Explicit solutions of generalized Cauchy-Riemann systems using the transplant operator

TL;DR

This paper explores how the transplant operator can transform solutions and formal powers between related Vekua equations, enabling explicit solutions for complex generalized Cauchy-Riemann systems in elliptic and hyperbolic contexts.

Contribution

It demonstrates that the transplant operator preserves formal powers across Vekua equations, facilitating explicit solution construction for complex systems.

Findings

Transplant operator maps formal powers between Vekua equations.

Allows derivation of solutions for complex Cauchy-Riemann systems.

Provides methods for constructing Cauchy kernels.

Abstract

In [8] it was shown that the transplant operator transforms solutions of one Vekua equation into solutions of another Vekua equation, related to the first via a Schr\"odinger equation. In this paper we demonstrate a fundamental property of this operator: it transforms formal powers of the first Vekua equation into formal powers of the same order for the second Vekua equation. This property allows us to obtain positive formal powers and a generating sequence of a "complicated" Vekua equation from positive formal powers and a generating sequence of "simpler" Vekua equation. Similar results is obtained regarding construction of Cauchy kernels. Elliptic and hyperbolic pseudoanalytic function theory are considered and examples are given to illustrate the procedure.