Geometrical Description of the Local Integrals of Motion of Maxwell-Bloch Equation

TL;DR

This paper provides a geometric framework for understanding the local integrals of motion of the Maxwell-Bloch equation, connecting it with affine Lie algebra structures and cohomology methods.

Contribution

It introduces a geometric description of Maxwell-Bloch integrals of motion using affine Lie algebra and cohomology, and explores quantization and latticization possibilities.

Findings

Maxwell-Bloch evolution is represented as an infinitesimal action of a nilpotent subalgebra.

Local integrals of motion are described via cohomology methods.

Hamiltonian flows are identified with an infinitesimal action of an abelian subalgebra.

Abstract

We represent a classical Maxwell-Bloch equation and related to it positive part of the AKNS hierarchy in geometrical terms. The Maxwell-Bloch evolution is given by an infinitesimal action of a nilpotent subalgebra $n_+$ of affine Lie algebra $\hat {sl}_2$ on a Maxwell-Bloch phase space treated as a homogeneous space of $n_+$. A space of local integrals of motion is described using cohomology methods. We show that hamiltonian flows associated to the Maxwell-Bloch local integrals of motion (i.e. positive AKNS flows) are identified with an infinitesimal action of an abelian subalgebra of the nilpotent subalgebra $n_+$ on a Maxwell- Bloch phase space. Possibilities of quantization and latticization of Maxwell-Bloch equation are discussed.