Gurevich-Pitaevskii problem and its development
Pattern Formation and Solitons
2021-05-26 v1
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.
Keywords
Cite
@article{arxiv.2104.14835,
title = {Gurevich-Pitaevskii problem and its development},
author = {A. M. Kamchatnov},
journal= {arXiv preprint arXiv:2104.14835},
year = {2021}
}
Comments
38 pages, 18 figures