φ is an oddfunction. (cid:3) A.1.
Asymptotic behavior of fundamental solutions of ´B x L p u q “ . The following com-putations are direct, but we include them by the sake of completeness. They are just simpleapplications of L’Hôpital’s rule.
Claim A.8.
The functions u , u , u and u found in Lemma A.1 and A.2 satisfy lim x Ñ´8 u p x q “ , lim x Ñ´8 u p x q “ `8 , lim x Ñ´8 u p x q “ , lim x Ñ´8 u p x q “ ´8 . Proof.
One has(1) lim x Ñ´8 u p x q “ lim x Ñ´8 Q p x q “ . (2) Second, lim x Ñ´8 u p x q “ lim x Ñ´8 Q p x q ż x p Q p r qq ´ dr “ lim x Ñ´8 ş x p Q p r qq ´ dr p Q p x qq ´ “ lim x Ñ´8 p Q p x qq ´ ´p Q p x qq ´ Q p x q “ lim x Ñ´8 ´ Q p x q “ lim x Ñ´8 Q p x qp Q p ´ p x q ´ q “ `8 . SYMPTOTICS FOR GOOD BOUSSINESQ SOLITONS 47 (3) Third, lim x Ñ´8 u p x q “ lim x Ñ´8 ´ p ´ „ ` p ` Q ż x Q ´ p r q dr “ ´ p ´ ´ p ` p ´ x Ñ´8 Q p x q ż x Q ´ p r q dr “ ´ p ´ ´ p ` p ´ x Ñ´8 ż x Q ´ p r q dr p Q p x qq ´ “ ´ p ´ ´ p ` p ´ x Ñ´8 Q ´ p x q´p Q p x qq ´ Q p x q “ ´ p ´ ` p ` p ´ x Ñ´8 Q ´ p x qp Q ´ p ` Q p ` q Q ´ Q p “ ´ p ´ ` p ` p ´ x Ñ´8 p ´ p ` Q p ´ q ´ Q p ´ “ . (4) Finally, lim x Ñ´8 u p x q “ lim x Ñ´8 Q p x q ż x p Q q ´ ˆ sQ p s q ´ ż s ´8 Q ˙ ds “ lim x Ñ´8 ż x p Q q ´ ˆ sQ p s q ´ ż s ´8 Q ˙ ds p Q p x qq ´ “ lim x Ñ´8 p Q p x qq ´ ˆ xQ p x q ´ ż x ´8 Q ˙ ´p Q p x qq ´ Q p x q“ lim x Ñ´8 ˆ xQ p x q ´ ż x ´8 Q ˙ ´ Q p x q “ lim x Ñ´8 ˆ xQ p x q ´ ż x ´8 Q ˙ Q p x qp Q p ´ p x q ´ q “ lim x Ñ´8 p xQ p x q ` Q p x q ´ Q p x qq Q p x qp pQ p ´ p x q ´ q“ lim x Ñ´8 xpQ p ´ p x q ´ “ ´8 . (cid:3) Appendix B. Proof of Claims 4.4 and 4.5
B.1.
Relation between B x z i and B x v i . We prove Claim 4.4. First, recall that z i “ χ A ζ B v i and B x z i “ p χ A ζ B q v i ` χ A ζ B B x v i . (B.1)Then pB x z i q “ pp χ A ζ B q v i q ` p χ A ζ B q χ A ζ B v i B x v i ` p χ A ζ B B x v i q . (B.2)For a function P p x q P C p R q , we consider ż P p x q χ A ζ B pB x v i q . Using (B.2), we obtain ż P p x q χ A ζ B pB x v i q “ ż P p x qpB x z i q ´ ż P p x qrp χ A ζ B q s v i ´ ż P p x qpp χ A ζ B q q B x p v i q“ ż P p x qpB x z i q ´ ż P p x qrp χ A ζ B q s v i ` ż r P p x qpp χ A ζ B q q s v i . (B.3)Now P p x qrp χ A ζ B q s “ P p x q ζ B „ p χ A q ` p χ A q ζ B ζ B ` P p x qp χ A ζ B q ˆ ζ B ζ B ˙ . Then, we have ż P p x qrp χ A ζ B q s v i “ ż P p x q ˆ ζ B ζ B ˙ z i ` ż P p x q „ p χ A q ` p χ A q ζ B ζ B ζ B v i . As for the third integral in the RHS of (B.3), we have r P p x qp χ A ζ B q s “ P p x qp χ A ζ B q ` P p x qp χ A ζ B q “ P p x q ζ B „ p χ A q ` χ A ˆ ζ B ζ B ˙ ` P p x q ζ B « p χ A q ` p χ A q ζ B ζ B ` χ A «ˆ ζ B ζ B ˙ ` ζ B ζ B ffff “ χ A ζ B « P p x q ζ B ζ B ` P p x q «ˆ ζ B ζ B ˙ ` ζ B ζ B ffff ` P p x q ζ B „ p χ A q ` p χ A q ζ B ζ B ` P p x q ζ B p χ A q . Then, we obtain ż r P p x qpp χ A ζ B q q s v i “ ż « P p x q ζ B ζ B ` P p x q «ˆ ζ B ζ B ˙ ` ζ B ζ B ffff z i ` ż P p x q „ p χ A q ` p χ A q ζ B ζ B ζ B v i ` ż P p x qp χ A q ζ B v i . We conclude in (B.3): ż P p x q χ A ζ B pB x v i q “ ż P p x qpB x z i q ´ ż P p x qrp χ A ζ B q s v i ` ż r P p x qpp χ A ζ B q q s v i “ ż P p x qpB x z i q ´ ż P p x q ˆ ζ B ζ B ˙ z i ´ ż P p x q „ p χ A q ` p χ A q ζ B ζ B ζ B v i ` ż « P p x q ζ B ζ B ` P p x q «ˆ ζ B ζ B ˙ ` ζ B ζ B ffff z i ` ż P p x q „ p χ A q ` p χ A q ζ B ζ B ζ B v i ` ż P p x qp χ A q ζ B v i “ ż P p x qpB x z i q ` ż „ P p x q ζ B ζ B ` P p x q ζ B ζ B z i ` ż E p P p x q , x q ζ B v i , (B.4)where E p P p x q , x q “ P p x q „ χ A χ A ` p χ A q ζ B ζ B ` P p x qp χ A q . (B.5)Finally, (4.13) follows directly from the definition of E p P p x q , x q and Remark 4.3.1 replacing B by A . This ends the proof of Claim 4.4.B.2. Relation between B x z i and B x v i . Now we prove Claim 4.5. The following relation isobtained from z i in (3.5): B x z i “ p χ A ζ B q v i ` p χ A ζ B q B x v i ` χ A ζ B B x v i , pB x z i q “ rp χ A ζ B q v i s ` rp χ A ζ B q B x v i s ` r χ A ζ B B x v i s ` p χ A ζ B q v i p χ A ζ B q B x v i ` p χ A ζ B q B x v i χ A ζ B B x v i ` p χ A ζ B q v i χ A ζ B B x v i “ rp χ A ζ B q v i s ` rp χ A ζ B q B x v i s ` r χ A ζ B B x v i s ` p χ A ζ B q p χ A ζ B q B x p v i q ` r χ A ζ B s B x rpB x v i q s ` p χ A ζ B q χ A ζ B v i B x v i . SYMPTOTICS FOR GOOD BOUSSINESQ SOLITONS 49
Then, p χ A ζ B B x v i q “ pB x z i q ´ pp χ A ζ B q v i q ´ pp χ A ζ B q B x v i q ´ p χ A ζ B q p χ A ζ B q B x p v i q ´ p χ A ζ B q χ A ζ B v i B x v i ´ r χ A ζ B s B x rpB x v i q s . (B.6)Now, ż R p x qp χ A ζ B B x v i q “ ż R p x qpB x z i q ` ż R p x q ` ´rp χ A ζ B q s v i ´ rp χ A ζ B q s pB x v i q ˘ ` ż R p x q “ ´ prp χ A ζ B q s q B x p v i q ´ rp χ A ζ B q s B x rpB x v i q s ´ p χ A ζ B q χ A ζ B v i B x v i ‰ “ ż R p x qpB x z i q ` ż R p x qp´rp χ A ζ B q s v i ´ rp χ A ζ B q s pB x v i q q` ż B x r R p x qprp χ A ζ B q s q s v i ` ż B x r R p x qrp χ A ζ B q s spB x v i q ´ ż R p x qp χ A ζ B q χ A ζ B v i B x v i . (B.7)Since ´ ż R p x qp χ A ζ B q χ A ζ B v i B x v i “ ´ ż B x r R p x qp χ A ζ B q χ A ζ B s v i ` ż R p x qp χ A ζ B q χ A ζ B pB x v i q , we get ż R p x qp χ A ζ B B x v i q “ ż R p x qpB x z i q ` ż R p x qp´rp χ A ζ B q s v i ´ rp χ A ζ B q s pB x v i q q` ż B x r R p x qprp χ A ζ B q s q s v i ` ż B x r R p x qrp χ A ζ B q s spB x v i q ´ ż B x r R p x qp χ A ζ B q χ A ζ B s v i ` ż R p x qp χ A ζ B q χ A ζ B pB x v i q “ ż R p x qpB x z i q ` ż r´B x r R p x qp χ A ζ B q χ A ζ B s ` B x r R p x qprp χ A ζ B q s q s ´ R p x qrp χ A ζ B q s s v i ` ż ” R p x qp χ A ζ B q χ A ζ B ` B x r R p x qrp χ A ζ B q s s ´ R p x qrp χ A ζ B q s ı pB x v i q . (B.8)Now we perform the following splitting: ż R p x qp χ A ζ B B x v i q “ ż R p x qpB x z i q ` ż „ B x r R p x qprp χ A ζ B q s q s ´ R p x qpp χ A ζ B q q ´ B x r R p x qp χ A ζ B q χ A ζ B s v i ` ż „ B x r R p x qp χ A ζ B q s ` R p x qp χ A ζ B q χ A ζ B ´ R p x qpp χ A ζ B q q pB x v i q “ : R ` R ` R . (B.9)Firstly, we will focus on R . The term that accompanies to v i , holds the following decomposition B x r R p x qprp χ A ζ B q s q s ´ R p x qpp χ A ζ B q q ´ B x r R p x qp χ A ζ B q χ A ζ B s“ χ A ζ B ˜ R p x q ` E p R p x q , x q ζ B , (B.10)where ˜ R p x q “ ´ R p x q « ζ p q B ζ B ` ζ B ζ B ζ B ζ B ff ´ R p x q ζ B ζ B ´ R p x q ζ B ζ B , (B.11) and E p R p x q , x q “ ´ R p x q ˆ χ p q A χ A ` χ A χ A ζ B ζ B ` χ A χ A ζ B ζ B ` p χ A q ζ B ζ B ˙ ´ R p x q ˆ χ A χ A ` χ A χ A ζ B ζ B ` χ A χ A ζ B ζ B ˙ ´ R p x q ˆ χ A χ A ` r χ A s ζ B ζ B ˙ . (B.12)Rewriting R , we obtain R “ ż ˜ R p x q z i ` ż E p R p x q , x q ζ B v i . (B.13)Secondly, for R , the term that accompanies to pB x v i q satisfies the following decomposition B x r R p x qp χ A ζ B q s ` R p x qp χ A ζ B q χ A ζ B ´ R p x qpp χ A ζ B q q “ P R p x q χ A ζ B ` E p R p x q , x q ζ B , (B.14)where P R p x q “ R p x q „ ζ B ζ B ´ ˆ ζ B ζ B ˙ ` R p x q ζ B ζ B , (B.15)and E p R p x q , x q “ R p x q „ χ A χ A ´ p χ A q ` ζ B ζ B p χ A q ` R p x qp χ A q . (B.16)Now, by Claim 4.4, we have ż P R p x q χ A ζ B pB x v i q “ ż P R p x qpB x z i q ` ż „ P R p x q ζ B ζ B ` P R p x q ζ B ζ B z i ` ż E p P R p x q , x q ζ B v i , (B.17)where E is given by (B.5). Finally, we obtain that R has the following decomposition R “ ż P R p x qpB x z i q ` ż „ P R p x q ζ B ζ B ` P R p x q ζ B ζ B z i ` ż E p P R p x q , x q ζ B v i ` ż E p R p x q , x q ζ B pB x v i q . (B.18)Collecting R , (B.13) and (B.18), we obtain ż R p x q χ A ζ B pB x v i q “ ż R p x qpB x z i q ` ż ˜ R p x q z i ` ż E p R p x q , x q ζ B v i ` ż P R p x qpB x z i q ` ż „ P R p x q ζ B ζ B ` P R p x q ζ B ζ B z i ` ż E p P R p x q , x q ζ B v i ` ż E p R p x q , x q ζ B pB x v i q , (B.19)where E , E , E and P R are given in (B.5), (B.12), (B.16) and (B.15), respectively. Finally, theproof of (4.20) is direct. This concludes the proof of the Claim 4.5. Appendix C. Proof of Lemma 6.1
Proof.
We claim that for all v P H p R q that satisfies x L φ , v y “ , one has x L v, v y ě . Then the conclusion is evident since x Q , v y “ . Suppose that for some nonzero u P H p R q with x L φ , u y “ , we have x L u, u y ă . Then, since φ satisfies (1.10), x L φ , φ y “ ν xB ´ x φ , φ y “ ν xB ´ x φ , B x B ´ x φ y “ ´ ν }B ´ x φ } L ă . SYMPTOTICS FOR GOOD BOUSSINESQ SOLITONS 51
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Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático (UMI 2807CNRS), Universidad de Chile, Casilla 170 Correo 3, Santiago, Chile.
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