# zbMATH — the first resource for mathematics

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