修订的黎曼-希尔伯特(Riemann-Hilbert)方法用于具有非消失边界条件的考普-纽厄尔方程:简单极点和高阶极点

IF 1.2 3区 物理与天体物理 Q3 PHYSICS, MATHEMATICAL Journal of Mathematical Physics Pub Date : 2024-08-21 DOI:10.1063/5.0205072
Yongshuai Zhang, Deqin Qiu, Shoufeng Shen, Jingsong He
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引用次数: 0

摘要

在非消失边界条件下,我们用黎曼-希尔伯特(Riemann-Hilbert)方法研究了考普-纽厄尔(Kaup-Newell,KN)方程(或导数非线性薛定谔方程)。我们的研究得出了 KN 方程的四种 Nth 阶解,它们分别对应于ρ圆(ρ 与非消失边界条件有关)上或不上的简单极点,以及黎曼-希尔伯特问题(Riemann-Hilbert problem,RHP)ρ圆上或不上的高阶极点。我们对通常的 RHP 进行了修订,引入了一个积分因子,确保 RHP 满足归一化条件。这一点非常重要,因为当谱参数达到无穷大时,约斯特解会进入积分因子,而不是单位矩阵。为了考虑高阶极点的情况,我们在不假设势具有紧凑支撑的情况下研究了 Jost 解之间的并行化条件,并提出了 RHP 的残差条件广义,这些条件在求解具有高阶极点的 RHP 时起着至关重要的作用。我们提供了四种 Nth 阶解的显式闭式公式,以显式一阶孤子和双极孤子为例,详细研究了它们的振幅、宽度和激发碰撞等性质。
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The revised Riemann–Hilbert approach to the Kaup–Newell equation with a non-vanishing boundary condition: Simple poles and higher-order poles
With a non-vanishing boundary condition, we study the Kaup–Newell (KN) equation (or the derivative nonlinear Schrödinger equation) using the Riemann–Hilbert approach. Our study yields four types of Nth order solutions of the KN equation that corresponding to simple poles on or not on the ρ circle (ρ related to the non-vanishing boundary condition), and higher-order poles on or not on the ρ circle of the Riemann–Hilbert problem (RHP). We make revisions to the usual RHP by introducing an integral factor that ensures the RHP satisfies the normalization condition. This is important because the Jost solutions go to an integral factor rather than the unit matrix when the spectral parameter goes to infinity. To consider the cases of higher-order poles, we study the parallelization conditions between the Jost solutions without assuming that the potential has compact support, and present the generalizations of residue conditions of the RHP, which play crucial roles in solving the RHP with higher-order poles. We provide explicit closed-form formulae for four types of Nth order solutions, display the explicit first-order and double-pole solitons as examples and study their properties in more detail, including amplitude, width, and exciting collisions.
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来源期刊
Journal of Mathematical Physics
Journal of Mathematical Physics 物理-物理:数学物理
CiteScore
2.20
自引率
15.40%
发文量
396
审稿时长
4.3 months
期刊介绍: Since 1960, the Journal of Mathematical Physics (JMP) has published some of the best papers from outstanding mathematicians and physicists. JMP was the first journal in the field of mathematical physics and publishes research that connects the application of mathematics to problems in physics, as well as illustrates the development of mathematical methods for such applications and for the formulation of physical theories. The Journal of Mathematical Physics (JMP) features content in all areas of mathematical physics. Specifically, the articles focus on areas of research that illustrate the application of mathematics to problems in physics, the development of mathematical methods for such applications, and for the formulation of physical theories. The mathematics featured in the articles are written so that theoretical physicists can understand them. JMP also publishes review articles on mathematical subjects relevant to physics as well as special issues that combine manuscripts on a topic of current interest to the mathematical physics community. JMP welcomes original research of the highest quality in all active areas of mathematical physics, including the following: Partial Differential Equations Representation Theory and Algebraic Methods Many Body and Condensed Matter Physics Quantum Mechanics - General and Nonrelativistic Quantum Information and Computation Relativistic Quantum Mechanics, Quantum Field Theory, Quantum Gravity, and String Theory General Relativity and Gravitation Dynamical Systems Classical Mechanics and Classical Fields Fluids Statistical Physics Methods of Mathematical Physics.
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