Dam break problem in shallow water model

The paper considers one-dimensional water flows that occur after the destruction of the dam. It is assumed that at the initial moment of time there is a resting liquid to the right of the dam, and a dry shore to the left. A system of shallow water equations is used as a mathematical model. In the system of shallow water equations, a self-similar feature is introduced into the variable x and for the resulting system, the Cauchy problem is posed with data on the sound characteristic. The solution of the initial boundary value problem is constructed in the form of a power series. The coefficients of the series are found by integrating ordinary differential equations. Next, to prove the convergence of this series, an initial boundary value problem is posed in the space of other independent variables. The solution to this initial boundary value problem is constructed in the form of its own convergent power series, and the equivalence of the solutions to the first and second initial boundary value problems is proven.