Exact Solutions of the Shallow Water System with an Inclined Bottom - Chirkunov, Yu. A. (Discontinuity, Nonlinearity, and Complexity, 2026)
In this paper, we study a system of shallow water equations with a rectilinear sloping bottom. Two fundamentally different cases are considered. In the first case, the Jacobi matrix of dependent variables with respect to independent variables is degenerate. This case has not been studied by anyone before. We have obtained all such solutions of the system. Each solution for specific values of its parameters is illustrated by a graph of the distribution of the excess of the free surface above the bottom and a graph of the distribution of the propagation velocity of a surface wave. The physical meaning of these solutions is indicated. In the second case, the Jacobi matrix of dependent variables with respect to independent variables is nondegenerate. This allows you to linearize the original system using a special hodograph transformation. Two exact solutions of this linear system are found: an invariant solution and a partially invariant solution. With the help of these solutions, we have obtained the exact solutions of the original system. Each solution for specific values of its parameters also is illustrated by a graph of the distribution of the excess of the free surface above the bottom and a graph of the distribution of the velocity of propagation of a surface wave. The physical meaning of these solutions is indicated. Using the method of $A$-operators, we have found all zero-order conservation laws for the original system with an inclined bottom.