具有全吸引力的非线性薛定谔系统
Nonlinear Schrödinger systems with all attractive forces
- KAIST（韩国科学技术院）
- Fujian Normal University（福建师范大学）
- Utah State University（犹他州立大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文系统研究全吸引力非线性薛定谔系统非负解结构，给出同步解的变分刻画、Morse指标及同步性判据，揭示相互作用矩阵谱与区域几何对解结构的影响。
AI中文摘要：
本文系统地研究了全吸引力区域中耦合非线性薛定谔系统的非负解的解结构。对于所有频率相等的情形，我们获得了同步正向量解和非负半向量解的Morse指标及其在这些解处的线性化系统的核的结果。我们建立了同步正向量解的变分刻画，并作为应用，给出了具有任意频率的一般系统的正向量解的一个新的存在性结果。我们还考察了正向量解的同步现象，证明了若相互作用矩阵恰好有一个正特征值，则任何正向量解都是同步的。最后，我们提供了当相互作用矩阵至少有两个正特征值时，正解的同步性不成立的区域例子。我们的结果揭示了耦合的相互作用矩阵的谱信息和区域的几何形状对解结构的影响。
英文摘要：
In this paper we investigate in a systematic way the solution structure of nonnegative solutions for coupled nonlinear Schrödinger systems in the all attractive regime. For all frequencies equal case we obtain results on Morse index of synchronized positive vector solutions and nonnegative semi-vector solutions as well as their kernel of linearized systems at these solutions. We establish a variational characterization of synchronized positive vector solutions and as applications we give a new existence result about positive vector solutions for the general systems with arbitrary frequencies. We also examine the synchronization phenomenon of positive vector solutions, and prove that if the interaction matrix has exactly one positive eigenvalue, any positive vector solution is synchronized. Finally we provide examples of domains for which synchronization of positive solutions fails to hold if the interaction matrix has at least two positive eigenvalues. Our results reveal the effect of the spectral information of the interaction matrix of the couplings and geometry of a domain on the solution structure.
