Solution asymptotics at large times for the non-linear Schr\"odinger equation

P. I. Naumkin

Izvestiya. Mathematics , Tome 61 (1997) no. 4, pp. 757-794.

Résumé

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.

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