2021/11/02 by Bryukhov, Dmitry
#30C65 #34A34 #35J46 #Complex Variables (math.CV) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Primary 35Q99 #Secondary 30G35
paper · doi:10.48550/arxiv.2111.01580
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 R3. Two families of generalizations of the Cauchy-Riemann system with variable coefficients in \mathbb R4 are presented in the context of non-Euclidean geometry. Analytic models of a wide range of potential meridional vector fields in \mathbb R4 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 α-meridional mappings of the first and second kind depending on the values of a real parameter α. In case α=2 tools of the radially holomorphic potential in \mathbb R4 are developed in the context of generalized axially symmetric potential theory (GASPT). Analytic models of potential meridional fields in \mathbb R4 generated by Bessel functions of the first kind of integer order and quaternionic argument are described in case α=2. In case α=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.