Topological discrete kinks

TL;DR

This paper introduces a novel spatial discretization of the nonlinear Klein-Gordon model that preserves topological properties, eliminating the Peierls-Nabarro barrier and enabling more accurate numerical simulations of kink dynamics.

Contribution

A new discretization method for Klein-Gordon models that maintains topological lower bounds and removes kink pinning, improving numerical simulations of kink behavior.

Findings

Kinks can be centered anywhere relative to the lattice with no energy barrier.

Numerical simulations show reduced kink pinning and radiation.

The method works well even on coarse lattices, mimicking continuum behavior.

Abstract

A spatially discrete version of the general kink-bearing nonlinear Klein-Gordon model in (1+1) dimensions is constructed which preserves the topological lower bound on kink energy. It is proved that, provided the lattice spacing h is sufficiently small, there exist static kink solutions attaining this lower bound centred anywhere relative to the spatial lattice. Hence there is no Peierls-Nabarro barrier impeding the propagation of kinks in this discrete system. An upper bound on h is derived and given a physical interpretation in terms of the radiation of the system. The construction, which works most naturally when the nonlinear Klein-Gordon model has a squared polynomial interaction potential, is applied to a recently proposed continuum model of polymer twistons. Numerical simulations are presented which demonstrate that kink pinning is eliminated, and radiative kink deceleration greatly reduced in comparison with the conventional discrete system. So even on a very coarse lattice, kinks behave much as they do in the continuum. It is argued, therefore, that the construction provides a natural means of numerically simulating kink dynamics in nonlinear Klein-Gordon models of this type. The construction is compared with the inverse method of Flach, Zolotaryuk and Kladko. Using the latter method, alternative spatial discretizations of the twiston and sine-Gordon models are obtained which are also free of the Peierls-Nabarro barrier.