Approximate conservation laws of partial differential equations with a small parameter
Date
2025-03
Authors
Cheviakov, Alexei
Tarayrah, Mahmood Rajih
Yang, Zhengzheng
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The Royal Society Publishing
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Article
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Abstract
Partial differential equations (PDEs) involving perturbation terms with a small parameter often have less analytical structure, in particular, fewer symmetries and conservation laws, compared to the unperturbed PDEs. For such perturbed PDEs, approximate conservation laws can be consistently defined. The set of approximate conservation laws comprises equivalence classes where members of each class differ by a trivial approximate conservation law. Similar to exact ones, approximate conservation laws can be systematically constructed using the characteristic approach with approximate multipliers. Examples of new approximate conservation laws are presented for perturbed nonlinear heat and wave equations. For approximately variational problems, an analogue of the first Noether’s theorem relates approximate multipliers to evolutionary components of approximate local Lie symmetry generators. The multiplier method used to obtain approximate conservation laws includes the Noether approach and generalizes it to a non-variational system. The procedure to use approximate local symmetries to obtain new approximate conservation laws from known ones, in terms of fluxes and multipliers, is established and illustrated. It is shown that approximate conservation laws lead to potential systems that can be used to obtain new approximate potential symmetries of the given PDE system with a small parameter.
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Keywords
approximate conservation laws, approximate multipliers, approximate variational symmetries, partial differential equations
Citation
Cheviakov, A., Rajih Tarayrah, M., & Yang, Z. (2025). Approximate conservation laws of partial differential equations with a small parameter. Proceedings of the Royal Society. A, Mathematical, Physical, and Engineering Sciences, 481(2310). https://doi.org/10.1098/rspa.2024.0159
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DOI
https://doi.org/10.1098/rspa.2024.0159