Differential invariant method for seeking nonlocally related systems and nonlocal symmetries. I: General theory and examples

Nonlocally related systems, obtained through conservation law and symmetry-based methods, have proved to be useful for determining nonlocal symmetries, nonlocal conservation laws, non-invertible mappings and new exact solutions of a given partial differential equation (PDE) system. In this paper, it...

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Bibliographic Details
Published in:Proceedings of the Royal Society. A, Mathematical, physical, and engineering sciences Mathematical, physical, and engineering sciences, 2021-06, Vol.477 (2250)
Main Authors: Bluman, George W., de la Rosa, Rafael, Santos Bruzón, María, Luz Gandarias, María
Format: Article
Language:English
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Summary:Nonlocally related systems, obtained through conservation law and symmetry-based methods, have proved to be useful for determining nonlocal symmetries, nonlocal conservation laws, non-invertible mappings and new exact solutions of a given partial differential equation (PDE) system. In this paper, it is shown that the symmetry-based method is a differential invariant-based method. It is shown that this allows one to naturally extend the symmetry-based method to ordinary differential equation (ODE) systems and to PDE systems with at least three independent variables. In particular, we present the situations for ODE systems, PDE systems with two independent variables and PDE systems with three or more independent variables, separately, and show that these three situations are directly connected. Examples are exhibited for each of the three situations.
ISSN:1471-2946
1471-2946
DOI:10.1098/rspa.2020.0908