Abstract

We obtain exact extended traveling-wave and spatiotemporal soliton solutions to the generalized (3+1)-dimensional nonlinear Schrödinger equations for both the normal and the anomalous dispersion.

© 2009 Optical Society of America

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References

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  1. N. N. Akhmediev and A. A. Ankiewicz, Solitons (Chapman and Hall, 1997).
  2. C. Sulem and P. Sulem, The Nonlinear Schrödinger Equation: Self-Focusing and Wave Collapse (Springer, 2000).
  3. Y. S. Kivshar and G. P. Agrawal, Optical Solitons, from Fibers to Photonic Crystals (Academic, 2003).
  4. B. A. Malomed, Soliton Management in Periodic Systems (Springer, 2006).
  5. A. Hasegava and M. Matsumoto, Optical Solitons in Fibers (Springer, 2003).
  6. B. A. Malomed, D. Michalache, F. Wise, and L. Torner, J. Opt. B 7, R53 (2005).
    [CrossRef]
  7. W. P. Zhong, R. H. Xie, M. Belić, N. Petrović, and G. Chen, Phys. Rev. A 78, 023821 (2008).
    [CrossRef]
  8. M. Belić, N. Petrović, W. P. Zhong, R. H. Xie, and G. Chen, Phys. Rev. Lett. 101, 123904 (2008).
    [CrossRef] [PubMed]
  9. G. G. Luther, A. C. Newell, and J. V. Moloney, Physica D 74, 59 (1994).
    [CrossRef]
  10. L. Yang, J. Liu, and K. Yang, Phys. Lett. A 278, 267 (2001).
    [CrossRef]
  11. Z. Yan and H. Q. Zhang, Phys. Lett. A 285, 355 (2001).
    [CrossRef]
  12. Y. B. Zhou, M. L. Wang, and Y. M. Wang, Phys. Lett. A 308, 31 (2003).
    [CrossRef]
  13. Y. B. Zhou, M. L. Wang, and T. D. Miao, Phys. Lett. A 323, 77 (2004).
    [CrossRef]
  14. V. I. Kruglov, A. C. Peacock, and J. D. Harvey, Phys. Rev. E 71, 056619 (2005).
    [CrossRef]

2008 (2)

W. P. Zhong, R. H. Xie, M. Belić, N. Petrović, and G. Chen, Phys. Rev. A 78, 023821 (2008).
[CrossRef]

M. Belić, N. Petrović, W. P. Zhong, R. H. Xie, and G. Chen, Phys. Rev. Lett. 101, 123904 (2008).
[CrossRef] [PubMed]

2005 (2)

B. A. Malomed, D. Michalache, F. Wise, and L. Torner, J. Opt. B 7, R53 (2005).
[CrossRef]

V. I. Kruglov, A. C. Peacock, and J. D. Harvey, Phys. Rev. E 71, 056619 (2005).
[CrossRef]

2004 (1)

Y. B. Zhou, M. L. Wang, and T. D. Miao, Phys. Lett. A 323, 77 (2004).
[CrossRef]

2003 (1)

Y. B. Zhou, M. L. Wang, and Y. M. Wang, Phys. Lett. A 308, 31 (2003).
[CrossRef]

2001 (2)

L. Yang, J. Liu, and K. Yang, Phys. Lett. A 278, 267 (2001).
[CrossRef]

Z. Yan and H. Q. Zhang, Phys. Lett. A 285, 355 (2001).
[CrossRef]

1994 (1)

G. G. Luther, A. C. Newell, and J. V. Moloney, Physica D 74, 59 (1994).
[CrossRef]

Agrawal, G. P.

Y. S. Kivshar and G. P. Agrawal, Optical Solitons, from Fibers to Photonic Crystals (Academic, 2003).

Akhmediev, N. N.

N. N. Akhmediev and A. A. Ankiewicz, Solitons (Chapman and Hall, 1997).

Ankiewicz, A. A.

N. N. Akhmediev and A. A. Ankiewicz, Solitons (Chapman and Hall, 1997).

Belic, M.

M. Belić, N. Petrović, W. P. Zhong, R. H. Xie, and G. Chen, Phys. Rev. Lett. 101, 123904 (2008).
[CrossRef] [PubMed]

W. P. Zhong, R. H. Xie, M. Belić, N. Petrović, and G. Chen, Phys. Rev. A 78, 023821 (2008).
[CrossRef]

Chen, G.

W. P. Zhong, R. H. Xie, M. Belić, N. Petrović, and G. Chen, Phys. Rev. A 78, 023821 (2008).
[CrossRef]

M. Belić, N. Petrović, W. P. Zhong, R. H. Xie, and G. Chen, Phys. Rev. Lett. 101, 123904 (2008).
[CrossRef] [PubMed]

Harvey, J. D.

V. I. Kruglov, A. C. Peacock, and J. D. Harvey, Phys. Rev. E 71, 056619 (2005).
[CrossRef]

Hasegava, A.

A. Hasegava and M. Matsumoto, Optical Solitons in Fibers (Springer, 2003).

Kivshar, Y. S.

Y. S. Kivshar and G. P. Agrawal, Optical Solitons, from Fibers to Photonic Crystals (Academic, 2003).

Kruglov, V. I.

V. I. Kruglov, A. C. Peacock, and J. D. Harvey, Phys. Rev. E 71, 056619 (2005).
[CrossRef]

Liu, J.

L. Yang, J. Liu, and K. Yang, Phys. Lett. A 278, 267 (2001).
[CrossRef]

Luther, G. G.

G. G. Luther, A. C. Newell, and J. V. Moloney, Physica D 74, 59 (1994).
[CrossRef]

Malomed, B. A.

B. A. Malomed, D. Michalache, F. Wise, and L. Torner, J. Opt. B 7, R53 (2005).
[CrossRef]

B. A. Malomed, Soliton Management in Periodic Systems (Springer, 2006).

Matsumoto, M.

A. Hasegava and M. Matsumoto, Optical Solitons in Fibers (Springer, 2003).

Miao, T. D.

Y. B. Zhou, M. L. Wang, and T. D. Miao, Phys. Lett. A 323, 77 (2004).
[CrossRef]

Michalache, D.

B. A. Malomed, D. Michalache, F. Wise, and L. Torner, J. Opt. B 7, R53 (2005).
[CrossRef]

Moloney, J. V.

G. G. Luther, A. C. Newell, and J. V. Moloney, Physica D 74, 59 (1994).
[CrossRef]

Newell, A. C.

G. G. Luther, A. C. Newell, and J. V. Moloney, Physica D 74, 59 (1994).
[CrossRef]

Peacock, A. C.

V. I. Kruglov, A. C. Peacock, and J. D. Harvey, Phys. Rev. E 71, 056619 (2005).
[CrossRef]

Petrovic, N.

M. Belić, N. Petrović, W. P. Zhong, R. H. Xie, and G. Chen, Phys. Rev. Lett. 101, 123904 (2008).
[CrossRef] [PubMed]

W. P. Zhong, R. H. Xie, M. Belić, N. Petrović, and G. Chen, Phys. Rev. A 78, 023821 (2008).
[CrossRef]

Sulem, C.

C. Sulem and P. Sulem, The Nonlinear Schrödinger Equation: Self-Focusing and Wave Collapse (Springer, 2000).

Sulem, P.

C. Sulem and P. Sulem, The Nonlinear Schrödinger Equation: Self-Focusing and Wave Collapse (Springer, 2000).

Torner, L.

B. A. Malomed, D. Michalache, F. Wise, and L. Torner, J. Opt. B 7, R53 (2005).
[CrossRef]

Wang, M. L.

Y. B. Zhou, M. L. Wang, and T. D. Miao, Phys. Lett. A 323, 77 (2004).
[CrossRef]

Y. B. Zhou, M. L. Wang, and Y. M. Wang, Phys. Lett. A 308, 31 (2003).
[CrossRef]

Wang, Y. M.

Y. B. Zhou, M. L. Wang, and Y. M. Wang, Phys. Lett. A 308, 31 (2003).
[CrossRef]

Wise, F.

B. A. Malomed, D. Michalache, F. Wise, and L. Torner, J. Opt. B 7, R53 (2005).
[CrossRef]

Xie, R. H.

M. Belić, N. Petrović, W. P. Zhong, R. H. Xie, and G. Chen, Phys. Rev. Lett. 101, 123904 (2008).
[CrossRef] [PubMed]

W. P. Zhong, R. H. Xie, M. Belić, N. Petrović, and G. Chen, Phys. Rev. A 78, 023821 (2008).
[CrossRef]

Yan, Z.

Z. Yan and H. Q. Zhang, Phys. Lett. A 285, 355 (2001).
[CrossRef]

Yang, K.

L. Yang, J. Liu, and K. Yang, Phys. Lett. A 278, 267 (2001).
[CrossRef]

Yang, L.

L. Yang, J. Liu, and K. Yang, Phys. Lett. A 278, 267 (2001).
[CrossRef]

Zhang, H. Q.

Z. Yan and H. Q. Zhang, Phys. Lett. A 285, 355 (2001).
[CrossRef]

Zhong, W. P.

W. P. Zhong, R. H. Xie, M. Belić, N. Petrović, and G. Chen, Phys. Rev. A 78, 023821 (2008).
[CrossRef]

M. Belić, N. Petrović, W. P. Zhong, R. H. Xie, and G. Chen, Phys. Rev. Lett. 101, 123904 (2008).
[CrossRef] [PubMed]

Zhou, Y. B.

Y. B. Zhou, M. L. Wang, and T. D. Miao, Phys. Lett. A 323, 77 (2004).
[CrossRef]

Y. B. Zhou, M. L. Wang, and Y. M. Wang, Phys. Lett. A 308, 31 (2003).
[CrossRef]

J. Opt. B (1)

B. A. Malomed, D. Michalache, F. Wise, and L. Torner, J. Opt. B 7, R53 (2005).
[CrossRef]

Phys. Lett. A (4)

L. Yang, J. Liu, and K. Yang, Phys. Lett. A 278, 267 (2001).
[CrossRef]

Z. Yan and H. Q. Zhang, Phys. Lett. A 285, 355 (2001).
[CrossRef]

Y. B. Zhou, M. L. Wang, and Y. M. Wang, Phys. Lett. A 308, 31 (2003).
[CrossRef]

Y. B. Zhou, M. L. Wang, and T. D. Miao, Phys. Lett. A 323, 77 (2004).
[CrossRef]

Phys. Rev. A (1)

W. P. Zhong, R. H. Xie, M. Belić, N. Petrović, and G. Chen, Phys. Rev. A 78, 023821 (2008).
[CrossRef]

Phys. Rev. E (1)

V. I. Kruglov, A. C. Peacock, and J. D. Harvey, Phys. Rev. E 71, 056619 (2005).
[CrossRef]

Phys. Rev. Lett. (1)

M. Belić, N. Petrović, W. P. Zhong, R. H. Xie, and G. Chen, Phys. Rev. Lett. 101, 123904 (2008).
[CrossRef] [PubMed]

Physica D (1)

G. G. Luther, A. C. Newell, and J. V. Moloney, Physica D 74, 59 (1994).
[CrossRef]

Other (5)

N. N. Akhmediev and A. A. Ankiewicz, Solitons (Chapman and Hall, 1997).

C. Sulem and P. Sulem, The Nonlinear Schrödinger Equation: Self-Focusing and Wave Collapse (Springer, 2000).

Y. S. Kivshar and G. P. Agrawal, Optical Solitons, from Fibers to Photonic Crystals (Academic, 2003).

B. A. Malomed, Soliton Management in Periodic Systems (Springer, 2006).

A. Hasegava and M. Matsumoto, Optical Solitons in Fibers (Springer, 2003).

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Figures (4)

Fig. 1
Fig. 1

Traveling-wave solutions as functions of the propagation distance for a 0 = 0 (without chirp) and ϵ = 0 . (a) Intensity u 2 for F = sn and (b) for F = cn , presented as functions of k 0 x + l 0 y + m 0 t and z. Coefficients: β ( z ) = cos ( z ) , γ ( z ) = γ 0 = 0.05 , M = 0.9999 , b 0 = 1 , e 0 = 0 , k 0 = l 0 = m 0 = 1 , ω 0 = 0 .

Fig. 2
Fig. 2

Traveling-wave solutions with the chirp as functions of the propagation distance. The setup and parameters are the same as in Fig. 1 except for a 0 = 0.1 .

Fig. 3
Fig. 3

Soliton solutions without chirp. The setup and parameters are as in Fig. 1 except for M = 1 .

Fig. 4
Fig. 4

Soliton solutions with chirp. Setup is the same as in Fig. 3 except for a 0 = 0.1 .

Equations (25)

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i z u + β ( z ) 2 ( Δ u + s t 2 u ) + χ ( z ) u 2 u = i γ ( z ) u ,
u ( z , x , y , t ) = A ( z , x , y , t ) exp [ i B ( z , x , y , t ) ] .
z A + 1 2 β [ 2 x A x B + 2 y A y B + 2 s t A t B + A ( Δ + s t 2 ) B ] = γ A ,
A z B + 1 2 β [ ( Δ + s t 2 ) A A ( x B ) 2 A ( y B ) 2 s A ( t B ) 2 ] + χ A 3 = 0 .
A = f ( z ) F ( θ ) + g ( z ) F 1 ( θ ) ,
θ = k ( z ) x + l ( z ) y + m ( z ) t + ω ( z ) ,
B = a ( z ) ( x 2 + y 2 + s t 2 ) + b ( z ) ( x + y + t ) + e ( z ) ,
d f j d z + 3 a β f j γ f j = 0 ,
f j ( d k d z + 2 k a β ) = 0 ,
f j ( d l d z + 2 l a β ) = 0 ,
f j ( d m d z + 2 m a β ) = 0 ,
f j ( d a d z + 2 β a 2 ) = 0 ,
f j ( d b d z + 2 β a b ) = 0 ,
f j ( d ω d z + β ( k + l + s m ) b ) = 0 ,
f j ( d e d z + β 2 [ ( 2 + s ) b 2 ( k 2 + l 2 + s m 2 ) c 2 ] 3 χ f 1 f 2 ) = 0 ,
f 1 [ β ( k 2 + l 2 + s m 2 ) c 4 + χ f 1 2 ] = 0 ,
f 2 [ β ( k 2 + l 2 + s m 2 ) c 0 + χ f 2 2 ] = 0 ,
f = ( α ) 3 2 f 0 exp ( 0 z γ d z ) , g = ϵ c 0 c 4 f ,
k = α k 0 , l = α l 0 , m = α m 0 ,
ω = ω 0 α ( k 0 + l 0 + s m 0 ) b 0 0 z β d z ,
a = α a 0 , b = α b 0 ,
e = e 0 + α 2 [ ( k 0 2 + l 0 2 + s m 0 2 ) ( c 2 6 ϵ c 0 c 4 ) ( 2 + s ) b 0 2 ] 0 z β d z ;
χ = β c 4 ( k 0 2 + l 0 2 + s m 0 2 ) f 0 2 exp ( 2 0 z γ d z ) α .
u = ( α ) 3 2 f 0 exp ( 0 z γ d z ) [ F ( θ ) + ϵ c 0 c 4 1 F ( θ ) ] × exp i [ a ( x 2 + y 2 + s t 2 ) + b ( x + y + t ) + e ] ,
θ = ω 0 + k x + l y + m t ( k + l + s m ) b 0 0 z β d z .

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