Breather gas and shielding for the focusing nonlinear Schr\"odinger equation with nonzero backgrounds

TL;DR

This paper studies the behavior of breather solutions in the focusing nonlinear Schrödinger equation with nonzero backgrounds, revealing phenomena like breather coagulation and shielding, with implications for experimental design.

Contribution

It introduces the concept of breather shielding and analyzes the spectral concentration effects in breather gases using inverse scattering and Riemann-Hilbert methods.

Findings

Breather gas coagulates into single or multi-breather solutions.

Spectral eigenvalues concentrate in specific domains, affecting solution structure.

Results connect spectral distribution to physical breather configurations.

Abstract

Breathers have been experimentally and theoretically found in many physical systems -- in particular, in integrable nonlinear-wave models. A relevant problem is to study the \textit{breather gas}, which is the limit, for $N\rightarrow \infty $, of $N$-breather solutions. In this paper, we investigate the breather gas in the framework of the focusing nonlinear Schr\"{o}dinger (NLS) equation with nonzero boundary conditions, using the inverse scattering transform and Riemann-Hilbert problem. We address aggregate states in the form of $N$-breather solutions, when the respective discrete spectra are concentrated in specific domains. We show that the breather gas coagulates into a single-breather solution whose spectral eigenvalue is located at the center of the circle domain, and a multi-breather solution for the higher-degree quadrature concentration domain. These coagulation phenomena in the breather gas are called \textit{breather shielding}. In particular, when the nonzero boundary conditions vanish, the breather gas reduces to an $n$-soliton solution. When the discrete eigenvalues are concentrated on a line, we derive the corresponding Riemann-Hilbert problem. When the discrete spectrum is uniformly distributed within an ellipse, it is equivalent to the case of the line domain. These results may be useful to design experiments with breathers in physical settings.