Focusing NLS Equation: Long-Time Dynamics of Step-Like Initial Data

We consider the initial value problem for the focusing nonlinear Schrodinger equation with "step-like" initial data: q(x, 0) = 0 for x <= 0 and q(x, 0) = Aexp (-2iBx) for x > 0, where A> 0 and B is an element of R are constants. The paper aims at studying the long-time asymptotics of the solution to this problem. We show that there are three regions in the halfplane -infinity < x < infinity, t > 0, where the asymptotics has qualitatively different forms: a slowly decaying self-similar wave of Zakharov-Manakov type for x < -4Bt, a modulated elliptic wave for -4Bt < x < -4(B -A root 2)t, and a plane wave for x > -4(B -A root 2)t. The main tool is the asymptotic analysis of an associated matrix Riemann-Hilbert problem.