Method of averaging for two-dimensional ''integrable'' equations (Q1263744)

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: \[ (3/4)\sigma^ 2u_{yy}+(u_ t-(3/2)uu_ x)_ x=0. \]