Direct determination of the underlying Lie algebra in nonlinear optics
Article first published online: 7 DEC 2012
Copyright 1991 by the American Geophysical Union.
Volume 26, Issue 1, pages 299–302, January-February 1991
How to Cite
1991), Direct determination of the underlying Lie algebra in nonlinear optics, Radio Sci., 26(1), 299–302, doi:10.1029/90RS01149.(
- Issue published online: 7 DEC 2012
- Article first published online: 7 DEC 2012
- Manuscript Accepted: 17 MAY 1990
- Manuscript Received: 29 DEC 1989
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.