Bokhari, Ashfaque H.; Al-Dweik, Ahmad Y.; Zaman, F. D.; Kara, A. H.; Mahomed, F. M. Generalization of the double reduction theory. (English) Zbl 1201.35014 Nonlinear Anal., Real World Appl. 11, No. 5, 3763-3769 (2010). Summary: Generalization of the double reduction theory to partial differential equations of higher dimensions is still an open problem. In this note we have attempted to provide this generalization to find invariant solutions for a nonlinear system of \(q\)th order partial differential equations with \(n\) independent and \(m\) dependent variables provided that the nonlinear system of partial differential equations admits a nontrivial conserved form which has at least one associated symmetry in every reduction. In order to give an application of the procedure we apply it to the nonlinear \((2+1)\) wave equation for arbitrary function \(f(u)\) and \(g(u)\). Cited in 27 Documents MSC: 35A25 Other special methods applied to PDEs 35B06 Symmetries, invariants, etc. in context of PDEs 35L70 Second-order nonlinear hyperbolic equations Keywords:double reduction theory; conservation laws; associated symmetry; invariant solutions PDFBibTeX XMLCite \textit{A. H. Bokhari} et al., Nonlinear Anal., Real World Appl. 11, No. 5, 3763--3769 (2010; Zbl 1201.35014) Full Text: DOI arXiv References: [1] Sjöberg, A., Double reduction of PDEs from the association of symmetries with conservation laws with applications, Appl. Math. Comput., 184, 608-616 (2007) · Zbl 1116.35004 [2] Sjöberg, A., On double reductions from symmetries and conservation laws, Nonlinear Anal. RWA, 10, 6, 3472-3477 (2009) · Zbl 1179.35038 [3] Kara, A.; Mahomed, F., The relationship between symmetries and conservation laws, Int. J. Theor. Phys., 39, 1, 23-40 (2000) · Zbl 0962.35009 [4] Kara, A.; Mahomed, F., A basis of conservation laws for partial differential equations, J. Nonlinear Math. Phys., 9, Suppl. 2, 60-72 (2002) · Zbl 1362.35024 [5] Steeb, W. H.; Strampp, W., Diffusion equations and Lie and Lie-Backlund transformation groups, Physica, 114A, 95-99 (1982) · Zbl 0513.58045 [6] A. Kara, F. Mahomed, Action of Lie-Backlund symmetries on conservation laws, in: Modern Group Analysis, vol. VII, Norway, 1997.; A. Kara, F. Mahomed, Action of Lie-Backlund symmetries on conservation laws, in: Modern Group Analysis, vol. VII, Norway, 1997. [7] Anco, S. C.; Bluman, G. W., New conservation laws obtained directly from symmetry action on a known conservation law, J. Math. Anal. Appl., 322, 233-250 (2006) · Zbl 1129.35067 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.