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

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.

Reductions and Conservation Laws of a Generalized Third-Order PDE via Multi-Reduction Method

“…Most importantly, the reduced ODE is separable, yielding a straightforward quadrature which gives the general traveling wave solution of PDE (2) for arbitrary f (u) and g(u). Thus, the results obtained in this paper on traveling wave solutions generalize those included in [33].…”

“…Proposition 2. The multipliers of the generalized third-order PDE (2) which are invariant under the translation symmetry (33), with c arbitrary constant, are Q 1 and Q 2 , given respectively by (7) and (8).…”

“…. , 7, the constraint (35) is satisfied for different specific values of the constant c. Therefore, the only multipliers invariant under the translation symmetry (33), with c arbitrary constant, are Q 1 and Q 2 .…”

“…Proof. The proof is immediately followed by taking into account that a conservation law is invariant under the translation symmetry (33) if, and only if, the corresponding multiplier is invariant with respect to the translation symmetry [32], and using the fact that from Proposition 2, only multipliers Q 1 and Q 2 , given respectively by ( 7) and ( 8), are invariant under the translation symmetry (33) for c arbitrary constant. Substitution of the traveling wave expression (32) into PDE (2) yields a nonlinear third-order ODE g U + U (3g U + g (U ) 2 + f + c) = 0.…”

“…In [33] a third-order PDE was analyzed from the point of view of point symmetries and reductions. In particular, the three-and four-dimensional solvable symmetry algebras admitted by the considered family depending on its arbitrary functions were determined.…”