变分形式不变约化的计算算法
Computational Algorithms for Invariant Reduction of Variational Forms
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中文总结 AI 辅助
本文提出三类计算算法，实现偏微分方程变分形式的不变约化，已在Maple中实现部分算法，示例涵盖多类方程，可将守恒律等结构约化归为单一流程。
中文摘要 AI 辅助
偏微分方程（PDE）的对称约化比其他类型的约化继承了更多几何结构：不变守恒律、变分结构，且在合适条件下，原模型的哈密顿型结构会通过不变约化机制传递到约化模型。本文开发了可显式执行此类约化的计算算法。我们将变分p-形式解释为扩阶系统（次数偏移的切系统）的守恒律，该系统由原方程及其线性化组成，其中扰动变量被视为反交换变量。这使我们能利用守恒律理论的知名概念来构建算法，并自然适配分次交换框架。由此，守恒律、变分1-形式和预辛结构的约化可归为单一算法流程。我们提出三类算法：（i）基于同伦的演化方程系统约化算法，已在Maple中实现；（ii）适用于p>0的一般ℓ-正规系统的下降约化算法；（iii）合适条件下适用于点对称的简单约化算法，包括用含更少独立变量的系统描述约化的方法。示例包括一维和二维空间的非线性演化方程、拉普拉斯方程、不可压缩欧拉方程，以及帕夫洛夫方程的余切系统。
英文摘要
Symmetry reductions of partial differential equations (PDEs) inherit more geometric structures than other types of reductions: invariant conservation laws, variational structures, and, under suitable conditions, Hamiltonian-type structures of the original model descend to the reduced model through the mechanism of invariant reduction. This paper develops computational algorithms that carry out such reductions explicitly. We use the interpretation of variational $p$-forms as conservation laws of an enlarged system --- the (degree-shifted) tangent system --- consisting of the original equations together with their linearizations, in which the perturbation variables are treated as anticommuting. This allows us to formulate the algorithms using well-known concepts from the theory of conservation laws, naturally adapted to the graded-commutative setting. The reduction of conservation laws, variational $1$-forms, and presymplectic structures thereby becomes a single algorithmic procedure. We present (i) a homotopy-based reduction algorithm for systems of evolution equations, implemented in Maple; (ii) a descent reduction algorithm applicable to general $\ell$-normal systems for $p>0$; and (iii) a simple reduction algorithm available for point symmetries under suitable conditions, including the description of reductions in terms of systems involving fewer independent variables. Examples include nonlinear evolution equations in one and two spatial dimensions, the Laplace equation, the incompressible Euler equations, and the cotangent system of Pavlov's equation.
发表机构
- University of Saskatchewan（萨斯喀彻温大学）
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