High-speed kinks in a generalized discrete $\phi^4$ model

TL;DR

This paper introduces a generalized discrete $$ model that admits exact, stable moving kink solutions at arbitrary velocities, with a reduction in complexity from three-point to two-point maps enabling explicit solutions.

Contribution

It presents a novel generalized discrete $$ model supporting exact high-speed moving kinks and demonstrates the integrability of the solution-finding process at specific velocities.

Findings

Exact moving kink solutions exist at arbitrary velocities.

The solution-finding problem reduces from a three-point to a two-point map.

High-speed kinks are stable and resistant to perturbations.

Abstract

We consider a generalized discrete $\phi^4$ model and demonstrate that it can support exact moving kink solutions in the form of tanh with an arbitrarily large velocity. The constructed exact moving solutions are dependent on the specific value of the propagation velocity. We demonstrate that in this class of models, given a specific velocity, the problem of finding the exact moving solution is integrable. Namely, this problem originally expressed as a three-point map can be reduced to a two-point map, from which the exact moving solutions can be derived iteratively. It was also found that these high-speed kinks can be stable and robust against perturbations introduced in the initial conditions.