Multiple lie-poisson structures, reductions, and geometric phases for the Maxwell-Bloch travelling wave equations

Abstract

The real-valued Maxwell-Bloch equations on ℝ3 are investigated as a Hamiltonian dynamical system obtained by applying an S1 reduction to an invariant subsystem of a dynamical system on ℂ3. These equations on ℝ3 are bi-Hamiltonian and possess several inequivalent Lie-Poisson structures parametrized by classes of orbits in the group SL(2, ℝ). Each Lie-Poisson structure possesses an associated Casimir function. When reduced to level sets of these functions, the motion takes various symplectic forms, from that of the pendulum to that of the Duffing oscillator. The values of the geometric (Hannay-Berry) phases obtained in reconstructing the solutions are found to depend upon the choice of Casimir function, that is, upon the parametrization of the reduced symplectic space.