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首页> 外文期刊>Journal of Mathematical Physics >Framework for nonlocally related partial differential equation systems and nonlocal symmetries: Extension, simplification, and examples
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Framework for nonlocally related partial differential equation systems and nonlocal symmetries: Extension, simplification, and examples

机译：非局部相关的偏微分方程系统和非局部对称性的框架：扩展，简化和示例

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Any partial differential equation (PDE) system can be effectively analyzed through consideration of its tree of nonlocally related systems. If a given PDE system has n local conservation laws, then each conservation law yields potential equations and a corresponding nonlocally related potential system. Moreover, from these n conservation laws, one can directly construct 2(n)-1 independent nonlocally related systems by considering these potential systems individually (n singlets), in pairs (n(n-1)/2 couplets),..., taken all together (one n-plet). In turn, any one of these 2(n)-1 systems could lead to the discovery of new nonlocal symmetries and/or nonlocal conservation laws of the given PDE system. Moreover, such nonlocal conservation laws could yield further nonlocally related systems. A theorem is proved that simplifies this framework to find such extended trees by eliminating redundant systems. The planar gas dynamics equations and nonlinear telegraph equations are used as illustrative examples. Many new local and nonlocal conservation laws and nonlocal symmetries are found for these systems. In particular, our examples illustrate that a local symmetry of a k-plet is not always a local symmetry of its "completed" n-plet (k < n). A new analytical solution, arising as an invariant solution for a potential Lagrange system, is constructed for a generalized polytropic gas. (c) 2006 American Institute of Physics.
机译：通过考虑非局部相关系统的树，可以有效地分析任何偏微分方程（PDE）系统。如果给定的PDE系统具有n个局部守恒律，则每个守恒律都会得出势方程和相应的非局部相关势系统。此外，从这n个守恒定律中，可以通过成对考虑（n（n-1）/ 2对），分别考虑这些潜在系统，直接构建2（n）-1个独立的非局部相关系统。 ，合计（一n-plet）。反过来，这些2（n）-1系统中的任何一个都可能导致发现给定PDE系统的新的非局部对称性和/或非局部守恒律。而且，这样的非本地保护法可能会产生更多与本地无关的系统。证明了一个定理，该定理通过消除冗余系统简化了查找此类扩展树的框架。平面气体动力学方程和非线性电报方程用作说明性示例。这些系统发现了许多新的本地和非本地保护律以及非本地对称性。特别地，我们的示例说明了k-plet的局部对称性并不总是其“完成” n-plet（k

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