The focusing NLS equation with step-like oscillating background: scenarios of long-time asymptotics

Abstract

We consider the Cauchy problem for the focusing nonlinear Schrödinger equation with initial data approaching two different plane waves $A_j{\mathrm{e}}^{\mathrm{i}{\varphi }_j}{\mathrm{e}}^{-2\mathrm{i}{B}_{j}x}$, $j=1,2$ as $x\to \pm \infty$. Using Riemann-Hilbert techniques and Deift-Zhou steepest descent arguments, we study the long-time asymptotics of the solution. We detect that each of the cases $B_1<B_2$, $B_1>B_2$, and $B_1=B_2$ deserves a separate analysis. Focusing mainly on the first case, the so-called shock case, we show that there is a wide range of possible asymptotic scenarios. We also propose a method for rigorously establishing the existence of certain higher-genus asymptotic sectors.

50 pages, 52 figures

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