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

Abstract

We consider the generalized Korteweg-de Vries equation ${\partial }_{t}u=-{\partial }_x({\partial }_x^{2}u+f(u))$, where $f(u)$ is an odd function of class $C^3$. Under some assumptions on $f$, this equation admits \emph{solitary waves}, that is solutions of the form $u(t,x)=Q_v(x-vt-x_0)$, for $v$ in some range $(0,v_{\ast })$. We study pure two-solitons in the case of the same limit speed, in other words global solutions $u(t)$ such that \begin{equation} \label{eq:abstract} \tag{$\ast$} \lim_{t\to\infty}\|u(t) - (Q_v(\cdot - x_1(t)) \pm Q_v(\cdot - x_2(t)))\|_{H^1} = 0, \qquad \text{with}\quad\lim_{t \to \infty}x_2(t) - x_1(t) = \infty. \end{equation} Existence of such solutions is known for $f(u)=|u{|}^{p-1}u$ with $p\in \mathbb{Z}\setminus \{5\}$ and $p>2$. We describe the~dynamical behavior of any solution satisfying \eqref{eq:abstract} 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 \eqref{eq:abstract} 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 $\kappa =\kappa (v)>0$.

45 pages. The new version takes into account the corrections of the referees. To appear in Annales de l'Institut Fourier