Waves in Strongly Nonlinear Gardner-like Equations on a Lattice

TL;DR

This paper introduces a family of strongly nonlinear lattice equations extending the Gardner equation, revealing complex solitary wave patterns unique to the discrete setting, including interlaced and reversing waves.

Contribution

It presents a new class of lattice equations with rich solitary wave dynamics, highlighting phenomena exclusive to discrete systems not observed in continuum models.

Findings

Discovery of complex solitary wave patterns in the lattice equations

Identification of interlaced solitary wave pairs

Observation of waves reversing direction spontaneously or after collisions

Abstract

We introduce and study a family of lattice equations which may be viewed either as a strongly nonlinear discrete extension of the Gardner equation, or a non-convex variant of the Lotka-Volterra chain. Their deceptively simple form supports a very rich family of complex solitary patterns. Some of these patterns are also found in the quasi-continuum rendition, but the more intriguing ones, like interlaced pairs of solitary waves, or waves which may reverse their direction either spontaneously or due a collision, are an intrinsic feature of the discrete realm.