Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 2 (2006), 038, 14 pages      math-ph/0603071      https://doi.org/10.3842/SIGMA.2006.038

On the Generalized Maxwell-Bloch Equations

Pavle Saksida
Department of Mathematics, Faculty of Mathematics and Physics, University of Ljubljana, Slovenia

Received December 01, 2005, in final form March 05, 2006; Published online March 27, 2006

Abstract
A new Hamiltonian structure of the Maxwell-Bloch equations is described. In this setting the Maxwell-Bloch equations appear as a member of a family of generalized Maxwell-Bloch systems. The family is parameterized by compact semi-simple Lie groups, the original Maxwell-Bloch system being the member corresponding to SU(2). The Hamiltonian structure is then used in the construction of a new family of symmetries and the associated conserved quantities of the Maxwell-Bloch equations.

Key words: Maxwell-Bloch equations; Hamiltonian structures; symmetries; conserved quantities.

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References

  1. Reyman A.G., Semenov-Tian-Shansky M.A., Reduction of Hamiltonian systems, affine Lie algebras and Lax equations I, Invent. Math., 1979, V.54, 81-100.
  2. Reyman A.G., Semenov-Tian-Shansky M.A., Integrable systems II, in Encyclopaedia of Mathematical Sciences, Vol. 16, Editors V.I. Arnold and S.P. Novikov, Berlin, Springer, 1994, 116-259.
  3. Saksida P., Nahm's equations and generalizations of the Neumann system, Proc. Lond. Math. Soc., 1999, V.78, 701-720.
  4. Novikov S.P., The Hamiltonian formalism and multivalued analogue of Morse theory, Uspekhi Mat. Nauk, 1982, V.37, 3-49 (in Russian).
  5. Marsden J.E., Lectures on mechanics, London Mathematical Society Lecture Note Series, Vol.174, Cambridge, Cambridge University Press, 1992.
  6. Marsden J.E., Ratiu T.S., Introduction to mechanics and symmetry, New York, Springer, 1994.
  7. Saksida P., Maxwell-Bloch equations, C. Neumann systems and Kaluza-Klein theory, J. Phys. A: Math. Gen., 2005, V.38, 10321-10344.
  8. Kostant B., Quantization and unitary representations, in Lectures in Modern Analysis and Applications III. Lecture Notes in Math., Vol.179, Editor C.T. Taam, Berlin, Springer, 1970, 87-208.
  9. Park Q-Han, Shin H.J., Complex sine-Gordon equation in coherent optical pulse propagation, J. Korean Phys. Soc., 1997, V.30, 336-340, solv-int/9904007.
  10. Park Q-Han, Shin H.J., Field theory for coherent optical pulse propagation, Phys. Rev. A, 1998, V.57, 4621-4642, solv-int/9709002.
  11. Abraham R., Marsden J.E., Foundations of Mechanics, 2nd ed., Reading MA, Benjamin-Cummings, 1978.
  12. Lamb G.L., Phase variation in coherent-optical-pulse propagation, Phys. Rev. Lett., 1973, V.31, 196-199.
  13. Lamb G.L., Coherent-optical-pulse propagation as an inverse problem, Phys. Rev. A, 1974, V.9, 422-430.
  14. Caudrey P.J., Eilbeck J.C., Gibbon J.D., An N-soliton solution of a nonlinear optics equation derived by inverse method, Lett. Nuovo Cimento, 1973, V.8, 773-779.
  15. Gabitov I.R., Zakharov V.E., Mikhailov A.V., The Maxwell-Bloch equation and the method of the inverse scattering problem, Teoret. Mat. Fiz., 1985, V.63, 11-31 (in Russian).
  16. Saksida P., Neumann system, spherical pendulum and magnetic fields, J. Phys. A: Math. Gen., 2002, V.35, 5237-5253.
  17. Saksida P., Integrable anharmonic oscillators on spheres and hyperbolic spaces, Nonlinearity, 2001, V.14, 977-994.
  18. Holm D., Kovacic G., Homoclinic chaos in a laser-matter system, Phys. D, 1992, V.56, 270-300.
  19. Naudts J., Kuna M., Special solutions of nonlinear von Neumann equations, math-ph/0506020.
  20. Czachor M., Kuna M., Leble S.B., Naudts J., Nonlinear von Neumann-type equations, in Trends in Quantum Mechanics (1998, Goslar), River Edge, NJ, World Sci. Publishing, 2000, 209-226, quant-ph/9904110.
  21. Nambu Y., Generalized Hamiltonian dynamics, Phys. Rev. D, 1973, V.7, 2405-2412.

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