Existence of two-solitary waves with logarithmic distance for the nonlinear Klein-Gordon equation

Abstract

We consider the focusing nonlinear Klein-Gordon (NLKG) equation \begin{equation*} \partial_{tt}u - Δu + u - |u|^{p-1}u = 0,\quad (t,x)\in \mathbb{R}\times \mathbb{R}^d \end{equation*} for $1\le d\le 5$ and $p>2$ subcritical for the ${\dot{H}}^1$ norm. In this paper we show the existence of a solution $u(t)$ of the equation such that \begin{equation*} \normo{u(t) - \sum_{k=1,2}Q_k(t)} + \normt{\partial_t u(t)} \to 0\quad \mbox{as $t\to +\infty$,} \end{equation*} where $Q_k(t,x)$ are two solitary waves of the equation with translations $z_k:\mathbb{R}\to {\mathbb{R}}^d$ satisfying \begin{equation*} |z_1(t) - z_2(t)| \sim 2\log(t)\quad \text{as } t\to +\infty. \end{equation*}

19 pages, minor update