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Breather gas and shielding for the focusing nonlinear Schrödinger equation with nonzero backgrounds

Weifang Weng, Guoqiang Zhang, Boris A. Malomed, Zhenya Yan

TL;DR

The paper tackles the problem of describing a breather gas for the focusing nonlinear Schrödinger equation with nonzero backgrounds by employing the inverse scattering transform and a Riemann-Hilbert (RH) framework. By organizing the discrete spectrum into continuum-density domains (quadrature, line, elliptic), the authors derive reduced RH problems that reveal shielding phenomena, whereby the N→∞ limit of N-breather solutions coagulates into a single-breather or into an $n$-breather state depending on the domain geometry. Key contributions include explicit formulations for the line- and elliptic-domain limits and the demonstration that the quasilinear interactions in a densely packed breather gas can be captured by effective low-dimensional states, with the zero-background case recovering $n$-solitons. These results provide analytical predictions for breather dynamics applicable to fiber optics and Bose-Einstein condensates, and they establish a methodological pathway to analyze similar shielding effects in other integrable systems.

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 Schrödinger (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.

Breather gas and shielding for the focusing nonlinear Schrödinger equation with nonzero backgrounds

TL;DR

The paper tackles the problem of describing a breather gas for the focusing nonlinear Schrödinger equation with nonzero backgrounds by employing the inverse scattering transform and a Riemann-Hilbert (RH) framework. By organizing the discrete spectrum into continuum-density domains (quadrature, line, elliptic), the authors derive reduced RH problems that reveal shielding phenomena, whereby the N→∞ limit of N-breather solutions coagulates into a single-breather or into an -breather state depending on the domain geometry. Key contributions include explicit formulations for the line- and elliptic-domain limits and the demonstration that the quasilinear interactions in a densely packed breather gas can be captured by effective low-dimensional states, with the zero-background case recovering -solitons. These results provide analytical predictions for breather dynamics applicable to fiber optics and Bose-Einstein condensates, and they establish a methodological pathway to analyze similar shielding effects in other integrable systems.

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 , of -breather solutions. In this paper, we investigate the breather gas in the framework of the focusing nonlinear Schrödinger (NLS) equation with nonzero boundary conditions, using the inverse scattering transform and Riemann-Hilbert problem. We address aggregate states in the form of -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 -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.
Paper Structure (7 sections, 6 theorems, 44 equations, 2 figures, 1 table)

This paper contains 7 sections, 6 theorems, 44 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Breather gas: the limit of the $N$-breather solution at $N\to\infty$
  4. Quadrature domains
  5. The line domain
  6. The elliptic domain
  7. Conclusions and discussions

Key Result

Proposition 1

Let $(x,t)$ be in compact subsets of $\mathbb{R}^2$. For any open set $B_+$ containing the domain $\Omega_1$, the following identities hold: uniformly for all $\mathbb{C}\setminus B_+$. The boundary $\partial\Omega_1$ has the counterclockwise orientation.

Figures (2)

  • Figure 1: The complex $k$-plane, showing the discrete spectrum $\{k_j,\, k_j^*,\, -q_0^2/k_j,\, -q_0^2/k_j^*\}_{j=1}^N$, and the shaded area indicates region $\mathbb{D}^+$, and the white area indicates region $\mathbb{D}^-$.
  • Figure 2: (a) The distribution of discrete spectrum $K$, the parameters being $s_{1}=2i-\frac{1}{4},s_{2}=\frac{1}{4},s_{3}=\frac{1}{2},m=1,q_{0}=1$. (b) The 1-breather solution with the same parameters and $r_1(k)=4$.

Theorems & Definitions (15)

  • Proposition 1
  • Proof
  • Proposition 2
  • Proof
  • Proposition 3
  • Remark 1
  • Proposition 4
  • Proof
  • Remark 2
  • Proposition 5
  • ...and 5 more