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Direct determination of the underlying Lie algebra in nonlinear optics
Author(s) -
Arnold J. M.
Publication year - 1991
Publication title -
radio science
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.371
H-Index - 84
eISSN - 1944-799X
pISSN - 0048-6604
DOI - 10.1029/90rs01149
Subject(s) - lie algebra , mathematics , lie conformal algebra , algebra over a field , integrable system , nonlinear system , eigenvalues and eigenvectors , graded lie algebra , mathematical physics , algebraic number , hamiltonian (control theory) , lie group , affine lie algebra , pure mathematics , super poincaré algebra , current algebra , physics , quantum mechanics , adjoint representation of a lie algebra , mathematical analysis , mathematical optimization
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.