Long-time asymptotics of the coupled nonlinear Schrödinger equation in a weighted Sobolev space

Abstract

We study the Cauchy problem for the focusing coupled nonlinear Schrödinger (CNLS) equation with initial data q₀ lying in the weighted Sobolev space and the scattering data having n simple zeros. Based on the corresponding 3 × 3 matrix spectral problem, we deduce the Riemann-Hilbert problem (RHP) for CNLS equation through inverse scattering transform. We remove discrete spectra of initial RHP using Darboux transformations. By applying the nonlinear steepest-descent method for RHP introduced by Deift and Zhou, we compute the long-time asymptotic expansion of the solution q(x, t) to an (optimal) residual error of order $O\left(t^{-3/4+1/\left(2p\right)}\right)$ where 2 ≤ p < ∞. The leading order term in this expansion is a multi-soliton whose parameters are modulated by soliton-soliton and soliton-radiation interactions. Our work strengthens and extends the earlier work regarding long-time asymptotics for solutions of the nonlinear Schrödinger equation with a delta potential and even initial data by Deift and Park.