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

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

1995-01-27

arxiv.org/abs/hep-th/9501128v1

Mod.Phys.Lett. A10 (1995) 1209-1224

Physics

High Energy Physics

High Energy Physics - Theory

16 pages, no figures, plain TeX, no macros

Scientific paper

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.

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