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.
Highlights
The study of integrable equations that model real-world phenomena has attracted a lot of attention from researchers in the last decades
A complete classification of the Lie point symmetries admitted by the family of third-order partial differential equations (PDEs) (3) involving arbitrary functions f (u), g(u), and h(u) have been determined
Taking into account the maximal symmetry groups, we have derived the solvable symmetry groups of dimension three or higher admitted by the family (3) for special forms of the functions f (u), g(u), and h(u)
Summary
The study of integrable equations that model real-world phenomena has attracted a lot of attention from researchers in the last decades. We focus our attention on deriving group-invariant solutions from admitted three-dimensional solvable symmetry subalgebras of Equation (3), which allow us to reduce the given third-order PDE into a first-order ODE. Many well-known classes of PDEs, which have been studied over the last years by using point symmetries, are included in family (3), the results obtained in this paper include numerous other equations which have not previously studied from the point of view of Lie symmetry reductions and allow a global analysis of the family considered.
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