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Double parametric resonance for matter-wave solitons in a time-modulated trap

Bakhtiyor Baizakov1, Giovanni Filatrella2, Boris Malomed3, and Mario Salerno1

  • 1Dipartimento di Fisica “E. R. Caianiello” and Istituto Nazionale di Fisica della Materia (INFM), Università di Salerno, I-84081 Baronissi (SA), Italy
  • 2INFM, Laboratorio Regionale di Salerno and Dipartimento di Scienze Biologiche ed Ambientali, Università del Sannio, via Port’Arsa 11, 82100 Benevento, Italy
  • 3Department of Interdisciplinary Studies, Faculty of Engineering, Tel Aviv University, Tel Aviv 69978, Israel

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Phys. Rev. E 71, 036619 – Published 23 March, 2005

DOI: https://doi.org/10.1103/PhysRevE.71.036619

Abstract

We analyze the motion of solitons in a self-attractive Bose-Einstein condensate, loaded into a quasi-one-dimensional parabolic potential trap, which is subjected to time-periodic modulation with an amplitude ε and frequency Ω. First, we apply the variational approximation, which gives rise to decoupled equations of motion for the center-of-mass coordinate of the soliton, ξ(t), and its width a(t). The equation for ξ(t) is the ordinary Mathieu equation (ME) (it is an exact equation that does not depend on the adopted ansatz), the equation for a(t) being a nonlinear generalization of the ME. Both equations give rise to the same map of instability zones in the (ε,Ω) plane, generated by the parametric resonances (PRs), if the instability is defined as the onset of growth of the amplitude of the parametrically driven oscillations. In this sense, the double PR is predicted. Direct simulations of the underlying Gross-Pitaevskii equation give rise to a qualitatively similar but quantitatively different stability map for oscillations of the soliton’s width a(t). In the direct simulations, we identify the soliton dynamics as unstable if the instability (again, realized as indefinite growth of the amplitude of oscillations) can be detected during a time comparable with, or smaller than, the lifetime of the condensate (therefore accessible to experimental detection). Two-soliton configurations are also investigated. It is concluded that multiple collisions between solitons are elastic, and they do not affect the instability borders.

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References (26)

  1. K. E. Strecker, G. B. Partridge, A. G. Truscott, and R. G. Hulet, Nature (London) 417, 150 (2002).
  2. L. Khaykovich, F. Schreck, G. Ferrari, T. Bourdel, J. Cubizolles, L. D. Carr, Y. Castin, and C. Salomon, Science 296, 1290 (2002).
  3. L. D. Carr and J. Brand, Phys. Rev. Lett. 92, 040401 (2004).
  4. H. Sakaguchi and B. A. Malomed, J. Phys. B 37, 1442 (2004).
  5. H. Sakaguchi and B. A. Malomed, J. Phys. B 37, 2225 (2004).
  6. J. J. García-Ripoll and V. M. Pérez-García, Phys. Rev. A 59, 2220 (1999).
  7. J. J. García-Ripoll and V. M. Pérez-García, Phys. Rev. Lett. 83, 1715 (1999).
  8. F. Kh. Abdullaev, R. M. Galimzyanov, M. Brtka, and R. A. Kraenkel, J. Phys. B 37, 3535 (2004).
  9. F. Kh. Abdullaev and R. Galimzyanov, J. Phys. B 36, 1099 (2003).
  10. F. Kh. Abdullaev and J. Garnier, Phys. Rev. A 70, 053604 (2004).
  11. Handbook of Mathematical Functions, edited by M. Abramowitz and I. A. Stegun (Dover Publications, New York, 1965).
  12. T. F. Scott, R. J. Ballagh, and K. Burnett, J. Phys. B 31, L329 (1998); V. M. Pérez-García, H. Michinel, and H. Herrero, Phys. Rev. A 57, 3837 (1998); L. Salasnich, A. Parola, and L. Reatto, ibid. 65, 043614 (2002); G. Theocharis, Z. Rapti, P. G. Kevrekidis, D. J. Frantzeskakis, and V. V. Konotop, ibid. 67, 063610 (2003); J. J. Garcia-Ripoll, V. V. Konotop, B. Malomed, and V. M. Perez-Garcia, Math. Comput. Simul. 62, 21 (2003).
  13. Y. B. Band, I. Towers, and B. A. Malomed, Phys. Rev. A 67, 023602 (2003).
  14. G. Baym and C. J. Pethick, Phys. Rev. Lett. 76, 6 (1996); F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, Rev. Mod. Phys. 71, 463 (1999).
  15. P. A. Ruprecht, M. J. Holland, K. Burnett, and M. Edwards, Phys. Rev. A 51, 4704 (1995); A. Eleftheriou and K. Huang, ibid. 61, 043601 (2000).
  16. L. D. Carr and Y. Castin, Phys. Rev. A 66, 063602 (2002).
  17. T. R. Taha and M. J. Ablowitz, J. Comput. Phys. 55, 203 (1984).
  18. S. K. Adhikari and P. Muruganandam, J. Phys. B 35, 2831 (2002).
  19. D. Anderson, Phys. Rev. A 27, 3135 (1983); B. A. Malomed, Prog. Opt. 43, 71 (2002).
  20. F. Kh. Abdullaev, J. C. Bronski, and G. Papanicolaou, Physica D 135, 369 (2000).
  21. V. M. Perez-Garcia, P. Torres, J. J. Garcia-Ripoll, and H. Michinel, J. Opt. B: Quantum Semiclassical Opt. 2, 353 (2000).
  22. R. Hasse, Phys. Rev. A 25, 583 (1982).
  23. B. V. Chirikov, Phys. Rep. 52, 263 (1979); A. J. Lichtenberg and M. A. Lieberman, Regular and Stochastic Motion (Springer-Verlag, New York, 1983).
  24. V. I. Karpman and E. M. Maslov, Zh. Eksp. Teor. Fiz. 75, 504 (1978) [Sov. Phys. JETP 48, 252 (1978)]; Yu. S. Kivshar and B. A. Malomed, Rev. Mod. Phys. 61, 763 (1989).
  25. W. L. Kath and N. F. Smyth, Phys. Rev. E 51, 1484 (1995).
  26. R. H. Rand, http://www.tam.cornell.edu/randdocs/nlvibe45.pdf.

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Phys. Rev. E 71, 036619– Published 23 March, 2005
  • Received 17 June 2004

© 2005 American Physical Society

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