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

TL;DR

This paper proves the existence of a solution to the focusing nonlinear Klein-Gordon equation that asymptotically resembles two solitary waves whose distance grows logarithmically over time.

Contribution

It demonstrates the existence of two-solitary wave solutions with logarithmic separation for the nonlinear Klein-Gordon equation, a novel configuration not previously established.

Findings

Existence of solutions with two solitary waves with logarithmic separation.

Asymptotic behavior of the solution approaching two solitary waves.

Logarithmic growth of the distance between the solitary waves.

Abstract

$\newcommand\normt[1]{\left\lVert#1\right\rVert_{L^2}} \newcommand\normo[1]{\left\lVert#1\right\rVert_{H^1}} \newcommand\normpro[1]{\left\lVert#1\right\rVert_{E}}$ We consider the focusing nonlinear Klein-Gordon (NLKG) equation \begin{equation*} \partial_{tt}u - \Delta u + u - |u|^{p-1}u = 0,\quad (t,x)\in \mathbb{R}\times \mathbb{R}^d \end{equation*} for $1\leq d\leq 5$ 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*}