Nonlinear representations for Poincare and Galilei algebras and nonlinear equations for electromagnetic fields

TL;DR

This paper develops nonlinear mathematical models for electromagnetic fields based on Poincare, Galilei, and conformal symmetries, introducing a new nonlinear Euler-type equation and identifying its invariance properties.

Contribution

It introduces nonlinear representations of fundamental symmetry algebras on electromagnetic field vectors and proposes a novel nonlinear Euler-type equation with known invariance.

Findings

Constructed nonlinear representations of Poincare, Galilei, and conformal algebras.

Proposed a new nonlinear Euler-type equation for electromagnetic fields.

Determined the invariance algebra of the proposed equation.

Abstract

We construct nonlinear representations of the Poincare, Galilei, and conformal algebras on a set of the vector-functions $\Psi =(\vec E, \vec H)$. A nonlinear complex equation of Euler type for the electromagnetic field is proposed. The invariance algebra of this equation is found.