The system of generalized Chaplygin gas equations with a coulomblike friction term has been investigated by using the famous Lie symmetry method. A direct and systematic algorithm based on the adjoint transformation and invariants of the admitted Lie algebras is then used to construct one-and two-dimensional optimal system of the Chaplygin gas equations. Inequivalent classes of group invariant solutions are then obtained using the one-dimensional optimal system. Further, the evolutionary behaviour of the weak discontinuity wave within the state characterized by one of the group invariant solutions is investigated in detail, and certain observations are noted in respect to their contrasting behaviour. KEYWORDSLie symmetries, first -order hyperbolic systems, Chaplygin gas equations, weak discontinuities MSC CLASSIFICATION 70G65; 35L40

A complete symmetry group classification for the system of shallow water equations with the horizontal temperature gradient, also known as Ripa system, is presented. A rigorous and systematic procedure based on the general invariants of the adjoint representation is used to construct the one-dimensional optimal system of the Lie algebra. The complete inequivalence class of the group invariant solutions are obtained by using the one-dimensional optimal system. One such solution of the Ripa system is used to study the evolutionary behaviour of the discontinuity wave.

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