Dynamics of strongly interacting unstable two-solitons for generalized Korteweg-de Vries equations

Jendrej, Jacek

doi:10.5802/aif.3708

Dynamics of strongly interacting unstable two-solitons for generalized Korteweg-de Vries equations
[Dynamique des 2-solitons instables en forte interaction pour les équations de Korteweg-de Vries généralisées]

Jendrej, Jacek ¹

¹ CNRS and Université Sorbonne Paris Nord LAGA, UMR 7539, 99 av J.-B. Clément 93430 Villetaneuse, France

Annales de l'Institut Fourier, Online first, 61 p.

Résumé
Abstract

On considère l’équation de Korteweg-de Vries généralisée $\partial_{t} u = - \partial_{x} (\partial_{x}^{2} u + f (u))$ , où $f$ est une fonction impaire de classe $C^{3}$ . Sous certaines hypothèses sur $f$ , cette équation possède des solutions de type onde progressive, c’est-à-dire $u (t, x) = Q_{v} (x - v t - x_{0})$ , pour tout $v$ dans un certain intervalle $(0, v_{*})$ . On étudie les paires de solitons pures dans le cas de la même vitesse limite, autrement dit les solutions globales $u (t)$ telles que

* lim_{t \to \infty} {∥ u (t) - (Q_{v} (\cdot - x_{1} (t)) \pm Q_{v} (\cdot - x_{2} (t))) ∥}_{H^{1}} = 0, lim_{t \to \infty} x_{2} (t) - x_{1} (t) = \infty .

L’existence de telles solutions est connue pour $f (u) = {| u |}^{p - 1} u$ avec $p \in ℤ ∖ {5}$ et $p > 2$ . On décrit le comportement dynamique de toute solution vérifiant $(*)$ , sous l’hypothèse que $Q_{v}$ soit linéairement instable (ce qui correspond à $p > 5$ si $f (u) = {| u |}^{p - 1} u$ ). On montre que dans ce cas le signe dans $(*)$ est “ $+$ ”, ce qui correspond à l’interaction attractive. On montre également que la distance $x_{2} (t) - x_{1} (t)$ entre les solitons vaut $\frac{2}{\sqrt{v}} log (κ t) + o (1)$ pour un certain $κ = κ (v) > 0$ .

We consider the generalized Korteweg-de Vries equation $\partial_{t} u = - \partial_{x} (\partial_{x}^{2} u + f (u))$ , where $f$ is an odd function of class $C^{3}$ . Under some assumptions on $f$ , this equation admits solitary waves, that is solutions of the form $u (t, x) = Q_{v} (x - v t - x_{0})$ , for $v$ in some range $(0, v_{*})$ . We study pure two-solitons in the case of the same limit speed, in other words global solutions $u (t)$ such that

* lim_{t \to \infty} {∥ u (t) - (Q_{v} (\cdot - x_{1} (t)) \pm Q_{v} (\cdot - x_{2} (t))) ∥}_{H^{1}} = 0, lim_{t \to \infty} x_{2} (t) - x_{1} (t) = \infty .

Existence of such solutions is known for $f (u) = {| u |}^{p - 1} u$ with $p \in ℤ ∖ {5}$ and $p > 2$ . We describe the dynamical behavior of any solution satisfying $(*)$ under the assumption that $Q_{v}$ is linearly unstable (which corresponds to $p > 5$ for power nonlinearities). We prove that in this case the sign in $(*)$ is necessarily “ $+$ ”, which corresponds to an attractive interaction. We also prove that the distance $x_{2} (t) - x_{1} (t)$ between the solitons equals $\frac{2}{\sqrt{v}} log (κ t) + o (1)$ for some $κ = κ (v) > 0$ .

Reçu le : 2018-02-25
Révisé le : 2020-07-21
Accepté le : 2024-03-21
Première publication : 2025-02-10

DOI : 10.5802/aif.3708

Classification : 35Q53, 35B40, 37K40
Mots-clés : multi-soliton, large-time asymptotics, strong interaction

Affiliations des auteurs :

Jendrej, Jacek ¹

¹ CNRS and Université Sorbonne Paris Nord LAGA, UMR 7539, 99 av J.-B. Clément 93430 Villetaneuse, France

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     title = {Dynamics of strongly interacting unstable two-solitons for generalized {Korteweg-de} {Vries} equations},
     journal = {Annales de l'Institut Fourier},
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     doi = {10.5802/aif.3708},
     language = {en},
     note = {Online first},
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Jendrej, Jacek. Dynamics of strongly interacting unstable two-solitons for generalized Korteweg-de Vries equations. Annales de l'Institut Fourier, Online first, 61 p.

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