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 $$A_j\mathrm {e}^{\mathrm {i}\phi _j}\mathrm {e}^{-2\mathrm {i}B_jx}$$ , $$j=1,2$$ as $$x\rightarrow \pm \infty $$ . 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 a recent paper where we develop the Riemann–Hilbert approach and detect different asymptotic scenarios, 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.