| We consider the one-dimensional focusing nonlinear Schrodinger equation (NLS) on R with a delta potential qdelta0( x) and even initial data. Due to the specific choice of the potential and initial data, the equation reduces to the initial boundary value (IBV) problem for the NLS equation on a half-line with homogeneous boundary conditions at x = 0: such problems are known to be integrable by an extension of the inverse scattering method.;Among a number of different ways to solve the IBV problem, we follow the method of Bikbaev and Tarasov which utilizes a Backlund transformation to extend the solution on the half-line to a solution of the NLS equation on the whole line, in such a way that the boundary condition is automatically satisfied. The long time asymptotics for the solutions of the equation can then be obtained by the nonlinear steepest-descent method for Riemann-Hilbert problems introduced by Deift and Zhou.;The main result of this thesis concerns the asymptotic stability of the stationary 1-soliton solution of the equation under perturbation. In particular, for perturbations comparable in strength to the delta potential, we show that it takes time of order q--2 for the solution to relax down to a pure soliton oscillation. The situation is different for q > 0 and q < 0. For q > 0, the asymptotic state as t → infinity is a 1-soliton oscillation: for q < 0, the asymptotic state is generically a 2-soliton oscillation. Our work strengthens, and extends, earlier work on the problem by Holmer and Zworski. |
