Solving a characteristic Cauchy problem

TL;DR

This paper introduces a novel approach to solving the nonlinear characteristic Cauchy problem for the Wave Equation by using generalized functions, establishing existence, uniqueness, and consistency with classical solutions.

Contribution

It develops a new method to interpret the nonlinear characteristic Cauchy problem via generalized functions, extending classical solutions to a broader context.

Findings

Existence of solutions in the generalized function framework

Solutions depend on the choice of the algebra of generalized functions

Classical solutions are recovered in the non-characteristic case

Abstract

In this paper we give a meaning to the nonlinear characteristic Cauchy problem for the Wave Equation in base form by replacing it by a family of non-characteristic problems in an appropriate algebra of generalized functions. We prove existence of a solution and we precise how it depends on the choice made. We also check that in the classical case (non-characteristic) our new solution coincides with the classical one.