Long-Time Asymptotics for Solutions of the NLS Equation with a Delta Potential and Even Initial Data

Details Long-Time Asymptotics for Solutions of the NLS Equation with a Delta Potential and Even Initial Data Long-Time Asymptotics for Solutions of the NLS Equation with a Delta Potential and Even Initial Data

2010-06-25

arxiv.org/abs/1006.4909v2

Mathematics

Analysis of PDEs

Scientific paper

We consider the one-dimensional focusing nonlinear Schr\"odinger equation (NLS) with a delta potential and even initial data. The problem is equivalent to the solution of the initial/boundary problem for NLS on a half-line with Robin boundary conditions at the origin. We follow the method of Bikbaev and Tarasov which utilizes a B\"acklund transformation to extend the solution on the half-line to a solution of the NLS equation on the whole line. We study the asymptotic stability of the stationary 1-soliton solution of the equation under perturbation by applying the nonlinear steepest-descent method for Riemann-Hilbert problems introduced by Deift and Zhou. Our work strengthens, and extends, earlier work on the problem by Holmer and Zworski.

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