Symmetries, conservation laws, and exact solutions to a one-dimensional system of shallow water equations over an uneven bottom

Abstract: The symmetries of a one-dimensional system of shallow water equations over an uneven bottom in Euler’s variables are classified. Based on the results of the group classification obtained, it is concluded that it is possible to reduce the one-dimensional system of shallow water equations to a linear system of equations using point transformations only in the cases of horizontal and inclined bottom profiles. We also classify the contact symmetries of the one-dimensional shallow water equation over an uneven bottom in Lagrangian’s variables.

The hydrodynamic conservation laws of a one-dimensional system of shallow water equations in Eulerian’s variables are classified. A new basic conservation law is obtained. The first-order conservation laws of the one-dimensional shallow water equation in Lagrangian’s variables are classified.

A three-parameter family of exact solutions of a one-dimensional system of shallow water equations over an inclined bottom is obtained and investigated, describing the ”step’’ wave's arrival on the shore and its reflection from it. The nonlinear the overwash effect and the effect of the amplification of the incoming wave when it is reflected from the shore are described.

Computer sciencemathematical physicsanalysis of PDEsexactly solvable and integrable systemsfluid dynamics

Audience: researchers in the topic

Mathematical models and integration methods