no. 5
Dynamics of strongly interacting unstable two-solitons for generalized Korteweg-de Vries equations
Jendrej, Jacek1
1CNRS and Université Sorbonne Paris Nord LAGA, UMR 7539, 99 av J.-B. Clément 93430 Villetaneuse, France
Annales de l'Institut Fourier, Volume 75 (2025) no. 5, pp. 1925-1985

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 $u(t, x) = Q_v(x - vt - x_0)$, 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

\begin{equation} * \lim _{t\rightarrow \infty }\Vert u(t) - (Q_v({\,\cdot \,} - x_1(t)) \pm Q_v({\,\cdot \,} - x_2(t)))\Vert _{H^1} = 0, \quad \lim _{t \rightarrow \infty }x_2(t) - x_1(t) = \infty . \end{equation}

Existence of such solutions is known for $f(u) = |u|^{p-1}u$ with $p \in \mathbb{Z} \setminus \lbrace 5\rbrace $ 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 $\frac{2}{\sqrt{v}}\log (\kappa t) + o(1)$ for some $\kappa = \kappa (v) > 0$.

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 $u(t, x) = Q_v(x - vt - x_0)$, 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

\begin{equation} * \lim _{t\rightarrow \infty }\Vert u(t) - (Q_v({\,\cdot \,} - x_1(t)) \pm Q_v({\,\cdot \,} - x_2(t)))\Vert _{H^1} = 0, \quad \lim _{t \rightarrow \infty }x_2(t) - x_1(t) = \infty . \end{equation}

L’existence de telles solutions est connue pour $f(u) = |u|^{p-1}u$ avec $p \in \mathbb{Z} \setminus \lbrace 5\rbrace $ 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 $\frac{2}{\sqrt{v}}\log (\kappa t) + o(1)$ pour un certain $\kappa = \kappa (v) > 0$.

Received:
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DOI: 10.5802/aif.3708
Classification: 35Q53, 35B40, 37K40
Keywords: multi-soliton, large-time asymptotics, strong interaction

Jendrej, Jacek  1

1 CNRS and Université Sorbonne Paris Nord LAGA, UMR 7539, 99 av J.-B. Clément 93430 Villetaneuse, France
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Jendrej, Jacek. Dynamics of strongly interacting unstable two-solitons for generalized Korteweg-de Vries equations. Annales de l'Institut Fourier, Volume 75 (2025) no. 5, pp. 1925-1985. doi: 10.5802/aif.3708

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