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Multiphase asymptotics of nonlinear partial differential equations with a small parameter. (English) Zbl 0551.35072

Multiphase asymptotic solutions which occur for nonlinear equations are in the form \((1)\quad u=y_ j(S_ 1(x,t)/h,...,S_ j(x,t)/h,x,t),\) where \(x=(x_ 1,...,x_ n)\) are spatial variables, t is the time variable, \(h>0\) is a small parameter, the phases \(S_ 1(x,t),...,S_ j(x,t)\) and \(y_ j(\tau_ 1,...,\tau_ j,x,t)\) are smooth functions while the \(y_ j\) are \(2\pi\)-periodic in each of the arguments \(\tau_ 1,...,\tau_ j\). Asymptotic solutions of the form (1) correspond to exact almost periodic solutions of the form \(u=y_ j(U_ 1x+V_ 1t,...,U_ jx+V_ jt)\) of certain ”model” nonlinear equations with constant coefficients \((U_ i\) and \(V_ i\) are constants). The subject of this paper is a method for constructing such asymptotics and a survey of previous results obtained by the authors. Solutions with real or complex phases are discussed. From that point of view, Korteweg-de Vries and sine-Gordon equations are studied in details.
Reviewer: D.Huet

MSC:

35Q99 Partial differential equations of mathematical physics and other areas of application
35C20 Asymptotic expansions of solutions to PDEs
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