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On the relationship between the invariance and conservation laws of differential equations. (English) Zbl 1487.35018

Summary: In this paper, we highlight the complimentary nature of the results of S. C. Anco and G. Bluman [“Direct construction method for conservation laws of partial differential equations Part I: Examples of conservation law classifications”, Eur. J. Appl. Math. 13, No. 5, 545–566 (2002; doi:10.1017/S095679250100465X); ibid. 13, No. 5, 567–585 (2002; doi:10.1017/S0956792501004661)] and N. H. Ibragimov [J. Math. Anal. Appl. 333, No. 1, 311–328 (2007; Zbl 1160.35008)] in the construction of conservation laws that whilst the former establishes the role of multipliers, the latter presents a formal procedure to determine the flows. Secondly, we show that there is an underlying relationship between the symmetries and conservation laws in a general setting, extending the results of A. H. Kara and F. M. Mahomed [Int. J. Theor. Phys. 39, No. 1, 23–40 (2000; Zbl 0962.35009)]. The results take apparently differently forms for point symmetry generators and higher-order symmetries. Similarities exist, to some extent, with a previously established result relating symmetries and multipliers of a differential equation. A number of examples are presented.

MSC:

35A30 Geometric theory, characteristics, transformations in context of PDEs
35K05 Heat equation
35Q53 KdV equations (Korteweg-de Vries equations)

Software:

GeM
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Full Text: DOI arXiv

References:

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