Symmetries of the hyperbolic shallow water equations and the Green-Naghdi model in Lagrangian coordinates

The observation that the hyperbolic shallow water equations and the Green-Naghdi equations in Lagrangian coordinates have the form of an Euler-Lagrange equation with a natural Lagrangian allows us to apply Noether's theorem for constructing conservation laws for these equations. In this study the complete group analysis of these equations is given: admitted Lie groups of point and contact transformations, classification of the point symmetries and all invariant solutions are studied. For the hyperbolic shallow water equations new conservation laws which have no analog in Eulerian coordinates are obtained. Using Noether's theorem a new conservation law of the Green-Naghdi equations is found. The dependence of solutions on the parameter is illustrated by self-similar solutions which are invariant solutions of both models.