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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
- 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.
- 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.
- Flaschka H., The Toda lattice I. Existence of
integrals, Phys. Rev. B, 1974, V.9, 1924-1925.
- Flaschka H., On the Toda lattice II. Inverse-scattering
solution, Progr. Theoret. Phys., 1974, V.51, 703-716.
- 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.
- Adler M., van Moerbeke P., Completely integrable systems,
Euclidean Lie algebras, and curves, Adv. Math., 1980, V.38,
267-317.
- Kostant B., The solution to a generalized Toda lattice and
representation theory, Adv. Math., 1979, V.34, 195-338.
- Symes W.W., Systems of Toda type, inverse spectral problems,
and representation theory, Invent. Math., 1980, V.59,
13-51.
- Flaschka H., Newell A.C., Ratiu T., Kac-Moody Lie
algebras and soliton equations, Phys. D, 1983, V.9,
300-323.
- McCall S.L., Hahn E.L., Self-induced transparency,
Phys. Rev., 1969, V.183, 457-486.
- Lamb G.L., Analytical description of utrashort
optical pulse propagation in a resonant medium, Rev. Modern
Phys., 1971, V.43, 99-124.
- 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.
- 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.
- Elyutin S.O., Dynamics of an extremely short pulse in a stark
medium, JETP, 2004, V.101, 11-21.
- Qing-Chun J.I., Darboux transformation and solitons for
reduced Maxwell-Bloch equations, Commun. Theor. Phys.,
2005, V.43, 983-986.
- Sazonov S.V., Ustinov N.V., Pulsed transparency of anisotropic
media with stark level splitting, Quantum Electronics, 2005,
V.35, 701-704.
- Sazonov S.V., Ustinov N.V., Nonlinear acoustic transparency
phenomena in strained paramagnetic crystals, JETP, 2006,
V.102, 741-752.
- 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.
- Glasgow S.A., Agrotis M.A., Ercolani N.M., An integrable
reduction of inhomogeneously broadended optical equations,
Phys. D, 2005, V.212, 82-99.
- Agrotis M.A., Hamiltonian flows for a reduced Maxwell-Bloch
system with permanent dipole, Phys. D, 2003, V.183,
141-158.
- Bäcklund A.V., Zur Theorie der
Flachentransformationen, Math. Ann., 1881, V.19, 387-422.
- Lamb G.L., Elements in soliton theory, Wiley-Interscience
Pub., 1980.
- Lonngren K., Alwyn S. (Editors), Solitons in action, Academic Press, 1978.
- Newell A.C., Solitons in mathematics and physics, CBMS-NSF Regional
Conference Series, Vol. 48, SIAM Press, 1985.
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