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

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 ${\widehat{sl}}_2$ 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.

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