Defocusing Nonlocal Nonlinear Schrodinger Equation with Step-like Boundary Conditions: Long-time Behavior for Shifted Initial Data

The present paper deals with the long-time asymptotic analysis of the initial value problem for the integrable defocusing nonlocal nonlinear Schrodinger equation iq(t)(x, t) + q(xx) (x, t) - 2q(2) (x, t)(q) over bar(-x, t) = 0 with a step-like initial data: q(x, 0) -> 0 as x -> -infinity and q(x, 0) -> A as x -> +infinity. Since the equation is not translation invariant, the solution of this problem is sensitive to shifts of the initial data. We consider a family of problems, parametrized by R > 0, with the initial data that can be viewed as perturbations of the "shifted step function" q(R,A)(x): q(R,A)(x) = 0 for x < R and q(R,A)(x) = A for x > R, where A > 0 and R > 0 are arbitrary constants. We show that the asymptotics is qualitatively different in sectors of the (x, t) plane, the number of which depends on the relationship between A and R : for a fixed A, the bigger R, the larger number of sectors.