Vector potential normal form classification for completely integrable solenoidal nilpotent singularities

TL;DR

This paper classifies a family of solenoidal, integrable vector fields with a triple zero singularity using normal forms that preserve their Lie algebra, vector potential, and Clebsch potentials, with computational implementation.

Contribution

It introduces a Lie algebra structure for solenoidal vector fields with a triple zero singularity and develops normal form classifications preserving key invariants and representations.

Findings

The vector fields form a Lie algebra structure.

Normal forms preserve vector potential and Clebsch potentials.

Implementation in Maple computes normal forms and coefficients.

Abstract

We introduce a sl_2-invariant family of nonlinear vector fields with a non-semisimple triple zero singularity. In this paper we are concerned with characterization and normal form classification of these vector fields. We show that the family constitutes a Lie algebra structure and each vector field from this family is solenoidal, completely integrable and rotational. All such vector fields share a common quadratic invariant. We provide a Poisson structure for the Lie algebra from which the second invariant for each vector field can be readily derived. We show that each vector field from this family can be uniquely characterized by two alternative representations, one uses a vector potential while the other uses two functionally independent Clebsch potentials. Our normal form results are designed to preserve these structures and representations. The results are implemented in Maple in order to compute vector potential and the Clebsch potential normal forms of a given vector field from this family. Some practical normal form coefficient formulas for degrees of up to four are presented.