ESTONIAN ACADEMY
PUBLISHERS
eesti teaduste
akadeemia kirjastus
PUBLISHED
SINCE 1952
 
Proceeding cover
proceedings
of the estonian academy of sciences
ISSN 1736-7530 (Electronic)
ISSN 1736-6046 (Print)
Impact Factor (2021): 1.024
Multisoliton interactions for the Manakov system under composite external potentials; pp. 368–378
PDF | doi: 10.3176/proc.2015.3S.07

Authors
Michail Todorov, Vladimir S. Gerdjikov, Assen V. Kyuldjiev
Abstract

The soliton interactions of Manakov soliton trains subjected to composite external potentials are modelled by the perturbed complex Toda chain (PCTC). The model is applied to several classes of potentials, such as: (i) harmonic, (ii) periodic, (iii) ‘wide well’-type potentials, and (iv) inter-channel interactions. We demonstrate that the potentials can change the asymptotic regimes of the soliton trains. Our results can be implemented, e.g., in experiments on Bose–Einstein condensates and can be used to control the soliton motion. In general, our numerical experiments demonstrate that the predictions of complex Toda chain (CTC) (respectively PCTC) match very well the Manakov (respectively perturbed Manakov) model numerics for long-time evolution, often much longer than expected. This means that both CTC and PCTC are reliable dynamical models for predicting the dynamics of the multisoliton trains of the Manakov model in adiabatic approximation. This extends our previous results on scalar soliton trains to the Manakov trains with compatible initial parameters.

References

 

  1. Anderson, D. and Lisak, M. Nonlinear asymmetric self-phase modulation and self-steepening of pulses in long optical waveguides. Phys. Rev. A, 1983, 27, 1393–1398.
http://dx.doi.org/10.1103/PhysRevA.27.1393

  2. Anderson, D., Lisak, M., and Reichel, T. Approximate analytical approaches to nonlinear pulse propagation in optical fibers: a comparison. Phys. Rev. A, 1988, 38, 1618–1620.
http://dx.doi.org/10.1103/PhysRevA.38.1618

  3. Brazhnyi, V. A. and Konotop, V. V. Theory of nonlinear matter waves in optical lattices. Mod. Phys. Lett. B, 2004, 18(14), 627–651.
http://dx.doi.org/10.1142/S0217984904007190

  4. Frantzeskakis, D. J. Dark solitons in atomic Bose–Einstein condensates: from theory to experiments. J. Phys. A: Math. Theor., 2010, 43, 213001,
http://dx.doi.org/10.1088/1751-8113/43/21/213001

  5. Gerdjikov, V. S., Baizakov, B. B., and Salerno, M. Modelling adiabatic N-soliton interactions and perturbations. Theor. Math. Phys.+, 2005, 144(2), 1138–1146.
http://dx.doi.org/10.1007/s11232-005-0143-5

  6. Gerdjikov, V. S., Evstatiev, E. G., Kaup, D. J., Diankov, G. L., and Uzunov, I. M. Stability and quasi-equidistant propagation of NLS soliton trains. Phys. Lett. A, 1998, 241, 323–328.
http://dx.doi.org/10.1016/S0375-9601(98)00152-2

  7. Gerdjikov, V. S., Kostov, N. A., Doktorov, E. V., and Matsuka, N. P. Generalized perturbed complex Toda chain for Manakov system and exact solutions of Bose–Einstein mixtures. Math. Comput. Simulat., 2009, 80, 112–119.
http://dx.doi.org/10.1016/j.matcom.2009.06.013

  8. Gerdjikov, V. S. and Todorov, M. D. N-soliton interactions for the Manakov system. Effects of external potentials. In Localized Excitations in Nonlinear Complex Systems. Nonlinear Systems and Complexity. Vol. 7. (Carretero-González, R., Cuevas-Maraver, J., Frantzeskakis, D., Karachalios, N., Kevrekidis, P., and Palmero-Acebedo, F., eds). Springer International Publishing, Switzerland, 2014, 147–169.
http://dx.doi.org/10.1007/978-3-319-02057-0_7

  9. Gerdjikov, V. S. and Todorov, M. D. On the effects of sech-like potentials on Manakov solitons. In Application of Mathematics in Technical and Natural Sciences. 5th International Conference for Promoting the Application of Mathematics in Technical and Natural Sciences – AmiTaNS’13. AIP Proceedings, Vol. 1561 (Todorov, M. D., ed.). AIP Publishing, Melville, NY, 2013, 75–83.
http://dx.doi.org/10.1063/1.4827216

10. Gerdjikov, V. S., Uzunov, I. M., Evstatiev, E. G., and Diankov, G. L. Nonlinear Schrödinger equation and N-soliton interactions: Generalized Karpman-Solov¢ev approach and the complex Toda chain. Phys. Rev. E, 1997, 55(5), 6039–6060.
http://dx.doi.org/10.1103/PhysRevE.55.6039

11. Griffin, A., Nikuni, T., and Zaremba, E. Bose-Condensed Gases at Finite Temperatures. Cambridge University Press, Cambridge, UK, 2009.
http://dx.doi.org/10.1017/CBO9780511575150

12. Ho, T.-L. Spinor Bose condensates in optical traps. Phys. Rev. Lett., 1998, 81, 742.
http://dx.doi.org/10.1103/PhysRevLett.81.742

13. Karpman, V. I. and Solov¢ev, V. V. A perturbational approach to the two-soliton systems. Physica D, 1981, 3, 487–502.
http://dx.doi.org/10.1016/0167-2789(81)90123-8

14. Kevrekidis, P. G. and Frantzeskakis, D. J. Pattern forming dynamical instabilities of Bose–Einstein condensates. Mod. Phys. Lett. B, 2004, 18, 173,
http://dx.doi.org/10.1142/S0217984904006809

15. Kevrekidis, P. G., Frantzeskakis, D. J., and Carretero-Gonzalez, R. (eds). Emergent Nonlinear Phenomena in Bose-Einstein Condensates: Theory and Experiment. Springer, Berlin, Heidelberg, New York, 2008, 45.
http://dx.doi.org/10.1007/978-3-540-73591-5

16. Lakoba, T. I. and Kaup, D. J. Perturbation theory for the Manakov soliton and its applications to pulse propagation in randomly birefringent fibers. Phys. Rev. E, 1997, 56, 6147–6165.
http://dx.doi.org/10.1103/PhysRevE.56.6147

17. Liang, Z. X., Zhang, Z. D., and Liu, W. M. Dynamics of a bright soliton in Bose-Einstein condensates with time-dependent atomic scattering length in an expulsive parabolic potential. Phys. Rev. Lett., 2005, 94, 050402,
http://dx.doi.org/10.1103/PhysRevLett.94.050402

18. Manakov, S. V. On the theory of two-dimensional stationary self-focusing of electromagnetic waves. Zh. Eksp. Teor. Fiz., 1974, 65, 505–516 (in Russian). English translation: Sov. Phys. JETP, 1974, 38, 248–253.

19. Modugno, M., Dalfovo, F., Fort, C., Maddaloni, P., and Minardi, F. Dynamics of two colliding Bose-Einstein condensates in an elongated magnetostatic trap. Phys. Rev. A, 2000, 62, 063607.
http://dx.doi.org/10.1103/PhysRevA.62.063607

20. Novikov, S. P., Manakov, S. V., Pitaevski, L. P., and Zakharov, V. E. Theory of Solitons, the Inverse Scattering Method. Consultant Bureau, New York, 1984.

21. Ohmi, T. and Machida, K. Bose-Einstein condensation with internal degrees of freedom in alkali atom gases. J. Phys. Soc. Jpn, 1998, 67, 1822.
http://dx.doi.org/10.1143/JPSJ.67.1822

22. Perez-Garcia, V. M., Michinel, H., Cirac, J. I., Lewenstein, M., and Zoller, P. Dynamics of Bose-Einstein condensates: variational solutions of the Gross-Pitaevskii equations. Phys. Rev. A, 1997, 56, 1424–1432.
http://dx.doi.org/10.1103/PhysRevA.56.1424

23. Perez-Garcia, V. M., Michinel, H., and Herrero, H. Bose-Einstein solitons in highly asymmetric traps. Phys. Rev. A, 1998, 57, 3837,
http://dx.doi.org/10.1103/PhysRevA.57.3837

24. Pitaevski, L. P. and Stringari, S. Bose-Einstein Condensation. Oxford University Press, Oxford, UK, 2003.

25. Sonnier, W. J. and Christov, C. I. Strong coupling of Scrödinger equations: conservative scheme approach. Math. Comput. Simulat., 2005, 69, 514–525.
http://dx.doi.org/10.1016/j.matcom.2005.03.016

26. Todorov, M. D. The effect of the elliptic polarization on the quasi-particle dynamics of linearly coupled systems of nonlinear Schrödinger equations. Math. Comput. Simulat., 2014, http://dx.doi.org/10.1016/j.matcom.2014.04.011 (accessed 18.07.2014).
http://dx.doi.org/10.1016/j.matcom.2014.04.011

27. Todorov, M. D. and Christov, C. I. Collision dynamics of polarized solitons in linearly CNSE. In International Workshop on Complex Structures, Integrability and Vector Fields. AIP Proceedings, Vol. 1340 (Sekigawa, K., ed.). AIP Publishing, Melville, NY, 2011, 144–153.

28. Todorov, M. D., Gerdjikov, V. S., and Kyuldjiev, A. V. Modeling interactions of soliton trains. Effects of external potentials. In Application of Mathematics in Technical and Natural Sciences. 6th International Conference for Promoting the Application of Mathematics in Technical and Natural Sciences – AmiTaNS’14. AIP Proceedings, Vol. 1629 (Todorov, M. D., ed.). AIP Publishing, Melville, NY, 2014, 186–200.

29. Uchiyama, M., Ieda, J., and Wadati, M. Multicomponent bright solitons in F = 2 spinor Bose-Einstein condensates. J. Phys. Soc. Jpn, 2007, 76(7), 74005.
http://dx.doi.org/10.1143/JPSJ.76.074005

30. Zakharov, V. E. and Shabat, A. B. Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media. Zh. Eksp. Teor. Fiz., 1972, 61, 118–134 (in Russian). English translation: Sov. Phys. JETP, 1972, 34, 62–69.

31. Zhang, X.-F., Hu, X.-H., Liu, X.-X., and Liu, W. M. Vector solitons in two-component Bose-Einstein condensates with tunable interactions and harmonic potential. Phys. Rev. A, 2009, 79, 033630,
http://dx.doi.org/10.1103/PhysRevA.79.033630

 

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