Soutenance mémoire
Cas des sommes de solitons
On se place dans une situation plus compliquée qu’un seul soliton. Le multi-soliton est une solution exacte de l’équation (LL) qui peut être vue comme une superposition non linéaire de plusieurs solitons découplés. Martel, Merle et Tsai [51] ont montré que si la donnée initiale est proche de la somme de N solitons alors la solution correspondante de (gKdV) converge fortement vers cette somme de soliton dans H1 (x ≥ c1t/10) où c1 est la vitesse du premier soliton en utilisant deux formule de monotonie, une sur la masse et l’autre sur l’énergie. On ne peut pas établir une convergence forte sur toute la droite réelle car dans ce cas la solution est exactement un multi-soliton [39]. Dans notre cas, on ne peut pas aboutir à une convergence forte comme pour les équations (gKdV) par la méthode de Martel, Merle et Tsai [51] à cause de l’absence d’une formule de monotonie sur l’énergie. La formule de monotonie sur le moment ne permet que d’analyser ce qui se passe autour de chaque soliton. Pour cela, on montre pour (LL) que les solitons s’éloignent l’un de l’autre de plus en plus c’est-à-dire qu’on ne peut pas avoir une interaction entre eux. On établit ensuite la stabilité asymptotique autour de chaque soliton puis celle entre les solitons.
Remarques. (i) Note that the case c = 0, that is black solitons, is excluded from the statement of
Theorem 2.1.1. In this case, the map uˇ0 vanishes and we cannot apply the Madelung transform and the subsequent arguments. Orbital and asymptotic stability remain open problems for this case. Note that, to our knowledge, there is currently no available proof of the local well-posedness of (LL) in the energy space, when u0 vanishes and so the hydrodynamical framework can no longer be used. (ii) Here, we state a weak convergence result and not a local strong convergence one, like the results given by Martel and Merle for the Korteweg-de Vries equation [45, 47]. In their situation, they can use two monotonicity formulae for the L 2 norm and the energy. This heuristically originates in the property that dispersion has negative speed in the context of the Korteweg de Vries equation. In contrast, the possible group velocities for the dispersion of the Landau-Lifshitz equation are given by vg(k)=±1+2k2√1+k2 , where k is the wave number. Dispersion has both negative and positive speeds. A monotonicity formula remains for the momentum due to the existence of a gap in the possible group velocities, which satisfy the condition |vg(k)| ≥ 1. However, there is no evidence that one can establish a monotonicity formula for the energy. Similar results were stated by Soffer and Weinstein in [58–60]. They provided the asymptotic stability of ground states for the nonlinear Schrödinger equation with a potential in a regime for which the nonlinear ground-state is a close continuation of the linear one. They rely on dispersive estimates for the linearized equation around the ground state in suitable weighted spaces, and they apply a fixed point argument. This strategy was successful extended in particular by Buslaev, Perelman, C. Sulem and Cuccagna to the nonlinear Schrödinger equations without potential (see e.g. [7–9, 11]) and with a potential (see e.g. [22]). We refer to the detailed historical survey by Cuccagna [12] for more details. Later, Cuccagna proved in [13] stronger result for the ground state satisfying the sufficient conditions for orbital stability of M. Weinstein, for seemingly generic Non-Linear Schrödinger equation which has a smooth short range nonlinearity with the presence of a very short range and smooth linear potential. In addition, asymptotic stability in spaces of exponentially localized perturbations was studied by Pego and Weinstein in [54] (see also [53] for perturbations with algebraic decay). Our strategy for establishing the asymptotic stability result in Theorem 2.1.1 is reminiscent from ideas developed by Martel and Merle for the Korteweg-de Vries equation [40, 45, 47], and successfully adapted by Béthuel, Gravejat and Smets in [5] for the Gross-Pitaevskii equation.
Plan of the paper
In the second section, we recall the orbital stability result for the multi-solitons, stated by de Laire and Gravejat in [19], which is an important tool to prove our results. In the third section, we prove the asymptotic stability around solitons. More precisely, we show that any solution close to the sum of N solitons is weakly convergent to a soliton in the translating neighbourhood of each soliton. We state that all other solitons stay far in the way that in this region the problem reduces to the asymptotic stability for a single soliton. This is the reason why we can use the Liouville type theorem proved in [1]. In the last section, we change the translation parameter to show that any solution, corresponding to an initial datum close to the sum of N solitons, converges weakly to zero when it is moving in the core of the region separating two solitons. For this, we establish a Liouville type theorem, which affirms that small solutions which are smooth and exponentially localized are zero solutions. As a consequence, (3.1.10) claims that there is no interaction between well separated solitons with ordered speed.
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Table des matières
1 Introduction
1.1 Motivation physique
1.2 État de l’art mathématique
1.2.1 Au sujet du problème de Cauchy
1.2.2 Les solitons
1.3 Stabilité asymptotique des solitons et multi-solitons de (LL) en dimension un
1.3.1 Cas d’un seul soliton
1.3.2 Cas des sommes de solitons
1.4 Conclusion
2 Asymptotic stability in the energy space for dark solitons of the Landau-Lifshitz equation
2.1 Introduction
2.2 Main steps for the proof of Theorem 2.1.1
2.2.1 The hydrodynamical framework
2.2.2 Orbital stability
2.2.3 Asymptotic stability for the hydrodynamical variables
2.2.4 Proof of Theorem 2.1.1
2.3 Proof of the orbital stability
2.4 Proofs of localization and smoothness of the limit profile
2.4.1 Proof of Proposition 2.2.2
2.4.2 Proof of Proposition 2.2.7
2.5 Proof of the Liouville theorem
2.5.1 Proof of Proposition 2.2.8
2.5.2 Proof of Lemma 2.2.1
2.5.3 Proof of Proposition 2.2.9
2.5.4 Proof of Proposition 2.2.10
2.5.5 Proof of Corollary 2.2.1
2.6 Appendix
2.6.1 Weak continuity of the hydrodynamical flow
2.6.2 Exponential decay of χc
3 On the asymptotic stability in the energy space for multi-solitons of the LandauLifshitz equation
3.1 Introduction
3.1.1 The hydrodynamical framework
3.1.2 Asymptotic stability in the original framework
3.1.3 Asymptotic stability in the hydrodynamical framework
3.1.4 Plan of the paper
3.2 Orbital stability in the hydrodynamical framework
3.3 Asymptotic stability around the solitons in the hydrodynamical variables
3.3.1 Proofs of (3.1.9) and (3.1.12)
3.3.2 Localization and smoothness of the limit profile
3.4 Asymptotic stability between the solitons in the hydrodynamical framework
3.4.1 Proof of (3.1.10)
3.4.2 Proof of the Liouville type theorem
3.4.3 Proof of Proposition 3.4.1
Bibliographie
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