A degeneration of two-phase solutions of focusing NLS via Riemann-Hilbert problems

TL;DR

This paper demonstrates how two-phase solutions of the focusing NLS equation degenerate into simpler soliton solutions in a specific limit, using Riemann-Hilbert problem techniques that avoid complex Riemann surface theory.

Contribution

It introduces a new approach to analyze the degeneration of two-phase solutions of focusing NLS via Riemann-Hilbert problems, simplifying the analysis by bypassing Riemann surface and theta-function complexities.

Findings

Two-phase solutions reduce to solitons on unstable condensate in a limit.

Riemann-Hilbert problem methods effectively analyze solution degeneration.

Analysis does not require Riemann surface or theta-function knowledge.

Abstract

Two-phase solutions of focusing NLS equation are classically constructed out of an appropriate Riemann surface of genus two, and expressed in terms of the corresponding theta-function. We show here that in a certain limiting regime such solutions reduce to some elementary ones called "Solitons on unstable condensate". This degeneration turns out to be conveniently studied by means of basic tools from the theory of Riemann-Hilbert problems. In particular no acquaintance with Riemann surfaces and theta-function is required for such analysis.