The focusing NLS equation with step-like oscillating background: the genus 3 sector

Abstract

We consider the Cauchy problem for the focusing nonlinear Schrödinger equation with initial data approaching different plane waves $A_j{\mathrm{e}}^{\mathrm{i}{\varphi }_j}{\mathrm{e}}^{-2\mathrm{i}{B}_{j}x}$, $j=1,2$ as $x\to \pm \infty$. The goal is to determine the long-time asymptotics of the solution, according to the value of $\xi =x/t$. The general situation is analyzed in [7] where we develop the Riemann-Hilbert approach and detect different scenarios of asymptotic analysis, depending on the relationships between the parameters $A_1$, $A_2$, $B_1$, and $B_2$. In particular, in the shock case $B_1<B_2$, some scenarios include genus $3$ sectors, i.e., ranges of values of $\xi$ where the leading term of the asymptotics is given in terms of hyperelliptic functions attached to a Riemann surface $M(\xi )$ of genus three. The present paper is devoted to the complete asymptotic analysis in such a sector.

56 pages, 21 figures