On asymptotic stability of N-solitons of the defocusing nonlinear Schrodinger equation

Abstract

We consider the Cauchy problem for the Gross-Pitaevskii (GP) equation. Using the DBAR generalization of the nonlinear steepest descent method of Deift and Zhou we derive the leading order approximation to the solution of the GP in the solitonic region of space time $|x|<2t$ for large times and provide bounds for the error which decay as $t\to \infty$ for a general class of initial data whose difference from the non-vanishing background possess's a fixed number of finite moments and derivatives. Using properties of the scattering map for (GP) we derive as a corollary an asymptotic stability result for initial data which are sufficiently close to the N-dark soliton solutions of (GP).

48 pages, 2 Figures, in previous version the title was given as, On asymptotic stability of N solitons of the Gross Pitaevskii equation

References in corpus (2)

• Long-time Asymptotics for the NLS equation via dbar methods
• On the evolution of scattering data under perturbations of the Toda lattice

Cited by in corpus (14)

• Soliton resolution for the Wadati-Konno-Ichikawa equation with weighted Sobolev initial data
• Long-Time Behavior of Solutions to the Derivative Nonlinear Schrödinger Equation for Soliton-Free Initial Data
• Long-time asymptotic behavior of the modified Schrödinger equation via Dbar-steepest descent method
• Soliton Resolution for the Short-pluse Equation
• On asymptotic approximation of the modified Camassa-Holm equation in different space-time solitonic regions
• Soliton resolution for a coupled generalized nonlinear Schrödinger equations with weighted Sobolev initial data
• Soliton resolution for the complex short pulse equation with weighted Sobolev initial data
• On the asymptotic stability of $N$-soliton solutions of the three-wave resonant interaction equation
• Long time asymptotic behavior for the derivative Schrödinger equation with nonzero boundary conditions
• Long time asymptotic for the Wadati-Konno-Ichikawa equation with finite density initial data
• On the asymptotic stability of $N$-soliton solutions of the modified nonlinear Schrödinger equation
• Long-time asymptotic behavior for the Novikov equation in solitonic regions of space time
• Long time asymptotics for the focusing nonlinear Schrödinger equation in the solitonic region with the presence of high-order discrete spectrum
• Orbital stability in the cubic defocusing NLS equation: II. The black soliton