Discrete equations and auto-traveling kinks of the $\phi^6$ model

TL;DR

This paper develops systematic lattice discretizations of the $^6$ model that support static, translationally invariant kinks and explores their dynamic properties, including auto-traveling and radiation emission.

Contribution

The authors introduce a systematic method for discretizing the $^6$ model to support static kinks and analyze their spectral and dynamic behaviors.

Findings

Discrete kinks lack internal modes, aligning with continuum theory.

Generic discretizations do not support static kinks, leading to auto-traveling solutions.

Auto-traveling kinks propagate at maximum group velocity and emit radiation.

Abstract

We study the $\phi^{6}$ model and derive two broad classes of lattice discretizations that admit static, translationally invariant kinks; that is, stationary kink profiles that can be centered at an arbitrary position relative to the lattice. These discretizations are constructed using a one-dimensional map, $\phi_{n+1}=F(\phi_{n})$, which provides a direct and systematic algorithm for generating such models. Numerical computations for two representative cases show that the discrete kinks do not possess internal modes, consistent with the continuum theory, although an additional high-frequency mode may appear above the phonon band. We also show that generic discretizations of the $\phi^{6}$ model do not support static kink solutions. Instead, the resulting dynamics produce auto-traveling and self-accelerating kinks that propagate at the maximal group velocity while continuously emitting radiation.