Approximate kink-kink solutions for the $\phi^{6}$ model in the low-speed limit

TL;DR

This paper constructs approximate solutions for low-speed kink-kink collisions in the $^{6}$ model, demonstrating convergence to true solutions and providing a foundation for analyzing elastic interactions in nonlinear wave equations.

Contribution

It introduces a sequence of approximate solutions for low-speed kink-kink collisions in the $^{6}$ model, applicable to broader nonlinear wave equations.

Findings

Approximate solutions converge to true kink-kink solutions as time progresses.

Methodology is adaptable beyond the $^{6}$ model.

Provides a framework for studying elastic collision behavior.

Abstract

This manuscript is the first of a series of two papers that study the problem of elasticity and stability of the collision of two kinks with low speed $v$ for the nonlinear wave equation known as the $\phi^{6}$ model in dimension $1+1$. In this paper, we construct a sequence of approximate solutions $(\phi_{k}(v,t,x))_{k\in\mathbb{N}_{\geq 2}}$ for this nonlinear wave equation such that each function $\phi_{k}(v,t,x)$ converges in the energy norm to the traveling kink-kink with speed $v$ when $t$ goes to $+\infty.$ The methods used in this paper are not restricted only to the $\phi^{6}$ model.