TY - JOUR

T1 - Long-Time Asymptotics for Solutions of the NLS Equation with a Delta Potential and Even Initial Data

T2 - Announcement of Results

AU - Deift, Percy

AU - Park, Jungwoon

N1 - Funding Information:
The work of the first author was supported in part by NSF Grant DMS-0500923 and NSF Grant DMS-1001886. The authors would like to thank Alexander Its, Maciej Zworski and Justin Holmer for very useful information and discussions.

PY - 2011/6

Y1 - 2011/6

N2 - We consider the one-dimensional focusing nonlinear Schrödinger 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 Bäcklund 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, the earlier work on the problem by Holmer and Zworski.

AB - We consider the one-dimensional focusing nonlinear Schrödinger 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 Bäcklund 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, the earlier work on the problem by Holmer and Zworski.

KW - Riemann-Hilbert problem

KW - initial-boundary value problem

KW - long-time behavior

KW - nonlinear Schrödinger equation

KW - nonlinear steepest descent

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U2 - 10.1007/s11005-010-0458-5

DO - 10.1007/s11005-010-0458-5

M3 - Article

AN - SCOPUS:79954756914

VL - 96

SP - 143

EP - 156

JO - Letters in Mathematical Physics

JF - Letters in Mathematical Physics

SN - 0377-9017

IS - 1-3

ER -