Standing and Traveling Waves in a Nonlinearly Dispersive Lattice Model

TL;DR

This paper systematically studies standing and traveling wave solutions in a nonlinearly dispersive lattice model, revealing discrete compactons, localized trapped states, and stable antidark traveling waves, advancing understanding of nonlinear lattice dynamics.

Contribution

It introduces a detailed analysis of coherent structures in a nonlinear dispersive lattice, including exact traveling solutions and their stability, extending prior minimal models.

Findings

Discovery of discrete compactons as standing waves.

Identification of localized and traveling solutions in the dimer model.

Confirmation of stable antidark traveling waves on non-vanishing backgrounds.

Abstract

In the work of Colliander et al. (2010), a minimal lattice model was constructed describing the transfer of energy to high frequencies in the defocusing nonlinear Schr\"odinger equation. In the present work, we present a systematic study of the coherent structures, both standing and traveling, that arise in the context of this model. We find that the nonlinearly dispersive nature of the model is responsible for standing waves in the form of discrete compactons. On the other hand, analysis of the dynamical features of the simplest nontrivial variant of the model, namely the dimer case, yields both solutions where the intensity is trapped in a single site and solutions where the intensity moves between the two sites, which suggests the possibility of moving excitations in larger lattices. Such excitations are also suggested by the dynamical evolution associated with modulational instability. Our numerical computations confirm this expectation, and we systematically construct such traveling states as exact solutions in lattices of varying size, as well as explore their stability. A remarkable feature of these traveling lattice waves is that they are of "antidark" type, i.e., they are mounted on top of a non-vanishing background. These studies shed light on the existence, stability and dynamics of such standing and traveling states in $1+1$ dimensions, and pave the way for exploration of corresponding configurations in higher dimensions.