Long-time asymptotics for the integrable nonlocal nonlinear Schrödinger equation

Abstract

We study the initial value problem for the integrable nonlocal nonlinear Schrödinger (NNLS) equation iqt(x,t)+qxx(x,t)+2σq2(x,t) q¯ (−x,t)=0 with decaying (as x → ±∞) boundary conditions. The main aim is to describe the long-time behavior of the solution of this problem. To do this, we adapt the nonlinear steepest-decent method to the study of the Riemann-Hilbert problem associated with the NNLS equation. Our main result is that, in contrast to the local NLS equation, where the main asymptotic term (in the solitonless case) decays to 0 as O(t−1/2) along any ray x/t = const, the power decay rate in the case of the NNLS depends, in general, on x/t and can be expressed in terms of the spectral functions associated with the initial data.