The long-time behaviour for perturbed wave-equations and related problems. (English) Zbl 0629.35085

Eckhaus, Wiktor

Trends in applications of pure mathematics to mechanics, Proc. 6th Symp., Bad Honnef/FRG 1985, Lect. Notes Phys. 249, 168-194 (1986).

[For the entire collection see Zbl 0591.00026.]
The author describes a transformation method in order to derive various nonlinear evolution equations as formal limits of perturbed equations which one can get as the results of certain transformations of the variables from other perturbed equations. This method of deriving some of the important nonlinear evolution equations has the advantage of avoiding long formal computations when using the procedure of formal expansions and multiple scaling. The transformation method used here is demonstrated by various explicit examples, e.g. for the derivation of the KdV equation from a perturbed wave equation, the KdV and BBM equations for water waves and the KdV-Burgers equation for ion acoustic waves from perturbed hyperbolic systems.

Reviewer: H.Lange

MSC:

35L70	Second-order nonlinear hyperbolic equations
35Q99	Partial differential equations of mathematical physics and other areas of application
35B20	Perturbations in context of PDEs

Keywords:

transformation method; nonlinear evolution equations; perturbed equations; formal expansions; multiple scaling; KdV equation; water waves; ion acoustic waves

Citations:

Zbl 0591.00026