偏微分方程不变性缩减。II：一般框架
Invariant Reduction for Partial Differential Equations. II: The General Framework
AI总结：
本文提出一个框架，用于系统计算PDE系统的不变形式，展示Noether定理如何继承到缩减系统，并通过实例展示点和高阶对称性缩减的应用。
摘要：
对于允许局部（点、接触或更高）对称性X的偏微分方程系统F=0，不变解满足缩减系统F=φ=0。我们提出一个框架，允许对于给定PDE系统F=0的每一个X-不变守恒律、预辛结构、变分原理或其他几何结构，系统计算其对应的缩减形式，描述缩减系统F=φ=0的相应结构。特别是，我们展示了Noether定理如何在给定PDE系统中继承到缩减PDE系统。我们考虑了多个详细例子，包括点和高阶对称性不变性的情况。所提出的框架可直接应用于广泛的PDE模型，包括跨学科的当代复杂PDE系统，其中对称性缩减对分析和模拟至关重要，以及可积、拉格朗日和规范系统。
英文摘要：
For a system of partial differential equations (PDEs) $F = 0$ admitting a local (point, contact, or higher) symmetry $X$ with the characteristic $φ$, invariant solutions satisfy the reduced system $F = φ= 0$. We propose a framework that allows, for every $X$-invariant conservation law, presymplectic structure, variational principle, or another geometric structure of the given PDE system $F = 0$, to systematically calculate its corresponding reduced form that describes the corresponding structure for the reduced system $F = φ= 0$. In particular, we show in what way Noether's theorem holding for the given PDE system is inherited by the reduced PDE system. We consider several detailed examples, including cases of point and higher symmetry invariance. The proposed framework is directly applicable to a wide range of PDE models, including complex PDE systems of contemporary interest arising across disciplines, where symmetry reduction is essential for analysis and simulation, as well as to integrable, Lagrangian, and gauge systems.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
