Non-Associative Structures and Their Applications in Differential Equations
Non-Associative Structures and Their Applications in Differential Equations
复制标题
非结合结构及其在微分方程中的应用
DOI：
10.3390/math11081790
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发表时间：
2023
期刊：
影响因子：
2.4
通讯作者：
H. Munthe
中科院分区：
文献类型：
--
作者：
Alexander Lundervold;H. Munthe
本文一方面建立了非线性微分方程与线性偏微分方程之间的联系，另一方面建立了非结合代数结构之间的联系。这种联系简化了微分方程许多结果的公式化及其求解方法。这些理论之间的主要联系是为代数和齐次微分方程发展的非线性谱理论。利用非线性谱方法证明了代数首次积分的存在性，各种相区的解释，以及常微分方程的分离线的构造。在代数中，同样的方法利用子代数构造和解释融合规则。总之，扰动方法也可以解释为近约旦代数的建设。
This article establishes a connection between nonlinear DEs and linear PDEs on the one hand, and non-associative algebra structures on the other. Such a connection simplifies the formulation of many results of DEs and the methods of their solution. The main link between these theories is the nonlinear spectral theory developed for algebra and homogeneous differential equations. A nonlinear spectral method is used to prove the existence of an algebraic first integral, interpretations of various phase zones, and the separatrices construction for ODEs. In algebra, the same methods exploit subalgebra construction and explain fusion rules. In conclusion, perturbation methods may also be interpreted for near-Jordan algebra construction.