k-Dirac operator and Cartan-Kahler theorem

TL;DR

This paper explores the properties of the k-Dirac operator using the Cartan-Kahler theorem, revealing involutivity in certain cases and providing new insights into initial conditions for the 2-Dirac operator.

Contribution

It introduces a novel application of the Cartan-Kahler theorem to the k-Dirac operator and characterizes initial conditions for the 2-Dirac operator.

Findings

Tableaux of first prolongations are involutive for k=2

Provides a new characterization of initial conditions for the 2-Dirac operator

Extends analysis to the parabolic version of the operator

Abstract

We apply the Cartan-Kahler theorem for the k-Dirac operator studied in Clifford analysis and to the parabolic version of this operator. We show that for k = 2 the tableaux of the first prolongations of these two operators are involutive. This gives us a new characterization of the set of initial conditions for the 2-Dirac operator.