Advertisement

Journal of Russian Laser Research

, Volume 33, Issue 1, pp 63–83 | Cite as

Soliton-like solutions for the nonlinear schrödinger equation with variable quadratic hamiltonians

  • Erwin Suazo
  • Sergei K. Suslov
Article

Abstract

We construct one-soliton solutions for the nonlinear Schr¨odinger equation with variable quadratic Hamiltonians in a unified form by taking advantage of the complete (super) integrability of generalized harmonic oscillators. The soliton-wave evolution in external fields with variable quadratic potentials is totally determined by the linear problem, like motion of a classical particle with acceleration, and the (self-similar) soliton shape is due to a subtle balance between the linear Hamiltonian (dispersion and potential) and nonlinearity in the Schr¨odinger equation by the standards of soliton theory. Most linear (hypergeometric, Bessel) and a few nonlinear (Jacobian elliptic, second Painlev´e transcendental) classical special functions of mathematical physics are linked together through these solutions, thus providing a variety of nonlinear integrable cases. Examples include bright and dark solitons and Jacobi elliptic and second Painlev´e transcendental solutions for several variable Hamiltonians that are important for research in nonlinear optics, plasma physics, and Bose–Einstein condensation. The Feshbach-resonance matter-wave-soliton management is briefly discussed from this new perspective.

Keywords

nonlinear Schrödinger equation Gross–Pitaevskii equation Bose–Einstein condensation Feshbach resonance fiber optics generalized harmonic oscillators soliton-like solutions Jacobian elliptic functions Painlevé II transcendents 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    U. Al Khawaja, H. T. C. Stoof, R. E. Hulet, et al., Phys. Rev. Lett., 89, 200404 (2002).ADSCrossRefGoogle Scholar
  2. 2.
    T. Brugarino and M. Sciacca, J. Math. Phys., 51, 093503 (2010).MathSciNetADSCrossRefGoogle Scholar
  3. 3.
    N. A. Kudryashov, Methods of Nonlinear Mathematical Physics [in Russian], Intellect, Dolgoprudny, Moscow Region, Russia (2010).Google Scholar
  4. 4.
    Z. Yan and V. V. Konotop, Phys. Rev. E, 80, 036607 (2009).ADSCrossRefGoogle Scholar
  5. 5.
    V. E. Zakharov and A. B. Shabat, Zh. Éksp. Teor. Fiz., 61, 118 (1971) [Sov. Phys. JETP 34, 62 (1972)].Google Scholar
  6. 6.
    F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, Rev. Mod. Phys., 71, 463 (1999).ADSCrossRefGoogle Scholar
  7. 7.
    L. Pitaevskii and S. Stringari, Bose–Einstein Condensation, Oxford University Press (2003).Google Scholar
  8. 8.
    K. Bongs and K. Sengstock, Rep. Prog. Phys., 67, 907 (2004).ADSCrossRefGoogle Scholar
  9. 9.
    Yu. Kagan, E. L. Surkov, and G. V. Shlyapnikov, Phys. Rev. A, 54, R1753 (1996).ADSCrossRefGoogle Scholar
  10. 10.
    Yu. Kagan, E. L. Surkov, and G. V. Shlyapnikov, Phys. Rev. A, 55, R18 (1997).ADSCrossRefGoogle Scholar
  11. 11.
    Yu. S. Kivshar, T. J. Alexander, and S. K. Turitsyn, Phys. Lett. A, 278, 225 (2001).MathSciNetADSMATHCrossRefGoogle Scholar
  12. 12.
    A. N. Oraevsky, Quantum Electron., 31, 1038 (2001).ADSCrossRefGoogle Scholar
  13. 13.
    V. M. Pérez-García, P. Torres, and G. D. Montesinos, SIAM J. Appl. Math. 67, 990 (2007).Google Scholar
  14. 14.
    S. Burger, K. Bongs, S. Dettmer, et al., Phys. Rev. Lett., 83 , 5198 (1999).ADSCrossRefGoogle Scholar
  15. 15.
    F. S. Cataliotti, S. Burger, C. Fort, et al., Science, 293, 843 (2001).ADSCrossRefGoogle Scholar
  16. 16.
    J. Denschlag, J. E. Simsarian, H. Haffner, et al., Science, 287, 97 (2000).ADSCrossRefGoogle Scholar
  17. 17.
    L. Khaykovich, F. Schreck, G. Ferrari, et al., Science, 296, 1290 (2002).ADSCrossRefGoogle Scholar
  18. 18.
    K. E. Strecker, G. B. Partridge, A. G. Truscott, and R. G. Hulet, Nature, 417, 150 (2002).ADSCrossRefGoogle Scholar
  19. 19.
    K. E. Strecker, G. B. Partridge, A. G. Truscott, et al., New J. Phys., 5, 73.1 (2003).Google Scholar
  20. 20.
    D. J. Frantzeskakis, J. Phys. A: Math. Gen., 43, 213001 (2010).MathSciNetADSCrossRefGoogle Scholar
  21. 21.
    V. A. Brazhnyi, V. V. Konotop, and L. P. Pitaevskii, Phys. Rev. A, 73, 053601 (2006).ADSCrossRefGoogle Scholar
  22. 22.
    V. V. Konotop and L. Pitaevskii, Phys. Rev. Lett., 93, 240403 (2004).ADSCrossRefGoogle Scholar
  23. 23.
    M. Ablowitz, B. Prinari, and A. D. Trubatch, Discrete and Continuous Schrödinger Systems, Cambridge University Press (2004).Google Scholar
  24. 24.
    G. P. Agrawal, Nonlinear Fiber Optics, 4th ed., Academic Press, New York (2007).Google Scholar
  25. 25.
    A. Desgasperis, J. Phys. A: Math. Theor., 43, 434001 (2010).ADSCrossRefGoogle Scholar
  26. 26.
    A. Hasegawa, Optical Solitons in Fibers, Springer, Berlin (1989).CrossRefGoogle Scholar
  27. 27.
    Yu. S. Kivshar and B. Luther-Davies, Phys. Rep., 298, 81 (1998).ADSCrossRefGoogle Scholar
  28. 28.
    Y. I. Kruglov, A. C. Peacock, and J. D. Harvey, Phys. Rev. Lett., 90, 113902 (2003).ADSCrossRefGoogle Scholar
  29. 29.
    Y. I. Kruglov, A. C. Peacock, and J. D. Harvey, Phys. Rev. E, 71, 1056619 (2005).MathSciNetADSCrossRefGoogle Scholar
  30. 30.
    J. D. Moores, Opt. Lett., 21, 555 (1996).ADSCrossRefGoogle Scholar
  31. 31.
    J. D. Moores, Opt. Lett., 26, 87 (2001).ADSCrossRefGoogle Scholar
  32. 32.
    S. Ponomarenko and G. P. Agrawal, Opt. Lett., 32, 1659 (2007).ADSCrossRefGoogle Scholar
  33. 33.
    V. N. Serkin and A. Hasegawa, Phys. Rev. Lett., 85, 4502 (2000).ADSCrossRefGoogle Scholar
  34. 34.
    V. N. Serkin, A. Hasegawa, and T. L. Belyeva, Phys. Rev. Lett., 92, 199401 (2004).ADSCrossRefGoogle Scholar
  35. 35.
    R. Balakrishnan, Phys. Rev. A, 32, 1144 (1985).ADSCrossRefGoogle Scholar
  36. 36.
    H.-H. Chen and Ch.-Sh. Liu, Phys. Rev. Lett., 37, 693 (1976).MathSciNetADSCrossRefGoogle Scholar
  37. 37.
    H.-H. Chen and Ch.-Sh. Liu, Phys. Fluids, 21, 377 (1978).ADSMATHCrossRefGoogle Scholar
  38. 38.
    A. C. Newell, J. Math. Phys., 19, 1126 (1978).MathSciNetADSMATHCrossRefGoogle Scholar
  39. 39.
    T. Xu, B. Tian, Li-Li Li, X. L¨u, and Ch. Zhang, Phys. Plasmas, 15, 102307 (2008).ADSCrossRefGoogle Scholar
  40. 40.
    U. Al Khawaja, J. Math. Phys., 51, 053506 (2010).MathSciNetADSCrossRefGoogle Scholar
  41. 41.
    R. Conte, Phys. Lett. A, 140, 383 (1989).MathSciNetADSCrossRefGoogle Scholar
  42. 42.
    R. Conte, “The Painlevé approach to nonlinear ordinary differential equations,” in: R. Conte (Ed.), The Painlevé Property, One Century Later, CRM Series in Mathematical Physics, Springer, New York (1991), p. 77.Google Scholar
  43. 43.
    R. Conte, A. P. Fordy, and A. Pickering, Physica D, 69, 33 (1993).MathSciNetADSMATHCrossRefGoogle Scholar
  44. 44.
    R. Conte and M. Musette, Stud. Appl. Math., 123, 63 (2009).MathSciNetMATHCrossRefGoogle Scholar
  45. 45.
    X-G. He, D. Zhao, L. Lee, and H-G. Luo, Phys. Rev. E, 79, 056610 (2009).ADSCrossRefGoogle Scholar
  46. 46.
    M. Musette and R. Conte, Physica D, 181, 70 (2003).MathSciNetADSMATHCrossRefGoogle Scholar
  47. 47.
    J. Weiss, M. Tabor, and G. Carnevalle, J. Math. Phys., 24, 522 (1983).MathSciNetADSMATHCrossRefGoogle Scholar
  48. 48.
    R. Hirota, Phys. Rev. Lett., 27, 1192 (1971).ADSMATHCrossRefGoogle Scholar
  49. 49.
    R. Hirota, The Direct Method in Soliton Theory, Cambridge University Press (2004).Google Scholar
  50. 50.
    P. D. Lax, Commun. Pure Appl. Math., 21, 467 (1968).MathSciNetMATHCrossRefGoogle Scholar
  51. 51.
    M. Ablowitz and P. A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, Cambridge University Press (1991).Google Scholar
  52. 52.
    M. J. Ablowitz, D. J. Kaup, A. C. Newell, and H. Segur, Phys. Rev. Lett., 31, 125 (1973).MathSciNetADSCrossRefGoogle Scholar
  53. 53.
    M. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform, SIAM, Philadelphia (1981).MATHCrossRefGoogle Scholar
  54. 54.
    L. D. Faddeev and L. A. Takhtajan, Hamiltonian Methods in the Theory of Solitons, Springer, Berlin, New York (1987).MATHGoogle Scholar
  55. 55.
    C. S. Gardner, J. M. Green, M. D. Kruskai, and R. M. Miura, Phys. Rev. Lett., 19, 1095 (1967).ADSMATHCrossRefGoogle Scholar
  56. 56.
    S. Novikov, S. V. Manakov, L. P. Pitaevskii, and V. E. Zakharov, Theory of Solitons: The Inverse Scattering Method, Kluwer, Dordrecht (1984).Google Scholar
  57. 57.
    H.-H. Chen, Phys. Rev. Lett., 33, 925 (1974).MathSciNetADSCrossRefGoogle Scholar
  58. 58.
    A. Desgasperis, Am. J. Phys., 66, 486 (1998).ADSCrossRefGoogle Scholar
  59. 59.
    B. A. Dubrovin, V. B. Matveev, and S. P. Novikov, Russ. Math. Surv., 31, 59 (1976) [Integrable System: Selected Papers, London Mathematical Society Lecture Note Series (1981), vol. 60, p. 53].Google Scholar
  60. 60.
    V. B. Matveev and M. A. Salle, Darboux Transformation and Solitons, Springer, Berlin (1991).Google Scholar
  61. 61.
    P. J. Olver, Applications of Lie Group to Differential Equations, Springer, Berlin (1991).Google Scholar
  62. 62.
    C. Rogers and W. K. Scheif, Bäcklund Transformation and Darboux Transformation, Cambridge University Press (2002).Google Scholar
  63. 63.
    V. N. Serkin, A. Hasegawa, and T. L. Belyeva, Phys. Rev. Lett., 98, 074102 (2007).ADSCrossRefGoogle Scholar
  64. 64.
    V. N. Serkin, A. Hasegawa, and T. L. Belyeva, Phys. Rev. A, 81, 023610 (2010).ADSCrossRefGoogle Scholar
  65. 65.
    P. A. Clarkson, Proc. Roy. Soc. Edinburgh, 109A, 109 (1988).MathSciNetCrossRefGoogle Scholar
  66. 66.
    J. He and Y. Li, Studies in Applied Mathematics (2010), doi: 10.1111/j.1467-9590.2010.00495.x.
  67. 67.
    A. Kundu, Phys. Rev. E, 79, 015601(R) (2009).MathSciNetADSCrossRefGoogle Scholar
  68. 68.
    V. M. Pérez-García, P. Torres, and V. V. Konotop, Physica D, 221, 31 (2006).MathSciNetADSMATHCrossRefGoogle Scholar
  69. 69.
    A. V. Zhukov, Phys. Lett. A, 256, 325 (1999).ADSCrossRefGoogle Scholar
  70. 70.
    R. Atre, P. K. Panigrahi, and G. S. Agarwal, Phys. Rev. E., 73, 056611 (2006).MathSciNetADSCrossRefGoogle Scholar
  71. 71.
    A. Ebaid and S. M. Khaled, J. Comput. Appl. Math., doi: 10.1016/j.cam.2010.09.024.
  72. 72.
    Z. X. Liang, Z. D. Zhang, and W. M. Liu, Phys. Rev. Lett., 94, 050402 (2005).ADSCrossRefGoogle Scholar
  73. 73.
    Sh. Chen and L. Yi, Phys. Rev. E, 61, 016606 (2005).ADSCrossRefGoogle Scholar
  74. 74.
    N. A. Kudryashov, Commun. Nonlin. Sci. Numer. Simul., 14, 3507 (2009).MathSciNetMATHCrossRefGoogle Scholar
  75. 75.
    O. S. Rozanova, Proc. Am. Math. Soc., 133, 2347 (2005).MathSciNetMATHCrossRefGoogle Scholar
  76. 76.
    V. N. Serkin and A. Hasegawa, JETP Lett., 72, 89 (2000).ADSCrossRefGoogle Scholar
  77. 77.
    C. Trallero-Giner, J. Drake, V. Lopez-Richard, et al., Phys. Lett. A, 354, 115 (2006).ADSMATHCrossRefGoogle Scholar
  78. 78.
    Z. Yan, Comput. Phys. Commun., 153, 145 (2003).ADSMATHCrossRefGoogle Scholar
  79. 79.
    Z. Yan, Chaos, Solitons Fractals, 21, 1013 (2004).ADSMATHCrossRefGoogle Scholar
  80. 80.
    Z. Yan, Phys. Lett. A, doi: 10.1016/j.physleta.2010.09.070.
  81. 81.
    T. Tao, Bull. Am. Math. Soc., 46, 1 (2009).MATHCrossRefGoogle Scholar
  82. 82.
    V. E. Zakharov and A. B. Shabat, Funct. Anal. Appl., 8, 226 (1974).MATHCrossRefGoogle Scholar
  83. 83.
    V. E. Zakharov and A. B. Shabat, Funct. Anal. Appl., 13, 166 (1979).MathSciNetGoogle Scholar
  84. 84.
    X.-F. Zhang, Q. Yang, J.-F. Zhang, et al., Phys. Rev. A, 77, 023613 (2008).ADSCrossRefGoogle Scholar
  85. 85.
    N. I. Akhiezer, Elements of the Theory of Elliptic Functions, Translations of Mathematical Monographs, American Mathematical Society, Providence, Rhode Island (1980), Vol. 79.Google Scholar
  86. 86.
    A. Erdélyi (Ed.), Higher Transcendental Functions, McGraw-Hill (1953), vol. III.Google Scholar
  87. 87.
    E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, 4th ed., Cambridge University Press (1952).Google Scholar
  88. 88.
    P. A. Clarkson, “Painlevé transcendents,” in: F. W. J. Olwer and D. M. Lozier (Eds.), NIST Handbook of Mathematical Functions, Cambridge University Press (2010) [http://dlmf.nist.gov/32].
  89. 89.
    M. Tajiri, J. Phys. Soc. Jpn, 52, 1908 (1983).MathSciNetADSCrossRefGoogle Scholar
  90. 90.
    R. Cordero-Soto, R. M. Lopez, E. Suazo, and S. K. Suslov, Lett. Math. Phys., 84, 159 (2008).MathSciNetADSMATHCrossRefGoogle Scholar
  91. 91.
    R. Cordero-Soto, E. Suazo, and S. K. Suslov, Ann. Phys., 325, 1884 (2010) [arXiv:0912.4900v9 [math-ph] 19 Mar 2010].MathSciNetADSMATHCrossRefGoogle Scholar
  92. 92.
    M. V. Berry, J. Phys. A: Math. Gen, 18, 15 (1985).ADSMATHCrossRefGoogle Scholar
  93. 93.
    V. V. Dodonov, I. A. Malkin, and V. I. Man’ko, Int. J. Theor. Phys., 14, 37 (1975).MathSciNetCrossRefGoogle Scholar
  94. 94.
    J. H. Hannay, J. Phys. A: Math. Gen, 18, 221(1985).MathSciNetADSCrossRefGoogle Scholar
  95. 95.
    K. B. Wolf, SIAM J. Appl. Math., 40, 419 (1981).MathSciNetMATHCrossRefGoogle Scholar
  96. 96.
    K-H. Yeon, K-K. Lee, Ch-I. Um, et al., Phys. Rev. A, 48, 2716 (1993).ADSCrossRefGoogle Scholar
  97. 97.
    E. Suazo and S. K. Suslov, “Cauchy problem for Schrödinger equation with variable quadratic Hamiltonians” (in preparation).Google Scholar
  98. 98.
    S. K. Suslov, Phys. Scr., 81, 055006 (2010) [arXiv:1002.0144v6 [math-ph] 11 Mar 2010].ADSCrossRefGoogle Scholar
  99. 99.
    V. V. Dodonov and V. I. Man’ko, “Invariants and correlated states of nonstationary quantum systems” [in Russian], Invariants and the Evolution of Nonstationary Quantum Systems, Proceedings of the Lebedev Physical Institute, Nauka, Moscow (1987), vol. 183, p. 71 [English translation published by Nova Science, Commack, New York (1989), p. 103].Google Scholar
  100. 100.
    I. A. Malkin and V. I. Man’ko, Dynamic Symmetries and Coherent States of Quantum Systems [in Russian], Nauka, Moscow (1979).Google Scholar
  101. 101.
    S. K. Suslov, “On integrability of nonautonomous nonlinear Schr¨odinger equations,” Proc. Am. Math. Soc.,  http://dx.doi.org/10.1090/S0002-9939-2011-11176-6; Posted: December 30, 2011 [arXiv:1012.3661v3 [math-ph] 16 Apr 2011].
  102. 102.
    G. V. Shlyapnikov (private communication).Google Scholar
  103. 103.
    E. Merzbacher, Quantum Mechanics, 3 rd ed., John Wiley & Sons, New York (1998).Google Scholar
  104. 104.
    E. D. Rainville, Special Functions, Macmillan, New York (1960).MATHGoogle Scholar
  105. 105.
    G. E. Andrews, R. A. Askey, and R. Roy, Special Functions, Cambridge University Press (1999).Google Scholar
  106. 106.
    W. Magnus and S. Winkler, Hill’s Equation, Dover, New York (1966).MATHGoogle Scholar
  107. 107.
    A. F. Nikiforov and V. B. Uvarov, Special Functions of Mathematical Physics, Birkhäuser, Basel, Boston (1988).MATHGoogle Scholar
  108. 108.
    G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed., Cambridge University Press (1944).Google Scholar
  109. 109.
    R. Cordero-Soto, E. Suazo, and S. K. Suslov, J. Phys. Math., 1, S090603 (2009).CrossRefGoogle Scholar
  110. 110.
    R. Cordero-Soto and S. K. Suslov, Theor. Math. Phys., 162, 286 (2010) [arXiv:0808.3149v9 [math-ph] 8 Mar 2009].MathSciNetMATHCrossRefGoogle Scholar
  111. 111.
    P. O. Fedichev, A. E. Muryshev, and G. V. Shlyapnikov, Phys. Rev. A, 60, 3220 (1999).ADSCrossRefGoogle Scholar
  112. 112.
    M. Ablowitz and H. Segur, Phys. Rev. Lett., 38, 1103 (1977).MathSciNetADSCrossRefGoogle Scholar
  113. 113.
    H. Segur and M. J. Ablowitz, Physica D, 3, 165 (1981).ADSMATHCrossRefGoogle Scholar
  114. 114.
    P. Deift and X. Zhou, Ann. Math., 137, 295 (1993).MathSciNetMATHCrossRefGoogle Scholar
  115. 115.
    P. Deift and X. Zhou, Commun. Pure Appl. Math., 48, 227 (1995).MathSciNetGoogle Scholar
  116. 116.
    A. P. Bassom, P. A. Clarkson, C. K. Law, and J. B. McLeod, Arch. Ration Mech. Anal., 103, 241 (1998).MathSciNetCrossRefGoogle Scholar
  117. 117.
    P. A. Clarkson, J. Comput. Appl. Math., 153, 127(2003).MathSciNetADSMATHCrossRefGoogle Scholar
  118. 118.
    P. A. Clarkson and J. B. McLeod, Arch. Ration Mech. Anal., 103, 97 (1988).MathSciNetMATHCrossRefGoogle Scholar
  119. 119.
    Y. Takei, ANZIAM J., 44, 111 (2002).MathSciNetMATHCrossRefGoogle Scholar
  120. 120.
    N. Lanfear, R. M. Lopez, and S. K. Suslov, J. Russ. Laser Res., 32, 352 (2011) [arXiv:11002.5119v1 [math-ph] 24 Feb 2011].CrossRefGoogle Scholar
  121. 121.
    N. Lanfear and S. K. Suslov, “The time-dependent Schrödinger equation, Riccati equation, and Airy functions,” arXiv:0903.3608v5 [math-ph] 22 Apr 2009.Google Scholar
  122. 122.
    P. Caldirola, Nuovo Cimento, 18, 393 (1941).CrossRefGoogle Scholar
  123. 123.
    H. Dekker, Phys. Rep., 80, 1 (1981).MathSciNetADSCrossRefGoogle Scholar
  124. 124.
    E. Kanai, Prog. Theor. Phys., 3, 440 (1948).ADSCrossRefGoogle Scholar
  125. 125.
    Ch-I. Um, K-H. Yeon, and T. F. George, Phys. Rep., 362, 63 (2002).MathSciNetADSCrossRefGoogle Scholar
  126. 126.
    M. Meiler, R. Cordero-Soto, and S. K. Suslov, J. Math. Phys., 49, 072102 (2008) [arXiv:0711.0559v4 [math-ph] 5 Dec 2007].MathSciNetADSCrossRefGoogle Scholar
  127. 127.
    R. Cordero-Soto and S. K. Suslov, J. Phys. A: Math. Theor., 44, 015101 (2011) [arXiv:1006.3362v3 [math-ph] 2 Jul 2010].MathSciNetADSCrossRefGoogle Scholar
  128. 128.
    D. Chruściński and J. Jurkowski, “Memory in a nonlocally damped oscillator,” arXiv:0707.1199v2 [quant-ph] 7 Dec 2007.Google Scholar
  129. 129.
    T. S. Raju, P. K. Panigrahi, and K. Porsezian, Phys. Rev. E, 71, 026608 (2005).ADSCrossRefGoogle Scholar
  130. 130.
    L. Gagnon and P. Winternitz, J. Phys. A: Math. Gen., 26, 7061 (1993).MathSciNetADSMATHCrossRefGoogle Scholar
  131. 131.
    M. Musette, “Painlevé analysis for nonlinear partial differential equations,” in: R. Conte (Ed.), The Painlevé Property, One Century Later, CRM Series in Mathematical Physics, Springer, New York (1999), p. 517.Google Scholar
  132. 132.
    E. A. Cornell and C. E. Wieman, Rev. Mod. Phys., 74, 875 (2002).ADSCrossRefGoogle Scholar
  133. 133.
    W. Ketterle, Rev. Mod. Phys., 74, 1131 (2002).ADSCrossRefGoogle Scholar
  134. 134.
    L. Erdös, B. Schlein, and H.-T. Yau, Phys. Rev. Lett., 98, 040404 (2007).ADSCrossRefGoogle Scholar
  135. 135.
    E. H. Lieb, R. Seiringer, and J. Yngvason, Phys. Rev. A, 61, 043602 (2000).ADSCrossRefGoogle Scholar
  136. 136.
    L. Salasnich, A. Parola, and L. Reatto, Phys. Rev. A, 65, 043614 (2002).ADSCrossRefGoogle Scholar
  137. 137.
    A. D. Jackson, G. M. Kavoulakis, and C. J. Pethick, Phys. Rev. A, 58, 2417 (1998).ADSCrossRefGoogle Scholar
  138. 138.
    C. Menotti and S. Stringari, Phys. Rev. A, 66, 043610 (2002).ADSCrossRefGoogle Scholar
  139. 139.
    A. Muñoz Mateo and V. Delgado, Phys. Rev. A, 75, 063610 (2007).ADSCrossRefGoogle Scholar
  140. 140.
    A. Muñoz Mateo and V. Delgado, Phys. Rev. A, 77, 013617 (2008).ADSCrossRefGoogle Scholar
  141. 141.
    A. Muñoz Mateo and V. Delgado, Ann. Phys., 324, 709 (2009).ADSMATHCrossRefGoogle Scholar
  142. 142.
    V. M. Pérez-García and H. Michinel, Phys. Rev. A, 57, 3837 (1998).ADSCrossRefGoogle Scholar
  143. 143.
    F. Kh. Abdullaev, A. M. Kamchatov, V. V. Konotop, and V. A. Brazhnyi, Phys. Rev. Lett., 90, 230402 (2003).ADSCrossRefGoogle Scholar
  144. 144.
    J. K. Chin, J. M. Vogels, and W. Ketterle, Phys. Rev. Lett., 90, 160405 (2003).ADSCrossRefGoogle Scholar
  145. 145.
    S. L. Cornish, N. R. Claussen, J. L. Roberts, et al., Phys. Rev. Lett., 85, 1795 (2000).ADSCrossRefGoogle Scholar
  146. 146.
    Ph. Courteille, R. S. Freeland, D. J. Heinzen, et al., Phys. Rev. Lett., 81, 69 (1998).ADSCrossRefGoogle Scholar
  147. 147.
    P. O. Fedichev, Yu. Kagan, G. V. Shlyapnikov and J. T. M. Walraven, Phys. Rev. Lett., 77, 2913 (1996).ADSCrossRefGoogle Scholar
  148. 148.
    M. Houbier, H. T. C. Stoof, W. I. McAlexander, and R. G. Hulet, Phys. Rev. A, 54, R1497 (1998).ADSCrossRefGoogle Scholar
  149. 149.
    S. Inouye, M. R. Andrews, J. Stenger, et al., Nature, 392, 151 (1998).ADSCrossRefGoogle Scholar
  150. 150.
    Yu. Kagan, E. L. Surkov, and G. V. Shlyapnikov, Phys. Rev. Lett., 79, 2604 (1997).ADSCrossRefGoogle Scholar
  151. 151.
    P. G. Kevrekidis, G. Theocharis, D. J. Frantzeskakis, and B. A. Malomed, Phys. Rev. Lett., 90, 230401 (2003).ADSCrossRefGoogle Scholar
  152. 152.
    M. Matuszewski, E. Infeld, B. A. Malomed, and M. Trippenbach, Phys. Rev. Lett., 95, 05403 (2005).CrossRefGoogle Scholar
  153. 153.
    A. J. Moerdijk, B. J. Verhaar, and A. Axelsson, Phys. Rev. A, 75, 4852 (1995).ADSCrossRefGoogle Scholar
  154. 154.
    D. J. Papoular, G. V. Shlyapnikov, and J. Dalibard, Phys. Rev. A, 81, 041603(R) (2010). ADSCrossRefGoogle Scholar
  155. 155.
    D. E. Pelinovsky, P. G. Kevrekidis, and D. J. Frantzeskakis, Phys. Rev. Lett., 91, 240201 (2003).ADSCrossRefGoogle Scholar
  156. 156.
    V. M. Pérez-García, V. V. Konotop, and V. A. Brazhnyi, Phys. Rev. Lett., 92, 220403 (2004).CrossRefGoogle Scholar
  157. 157.
    J. L. Roberts, N. R. Claussen, J. P. Burke, Jr., et al., Phys. Rev. Lett., 81, 5109 (1998).ADSCrossRefGoogle Scholar
  158. 158.
    J. Stenger, S. Inouye, M. R. Andrews, et al., Phys. Rev. Lett., 82, 2422 (1999).ADSCrossRefGoogle Scholar
  159. 159.
    W. C. Stwalley, Phys. Rev. Lett., 37, 1628 (1976).ADSCrossRefGoogle Scholar
  160. 160.
    E. Timmermans, P. Tommasi, M. Hussein, and A. Kerman, Phys. Rep., 315, 199 (1999).ADSCrossRefGoogle Scholar
  161. 161.
    E. Tiesinga, B. J. Verhaar, and H. T. C. Stoof, Phys. Rev. A, 47, 4114 (1993).ADSCrossRefGoogle Scholar
  162. 162.
    K. M. O’Hara, S. L. Hemmer, S. R. Granade, et al., Phys. Rev. A, 66, 041401(R) (2002).ADSCrossRefGoogle Scholar
  163. 163.
    F. D. Tappert and N. J. Zabusky, Phys. Rev. Lett., 26, 1774 (1971).ADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2012

Authors and Affiliations

  1. 1.Department of Mathematical SciencesUniversity of Puerto RicoMayagüezPuerto Rico
  2. 2.School of Mathematical & Statistical Sciences and Mathematical & Computational Modeling Sciences CenterArizona State UniversityTempeU.S.A.

Personalised recommendations