Exact solutions of nonlinear shallow water equations over square bottom

Abstract

Exact solutions for each mathematical model are important. They allow us to assess, the degree of the adequacy of the mathematical model of real physical processes, after carrying out experiments appropriate to these solutions, and an evaluation of the arising deviations. Exact solutions can be used to describe of some physical processes. Exact solutions are good tests to check the approximate numerical solutions. In this paper we performed a group analysis of the one-parameter family of the equations, describing within the framework of the nonlinear one-dimensional shallow water model, the propagation of surface waves above a straight bottom. A parameter of this family is an angular coefficient of inclination of the bottom. As a result of a special nonlocal hodograph transformation, associated with the group property of the original non-linear system, this system is reduced to a linear system. Using the group properties of these systems, we obtained the formulas of the production ( "reproduction") of the solutions of nonlinear system. With a help of the invariant and partially invariant solutions of the linear system and found formulas of the production of the solutions, we obtained an infinite set of non-singular solutions of nonlinear system. We found all degenerate solutions of this system. These solutions can be used in the study of waves rolling on shore, and also in the study of the spread of liquid in the channels.