On the Klein-Gordon equation and hyperbolic pseudoanalytic function theory.

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 the application of the proposed procedure. [ABSTRACT FROM AUTHOR]