Integrable G-Strands on semisimple Lie groups

TL;DR

This paper develops integrable G-strand equations on semisimple Lie groups, providing their zero curvature representations, Hamiltonian formulations, and stability analysis for specific Lie algebras.

Contribution

It introduces a general framework for integrable G-strands on semisimple Lie groups, including zero curvature conditions and Hamiltonian structures.

Findings

Derived PDE systems with quadratic zero curvature representations.

Established Hamiltonian principles and Hamiltonians for these systems.

Analyzed linear stability of equilibrium solutions in specific Lie algebra examples.

Abstract

The present paper derives systems of partial differential equations that admit a quadratic zero curvature representation for an arbitrary real semisimple Lie algebra. It also determines the general form of Hamilton's principles and Hamiltonians for these systems and analyzes the linear stability of their equilibrium solutions in the examples of $\mathfrak{so}(3)$ and $\mathfrak{sl}(2,\mathbb{R})$.