On the generalized Fourier transform for the modified Hunter-Saxton equation

IF 1.2 3区数学 Q1 MATHEMATICS

Journal of Geometry and Physics Pub Date : 2025-09-11 DOI:10.1016/j.geomphys.2025.105647

Miao-Miao Xie, Shou-Fu Tian, Xing-Jie Yan

引用次数: 0

Abstract

In this work, by the squared eigenfunctions of the spectral problem for the modified Hunter-Saxton equation, we derive the generalized Fourier transform and the symplectic basis for the equation. First, we present the symmetry and the asymptotic behavior of the Jost solutions and the scattering data from the inverse scattering transform. Then the completeness relations of the Jost solutions and the squared eigenfunctions are derived by constructing two meromorphic functions, from which we can derive the generalized Fourier transform. Finally, we verified that a set of variables defined by the scattering data and the squared eigenfunctions form the symplectic basis of the phase space, which gives the description in symplectic geometry for the modified Hunter-Saxton equation.

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修正Hunter-Saxton方程的广义傅里叶变换

本文利用改进Hunter-Saxton方程的谱问题的平方特征函数，导出了该方程的广义傅里叶变换和辛基。首先，我们给出了Jost解的对称性和渐近性，并从散射逆变换中得到了散射数据。然后通过构造两个亚纯函数，导出了约斯特解与特征函数平方的完备关系，并由此导出了广义傅里叶变换。最后，我们验证了一组由散射数据和平方特征函数定义的变量构成相空间的辛基，给出了修正的hunt - saxton方程在辛几何中的描述。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Journal of Geometry and Physics 物理-物理：数学物理

CiteScore

2.90

自引率

6.70%

发文量

205

审稿时长

64 days

期刊介绍： The Journal of Geometry and Physics is an International Journal in Mathematical Physics. The Journal stimulates the interaction between geometry and physics by publishing primary research, feature and review articles which are of common interest to practitioners in both fields. The Journal of Geometry and Physics now also accepts Letters, allowing for rapid dissemination of outstanding results in the field of geometry and physics. Letters should not exceed a maximum of five printed journal pages (or contain a maximum of 5000 words) and should contain novel, cutting edge results that are of broad interest to the mathematical physics community. Only Letters which are expected to make a significant addition to the literature in the field will be considered. The Journal covers the following areas of research: Methods of: • Algebraic and Differential Topology • Algebraic Geometry • Real and Complex Differential Geometry • Riemannian Manifolds • Symplectic Geometry • Global Analysis, Analysis on Manifolds • Geometric Theory of Differential Equations • Geometric Control Theory • Lie Groups and Lie Algebras • Supermanifolds and Supergroups • Discrete Geometry • Spinors and Twistors Applications to: • Strings and Superstrings • Noncommutative Topology and Geometry • Quantum Groups • Geometric Methods in Statistics and Probability • Geometry Approaches to Thermodynamics • Classical and Quantum Dynamical Systems • Classical and Quantum Integrable Systems • Classical and Quantum Mechanics • Classical and Quantum Field Theory • General Relativity • Quantum Information • Quantum Gravity