Long-time asymptotics for the integrable nonlocal nonlinear Schr\"odinger equation

We study the initial value problem for the integrable nonlocal nonlinear Schr"odinger (NNLS) equation $$i{q}_t(x,t)+q_{xx}(x,t)+2\sigma q^2(x,t)\bar{q}(-x,t)=0$$ with decaying (as $x\to \pm \mathrm{\infty }$) 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 \cite{DZ} 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 $O(t^{-1\mathrm{/}2})$ along any ray $x\mathrm{/}t=const$, the power decay rate in the case of the NNLS depends, in general, on $x\mathrm{/}t$, and can be expressed in terms of the spectral functions associated with the initial data.