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

Kalyakin, L. A.

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

MSC:

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

Keywords:

asymptotic limits; nonlinear systems; small-amplitude expansions; long- wave expansions