Long-time asymptotics for the nonlocal nonlinear Schrödinger equation with step-like initial data

Abstract

We study the Cauchy problem for the integrable nonlocal nonlinear Schrödinger (NNLS) equationiqt(x,t)+qxx(x,t)+2q2(x,t)q¯(−x,t)=0 with a step-like initial data: q(x,0)=q0(x), where q0(x)=o(1) as x→−∞ and q0(x)=A+o(1) as x→∞, with an arbitrary positive constant A>0. The main aim is to study the long-time behavior of the solution of this problem. We show that the asymptotics has qualitatively different form in the quarter-planes of the half-plane −∞<x<∞, t>0: (i) for x<0, the solution approaches a slowly decaying, modulated wave of the Zakharov-Manakov type; (ii) for x>0, the solution approaches the “modulated constant”. The main tool is the representation of the solution of the Cauchy problem in terms of the solution of an associated matrix Riemann-Hilbert (RH) problem and the consequent asymptotic analysis of this RH problem.