The purpose of this paper is to analyze in detail a special nonlinear partial differential equation (nPDE) of the second order which is important in physical, chemical and technical applications. The present nPDE describes nonlinear diffusion and is of interest in several parts of physics, chemistry and engineering problems alike. Since nature is not linear intrinsically the nonlinear case is t...

We present the Lie symmetry analysis for a hyperbolic partial differential system known as one-dimensional Saint-Venant-Exner model. The model describes shallow-water systems with bed evolution given by Exner terms. sediment flux is considered to be power-law function of velocity fluid. admitted symmetries are classified according power index flux. Furthermore, optimal determined in all cases. ...