On the Klein-Gordon equation and hyperbolic pseudoanalytic function theory

TL;DR

This paper develops a hyperbolic pseudoanalytic function theory using hyperbolic numbers to analyze the Klein-Gordon equation with potential, providing explicit solution constructions and applications.

Contribution

It introduces a hyperbolic analogue of pseudoanalytic function theory and applies it to factorize and solve the Klein-Gordon equation with potential.

Findings

Factorization of Klein-Gordon operator using Vekua-type operators

Explicit construction of infinite solution systems

Application examples demonstrating the method

Abstract

Elliptic pseudoanalytic function theory was considered independently by Bers and Vekua decades ago. In this paper we develop a hyperbolic analogue of pseudoanalytic function theory using the algebra of hyperbolic numbers. We consider the Klein-Gordon equation with a potential. With the aid of one particular solution we factorize the Klein-Gordon operator in terms of two Vekua-type operators. We show that real parts of the solutions of one of these Vekua-type operators are solutions of the considered Klein-Gordon equation. Using hyperbolic pseudoanalytic function theory, we then obtain explicit construction of infinite systems of solutions of the Klein-Gordon equation with potential. Finally, we give some examples of application of the proposed procedure.