Mathematics

We study the initial value problem for  the integrable nonlocal nonlinear Schrodinger (NNLS) equation with the initial conditions of two types: (i) decaying  at infinity  initial conditions;(ii) step-like initial data:  Our main tool is the adaptation of the nonlinear steepest-decent method  to the study of  Riemann-Hilbert problems associated with the  NNLS equation with the specified boundary conditions.In case (i), our main result is that, in contrast to the conventional (local) NLS equation,  the power decay rate as t goes to infinity depends  on the ratio x/t.For case (ii), since our equation is not translation invariant, we explore the dependence of the asymptotic scenarios on shifts of the  step-like initial data.