Similarity transformations for modified shallow water equations with density dependence on the average temperature

Abstract The Lie symmetry analysis is applied for the study of a modified one-dimensional Saint–Venant system in which the density depends on the average temperature of the fluid. The geometry of the bottom we assume that is a plane, while the viscosity term is considered to be nonzero, as the gravitational force is included. The modified shallow water system is consisted by three hyperbolic first-order partial differential equations. The admitted Lie symmetries form a four-dimensional Lie algebra, the A 3,3 ⊕ A 1. However, for the viscosity free model, the admitted Lie symmetries are six and form the A 5,19 ⊕ A 1 Lie algebra. For each Lie algebra we determine the one-dimensional optimal system and we present all the possible independent reductions provided by the similarity transformations. New exact and analytic solutions are calculated for the modified Saint–Venant system.