This article is cited in 9 papers

Solution asymptotics at large times for the non-linear Schrödinger equation

P. I. Naumkin

M. V. Lomonosov Moscow State University

Abstract: We consider a spatially uniform asymptotic representation at large times of the solution to the Cauchy problem for the non-linear Schrödinger equation. If the non-linear term decreases in time faster than the linear terms, then the asymptotics are quasi-linear. Of particular interest is the case in which the non-linearity decreases in time at the same rate as or even more slowly then the linear terms and thus has a stronger effect on the solution asymptotics at large times. In this paper we employ an appropriate change of variables to reduce this case to the quasi-linear one. Namely, we derive an integral equation with rapidly decreasing non-linearity for the new unknown function, which can be solved by the method of successive approximations. Thus, we have a constructive algorithm for calculating the asymptotics of the solution to the Cauchy problem for the non-linear Schrödinger equation from the initial data.

MSC: 35Q55, 58F07

Received: 12.09.1995

DOI: 10.4213/im137

 English version: Izvestiya: Mathematics, 1997, 61:4, 757–794

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