On Symmetry Reductions of a Third-Order Partial Differential Equation

This work is devoted to perform symmetry reductions of a third-order partial differential equation belonging to a wide class which models many real- world phenomena. In particular, many third-order partial differential equations of this class appear in different macroscopic models for semiconductors, which consider quantum effects, for instance, quantum hydrodynamic models; or models for the transmission of electrical lines, among others. The symmetry group of the considered equation has been determined. We prove that this group constitutes a three-dimensional solvable Lie group which allows us to reduce the equation to a first-order nonlinear ODE. Furthermore, a solution of the equation is determined by quadrature in a particular case.