Long-time asymptotics of (1,3)-sign solitary waves for the damped nonlinear Klein-Gordon equation

TL;DR

This paper analyzes the long-term behavior of specific solitary wave solutions to the damped nonlinear Klein-Gordon equation, showing how multi-soliton configurations evolve over time.

Contribution

It provides the first detailed description of the asymptotic dynamics of (1,3)-sign solitary waves in this damped nonlinear setting.

Findings

Three like-signed solitons spread out in an equilateral triangle.

The solution asymptotically approaches a superposition of four solitons.

The configuration persists over long times, demonstrating stability.

Abstract

We consider the damped nonlinear Klein-Gordon equation: \begin{align*} \partial_{t}^2u-\Delta u+2\alpha \partial_{t}u+u-|u|^{p-1}u=0, \ & (t,x) \in \mathbb{R} \times \mathbb{R}^d, \end{align*} where $\alpha>0$, $2\leq d\leq 5$ and energy sub-critical exponents $p>2$. In this paper, we prove that any solution which is asymptotic to a superposition of four solitons with exactly one soliton of opposite sign evolves so that the three like-signed solitons spread out in an equilateral-triangle configuration centered at the oppositely signed soliton.