Direct determination of the underlying Lie algebra in nonlinear optics


  • J. M. Arnold


It is shown that the equations of resonant nonlinear optics can be studied entirely within the framework of an underlying Lie algebra, in which the 2x2 su(2) Hamiltonian and density matrices of the quantum mechanical description of the atomic system transform directly to the 2x2 sl(2,R) matrices of the Ablowitz-Kaup-Newell-Segur (AKNS) scheme, and the AKNS eigenvalue is introduced naturally as a free parameter. The Lie algebra sl(2,R) is also the symmetry algebra of transformations between equivalence classes of AKNS systems under SL(2,R) gauge transformations. The Lie algebra formalism condenses much algebraic manipulation, and provides a natural basis for the perturbation theory of “nearly integrable” nonlinear wave systems.