Complex Painlevé type transient asymptotics of the focusing NLS equation: Step-like oscillating background

Abstract

In this paper, we consider the Cauchy problem for the focusing nonlinear Schrödinger (NLS) equation with a step-like background. Based on a proposed Riemann-Hilbert (RH) representation of the Cauchy problem, with nonlinear steepest descent method and a double limit technique, we derive the long-time behavior to the solution of the focusing NLS equation in a transition space-time region with $x/t$ near 0. Especially we find that the sub-leading term can be described by the solution of a fourth-order Painlevé transcendent, which is a modified version of the second member of generalized Painlevé II hierarchy. The numerical comparisons demonstrate that the asymptotic solutions agree excellently with results from direct numerical simulations.