Abstract
Infinitely many Hamilton–Poisson realizations of the five-dimensional real valued Maxwell–Bloch equations with the rotating wave approximation are constructed and the energy-Casimir mapping is considered. Also, the image of this mapping is presented and connections with the equilibrium states of the considered system are studied. Using some fibers of the image of the energy-Casimir mapping, some special orbits are obtained. Finally, a Lax formulation of the system is given.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Birtea, P. and Caşu, I., The Stability Problem and Special Solutions for the 5-Components Maxwell–Bloch Equations, Appl. Math. Lett., 2013, vol. 26, no. 8, pp. 875–880.
Bînzar, T. and Lăzureanu, C., A Rikitake Type System with One Control, Discrete Contin. Dyn. Syst. Ser. B, 2013, vol. 18, no. 7, pp. 1755–1776.
Bînzar, T. and Lăzureanu, C., On Some Dynamical and Geometrical Properties of the Maxwell–Bloch Equations with a Quadratic Control, J. Geom. Phys., 2013, vol. 70, pp. 1–8.
Bolsinov, A.V., Borisov, A. V., and Mamaev, I. S., Topology and Stability of Integrable Systems, Russian Math. Surveys, 2010, vol. 65, no. 2, pp. 259–318; see also: Uspekhi Mat. Nauk, 2010, vol. 65, no. 2, pp. 71–132.
Borisov, A. V. and Mamaev, I. S., Topological Analysis of an Integrable System Related to the Rolling of a Ball on a Sphere, Regul. Chaotic Dyn., 2013, vol. 18, no. 4, pp. 356–371.
Caşu, I., Symmetries of the Maxwell–Bloch Equations with the Rotating Wave Approximation, Regul. Chaotic Dyn., 2014, vol. 19, no. 5, pp. 548–555.
David, D. and Holm, D.D., Multiple Lie–Poisson Structures, Reductions, and Geometric Phases for the Maxwell–Bloch Travelling Wave Equations, J. Nonlinear Sci., 1992, vol. 2, no. 2, pp. 241–262.
Holm, D.D. and Kovačič, G., Homoclinic Chaos in a Laser-Matter System, Phys. D, 1992, vol. 56, nos. 2–3, pp. 270–300.
Holm, D.D., Kovačič, G., and Sundaram, B., Chaotic Laser-Matter Interaction, Phys. Lett. A, 1991, vol. 154, nos. 7–8, pp. 346–352.
Holm, D.D., Marsden, J. E., Raţiu, T., and Weinstein, A., Nonlinear Stability of Fluid and Plasma Equilibria, Phys. Rep., 1985, vol. 123, nos. 1–2, 116 pp.
Huang, D., Bi-Hamiltonian Structure and Homoclinic Orbits of the Maxwell–Bloch Equations with RWA, Chaos Solitons Fractals, 2004, vol. 22, no. 1, pp. 207–212.
Libermann, P. and Marle, Ch.-M., Symplectic Geometry and Analytical Mechanics, Math. Appl., vol. 35, Dordrecht: Reidel, 1987.
Lyapunov, A. M., The General Problem of the Stability of Motion, London: Taylor & Francis, 1992.
Lax, P.D., Integrals of Nonlinear Equations of Evolution and Solitary Waves, Comm. Pure Appl. Math., 1968, vol. 21, pp. 467–490.
Lăzureanu, C. and Bînzar, T., A Rikitake Type System with Quadratic Control, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 2012, vol. 22, no. 11, 1250274, 14 pp.
Marsden, J. E. and Raţiu, T. S., Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems, 2nd ed., Texts Appl. Math., vol. 17, New York: Springer, 1999.
Tudoran, R. M., Aron, A., Nicoară, S., On a Hamiltonian Version of the Rikitake System, SIAM J. Appl. Dyn. Syst., 2009, vol. 8, no. 1, pp. 454–479.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Caşu, I., Lăzureanu, C. Stability and integrability aspects for the Maxwell–Bloch equations with the rotating wave approximation. Regul. Chaot. Dyn. 22, 109–121 (2017). https://doi.org/10.1134/S1560354717020010
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1560354717020010