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Applications of Solvable Lie Algebras to a Class of Third Order Equations

Author

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  • María S. Bruzón

    (Departamento de Matemáticas, Universidad de Cádiz, 11510 Puerto Real, Spain
    María S. Bruzón sadly passed away prior to the submission of this paper. This is one of her last works.)

  • Rafael de la Rosa

    (Departamento de Matemáticas, Universidad de Cádiz, 11510 Puerto Real, Spain)

  • María L. Gandarias

    (Departamento de Matemáticas, Universidad de Cádiz, 11510 Puerto Real, Spain)

  • Rita Tracinà

    (Dipartimento di Matematica e Informatica, Università di Catania, 95125 Catania, Italy)

Abstract

A family of third-order partial differential equations (PDEs) is analyzed. This family broadens out well-known PDEs such as the Korteweg-de Vries equation, the Gardner equation, and the Burgers equation, which model many real-world phenomena. Furthermore, several macroscopic models for semiconductors considering quantum effects—for example, models for the transmission of electrical lines and quantum hydrodynamic models—are governed by third-order PDEs of this family. For this family, all point symmetries have been derived. These symmetries are used to determine group-invariant solutions from three-dimensional solvable subgroups of the complete symmetry group, which allow us to reduce the given PDE to a first-order nonlinear ordinary differential equation (ODE). Finally, exact solutions are obtained by solving the first-order nonlinear ODEs or by taking into account the Type-II hidden symmetries that appear in the reduced second-order ODEs.

Suggested Citation

  • María S. Bruzón & Rafael de la Rosa & María L. Gandarias & Rita Tracinà, 2022. "Applications of Solvable Lie Algebras to a Class of Third Order Equations," Mathematics, MDPI, vol. 10(2), pages 1-19, January.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:2:p:254-:d:724908
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    References listed on IDEAS

    as
    1. Qiao, Zhijun & Liu, Liping, 2009. "A new integrable equation with no smooth solitons," Chaos, Solitons & Fractals, Elsevier, vol. 41(2), pages 587-593.
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