Generalization of the algebraic method of group classification with application to nonlinear wave and elliptic equations

Abstract

Enhancing and essentially generalizing previous results on a class of (1+1)-dimensional nonlinear wave and elliptic equations, we apply several new techniques to classify admissible point transformations within this class up to the equivalence generated by its equivalence group. This gives an exhaustive description of its equivalence groupoid. After extending the algebraic method of group classification to non-normalized classes of differential equations, we solve the complete group classification problem for the class under study up to both usual and general point equivalences. The solution includes the complete preliminary group classification of the class and the construction of singular Lie-symmetry extensions, which are not related to subalgebras of the equivalence algebra. The complete preliminary group classification is based on classifying appropriate subalgebras of the entire infinite-dimensional equivalence algebra whose projections are qualified as maximal extensions of the kernel invariance algebra. The results obtained can be used to construct exact solutions of nonlinear wave and elliptic equations.

arXiv comment40 pages, minor corrections and extensions

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Cited by in corpus (10)

• Extended symmetry analysis of remarkable (1+2)-dimensional Fokker-Planck equation
• Realizations of Lie algebras on the line and the new group classification of (1+1)-dimensional generalized nonlinear Klein-Gordon equations
• Point-symmetry pseudogroup, Lie reductions and exact solutions of Boiti-Leon-Pempinelli system
• Mapping method of group classification
• Point and contact equivalence groupoids of two-dimensional quasilinear hyperbolic equations
• Admissible transformations and Lie symmetries of linear systems of second-order ordinary differential equations
• Extended symmetry analysis of (1+2)-dimensional fine Kolmogorov backward equation
• A closer look at predicting turbulence statistics of arbitrary moments when based on a non-modelled symmetry approach
• Similarity reductions, new traveling wave solutions, conservation laws of (2+1)- dimensional Boiti-Leon-Pempinelli system
• Surprising symmetry properties and exact solutions of Kolmogorov backward equations with power diffusivity