Potential Vector Fields in $\mathbb R^4$ and New Generalizations of the Cauchy-Riemann System

TL;DR

This paper develops new generalizations of the Cauchy-Riemann system in four-dimensional space, introducing models of meridional vector fields, and explores their properties using Vekua systems, with applications to potential theory and harmonic mappings.

Contribution

It introduces two families of generalized Cauchy-Riemann systems in $\

Findings

Models of potential meridional vector fields in $\\mathbb{R}^4$ are characterized.

Four-dimensional $\\alpha$-meridional mappings are defined and analyzed.

Explicit solutions involving Bessel functions and quaternionic arguments are provided.

Abstract

This paper extends approach of recent author's paper devoted to special classes of exact solutions of the static Maxwell system in inhomogeneous isotropic media and new generalizations of the Cauchy-Riemann system in $\mathbb R^3$. Two families of generalizations of the Cauchy-Riemann system with variable coefficients in $\mathbb R^4$ are presented in the context of non-Euclidean geometry. Analytic models of a wide range of potential meridional vector fields in $\mathbb R^4$ are characterized using a family of Vekua type systems in cylindrical coordinates. The specifics of meridional fields allows us to introduce the concept of four-dimensional $\alpha$-meridional mappings of the first and second kind depending on the values of a real parameter $\alpha$. In case $\alpha=2$ tools of the radially holomorphic potential in $\mathbb R^4$ are developed in the context of generalized axially symmetric potential theory (GASPT). Analytic models of potential meridional fields in $\mathbb R^4$ generated by Bessel functions of the first kind of integer order and quaternionic argument are described in case $\alpha=2$. In case $\alpha =0$ the geometric specifics of four-dimensional harmonic meridional mappings of the second kind is demonstrated explicitly in the context of the theory of gradient dynamical systems with harmonic potential.