“Operator separation of variables” in problems of short-wave asymptotic behavior for differential equations with fast oscillating coefficients. (English. Russian original) Zbl 0671.35013

Berlyand, L. V.; Dobrokhotov, S. Yu.

Sov. Phys., Dokl. 32, No. 9, 714-716 (1987); translation from Dokl. Akad. Nauk SSSR 296, 80-84 (1987).

Summary: We propose a method for finding the overall asymptotic behavior (taking focal points into account) of solutions of the Cauchy problem for partial differential equations in the case where the initial data and the coefficients of the equation oscillate with frequencies \(h^{-1}\) and \(\epsilon^{-1}\), respectively, \(h\ll \epsilon \ll 1\). The discussion is based on the example of the Cauchy problem for the Schrödinger equation describing the scattering of a wave packet (wave train) by a fast oscillating potential. We use methods developed by Maslov.

MSC:

35G10	Initial value problems for linear higher-order PDEs
35B40	Asymptotic behavior of solutions to PDEs
35B05	Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs

Keywords:

asymptotic behavior; Cauchy problem; Schrödinger equation; scattering of a wave packet; wave train; fast oscillating potential