Lie Symmetries, One-Dimensional Optimal System and Group Invariant Solutions for the Ripa System

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.

SET IDEALS WITH ISOMORPHIC SYMMETRY GROUPS

A criterion of isomorphism for symmetry groups of set ideals is provided in terms of ideal quotients and cardinal invariants. Furthermore, set ideals with complete symmetry group are characterized. They form a wide class, comprising, for example, all uniform dense ideals and the ideals of ‘thin’ sets in separable metric spaces. If the symmetry group of a set ideal is not complete, then its outer automorphism group is shown to be cyclic of order 2.