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Asymptotics of the solution of the Cauchy problem for the perturbed Klein-Gordon-Fock equation. (Russian) Zbl 0659.35005

The following Cauchy problem for a perturbed Klein-Gordon-Fock equation is considered: \[ u_{tt}-u_{xx}+bu=\epsilon \cdot f(u);\quad u|_{t=0}=\phi (x,\xi),\quad u_ t|_{t=0}=\psi (x,\xi), \] where \(\phi\) (x,\(\xi)\) and \(\psi\) (x,\(\xi)\) are \(2\pi\)-periodic functions on x and \(\xi\), \(b>0\), \(\epsilon\) is a small parameter and f(u) has the form \(f(u)=\sum^{\infty}_{n=1}f_ nu^ n.\) The main term of the asymptotic solution of the above Cauchy problem uniformly for all \(t\leq O(\epsilon^{-1})\) is determined. The number of modes in the main term is increasing.
Reviewer: V.A.Yumaguzhin

MSC:

35B25 Singular perturbations in context of PDEs
35C20 Asymptotic expansions of solutions to PDEs
35L70 Second-order nonlinear hyperbolic equations