Dynamics of two interacting kinks for the $\phi^{6}$ model

TL;DR

This paper analyzes the long-term dynamics of two interacting kinks in the 1+1 dimensional $^{6}$ model, proving their orbital stability and providing detailed estimates of their behavior over time.

Contribution

It establishes the orbital stability of two moving kinks and describes their long-time behavior with explicit estimates, a novel result for the $^{6}$ model.

Findings

Proves orbital stability of two moving kinks.

Provides explicit long-time behavior estimates.

Achieves optimal energy norm estimates for the remainder.

Abstract

We consider the nonlinear wave equation known as the $\phi^{6}$ model in dimension 1+1. We describe the long time behavior of all the solutions of this model close to a sum of two kinks with energy slightly larger than twice the minimum energy of non constant stationary solutions. We prove orbital stability of two moving kinks. We show for low energy excess $\epsilon$ that these solutions can be described for long time less o equivalent than $-\ln{(\epsilon)}\epsilon^{-\frac{1}{2}}$ as the sum of two moving kinks such that each kink's center is close to an explicit function which is a solution of an ordinary differential system. We give an optimal estimate in the energy norm of the remainder $(g(t),\partial_{t}g(t))$ and we prove that this estimate is achieved during a finite instant $t=T\lesssim -\ln{(\epsilon)}\epsilon^{-\frac{1}{2}}.$