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.

Keywords: soliton, dispersive shock wave, Bose—Einstein condensate, nonlinear optics, Gurevich—Pitaevskii problem, Whitham method PACS: 03.75.Kk, 03.75.Lm, 05.45.Yv, 42.65.−k, 42.65.Tg, 47.35.Fg, 67.85.Fg (all) DOI: 10.3367/UFNe.2020.08.038815 URL: https://ufn.ru/en/articles/2021/1/d/ 000632470400003 2-s2.0-85095480354 2021PhyU...64...48K Citation: Kamchatnov A M "Gurevich—Pitaevskii problem and its development" Phys. Usp. 64 48–82 (2021) BibTexBibNote ® (generic)BibNote ® (RIS)MedlineRefWorks Received: 21st, July 2020, accepted: 11th, August 2020 Оригинал: Камчатнов А М «Задача Гуревича—Питаевского и её развитие» УФН 191 52–87 (2021); DOI: 10.3367/UFNr.2020.08.038815