Hodograph Method for Solving the Problem of Shallow Water under Solid Cover in the Case of Hyperbolic Equations

An exact two-parameter solution of the Cauchy problem of the flow of two-layer shallow water under a solid cover, i.e., two infinite contacting layers of liquid with a small density difference moving at different velocities in a horizontal channel with solid walls, is constructed. The distortion of the layer interface occurs due to the Kelvin–Helmholtz instability. The problem is described by a system of two quasi-linear first-order partial hyperbolic differential equations. The solution is constructed using a version of the hodograph method based on the conservation law. This method makes it possible to transform a system of first-order quasi-linear partial differential equations to a linear second-order partial differential equation with variable coefficients, for which the Riemann–Green function is known. A method is proposed for reconstructing the explicit solution of the Cauchy problem on the level lines of the implicit solution, which ultimately reduces the solution of the original problem to the solution of a certain Cauchy problem for a system of ordinary differential equations. The results of calculations for spatially periodic initial data are presented as an example.