一类 Schrödinger 方程组驻波解的存在性与渐近性——献给邓引斌教授 70 寿辰

数学物理学报 ›› 2026, Vol. 46 ›› Issue (4): 1513-1528.

一类 Schrödinger 方程组驻波解的存在性与渐近性——献给邓引斌教授 70 寿辰

时龙鸽¹(), 杨小龙²^,^*()

¹ 河南财经政法大学数学与信息科学学院郑州 450046
² 河南大学数学与统计学院河南开封 475004

收稿日期:2026-01-04 修回日期:2026-04-30 出版日期:2026-08-26 发布日期:2026-06-10
通讯作者: 杨小龙 E-mail:shilg321@163.com;xlyang@henu.edu.cn
作者简介:时龙鸽,E-mail: shilg321@163.com
基金资助:
国家自然科学基金(12401130)

Existence and Asymptotic Behavior of Standing Wave Solutions for a Class of Schrödinger Systems

Longge Shi¹(), Xiaolong Yang²^,^*()

¹ College of Mathematics and Information Sciences, Henan University of Economics and Law, Zhengzhou 450046
² School of Mathematics and Statistics, Henan University, Henan Kaifeng 475004

Received:2026-01-04 Revised:2026-04-30 Online:2026-08-26 Published:2026-06-10
Contact: Xiaolong Yang E-mail:shilg321@163.com;xlyang@henu.edu.cn
Supported by:
NSFC(12401130)

摘要/Abstract

摘要：

该文研究了描述等离子体中拉曼放大模型的一类非线性 Schrödinger 方程组 (含三波相互作用) 在非聚焦情形下的驻波解. 在空间维数$ N=4$ 的情形下, 运用变分方法和紧性分析建立了该方程组基态解的存在性和非存在性. 进一步, 还刻画了基态的质量同步渐近行为, 并建立了其与 Thomas-Fermi 极限之间的精确对应关系. 该工作是在文献 [Forcella L, Luo X, Yang T, et al. arXiv: 2210.07643] 基础上对高维情形的进一步研究与推广.

关键词: NLS 方程组, 驻波, 基态解, 渐近行为

Abstract:

This paper studies standing wave solutions for a class of nonlinear Schrödinger system (involving three-wave interactions) that describe the Raman amplification model in plasmas within the non-focusing regime. For spatial dimension $N=4$, the existence and nonexistence of ground state solutions are established by means of variational methods and compactness analysis. Furthermore, the asymptotic behavior of the ground states under synchronized mass variations is characterized, and a precise correspondence with the Thomas-Fermi limit is established. This work extends and generalizes the results of [Forcella L, Luo X, Yang T, et al. arXiv: 2210.07643] to higher dimensions.

Key words: NLS system, standing waves, ground state solution, asymptotic behavior

中图分类号:

O175.23

时龙鸽, 杨小龙. 一类 Schrödinger 方程组驻波解的存在性与渐近性——献给邓引斌教授 70 寿辰[J]. 数学物理学报, 2026, 46(4): 1513-1528.

Longge Shi, Xiaolong Yang. Existence and Asymptotic Behavior of Standing Wave Solutions for a Class of Schrödinger Systems[J]. Acta mathematica scientia,Series A, 2026, 46(4): 1513-1528.

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