Asymptotic stage of modulation instability for the nonlocal nonlinear Schrödinger equation

We study the initial value problem for 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. describe asymptotic stage of modulation instability NNLS by computing large-time asymptotics solution $q(x,t)$ this problem. shown that it exhibits a non-universal, in sense, behavior: $|q(x,t)|$ depends on details data $q(x,0)$. This sharp contrast local classical NLS equation, long-time through phase parameters only. The main tool used work inverse scattering transform method applied form matrix Riemann-Hilbert associated original analyzed asymptotically steepest decent method.