Reality conditions for the KdV equation and exact quasi-periodic solutions in finite phase spaces

IF 1.6 3区数学 Q1 MATHEMATICS

Journal of Geometry and Physics Pub Date : 2024-09-17 DOI:10.1016/j.geomphys.2024.105322

Julia Bernatska

引用次数: 0

Abstract

In the present paper reality conditions for quasi-periodic solutions of the KdV equation are determined completely. As a result, solutions in the form of non-linear waves can be plotted and investigated.

The full scope of obtaining finite-gap solutions of the KdV equation is presented. It is proven that the multiply periodic

℘_{1, 1}

-function on the Jacobian variety of a hyperelliptic curve of arbitrary genus serves as the finite-gap solution, the genus coincides with the number of gaps. The subspace of the Jacobian variety where

℘_{1, 1}

, as well as other ℘-functions, are bounded and real-valued is found in any genus. This result covers every finite phase space of the KdV hierarchy, and can be extended to other completely integrable equations. A method of effective computation of this type of solutions is suggested, and illustrated in genera 2 and 3.

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KdV 方程的现实条件和有限相空间中的精确准周期解

本文完全确定了 KdV 方程准周期解的现实条件。因此，可以绘制和研究非线性波形式的解。本文介绍了获得 KdV 方程有限间隙解的全部范围。研究证明，任意种属的超椭圆曲线雅各布曲线上的乘周期℘1,1-函数可作为有限间隙解，其种属与间隙数重合。在任意种属中，都可以找到雅各布变中℘1,1 以及其他℘函数都是有界实值的子空间。这一结果涵盖了 KdV 层次的每个有限相空间，并可扩展到其他完全可积分方程。我们提出了一种有效计算这类解的方法，并在属 2 和属 3 中进行了说明。

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