Evolution of a Whitham zone in the Korteweg-de Vries theory. (English. Russian original) Zbl 0655.65132

Avilov, V. V.; Krichever, I. M.; Novikov, S. P.

Sov. Phys., Dokl. 32, No. 7, 564-566 (1987); translation from Dokl. Akad. Nauk SSSR 295, 345-349 (1987).

Consider the Korteweg-de Vries-Burgers equation with low viscosity \(\mu >0\), \((1)\quad u_ t+u_{xxx}+uu_ x+\mu u_{xx}=0,\quad | \mu u_{xx}| \ll | u_{xxx}|,| uu_ x|.\) The boundary conditions are \(u\to A_{\pm}\), \(x\to \pm \infty\). The evolution of an oscillatory zone is described in the framework of the Bogolyubov-Whitham averaging method using a set of cnoidal travelling waves of (1). The present paper is devoted to a numerical study of the evolution of an oscillatory zone in the “step decay” when there is a low viscosity present.

Reviewer: J.H.Tian

MSC:

65Z05	Applications to the sciences
35Q99	Partial differential equations of mathematical physics and other areas of application

Keywords:

Whitham zone; Korteweg-de Vries theory; Korteweg-de Vries-Burgers equation; oscillatory zone; Bogolyubov-Whitham averaging method; travelling waves