On the Generation of Dispersive Shock Waves

TL;DR

This paper reviews analytical methods for understanding dispersive shock waves in integrable dispersive equations, highlighting their formation, structure, and universal wave patterns in the weak-dispersion regime.

Contribution

It provides a comprehensive overview of methods that rigorously analyze dispersive shock wave generation and structure in integrable systems.

Findings

Rigorous explanation of dispersive shock wave formation.

Detailed description of solution behavior near caustic curves.

Identification of universal wave patterns in dispersive shocks.

Abstract

We review various methods for the analysis of initial-value problems for integrable dispersive equations in the weak-dispersion or semiclassical regime. Some methods are sufficiently powerful to rigorously explain the generation of modulated wavetrains, so-called dispersive shock waves, as the result of shock formation in a limiting dispersionless system. They also provide a detailed description of the solution near caustic curves that delimit dispersive shock waves, revealing fascinating universal wave patterns.