Long wave asymptotics. Integrable equations as asymptotic limits of non- linear systems. (English. Russian original) Zbl 0683.35082

Russ. Math. Surv. 44, No. 1, 3-42 (1989); translation from Usp. Mat. Nauk 44, No. 1(265), 5-34 (1989).
In many cases exactly integrable PDE’s are derived by means of formal small-amplitude and/or long-wave expansions from complicated systems of equations arising in various physical problems. The paper gives a survey of results concerning rigorous mathematical substantiations of derivation of the exactly integrable asymptotic equations from the underlying “physical” systems. The substantiations are realized as theorems for estimating a difference between a solution of an asymptotic equation and that of an underlying system. The results surveyed pertain to systems for which the asymptotic exactly integrable equations are the Hopf, Burgers, Korteweg-de Vries, nonlinear Schrödinger equations, and also some two- dimensional ones.
Reviewer: B.A.Malomed


35Q99 Partial differential equations of mathematical physics and other areas of application
35A35 Theoretical approximation in context of PDEs
35B20 Perturbations in context of PDEs
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