Classical solution of the Cauchy problem for a quasi-linear wave equation with discontinuous initial conditions

We consider the Cauchy problem for a one-dimensional weakly quasi-linear wave equation given in the upper half-plane. The initial conditions have a first-kind discontinuity at one point. We construct the solution using the method of characteristics in implicit analytical form as a solution of some integro-differential equations. The solvability of these equations, as well the smoothness of their solutions, is studied. For the problem in question, we prove the uniqueness of the solution and establish the conditions, under which its classical solution exists.