Focusing Nonlocal Nonlinear Schrödinger Equation with Asymmetric Boundary Conditions: Large-Time Behavior

Abstract

AbstractWe consider the focusing integrable nonlocal nonlinear Schrödinger equation $$\displaystyle \mathrm{i} q_{t}(x,t)+q_{xx}(x,t)+2q^{2}(x,t)\bar {q}(-x,t)=0 $$ with asymmetric nonzero boundary conditions: \(q(x,t)\to \pm A\mathrm{e} ^{-2\mathrm{i} A^2t}\) as x →±∞, where A > 0 is an arbitrary constant. The goal of this work is to study the asymptotics of the solution of the initial value problem for this equation as t → +∞. For a class of initial values we show that there exist three qualitatively different asymptotic zones in the (x, t) plane. Namely, there are regions where the parameters are modulated (being dependent on the ratio x∕t) and a central region, where the parameters are unmodulated. This asymptotic picture is reminiscent of that for the defocusing classical nonlinear Schrödinger equation, but with some important differences. In particular, the absolute value of the solution in all three regions depends on details of the initial data.KeywordsNonlocal nonlinear Schrödinger equationRiemann–Hilbert problemLarge-time asymptoticsMathematics Subject Classification35Q5337K1535Q1535B4035Q5137K40