Research Article

Group properties and similarity transformations for generalized peakon equation with cubic nonlinearities

Published in: Quaestiones Mathematicae
Volume 47 , issue 3 , 2024 , pages: 489–499

DOI: 10.2989/16073606.2023.2229558

Author(s): P.G.L Leach Institute of Systems Science, Durban University of Technology, Republic of South Africa , Andronikos Paliathanasis Institute of Systems Science, Durban University of Technology, South Africa

Keywords: Primary: 35C05 , Secondary: 35F50 , Lie symmetries , invariant functions , Geng-Xue equation , nonlinear partial differential equations

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Abstract

We perform a detailed analysis of the point transformations which leave invariant the Geng-Xue equation. We find that the Geng-Xue equation admits six Lie point symmetries which possess two three-dimensional subalgebras, the A _2,1 ⊗s A ₁ and A _3,8 Lie algebras. For the Lie point symmetries we derive the one-dimensional optimal system and we perform a classification of the corresponding invariant transformations. We demonstrate the application of the Lie symmetries by deriving similarity solutions expressed by closed-form functions.

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