Nonlinear evolution equations, rescalings, model PDEs and their integrability. I. (English) Zbl 0645.35087

Calogero, Francesco; Eckhaus, Wiktor

Inverse Probl. 3, 229-262 (1987).

Summary: We study the problem of wave modulation for a large and quite general class of nonlinear evolution equations. We demonstrate that only a very limited number of ‘universal’ model equations, on relevant time and space scales, describe the phenomena of interest under all circumstances. Classical among the model equations is of course the nonlinear Schrödinger equation (NLS); however, under certain conditions, modulations occur on shorter time and space scales than those relevant for the NLS. On the other hand, if the NLS becomes linear by cancellation of terms, then appropriate model equations exist on longer time and space scales. The limited number of model equations suggest that they should have wide applicability.
Our method of analysis consists basically of Fourier decomposition followed by rescalings and appropriate limits. If one adheres to the principle that integrability properties are inherited through such limit procedures, then the model equations should be integrable (in some sense) under a wide range of conditions. Through our investigation of integrability properties we find this expectation largely confirmed.

MSC:

35Q99	Partial differential equations of mathematical physics and other areas of application
35G20	Nonlinear higher-order PDEs
35B40	Asymptotic behavior of solutions to PDEs

Keywords:

wave modulation; nonlinear evolution; nonlinear Schrödinger equation; Fourier decomposition; rescalings; appropriate limits