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UFN, 2021, Volume 191, Number 1, Pages 52–87 (Mi ufn6883)  

PHYSICS OF OUR DAYS

Gurevich–Pitaevskii problem and its development

A. M. Kamchatnovab

a Institute of Spectroscopy, Russian Academy of Sciences, Troitsk, Moscow
b Moscow Institute of Physics and Technology (National Research University), Dolgoprudny, Moscow Region

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., Vol. 65, p. 590 (1973) [Sov. Phys. JETP, Vol. 38, p. 291 (1974)]) based on Whitham's 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.

DOI: https://doi.org/10.3367/UFNr.2020.08.038815

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English version:
Physics–Uspekhi, 2021, 64:1, 48–82

Bibliographic databases:

PACS: 03.75.Kk, 03.75.Lm, 05.45.Yv, 42.65.-k, 42.65.Tg, 47.35.Fg, 67.85.Fg
Received: July 21, 2020
Accepted: August 11, 2020

Citation: A. M. Kamchatnov, “Gurevich–Pitaevskii problem and its development”, UFN, 191:1 (2021), 52–87; Phys. Usp., 64:1 (2021), 48–82

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