Paper detail

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

We consider the Cauchy problem for the focusing nonlinear Schrödinger equation with initial data approaching different plane waves $A_j\mathrm{e}^{\mathrm{i}ϕ_j}\mathrm{e}^{-2\mathrm{i}B_jx}$, $j=1,2$ as $x\to\pm\infty$. The goal is to determine the long-time asymptotics of the solution, according to the value of $ξ=x/t$. 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 $ξ$ where the leading term of the asymptotics is given in terms of hyperelliptic functions attached to a Riemann surface $M(ξ)$ of genus three. The present paper is devoted to the complete asymptotic analysis in such a sector.