Contact symmetries and conservation laws of the first order of the equation of one-dimensional shallow water over a rough bottom in Lagrange’s variables

Abstract

The systems of equations of one-dimensional shallow water over a rough bottom in Euler’s and Lagrange’s variables are considered. The problem of group classification of contact symmetries of the shallow water equation in Lagrange’s variables is solved. First order conservation laws of the shallow water equation in Lagrange’s variables obtained using the Noether’s theorem. Bottom profiles, providing additional conservation laws, are given.