Bose-Einstein condensation in an external potential at zero temperature: Solitary-wave theory

Journal of Mathematical Physics

Volumn 40, Issue 11, 1999, Pages 5522-5543

  Margetis, Dionisios ^a  

^a HARVARD UNIVERSITY   (United States)

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EID: 0033240081     PISSN: 00222488     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.533043     Document Type: Article

References (70)

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24. Some recent papers dealing with lime-dependent solutions to the nonlinear Schrödinger equation are: P. A. Ruprecht, M. J. Holland, K. Burnett, and M. Edwards, Phys. Rev. A 51, 4704 (1995); M. Holland and J. Cooper, ibid. 53, R1954 (1996); Yu. Kagan, E. L. Surkov, and G. V. Shlyapnikov, ibid. 54, R1753 (1996); Y. Castin and R. Dum, Phys. Rev. Lett. 77, 5315 (1996); A. Smerzi and S. Fantoni, ibid. 78, 3589 (1997); V. M. Pérez-García, H. Michinel, J. I. Chirac, M. Lewenstein, and P. Zoller, Phys. Rev. A 56, 1424 (1997); R. Dum, A. Sanpera, K.-A. Suominen, M. Brewczyk, M. Kuś, K. Rza̧zewski, and M. Lewenstein, Preprint quant-ph/9708009, xxx.lanl.gov.
25. Some recent papers dealing with lime-dependent solutions to the nonlinear Schrödinger equation are: P. A. Ruprecht, M. J. Holland, K. Burnett, and M. Edwards, Phys. Rev. A 51, 4704 (1995); M. Holland and J. Cooper, ibid. 53, R1954 (1996); Yu. Kagan, E. L. Surkov, and G. V. Shlyapnikov, ibid. 54, R1753 (1996); Y. Castin and R. Dum, Phys. Rev. Lett. 77, 5315 (1996); A. Smerzi and S. Fantoni, ibid. 78, 3589 (1997); V. M. Pérez-García, H. Michinel, J. I. Chirac, M. Lewenstein, and P. Zoller, Phys. Rev. A 56, 1424 (1997); R. Dum, A. Sanpera, K.-A. Suominen, M. Brewczyk, M. Kuś, K. Rza̧zewski, and M. Lewenstein, Preprint quant-ph/9708009, xxx.lanl.gov.
26. Some recent papers dealing with lime-dependent solutions to the nonlinear Schrödinger equation are: P. A. Ruprecht, M. J. Holland, K. Burnett, and M. Edwards, Phys. Rev. A 51, 4704 (1995); M. Holland and J. Cooper, ibid. 53, R1954 (1996); Yu. Kagan, E. L. Surkov, and G. V. Shlyapnikov, ibid. 54, R1753 (1996); Y. Castin and R. Dum, Phys. Rev. Lett. 77, 5315 (1996); A. Smerzi and S. Fantoni, ibid. 78, 3589 (1997); V. M. Pérez-García, H. Michinel, J. I. Chirac, M. Lewenstein, and P. Zoller, Phys. Rev. A 56, 1424 (1997); R. Dum, A. Sanpera, K.-A. Suominen, M. Brewczyk, M. Kuś, K. Rza̧zewski, and M. Lewenstein, Preprint quant-ph/9708009, xxx.lanl.gov.
27. Some recent papers dealing with lime-dependent solutions to the nonlinear Schrödinger equation are: P. A. Ruprecht, M. J. Holland, K. Burnett, and M. Edwards, Phys. Rev. A 51, 4704 (1995); M. Holland and J. Cooper, ibid. 53, R1954 (1996); Yu. Kagan, E. L. Surkov, and G. V. Shlyapnikov, ibid. 54, R1753 (1996); Y. Castin and R. Dum, Phys. Rev. Lett. 77, 5315 (1996); A. Smerzi and S. Fantoni, ibid. 78, 3589 (1997); V. M. Pérez-García, H. Michinel, J. I. Chirac, M. Lewenstein, and P. Zoller, Phys. Rev. A 56, 1424 (1997); R. Dum, A. Sanpera, K.-A. Suominen, M. Brewczyk, M. Kuś, K. Rza̧zewski, and M. Lewenstein, Preprint quant-ph/9708009, xxx.lanl.gov.
28. Some recent papers dealing with lime-dependent solutions to the nonlinear Schrödinger equation are: P. A. Ruprecht, M. J. Holland, K. Burnett, and M. Edwards, Phys. Rev. A 51, 4704 (1995); M. Holland and J. Cooper, ibid. 53, R1954 (1996); Yu. Kagan, E. L. Surkov, and G. V. Shlyapnikov, ibid. 54, R1753 (1996); Y. Castin and R. Dum, Phys. Rev. Lett. 77, 5315 (1996); A. Smerzi and S. Fantoni, ibid. 78, 3589 (1997); V. M. Pérez-García, H. Michinel, J. I. Chirac, M. Lewenstein, and P. Zoller, Phys. Rev. A 56, 1424 (1997); R. Dum, A. Sanpera, K.-A. Suominen, M. Brewczyk, M. Kuś, K. Rza̧zewski, and M. Lewenstein, Preprint quant-ph/9708009, xxx.lanl.gov.
29. Some recent papers dealing with lime-dependent solutions to the nonlinear Schrödinger equation are: P. A. Ruprecht, M. J. Holland, K. Burnett, and M. Edwards, Phys. Rev. A 51, 4704 (1995); M. Holland and J. Cooper, ibid. 53, R1954 (1996); Yu. Kagan, E. L. Surkov, and G. V. Shlyapnikov, ibid. 54, R1753 (1996); Y. Castin and R. Dum, Phys. Rev. Lett. 77, 5315 (1996); A. Smerzi and S. Fantoni, ibid. 78, 3589 (1997); V. M. Pérez-García, H. Michinel, J. I. Chirac, M. Lewenstein, and P. Zoller, Phys. Rev. A 56, 1424 (1997); R. Dum, A. Sanpera, K.-A. Suominen, M. Brewczyk, M. Kuś, K. Rza̧zewski, and M. Lewenstein, Preprint quant-ph/9708009, xxx.lanl.gov.
30. Some recent papers dealing with lime-dependent solutions to the nonlinear Schrödinger equation are: P. A. Ruprecht, M. J. Holland, K. Burnett, and M. Edwards, Phys. Rev. A 51, 4704 (1995); M. Holland and J. Cooper, ibid. 53, R1954 (1996); Yu. Kagan, E. L. Surkov, and G. V. Shlyapnikov, ibid. 54, R1753 (1996); Y. Castin and R. Dum, Phys. Rev. Lett. 77, 5315 (1996); A. Smerzi and S. Fantoni, ibid. 78, 3589 (1997); V. M. Pérez-García, H. Michinel, J. I. Chirac, M. Lewenstein, and P. Zoller, Phys. Rev. A 56, 1424 (1997); R. Dum, A. Sanpera, K.-A. Suominen, M. Brewczyk, M. Kuś, K. Rza̧zewski, and M. Lewenstein, Preprint quant-ph/9708009, xxx.lanl.gov.
31. 2(i≠j). However, even approximate analytical solutions to the resulting hyperangular equation seem to be intractable.
32. Some papers concerned with soliton solutions to nonlinear Schrödinger equations containing an external potential are the following: H.-H. Chen and C.-S. Liu, Phys. Rev. Lett. 37, 693 (1976); Phys. Fluids 21, 377 (1978); A. C. Newell, J. Math. Phys. 19, 1126 (1978); D. J. Kaup and A. C. Newell, ibid. 19, 798 (1978); Proc. R. Soc. London, Ser. A 361, 413 (1978); R. W. Hasse, Phys. Rev. A 25, 583 (1982); J. E. Herrera, J. Phys. A 17, 95 (1984); R. Balakrishnan, Phys. Rev. A 32, 1144 (1985); A. B. Nassar, ibid. 33, 3502 (1986); M. A. de Moura, ibid. 37, 4998 (1988); Y. Nogami and F. M. Toyama, Phys. Rev. E 49, 4497 (1994); M. A. de Moura, J. Phys. A 27, 7157 (1994).
33. Some papers concerned with soliton solutions to nonlinear Schrödinger equations containing an external potential are the following: H.-H. Chen and C.-S. Liu, Phys. Rev. Lett. 37, 693 (1976); Phys. Fluids 21, 377 (1978); A. C. Newell, J. Math. Phys. 19, 1126 (1978); D. J. Kaup and A. C. Newell, ibid. 19, 798 (1978); Proc. R. Soc. London, Ser. A 361, 413 (1978); R. W. Hasse, Phys. Rev. A 25, 583 (1982); J. E. Herrera, J. Phys. A 17, 95 (1984); R. Balakrishnan, Phys. Rev. A 32, 1144 (1985); A. B. Nassar, ibid. 33, 3502 (1986); M. A. de Moura, ibid. 37, 4998 (1988); Y. Nogami and F. M. Toyama, Phys. Rev. E 49, 4497 (1994); M. A. de Moura, J. Phys. A 27, 7157 (1994).
34. Some papers concerned with soliton solutions to nonlinear Schrödinger equations containing an external potential are the following: H.-H. Chen and C.-S. Liu, Phys. Rev. Lett. 37, 693 (1976); Phys. Fluids 21, 377 (1978); A. C. Newell, J. Math. Phys. 19, 1126 (1978); D. J. Kaup and A. C. Newell, ibid. 19, 798 (1978); Proc. R. Soc. London, Ser. A 361, 413 (1978); R. W. Hasse, Phys. Rev. A 25, 583 (1982); J. E. Herrera, J. Phys. A 17, 95 (1984); R. Balakrishnan, Phys. Rev. A 32, 1144 (1985); A. B. Nassar, ibid. 33, 3502 (1986); M. A. de Moura, ibid. 37, 4998 (1988); Y. Nogami and F. M. Toyama, Phys. Rev. E 49, 4497 (1994); M. A. de Moura, J. Phys. A 27, 7157 (1994).
35. Some papers concerned with soliton solutions to nonlinear Schrödinger equations containing an external potential are the following: H.-H. Chen and C.-S. Liu, Phys. Rev. Lett. 37, 693 (1976); Phys. Fluids 21, 377 (1978); A. C. Newell, J. Math. Phys. 19, 1126 (1978); D. J. Kaup and A. C. Newell, ibid. 19, 798 (1978); Proc. R. Soc. London, Ser. A 361, 413 (1978); R. W. Hasse, Phys. Rev. A 25, 583 (1982); J. E. Herrera, J. Phys. A 17, 95 (1984); R. Balakrishnan, Phys. Rev. A 32, 1144 (1985); A. B. Nassar, ibid. 33, 3502 (1986); M. A. de Moura, ibid. 37, 4998 (1988); Y. Nogami and F. M. Toyama, Phys. Rev. E 49, 4497 (1994); M. A. de Moura, J. Phys. A 27, 7157 (1994).
36. Some papers concerned with soliton solutions to nonlinear Schrödinger equations containing an external potential are the following: H.-H. Chen and C.-S. Liu, Phys. Rev. Lett. 37, 693 (1976); Phys. Fluids 21, 377 (1978); A. C. Newell, J. Math. Phys. 19, 1126 (1978); D. J. Kaup and A. C. Newell, ibid. 19, 798 (1978); Proc. R. Soc. London, Ser. A 361, 413 (1978); R. W. Hasse, Phys. Rev. A 25, 583 (1982); J. E. Herrera, J. Phys. A 17, 95 (1984); R. Balakrishnan, Phys. Rev. A 32, 1144 (1985); A. B. Nassar, ibid. 33, 3502 (1986); M. A. de Moura, ibid. 37, 4998 (1988); Y. Nogami and F. M. Toyama, Phys. Rev. E 49, 4497 (1994); M. A. de Moura, J. Phys. A 27, 7157 (1994).
37. Some papers concerned with soliton solutions to nonlinear Schrödinger equations containing an external potential are the following: H.-H. Chen and C.-S. Liu, Phys. Rev. Lett. 37, 693 (1976); Phys. Fluids 21, 377 (1978); A. C. Newell, J. Math. Phys. 19, 1126 (1978); D. J. Kaup and A. C. Newell, ibid. 19, 798 (1978); Proc. R. Soc. London, Ser. A 361, 413 (1978); R. W. Hasse, Phys. Rev. A 25, 583 (1982); J. E. Herrera, J. Phys. A 17, 95 (1984); R. Balakrishnan, Phys. Rev. A 32, 1144 (1985); A. B. Nassar, ibid. 33, 3502 (1986); M. A. de Moura, ibid. 37, 4998 (1988); Y. Nogami and F. M. Toyama, Phys. Rev. E 49, 4497 (1994); M. A. de Moura, J. Phys. A 27, 7157 (1994).
38. Some papers concerned with soliton solutions to nonlinear Schrödinger equations containing an external potential are the following: H.-H. Chen and C.-S. Liu, Phys. Rev. Lett. 37, 693 (1976); Phys. Fluids 21, 377 (1978); A. C. Newell, J. Math. Phys. 19, 1126 (1978); D. J. Kaup and A. C. Newell, ibid. 19, 798 (1978); Proc. R. Soc. London, Ser. A 361, 413 (1978); R. W. Hasse, Phys. Rev. A 25, 583 (1982); J. E. Herrera, J. Phys. A 17, 95 (1984); R. Balakrishnan, Phys. Rev. A 32, 1144 (1985); A. B. Nassar, ibid. 33, 3502 (1986); M. A. de Moura, ibid. 37, 4998 (1988); Y. Nogami and F. M. Toyama, Phys. Rev. E 49, 4497 (1994); M. A. de Moura, J. Phys. A 27, 7157 (1994).
39. Some papers concerned with soliton solutions to nonlinear Schrödinger equations containing an external potential are the following: H.-H. Chen and C.-S. Liu, Phys. Rev. Lett. 37, 693 (1976); Phys. Fluids 21, 377 (1978); A. C. Newell, J. Math. Phys. 19, 1126 (1978); D. J. Kaup and A. C. Newell, ibid. 19, 798 (1978); Proc. R. Soc. London, Ser. A 361, 413 (1978); R. W. Hasse, Phys. Rev. A 25, 583 (1982); J. E. Herrera, J. Phys. A 17, 95 (1984); R. Balakrishnan, Phys. Rev. A 32, 1144 (1985); A. B. Nassar, ibid. 33, 3502 (1986); M. A. de Moura, ibid. 37, 4998 (1988); Y. Nogami and F. M. Toyama, Phys. Rev. E 49, 4497 (1994); M. A. de Moura, J. Phys. A 27, 7157 (1994).
40. Some papers concerned with soliton solutions to nonlinear Schrödinger equations containing an external potential are the following: H.-H. Chen and C.-S. Liu, Phys. Rev. Lett. 37, 693 (1976); Phys. Fluids 21, 377 (1978); A. C. Newell, J. Math. Phys. 19, 1126 (1978); D. J. Kaup and A. C. Newell, ibid. 19, 798 (1978); Proc. R. Soc. London, Ser. A 361, 413 (1978); R. W. Hasse, Phys. Rev. A 25, 583 (1982); J. E. Herrera, J. Phys. A 17, 95 (1984); R. Balakrishnan, Phys. Rev. A 32, 1144 (1985); A. B. Nassar, ibid. 33, 3502 (1986); M. A. de Moura, ibid. 37, 4998 (1988); Y. Nogami and F. M. Toyama, Phys. Rev. E 49, 4497 (1994); M. A. de Moura, J. Phys. A 27, 7157 (1994).
41. Some papers concerned with soliton solutions to nonlinear Schrödinger equations containing an external potential are the following: H.-H. Chen and C.-S. Liu, Phys. Rev. Lett. 37, 693 (1976); Phys. Fluids 21, 377 (1978); A. C. Newell, J. Math. Phys. 19, 1126 (1978); D. J. Kaup and A. C. Newell, ibid. 19, 798 (1978); Proc. R. Soc. London, Ser. A 361, 413 (1978); R. W. Hasse, Phys. Rev. A 25, 583 (1982); J. E. Herrera, J. Phys. A 17, 95 (1984); R. Balakrishnan, Phys. Rev. A 32, 1144 (1985); A. B. Nassar, ibid. 33, 3502 (1986); M. A. de Moura, ibid. 37, 4998 (1988); Y. Nogami and F. M. Toyama, Phys. Rev. E 49, 4497 (1994); M. A. de Moura, J. Phys. A 27, 7157 (1994).
42. Some papers concerned with soliton solutions to nonlinear Schrödinger equations containing an external potential are the following: H.-H. Chen and C.-S. Liu, Phys. Rev. Lett. 37, 693 (1976); Phys. Fluids 21, 377 (1978); A. C. Newell, J. Math. Phys. 19, 1126 (1978); D. J. Kaup and A. C. Newell, ibid. 19, 798 (1978); Proc. R. Soc. London, Ser. A 361, 413 (1978); R. W. Hasse, Phys. Rev. A 25, 583 (1982); J. E. Herrera, J. Phys. A 17, 95 (1984); R. Balakrishnan, Phys. Rev. A 32, 1144 (1985); A. B. Nassar, ibid. 33, 3502 (1986); M. A. de Moura, ibid. 37, 4998 (1988); Y. Nogami and F. M. Toyama, Phys. Rev. E 49, 4497 (1994); M. A. de Moura, J. Phys. A 27, 7157 (1994).
43. Some papers concerned with soliton solutions to nonlinear Schrödinger equations containing an external potential are the following: H.-H. Chen and C.-S. Liu, Phys. Rev. Lett. 37, 693 (1976); Phys. Fluids 21, 377 (1978); A. C. Newell, J. Math. Phys. 19, 1126 (1978); D. J. Kaup and A. C. Newell, ibid. 19, 798 (1978); Proc. R. Soc. London, Ser. A 361, 413 (1978); R. W. Hasse, Phys. Rev. A 25, 583 (1982); J. E. Herrera, J. Phys. A 17, 95 (1984); R. Balakrishnan, Phys. Rev. A 32, 1144 (1985); A. B. Nassar, ibid. 33, 3502 (1986); M. A. de Moura, ibid. 37, 4998 (1988); Y. Nogami and F. M. Toyama, Phys. Rev. E 49, 4497 (1994); M. A. de Moura, J. Phys. A 27, 7157 (1994).
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52. For example, see the classical papers V. E. Zakharov and A. B. Shabat, Zh. Eksp. Teor. Fiz. 61, 118 (1971) [ Sov. Phys. JETP 34, 62 (1972)]; Zh. Eksp. Teor. Fiz. 64, 1627 (1973) [ Sov. Phys. JETP 37, 823 (1973)]; R. Hirota, J. Math. Phys. 14, 805 (1973).
53. For example, see the classical papers V. E. Zakharov and A. B. Shabat, Zh. Eksp. Teor. Fiz. 61, 118 (1971) [ Sov. Phys. JETP 34, 62 (1972)]; Zh. Eksp. Teor. Fiz. 64, 1627 (1973) [ Sov. Phys. JETP 37, 823 (1973)]; R. Hirota, J. Math. Phys. 14, 805 (1973).
54. For example, see the classical papers V. E. Zakharov and A. B. Shabat, Zh. Eksp. Teor. Fiz. 61, 118 (1971) [ Sov. Phys. JETP 34, 62 (1972)]; Zh. Eksp. Teor. Fiz. 64, 1627 (1973) [ Sov. Phys. JETP 37, 823 (1973)]; R. Hirota, J. Math. Phys. 14, 805 (1973).
55. For example, see the classical papers V. E. Zakharov and A. B. Shabat, Zh. Eksp. Teor. Fiz. 61, 118 (1971) [ Sov. Phys. JETP 34, 62 (1972)]; Zh. Eksp. Teor. Fiz. 64, 1627 (1973) [ Sov. Phys. JETP 37, 823 (1973)]; R. Hirota, J. Math. Phys. 14, 805 (1973).
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62. 0.
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64. For three dimensions, see the classical book O. D. Kellogg, Foundations of Potential Theory (Dover, New York, 1953), p. 223; for two dimensions, see L. V. Ahlfors, Complex Analysis (McGraw-Hill, Singapore, 19791, p. 165; for generalization to higher space dimensions, see, for instance, D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order (Springer, New York, 1983), p. 21.
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69. This is an elementary result in quantum mechanics. For generalizations, see K. Husimi, Prog. Theor. Phys. 9, 381 (1953); E. H. Kerner, Can. J. Phys. 36, 371 (1958).
70. This is an elementary result in quantum mechanics. For generalizations, see K. Husimi, Prog. Theor. Phys. 9, 381 (1953); E. H. Kerner, Can. J. Phys. 36, 371 (1958).