Gurevich-Pitaevskii problem and its development

TL;DR

This paper introduces the theory of dispersive shock waves using Whitham modulation theory, illustrating its application to the Korteweg-de Vries and Gross-Pitaevskii equations, and discusses generalizations to systems with dissipation.

Contribution

It provides an accessible overview of Gurevich and Pitaevskii's approach, deriving Whitham equations for periodic solutions and extending the theory to dissipative and Bose-Einstein condensate systems.

Findings

Derivation of Whitham equations for KdV equation

Solutions to classical dispersive shock wave problems

Extension of theory to dissipative and Gross-Pitaevskii systems

Abstract

We present an introduction to the theory of dispersive shock waves in the framework of the approach proposed by Gurevich and Pitaevskii (Zh. Eksp. Teor. Fiz., 65, 590 (1973) [Sov. Phys. JETP, 38, 291 (1974)]) based on the Whitham theory of modulation of nonlinear waves. We explain how Whitham equations for a periodic solution can be derived for the Korteweg-de Vries equation and outline some elementary methods to solve them. We illustrate this approach with solutions to the main problems discussed by Gurevich and Pitaevskii. We consider a generalization of the theory to systems with weak dissipation and discuss the theory of dispersive shock waves for the Gross-Pitaevskii equation.