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

SIGMA 2 (2006), 075, 15 pages      math-ph/0611018      https://doi.org/10.3842/SIGMA.2006.075

Prolongation Loop Algebras for a Solitonic System of Equations

Maria A. Agrotis
Department of Mathematics and Statistics, University of Cyprus, Nicosia 1678, Cyprus

Received September 13, 2006, in final form November 01, 2006; Published online November 08, 2006

Abstract
We consider an integrable system of reduced Maxwell-Bloch equations that describes the evolution of an electromagnetic field in a two-level medium that is inhomogeneously broadened. We prove that the relevant Bäcklund transformation preserves the reality of the n-soliton potentials and establish their pole structure with respect to the broadening parameter. The natural phase space of the model is embedded in an infinite dimensional loop algebra. The dynamical equations of the model are associated to an infinite family of higher order Hamiltonian systems that are in involution. We present the Hamiltonian functions and the Poisson brackets between the extended potentials.

Key words: loop algebras; Bäcklund transformation; soliton solutions.

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References

  1. Ablowitz M.J., Kaup D.J., Newell A.C., Segur H., The inverse scattering transform-Fourier analysis for nonlinear problems, Stud. Appl. Math., 1974, V.53, 249-315.
  2. Zakharov V.E., Shabat A.B., Integration of the nonlinear equations of mathematical physics by the method of the inverse scattering problem, Funct. Anal. Appl., 1979, V.13, N 3, 13-22.
  3. Flaschka H., The Toda lattice I. Existence of integrals, Phys. Rev. B, 1974, V.9, 1924-1925.
  4. Flaschka H., On the Toda lattice II. Inverse-scattering solution, Progr. Theoret. Phys., 1974, V.51, 703-716.
  5. Adler M., On a trace functional for formal pseudo-differential operators and the symplectic structure of the Korteweg-de Vries type equations, Invent. Math., 1979, V.50, 219-48.
  6. Adler M., van Moerbeke P., Completely integrable systems, Euclidean Lie algebras, and curves, Adv. Math., 1980, V.38, 267-317.
  7. Kostant B., The solution to a generalized Toda lattice and representation theory, Adv. Math., 1979, V.34, 195-338.
  8. Symes W.W., Systems of Toda type, inverse spectral problems, and representation theory, Invent. Math., 1980, V.59, 13-51.
  9. Flaschka H., Newell A.C., Ratiu T., Kac-Moody Lie algebras and soliton equations, Phys. D, 1983, V.9, 300-323.
  10. McCall S.L., Hahn E.L., Self-induced transparency, Phys. Rev., 1969, V.183, 457-486.
  11. Lamb G.L., Analytical description of utrashort optical pulse propagation in a resonant medium, Rev. Modern Phys., 1971, V.43, 99-124.
  12. Eilbeck J.C., Gibbon J.D., Caudrey P.J., Bullough R.K., Solitons in nonlinear optics I. A more accurate description of the 2p pulse in self-induced transparency, J. Phys. A: Math. Gen., 1973, V.6, 1337-1347.
  13. Caputo J., Maimistov A.I., Unidirectional propagation of an ultra-short electromagnetic pulse in a resonant medium with high frequency stark shift, Phys. Lett. A, 2002, V.296, 34-42, nlin.SI/0107040.
  14. Elyutin S.O., Dynamics of an extremely short pulse in a stark medium, JETP, 2004, V.101, 11-21.
  15. Qing-Chun J.I., Darboux transformation and solitons for reduced Maxwell-Bloch equations, Commun. Theor. Phys., 2005, V.43, 983-986.
  16. Sazonov S.V., Ustinov N.V., Pulsed transparency of anisotropic media with stark level splitting, Quantum Electronics, 2005, V.35, 701-704.
  17. Sazonov S.V., Ustinov N.V., Nonlinear acoustic transparency phenomena in strained paramagnetic crystals, JETP, 2006, V.102, 741-752.
  18. Bakhar N.V., Ustinov N.V., Dynamics of two-component electromagnetic and acoustic extremely short pulses, Proceedings of SPIE, 2006, 61810Q, 10 pages, nlin.SI/0512068.
  19. Glasgow S.A., Agrotis M.A., Ercolani N.M., An integrable reduction of inhomogeneously broadended optical equations, Phys. D, 2005, V.212, 82-99.
  20. Agrotis M.A., Hamiltonian flows for a reduced Maxwell-Bloch system with permanent dipole, Phys. D, 2003, V.183, 141-158.
  21. Bäcklund A.V., Zur Theorie der Flachentransformationen, Math. Ann., 1881, V.19, 387-422.
  22. Lamb G.L., Elements in soliton theory, Wiley-Interscience Pub., 1980.
  23. Lonngren K., Alwyn S. (Editors), Solitons in action, Academic Press, 1978.
  24. Newell A.C., Solitons in mathematics and physics, CBMS-NSF Regional Conference Series, Vol. 48, SIAM Press, 1985.

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