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Method of averaging for two-dimensional ”integrable” equations. (English. Russian original) Zbl 0688.35088

Funct. Anal. Appl. 22, No. 3, 200-213 (1988); translation from Funkts. Anal. Prilozh. 22, No. 3, 37-52 (1988).
The author deals with an averaging method usually called the nonlinear WKB method - a generalization of the classic Bogoljubov-Krylov averaging method to PDE’s. The goal of this paper is to generalize the WKB method to the case of the bidimensional “integrable”, analogous to the Lax’s equation: \([\partial_ y-L,\partial_ t-A]=0,\) where \(L=\sum^{n}_{i=0}u_ i(x,y,t)\partial^ i_ x,\) \(A=\sum^{m}_{j=0}w_ j(x,y,t)\partial^ j_ x,\) with scalar or matricial coefficients. One of the given examples is the Hochlov- Zabolowskij equation: \[ (3/4)\sigma^ 2u_{yy}+(u_ t-(3/2)uu_ x)_ x=0. \]
Reviewer: I.Toma

MSC:

35Q99 Partial differential equations of mathematical physics and other areas of application
35G20 Nonlinear higher-order PDEs
35C20 Asymptotic expansions of solutions to PDEs
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