Dispersive shock waves in periodic lattices

TL;DR

This paper investigates dispersive shock waves in periodic lattices modeled by a nonlinear Schrödinger equation with a periodic potential, using approximations and modulation theory to analyze complex wave phenomena.

Contribution

It introduces a systematic approach combining tight-binding approximation and Whitham theory to analyze dispersive shock waves in periodic lattice systems.

Findings

Tight-binding approximation accurately models deep periodic potentials.

Discrete NLS captures single-band dynamics of the continuum model.

Rich non-convex dispersive hydrodynamic phenomena are identified and analyzed.

Abstract

We introduce and systematically investigate the generation of dispersive shock waves, which arise naturally in physical settings such as optical waveguide arrays and superfluids confined within optical lattices. The underlying physically relevant model is a nonlinear Schr\"odinger (NLS) equation with a periodic potential. We consider the evolution of piecewise smooth initial data composed of two distinct nonlinear periodic eigenmodes. To begin interpreting the resulting wave dynamics, we employ the tight-binding approximation, reducing the continuous system to a discrete NLS (DNLS) model with piecewise constant initial data (i.e., a Riemann problem), where each constant state represents a discrete Floquet-Bloch mode at the continuum model level. The resulting tight-binding approximation is shown to display higher-fidelity for {deeper} periodic potentials. This reduced DNLS model effectively models the dynamics at the minima of the periodic potential of the original continuum NLS. Within such a single-band DNLS framework, we apply tools from Whitham modulation theory and long-wave quasi-continuum reductions to uncover and analyze a rich spectrum of non-convex, discrete dispersive hydrodynamic phenomena, comparing the resulting phenomenology with that of the periodic-potential-bearing continuum model.