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

María S. Bruzón, Rafael de la Rosa, María L. Gandarias and Rita Tracinà
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María S. Bruzón: Departamento de Matemáticas, Universidad de Cádiz, 11510 Puerto Real, Spain
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

Mathematics, 2022, vol. 10, issue 2, 1-19

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.

Keywords: third-order partial differential equations; lie symmetries; solvable symmetry algebras; group invariant solutions (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
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