On exact solution of a hyperbolic system of differential equations

The paper considers a hyperbolic system of two first-order partial differential equations with constant coefficients, one of which is nonlinear and contains the square of one of the unknown functions. Moreover, each equation contains two unknown functions which in turn depend on two variables. Exact solutions are found for this system: a traveling wave solution and a self-similar solution. There is also defined the type of initial-boundary conditions which allow to use the constructed general solutions in order to write out a solution of the initial-boundary value problem for the system of differential equations under consideration.