Abstract

Elliptical solitons in 2D nonlinear Schödinger equations are studied numerically with a more-generalized formulation. New families of solitons, vortices, and soliton rings with elliptical symmetry are found and investigated. With a suitable symmetry-breaking parameter, we show that perturbed elliptical solitons tend to move transversely owing to the existences of dipole excitation modes, which are totally suppressed in circularly symmetric solitons. Furthermore, by numerical evolutions we demonstrate that elliptical vortices and soliton rings collapse into a pair of stripes and clusters, respectively, revealing the experimental observations in the literature.

© 2008 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. Y. S. Kivshar and D. E. Pelinovsky, Phys. Rep. 331, 117 (2000); and references therein.
    [CrossRef]
  2. V. E. Zakharov and A. M. Rubenchik, Sov. Phys. JETP 38, 494 (1974).
  3. E. A. Kuznetsov and S. K. Turitsyn, Sov. Phys. JETP 67, 1583 (1988).
  4. J. J. Rasmussen and K. Rypdal, Phys. Scr. 33, 481 (1986).
    [CrossRef]
  5. N. N. Akhmediev, V. I. Korneev, and R. F. Nabiev, Phys. Rev. A 46, 430 (1992).
    [CrossRef] [PubMed]
  6. A. S. Desyatnikov, A. A. Sukhorukov, and Y. S. Kivshar, Phys. Rev. Lett. 95, 203904 (2005).
    [CrossRef] [PubMed]
  7. D. V. Skryabin and W. J. Firth, Phys. Rev. E 58, 3916 (1998).
    [CrossRef]
  8. C. Anastassiou, C. Pigier, M. Segev, D. Kip, E. D. Eugenieva, and D. N. Christodoulides, Opt. Lett. 26, 911 (2001).
    [CrossRef]
  9. M. A. Bandres and J. C. Gutiérrez-Vega, J. Opt. Soc. Am. A 21, 873 (2004).
    [CrossRef]
  10. S. Lopez-Aguayo and J. C. Gutierrez-Vega, Opt. Express 15, 18,326 (2007).
    [CrossRef]
  11. D. Deng and Q. Guo, Opt. Lett. 32, 3206 (2007).
    [CrossRef] [PubMed]
  12. P. Zhang, J. Zhao, C. Lou, X. Tan, Y. Gao, Q. Liu, D. Yang, J. Xu, and Z. Chen, Opt. Express 15, 536 (2007).
    [CrossRef] [PubMed]
  13. O. Katz, T. Carmon, T. Schwartz, M. Segev, and D. N. Christodoulides, Opt. Lett. 29, 1248 (2004).
    [CrossRef] [PubMed]
  14. D. Buccoliero, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, Phys. Rev. Lett. 98, 053901 (2007).
    [CrossRef] [PubMed]
  15. B. A. Malomed, in Progress in Optics, E.Wolf, ed. (North-Holland, 2002), Vol. 43, p. 71.
    [CrossRef]
  16. A. Lifschitz, Phys. Fluids 7, 1626 (1995).
    [CrossRef]

2007 (4)

S. Lopez-Aguayo and J. C. Gutierrez-Vega, Opt. Express 15, 18,326 (2007).
[CrossRef]

D. Deng and Q. Guo, Opt. Lett. 32, 3206 (2007).
[CrossRef] [PubMed]

P. Zhang, J. Zhao, C. Lou, X. Tan, Y. Gao, Q. Liu, D. Yang, J. Xu, and Z. Chen, Opt. Express 15, 536 (2007).
[CrossRef] [PubMed]

D. Buccoliero, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, Phys. Rev. Lett. 98, 053901 (2007).
[CrossRef] [PubMed]

2005 (1)

A. S. Desyatnikov, A. A. Sukhorukov, and Y. S. Kivshar, Phys. Rev. Lett. 95, 203904 (2005).
[CrossRef] [PubMed]

2004 (2)

2001 (1)

2000 (1)

Y. S. Kivshar and D. E. Pelinovsky, Phys. Rep. 331, 117 (2000); and references therein.
[CrossRef]

1998 (1)

D. V. Skryabin and W. J. Firth, Phys. Rev. E 58, 3916 (1998).
[CrossRef]

1995 (1)

A. Lifschitz, Phys. Fluids 7, 1626 (1995).
[CrossRef]

1992 (1)

N. N. Akhmediev, V. I. Korneev, and R. F. Nabiev, Phys. Rev. A 46, 430 (1992).
[CrossRef] [PubMed]

1988 (1)

E. A. Kuznetsov and S. K. Turitsyn, Sov. Phys. JETP 67, 1583 (1988).

1986 (1)

J. J. Rasmussen and K. Rypdal, Phys. Scr. 33, 481 (1986).
[CrossRef]

1974 (1)

V. E. Zakharov and A. M. Rubenchik, Sov. Phys. JETP 38, 494 (1974).

Akhmediev, N. N.

N. N. Akhmediev, V. I. Korneev, and R. F. Nabiev, Phys. Rev. A 46, 430 (1992).
[CrossRef] [PubMed]

Anastassiou, C.

Bandres, M. A.

Buccoliero, D.

D. Buccoliero, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, Phys. Rev. Lett. 98, 053901 (2007).
[CrossRef] [PubMed]

Carmon, T.

Chen, Z.

Christodoulides, D. N.

Deng, D.

Desyatnikov, A. S.

D. Buccoliero, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, Phys. Rev. Lett. 98, 053901 (2007).
[CrossRef] [PubMed]

A. S. Desyatnikov, A. A. Sukhorukov, and Y. S. Kivshar, Phys. Rev. Lett. 95, 203904 (2005).
[CrossRef] [PubMed]

Eugenieva, E. D.

Firth, W. J.

D. V. Skryabin and W. J. Firth, Phys. Rev. E 58, 3916 (1998).
[CrossRef]

Gao, Y.

Guo, Q.

Gutierrez-Vega, J. C.

S. Lopez-Aguayo and J. C. Gutierrez-Vega, Opt. Express 15, 18,326 (2007).
[CrossRef]

Gutiérrez-Vega, J. C.

Katz, O.

Kip, D.

Kivshar, Y. S.

D. Buccoliero, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, Phys. Rev. Lett. 98, 053901 (2007).
[CrossRef] [PubMed]

A. S. Desyatnikov, A. A. Sukhorukov, and Y. S. Kivshar, Phys. Rev. Lett. 95, 203904 (2005).
[CrossRef] [PubMed]

Y. S. Kivshar and D. E. Pelinovsky, Phys. Rep. 331, 117 (2000); and references therein.
[CrossRef]

Korneev, V. I.

N. N. Akhmediev, V. I. Korneev, and R. F. Nabiev, Phys. Rev. A 46, 430 (1992).
[CrossRef] [PubMed]

Krolikowski, W.

D. Buccoliero, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, Phys. Rev. Lett. 98, 053901 (2007).
[CrossRef] [PubMed]

Kuznetsov, E. A.

E. A. Kuznetsov and S. K. Turitsyn, Sov. Phys. JETP 67, 1583 (1988).

Lifschitz, A.

A. Lifschitz, Phys. Fluids 7, 1626 (1995).
[CrossRef]

Liu, Q.

Lopez-Aguayo, S.

S. Lopez-Aguayo and J. C. Gutierrez-Vega, Opt. Express 15, 18,326 (2007).
[CrossRef]

Lou, C.

Malomed, B. A.

B. A. Malomed, in Progress in Optics, E.Wolf, ed. (North-Holland, 2002), Vol. 43, p. 71.
[CrossRef]

Nabiev, R. F.

N. N. Akhmediev, V. I. Korneev, and R. F. Nabiev, Phys. Rev. A 46, 430 (1992).
[CrossRef] [PubMed]

Pelinovsky, D. E.

Y. S. Kivshar and D. E. Pelinovsky, Phys. Rep. 331, 117 (2000); and references therein.
[CrossRef]

Pigier, C.

Rasmussen, J. J.

J. J. Rasmussen and K. Rypdal, Phys. Scr. 33, 481 (1986).
[CrossRef]

Rubenchik, A. M.

V. E. Zakharov and A. M. Rubenchik, Sov. Phys. JETP 38, 494 (1974).

Rypdal, K.

J. J. Rasmussen and K. Rypdal, Phys. Scr. 33, 481 (1986).
[CrossRef]

Schwartz, T.

Segev, M.

Skryabin, D. V.

D. V. Skryabin and W. J. Firth, Phys. Rev. E 58, 3916 (1998).
[CrossRef]

Sukhorukov, A. A.

A. S. Desyatnikov, A. A. Sukhorukov, and Y. S. Kivshar, Phys. Rev. Lett. 95, 203904 (2005).
[CrossRef] [PubMed]

Tan, X.

Turitsyn, S. K.

E. A. Kuznetsov and S. K. Turitsyn, Sov. Phys. JETP 67, 1583 (1988).

Xu, J.

Yang, D.

Zakharov, V. E.

V. E. Zakharov and A. M. Rubenchik, Sov. Phys. JETP 38, 494 (1974).

Zhang, P.

Zhao, J.

J. Opt. Soc. Am. A (1)

Opt. Express (2)

Opt. Lett. (3)

Phys. Fluids (1)

A. Lifschitz, Phys. Fluids 7, 1626 (1995).
[CrossRef]

Phys. Rep. (1)

Y. S. Kivshar and D. E. Pelinovsky, Phys. Rep. 331, 117 (2000); and references therein.
[CrossRef]

Phys. Rev. A (1)

N. N. Akhmediev, V. I. Korneev, and R. F. Nabiev, Phys. Rev. A 46, 430 (1992).
[CrossRef] [PubMed]

Phys. Rev. E (1)

D. V. Skryabin and W. J. Firth, Phys. Rev. E 58, 3916 (1998).
[CrossRef]

Phys. Rev. Lett. (2)

A. S. Desyatnikov, A. A. Sukhorukov, and Y. S. Kivshar, Phys. Rev. Lett. 95, 203904 (2005).
[CrossRef] [PubMed]

D. Buccoliero, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, Phys. Rev. Lett. 98, 053901 (2007).
[CrossRef] [PubMed]

Phys. Scr. (1)

J. J. Rasmussen and K. Rypdal, Phys. Scr. 33, 481 (1986).
[CrossRef]

Sov. Phys. JETP (2)

V. E. Zakharov and A. M. Rubenchik, Sov. Phys. JETP 38, 494 (1974).

E. A. Kuznetsov and S. K. Turitsyn, Sov. Phys. JETP 67, 1583 (1988).

Other (1)

B. A. Malomed, in Progress in Optics, E.Wolf, ed. (North-Holland, 2002), Vol. 43, p. 71.
[CrossRef]

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (4)

Fig. 1
Fig. 1

(a) Radial function U ( μ ) , (b) angular function V ( θ ) , and (c) bifurcation curves of elliptical solitons at β = 0.5 for different semifocal separations: l = 0.05 (solid curve), l = 0.7 (dashed curve), and l = 1.4 (dotted curve). The corresponding wave functions Ψ are given separately in (d) for l = (d) 0.05, (e) 0.7, and (f) 1.4.

Fig. 2
Fig. 2

(a) Modulation instabilities of elliptical solitons for different semifocal separations, l. A, B, and C indicate three different eigenfunctions that are shown in the second row with l = 1.3 . Snapshots of propagation of the fundamental elliptical soliton simulated by the NLS are shown at z = (b) 0, (c) 0.5, (d) 2, and (e) 10, with the semifocal separation l = 1.7 .

Fig. 3
Fig. 3

(a) Radial function U ( μ ) and (b) angular function V ( θ ) (black for envelopes; the diagonal line is for phases) of an elliptical vortice at β = 0.5 for different semifocal separations: l = 1 (solid curve), l = 5 (dashed curve), and l = 30 (dotted curve); evolution of elliptical vortices for (c)–(f) l = 1 and (g)–(i) l = 30 at the corresponding distances z = 0 , 1, 4, and 6.7 .

Fig. 4
Fig. 4

Radial function U ( μ ) (a) and angular function V ( θ ) (b) of elliptical soliton rings at β = 0.5 are shown in solid, dashed, and dotted curves for l = 1, 5, and 7, respectively. Evolution of elliptical soliton rings given l = 1 [(c)–(f)], l = 5 [(g)–(i)], and l = 7 [(j)–(n)] in a window of size 40 × 40 are plotted at z=0, 1.5, 3, and 4.5.

Equations (2)

Equations on this page are rendered with MathJax. Learn more.

[ β f 1 ( θ ) + f 2 ( θ ) 2 ] V + θ { f 3 ( θ ) 2 V θ } + f 4 ( θ ) V 2 V = 0 ,
[ β g 1 ( μ ) + g 2 ( μ ) 2 ] U + μ { g 3 ( μ ) 2 U μ } + g 4 ( μ ) U 2 U = 0 ,

Metrics