Dispersive Shock Waves in Lattices: A Dimension Reduction Approach

TL;DR

This paper introduces a dimension reduction method to analyze dispersive shock waves in lattice systems, making the complex Whitham modulation theory more tractable for discrete media.

Contribution

It demonstrates how DSW dynamics in lattices can be accurately described by a low-dimensional ODE, bridging the gap between continuous and discrete systems.

Findings

Low-dimensional ODE accurately captures DSW orbits in lattice systems.

Data-driven approaches identify the underlying structure of DSWs.

Connection established between the reduced model and Whitham modulation theory.

Abstract

Dispersive shock waves (DSWs), which connect states of different amplitude via a modulated wave train, form generically in nonlinear dispersive media subjected to abrupt changes in state. The primary tool for the analytical study of DSWs is Whitham's modulation theory. While this framework has been successfully employed in many space-continuous settings to describe DSWs, the Whitham modulation equations are virtually intractable in most spatially discrete systems. In this article, we illustrate the relevance of the reduction of the DSW dynamics to a planar ODE in a broad class of lattice examples. Solutions of this low-dimensional ODE accurately describe the orbits of the DSW in self-similar coordinates and the local averages in a manner consistent with the modulation equations. We use data-driven and quasi-continuum approaches within the context of a discrete system of conservation laws to demonstrate how the underlying low dimensional structure of DSWs can be identified and analyzed. The connection of these results to Whitham modulation theory is also discussed.