Classical solution of the initial-value problem for a quasilinear wave equation with discontinuous initial conditions

TL;DR

This paper provides an explicit analytical solution to a one-dimensional quasilinear wave equation with discontinuous initial conditions, using the method of characteristics and studying the solution's existence and uniqueness.

Contribution

It introduces a novel approach to solving a quasilinear wave equation with discontinuous initial data via implicit integro-differential equations.

Findings

Constructed explicit solution using characteristics

Proved uniqueness of the solution

Established conditions for classical solution existence

Abstract

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.