非零背景下的聚焦非线性薛定谔方程的呼吸子气体与屏蔽效应
Breather gas and shielding for the focusing nonlinear Schrödinger equation with nonzero backgrounds
AI总结：
本文利用逆散射变换和Riemann-Hilbert问题研究非零背景下聚焦NLS方程的呼吸子气体，发现其凝聚为单或多呼吸子解，即呼吸子屏蔽效应，并分析了谱分布的影响。
摘要：
呼吸子已在许多物理系统中被实验和理论发现，特别是在可积非线性波模型中。一个相关的问题是研究呼吸子气体，即当$N\rightarrow \infty$时，$N$-呼吸子解的极限。在本文中，我们利用逆散射变换和Riemann-Hilbert问题，在具有非零边界条件的聚焦非线性薛定谔（NLS）方程框架下研究呼吸子气体。我们处理当相应的离散谱集中在特定区域时，以$N$-呼吸子解形式的聚合态。我们证明，当离散谱集中在圆域时，呼吸子气体凝聚为单个呼吸子解，其谱特征值位于圆域中心；当离散谱集中在高次求积区域时，凝聚为多呼吸子解。呼吸子气体中的这些凝聚现象被称为呼吸子屏蔽。特别地，当非零边界条件消失时，呼吸子气体退化为$n$-孤子解。当离散特征值集中在一条线上时，我们推导出相应的Riemann-Hilbert问题。当离散谱在椭圆内均匀分布时，它等价于线域的情况。这些结果可能有助于在物理环境中设计涉及呼吸子的实验。
英文摘要：
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.
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