Long-time asymptotics for the integrable nonlocal nonlinear Schrodinger equation

We study the initial value problem for the integrable nonlocal nonlinear Schrodinger (NNLS) equation iq(t)(x, t) + q(xx)(x, t) + 2 sigma q(2) (x, t)(q) over bar (-x, t) = 0 with decaying (as x -> +/- infinity) 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 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(-1/2)) along any ray x/t = const, the power decay rate in the case of the NNLS depends, in general, on x/t and can be expressed in terms of the spectral functions associated with the initial data. Published under license by AIP Publishing.