Symmetries of the shallow water equations in the Boussinesq approximation

The shallow water equations in the Boussinesq approximation are studied in this paper. Two cases of these equations are studied: in Eulerian coordinates and Lagrangian coordinates. A detailed analysis of the admitted Lie groups is given. All invariant solutions of these two representations are presented. Using Noether's theorem, new conservation laws in Eulerian coordinate and Lagrangian coordinates are found.