Abstract

We derive analytical 3D spatiotemporal similaritons and solitons of the generalized (3+1)-dimensional nonlinear Schrödinger equation with distributed coefficients.

© 2010 Optical Society of America

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References

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  1. Y. S. Kivshar and G. P. Agrawal, Optical Solitons: from Fibers to Photonic Crystals (Academic, 2003).
  2. B. A. Malomed, D. Mihalache, F. Wise, and L. Torner,J. Opt. B 7, R53 (2005).
    [CrossRef]
  3. N. Petrović, M. Belić, W. P. Zhong, R. H. Xie, and G. Chen, Opt. Lett. 34, 1609 (2009).
    [CrossRef] [PubMed]
  4. S. H. Chen and J. M. Dudley, Phys. Rev. Lett. 102, 233903 (2009).
    [CrossRef] [PubMed]
  5. V. I. Kruglov, A. C. Peacock, and J. D. Harvey, Phys. Rev. Lett. 90, 113902 (2003).
    [CrossRef] [PubMed]
  6. L. Wu, J. F. Zhang, L. Li, Q. Tian, and K. Porsezian, Opt. Express 16, 6352 (2008).
    [CrossRef] [PubMed]
  7. C. Q. Dai, Y. Y. Wang, and C. J. Yan, Opt. Commun. 283, 1489 (2010).
    [CrossRef]
  8. J. F. Zhang, C. Q. Dai, Q. Yang, and J. M. Zhu, Opt. Commun. 252, 408 (2005).
    [CrossRef]
  9. S. L. Xu, J. C. Liang, and L. Yi, J. Opt. Soc. Am. B 27, 99 (2010).
    [CrossRef]
  10. W. P. Zhong, R. H. Xie, M. Belić, N. Petrović, G. Chen, and L. Yi, Phys. Rev. A 78, 023821 (2008).
    [CrossRef]
  11. M. Belić, N. Petrović, W. P. Zhong, R. H. Xie, and G. Chen, Phys. Rev. Lett. 101, 123904 (2008).
    [CrossRef] [PubMed]
  12. V. N. Serkin and A. Hasegawa, Phys. Rev. Lett. 85, 4502 (2000).
    [CrossRef] [PubMed]

2010 (2)

S. L. Xu, J. C. Liang, and L. Yi, J. Opt. Soc. Am. B 27, 99 (2010).
[CrossRef]

C. Q. Dai, Y. Y. Wang, and C. J. Yan, Opt. Commun. 283, 1489 (2010).
[CrossRef]

2009 (2)

2008 (3)

L. Wu, J. F. Zhang, L. Li, Q. Tian, and K. Porsezian, Opt. Express 16, 6352 (2008).
[CrossRef] [PubMed]

W. P. Zhong, R. H. Xie, M. Belić, N. Petrović, G. Chen, and L. Yi, 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. Mihalache, F. Wise, and L. Torner,J. Opt. B 7, R53 (2005).
[CrossRef]

J. F. Zhang, C. Q. Dai, Q. Yang, and J. M. Zhu, Opt. Commun. 252, 408 (2005).
[CrossRef]

2003 (1)

V. I. Kruglov, A. C. Peacock, and J. D. Harvey, Phys. Rev. Lett. 90, 113902 (2003).
[CrossRef] [PubMed]

2000 (1)

V. N. Serkin and A. Hasegawa, Phys. Rev. Lett. 85, 4502 (2000).
[CrossRef] [PubMed]

Agrawal, G. P.

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

Belic, M.

N. Petrović, M. Belić, W. P. Zhong, R. H. Xie, and G. Chen, Opt. Lett. 34, 1609 (2009).
[CrossRef] [PubMed]

W. P. Zhong, R. H. Xie, M. Belić, N. Petrović, G. Chen, and L. Yi, 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]

Chen, G.

N. Petrović, M. Belić, W. P. Zhong, R. H. Xie, and G. Chen, Opt. Lett. 34, 1609 (2009).
[CrossRef] [PubMed]

W. P. Zhong, R. H. Xie, M. Belić, N. Petrović, G. Chen, and L. Yi, 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]

Chen, S. H.

S. H. Chen and J. M. Dudley, Phys. Rev. Lett. 102, 233903 (2009).
[CrossRef] [PubMed]

Dai, C. Q.

C. Q. Dai, Y. Y. Wang, and C. J. Yan, Opt. Commun. 283, 1489 (2010).
[CrossRef]

J. F. Zhang, C. Q. Dai, Q. Yang, and J. M. Zhu, Opt. Commun. 252, 408 (2005).
[CrossRef]

Dudley, J. M.

S. H. Chen and J. M. Dudley, Phys. Rev. Lett. 102, 233903 (2009).
[CrossRef] [PubMed]

Harvey, J. D.

V. I. Kruglov, A. C. Peacock, and J. D. Harvey, Phys. Rev. Lett. 90, 113902 (2003).
[CrossRef] [PubMed]

Hasegawa, A.

V. N. Serkin and A. Hasegawa, Phys. Rev. Lett. 85, 4502 (2000).
[CrossRef] [PubMed]

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. Lett. 90, 113902 (2003).
[CrossRef] [PubMed]

Li, L.

Liang, J. C.

Malomed, B. A.

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

Mihalache, D.

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

Peacock, A. C.

V. I. Kruglov, A. C. Peacock, and J. D. Harvey, Phys. Rev. Lett. 90, 113902 (2003).
[CrossRef] [PubMed]

Petrovic, N.

N. Petrović, M. Belić, W. P. Zhong, R. H. Xie, and G. Chen, Opt. Lett. 34, 1609 (2009).
[CrossRef] [PubMed]

W. P. Zhong, R. H. Xie, M. Belić, N. Petrović, G. Chen, and L. Yi, 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]

Porsezian, K.

Serkin, V. N.

V. N. Serkin and A. Hasegawa, Phys. Rev. Lett. 85, 4502 (2000).
[CrossRef] [PubMed]

Tian, Q.

Torner, L.

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

Wang, Y. Y.

C. Q. Dai, Y. Y. Wang, and C. J. Yan, Opt. Commun. 283, 1489 (2010).
[CrossRef]

Wise, F.

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

Wu, L.

Xie, R. H.

N. Petrović, M. Belić, W. P. Zhong, R. H. Xie, and G. Chen, Opt. Lett. 34, 1609 (2009).
[CrossRef] [PubMed]

W. P. Zhong, R. H. Xie, M. Belić, N. Petrović, G. Chen, and L. Yi, 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]

Xu, S. L.

Yan, C. J.

C. Q. Dai, Y. Y. Wang, and C. J. Yan, Opt. Commun. 283, 1489 (2010).
[CrossRef]

Yang, Q.

J. F. Zhang, C. Q. Dai, Q. Yang, and J. M. Zhu, Opt. Commun. 252, 408 (2005).
[CrossRef]

Yi, L.

S. L. Xu, J. C. Liang, and L. Yi, J. Opt. Soc. Am. B 27, 99 (2010).
[CrossRef]

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

Zhang, J. F.

L. Wu, J. F. Zhang, L. Li, Q. Tian, and K. Porsezian, Opt. Express 16, 6352 (2008).
[CrossRef] [PubMed]

J. F. Zhang, C. Q. Dai, Q. Yang, and J. M. Zhu, Opt. Commun. 252, 408 (2005).
[CrossRef]

Zhong, W. P.

N. Petrović, M. Belić, W. P. Zhong, R. H. Xie, and G. Chen, Opt. Lett. 34, 1609 (2009).
[CrossRef] [PubMed]

W. P. Zhong, R. H. Xie, M. Belić, N. Petrović, G. Chen, and L. Yi, 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]

Zhu, J. M.

J. F. Zhang, C. Q. Dai, Q. Yang, and J. M. Zhu, Opt. Commun. 252, 408 (2005).
[CrossRef]

J. Opt. B (1)

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

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

Opt. Commun. (2)

C. Q. Dai, Y. Y. Wang, and C. J. Yan, Opt. Commun. 283, 1489 (2010).
[CrossRef]

J. F. Zhang, C. Q. Dai, Q. Yang, and J. M. Zhu, Opt. Commun. 252, 408 (2005).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Phys. Rev. A (1)

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

Phys. Rev. Lett. (4)

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

V. N. Serkin and A. Hasegawa, Phys. Rev. Lett. 85, 4502 (2000).
[CrossRef] [PubMed]

S. H. Chen and J. M. Dudley, Phys. Rev. Lett. 102, 233903 (2009).
[CrossRef] [PubMed]

V. I. Kruglov, A. C. Peacock, and J. D. Harvey, Phys. Rev. Lett. 90, 113902 (2003).
[CrossRef] [PubMed]

Other (1)

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

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Figures (4)

Fig. 1
Fig. 1

Bright and dark similariton pairs with chirp for s 0 = 0.4 . Parameters β 0 = 0.1 , γ 0 = 0.05 , η 1 = ξ 2 = 1.5 , ξ 1 = η 2 = 1.4 , θ 10 = X 10 = θ 20 = X 20 = r = 2 , and ϵ = ± 1 for bright and dark similaritons, respectively. Other parameters are all chosen as 1.

Fig. 2
Fig. 2

Bright and dark soliton pairs without chirp for s 0 = 0 . The parameters are the same as in Fig. 1.

Fig. 3
Fig. 3

Periodic attraction and repulsion between bright similariton pairs with zigzag propagation. The parameters are the same as in Fig. 1, except for (a) ξ 1 = 0.5 , ξ 2 = 5 and (b) ξ 1 = ξ 2 = 4 with η 1 = η 2 = 1 .

Fig. 4
Fig. 4

Periodic attraction and repulsion between dark similariton pairs with zigzag propagation. The parameters are the same as in Fig. 1, except for (a) ξ 1 = 0 , η 1 = 5 , ξ 2 = 3 , η 2 = 4 and (b) ξ 1 = ξ 2 = 4 , η 1 = η 2 = 3 with X 10 = X 20 = 2.5 .

Tables (1)

Tables Icon

Table 1 Existence Conditions

Equations (18)

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i u z + β ( z ) 2 ( Δ u + σ u t t ) + χ ( z ) | u | 2 u = i γ ( z ) u ,
i U Z + ϵ 2 U X X + | U | 2 U = 0 ,
u ( z , x , y , t ) = A ( z ) e i φ ( z , x , y , t ) U ( Z , X ) ,
A z + 1 2 β A ( φ x x + φ y y + σ φ t t ) γ A = 0 ,
X z + β ( X x φ x + X y φ y + σ X t φ t ) = 0 ,
φ z + 1 2 β ( φ x 2 + φ y 2 + σ φ t 2 ) = 0 ,
X x x + X y y + σ X t t = 0 ,
β ( X x 2 + X y 2 + σ X t 2 ) = ϵ Z z , A 2 χ = Z z .
A = A 0 α 3 2 exp [ 0 z γ ( s ) d s ] ,
φ = s 0 α 2 ( x 2 + y 2 + t 2 σ ) d 0 α 3 ( x p + y q + t r σ ) d 0 2 α D ( z ) 18 ( 1 p 2 + 1 q 2 + 1 r 2 σ ) ,
Z = ( p 2 + q 2 + r 2 σ ) α D ( z ) ϵ w 0 2 , X = ζ ζ c ( z ) w ( z ) ,
ζ = p x + q y + r t , w ( z ) = w 0 α ,
ζ c ( z ) = ζ 0 ( d 0 + s 0 ζ 0 ) D ( z ) ,
χ = β ( p 2 + q 2 + r 2 σ ) ϵ α ( w 0 A 0 ) 2 exp [ 2 0 z γ ( s ) d s ] ,
u = ϵ A ( z ) e i φ ( z , x , y , t ) G 1 F 1 ,
u = ϵ μ A ( z ) e i φ ( z , x , y , t ) + i ψ ( Z , X ) ( 1 + G 2 F 2 ) ,
G 2 = 4 μ ( ρ 1 + ρ 2 2 μ ) 4 i λ 1 + λ 2 η 1 + η 2 ϱ ,
F 2 = 4 μ 2 + ( λ 1 + λ 2 η 1 + η 2 ) 2 ϱ ,

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