  For a one-dimensional mildly quasilinear wave equation given in the upper
half-plane, we consider the Cauchy problem. The initial conditions have
discontinuity of the first kind at one point. We construct the solution using
the method of characteristics in an 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.
