On Bers generating functions for first order systems of mathematical physics

TL;DR

This paper generalizes Bers' pseudoanalytic function theory to biquaternionic equations, enabling new solution methods for first-order systems in mathematical physics, including Maxwell, magnetic fields, and Dirac equations.

Contribution

It introduces a generalized generating pair concept for biquaternionic equations, linking solutions of physical systems to pseudoanalytic functions and expanding their analytical framework.

Findings

Derived relations between Maxwell systems and hyperbolic pseudoanalytic functions

Constructed infinite solutions for Maxwell equations in inhomogeneous media

Connected force-free magnetic fields and Dirac systems to pseudoanalytic function theory

Abstract

Considering one of the fundamental notions of Bers' theory of pseudoanalytic functions the generating pair via an intertwining relation we introduce its generalization for biquaternionic equations corresponding to different first-order systems of mathematical physics with variable coefficients. We show that the knowledge of a generating set of solutions of a system allows one to obtain its different form analogous to the complex equation describing pseudoanalytic functions of the second kind and opens the way for new results and applications of pseudoanalytic function theory. As one of the examples the Maxwell system for an inhomogeneous medium is considered, and as one of the consequences of the introduced approach we find a relation between the time-dependent one-dimensional Maxwell system and hyperbolic pseudoanalytic functions and obtain an infinite system of solutions of the Maxwell system. Other considered examples are the system describing force-free magnetic fields and the Dirac system from relativistic quantum mechanics.