On the kink-kink collision problem of for the $\phi^{6}$ model with low speed

TL;DR

This paper proves that low-speed kink collisions in the 1+1 dimensional $\

Contribution

It introduces a method to precisely analyze low-speed kink collisions in the $\

Findings

Post-collision velocities are close to initial velocities within a small error.

The energy of the residual after collision is very small.

The approach extends previous approximate solutions to analyze collision dynamics.

Abstract

We study the elasticity of the collision of two kinks with an incoming low speed $v\in (0,1)$ for the nonlinear wave equation in dimension $1+1$ known as the $\phi^{6}$ model. We prove for any $k\in\mathbb{N}$ that if the incoming speed $v$ is small enough, then, after the collision, the two kinks will move away with a velocity $v_{f}$ such that $\vert v_{f}-v\vert\leq v^{k}$ and the energy of the remainder will also be smaller than $v^{k}.$ This manuscript is the continuation of our previous paper where we constructed a sequence $\phi_{k}$ of approximate solutions for the $\phi^{6}$ model. The proof of our main result relies on the use of the set of approximate solutions from our previous work, modulation analysis, and a refined energy estimate method to evaluate the precision of our approximate solutions during a large time interval.