Asymptotic stage of modulation instability for the nonlocal nonlinear Schr\"odinger equation

Abstract

We study the initial value problem for the integrable nonlocal nonlinear Schr\"odinger (NNLS) equation \[ iq_{t}(x,t)+q_{xx}(x,t)+2 q^{2}(x,t)\bar{q}(-x,t)=0 \] with symmetric boundary conditions: $q(x,t)\to Ae^{2iA^2t}$ as $x\to\pm\infty$, where $A>0$ is an arbitrary constant. We describe the asymptotic stage of modulation instability for the NNLS equation by computing the large-time asymptotics of the solution $q(x,t)$ of this initial value problem. We shown that it exhibits a non-universal, in a sense, behavior: the asymptotics of $|q(x,t)|$ depends on details of the initial data $q(x,0)$. This is in a sharp contrast with the local classical NLS equation, where the long-time asymptotics of the solution depends on the initial value through the phase parameters only. The main tool used in this work is the inverse scattering transform method applied in the form of the matrix Riemann-Hilbert problem. The Riemann-Hilbert problem associated with the original initial value problem is analyzed asymptotically by the nonlinear steepest decent method.