Abstract

We analyze interactions between moving dissipative solitons in one- and multidimensional cubic-quintic complex Ginzburg–Landau equations with a linear potential and effective viscosity. The interactions between the solitons are analyzed by using balance equations for the energy and momentum. We demonstrate that the separation between two solitons forming a bound state decreases with the increase of the slope of the linear potential.

© 2009 Optical Society of America

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  1. Y. V. Kartashov, A. A. Egorov, L. Torner, and D. N. Christodoulides, Opt. Lett. 29, 1918 (2004).
    [CrossRef] [PubMed]
  2. T. S. Ku, M.-F. Shih, A. A. Sukhorukov, and Y. S. Kivshar, Phys. Rev. Lett. 94, 063904 (2005).
    [CrossRef] [PubMed]
  3. W. J. Firth and A. J. Scroggie, Phys. Rev. Lett. 76, 1623 (1996).
    [CrossRef] [PubMed]
  4. L. Spinelli, G. Tissoni, M. Brambilla, F. Prati, and L. A. Lugiato, Phys. Rev. A 58, 2542 (1998).
    [CrossRef]
  5. M. Saffman, D. Montgomery, and D. Z. Anderson, Opt. Lett. 19, 518 (1994).
    [CrossRef] [PubMed]
  6. C. Cleff, B. Gütlich, and C. Denz, Phys. Rev. Lett. 100, 233902 (2008).
    [CrossRef] [PubMed]
  7. F. Pedaci, S. Barland, E. Gaboche, P. Genevet, M. Giudici, J. R. Tredicce, T. Ackemann, A. J. Scroggie, W. J. Firth, and G.-L. Oppo, Appl. Phys. Lett. 92, 011101 (2008).
    [CrossRef]
  8. D. V. Skryabin and A. G. Vladimirov, Phys. Rev. Lett. 89, 044101 (2002).
    [CrossRef] [PubMed]
  9. D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed, Phys. Rev. A 77, 033817 (2008).
    [CrossRef]
  10. J. M. Soto-Crespo, P. Grelu, and N. Akhmediev, Opt. Express 14, 4013 (2006).
    [CrossRef] [PubMed]
  11. V. Skarka and N. B. Aleksić, Phys. Rev. Lett. 96, 013903 (2006).
    [CrossRef] [PubMed]
  12. J. Lega, J. V. Moloney, and A. C. Newell, Phys. Rev. Lett. 73, 2978 (1994).
    [CrossRef] [PubMed]
  13. D. Mihalache, D. Mazilu, F. Lederer, Y. V. Kartashov, L.-C. Crasovan, L. Torner, and B. A. Malomed, Phys. Rev. Lett. 97, 073904 (2006).
    [CrossRef] [PubMed]
  14. D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed, Phys. Rev. A 75, 033811 (2007).
    [CrossRef]
  15. Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Opt. Express 15, 12409 (2007).
    [CrossRef] [PubMed]
  16. N. N. Akhmediev, A. Ankiewicz, and J. M. Soto-Crespo, J. Opt. Soc. Am. B 15, 515 (1998).
    [CrossRef]
  17. V. V. Afanasjev, B. A. Malomed, and P. L. Chu, Phys. Rev. E 56, 6020 (1997).
    [CrossRef]

2008

C. Cleff, B. Gütlich, and C. Denz, Phys. Rev. Lett. 100, 233902 (2008).
[CrossRef] [PubMed]

F. Pedaci, S. Barland, E. Gaboche, P. Genevet, M. Giudici, J. R. Tredicce, T. Ackemann, A. J. Scroggie, W. J. Firth, and G.-L. Oppo, Appl. Phys. Lett. 92, 011101 (2008).
[CrossRef]

D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed, Phys. Rev. A 77, 033817 (2008).
[CrossRef]

2007

D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed, Phys. Rev. A 75, 033811 (2007).
[CrossRef]

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Opt. Express 15, 12409 (2007).
[CrossRef] [PubMed]

2006

J. M. Soto-Crespo, P. Grelu, and N. Akhmediev, Opt. Express 14, 4013 (2006).
[CrossRef] [PubMed]

V. Skarka and N. B. Aleksić, Phys. Rev. Lett. 96, 013903 (2006).
[CrossRef] [PubMed]

D. Mihalache, D. Mazilu, F. Lederer, Y. V. Kartashov, L.-C. Crasovan, L. Torner, and B. A. Malomed, Phys. Rev. Lett. 97, 073904 (2006).
[CrossRef] [PubMed]

2005

T. S. Ku, M.-F. Shih, A. A. Sukhorukov, and Y. S. Kivshar, Phys. Rev. Lett. 94, 063904 (2005).
[CrossRef] [PubMed]

2004

2002

D. V. Skryabin and A. G. Vladimirov, Phys. Rev. Lett. 89, 044101 (2002).
[CrossRef] [PubMed]

1998

L. Spinelli, G. Tissoni, M. Brambilla, F. Prati, and L. A. Lugiato, Phys. Rev. A 58, 2542 (1998).
[CrossRef]

N. N. Akhmediev, A. Ankiewicz, and J. M. Soto-Crespo, J. Opt. Soc. Am. B 15, 515 (1998).
[CrossRef]

1997

V. V. Afanasjev, B. A. Malomed, and P. L. Chu, Phys. Rev. E 56, 6020 (1997).
[CrossRef]

1996

W. J. Firth and A. J. Scroggie, Phys. Rev. Lett. 76, 1623 (1996).
[CrossRef] [PubMed]

1994

J. Lega, J. V. Moloney, and A. C. Newell, Phys. Rev. Lett. 73, 2978 (1994).
[CrossRef] [PubMed]

M. Saffman, D. Montgomery, and D. Z. Anderson, Opt. Lett. 19, 518 (1994).
[CrossRef] [PubMed]

Ackemann, T.

F. Pedaci, S. Barland, E. Gaboche, P. Genevet, M. Giudici, J. R. Tredicce, T. Ackemann, A. J. Scroggie, W. J. Firth, and G.-L. Oppo, Appl. Phys. Lett. 92, 011101 (2008).
[CrossRef]

Afanasjev, V. V.

V. V. Afanasjev, B. A. Malomed, and P. L. Chu, Phys. Rev. E 56, 6020 (1997).
[CrossRef]

Akhmediev, N.

Akhmediev, N. N.

Aleksic, N. B.

V. Skarka and N. B. Aleksić, Phys. Rev. Lett. 96, 013903 (2006).
[CrossRef] [PubMed]

Anderson, D. Z.

Ankiewicz, A.

Barland, S.

F. Pedaci, S. Barland, E. Gaboche, P. Genevet, M. Giudici, J. R. Tredicce, T. Ackemann, A. J. Scroggie, W. J. Firth, and G.-L. Oppo, Appl. Phys. Lett. 92, 011101 (2008).
[CrossRef]

Brambilla, M.

L. Spinelli, G. Tissoni, M. Brambilla, F. Prati, and L. A. Lugiato, Phys. Rev. A 58, 2542 (1998).
[CrossRef]

Christodoulides, D. N.

Chu, P. L.

V. V. Afanasjev, B. A. Malomed, and P. L. Chu, Phys. Rev. E 56, 6020 (1997).
[CrossRef]

Cleff, C.

C. Cleff, B. Gütlich, and C. Denz, Phys. Rev. Lett. 100, 233902 (2008).
[CrossRef] [PubMed]

Crasovan, L.-C.

D. Mihalache, D. Mazilu, F. Lederer, Y. V. Kartashov, L.-C. Crasovan, L. Torner, and B. A. Malomed, Phys. Rev. Lett. 97, 073904 (2006).
[CrossRef] [PubMed]

Denz, C.

C. Cleff, B. Gütlich, and C. Denz, Phys. Rev. Lett. 100, 233902 (2008).
[CrossRef] [PubMed]

Egorov, A. A.

Firth, W. J.

F. Pedaci, S. Barland, E. Gaboche, P. Genevet, M. Giudici, J. R. Tredicce, T. Ackemann, A. J. Scroggie, W. J. Firth, and G.-L. Oppo, Appl. Phys. Lett. 92, 011101 (2008).
[CrossRef]

W. J. Firth and A. J. Scroggie, Phys. Rev. Lett. 76, 1623 (1996).
[CrossRef] [PubMed]

Gaboche, E.

F. Pedaci, S. Barland, E. Gaboche, P. Genevet, M. Giudici, J. R. Tredicce, T. Ackemann, A. J. Scroggie, W. J. Firth, and G.-L. Oppo, Appl. Phys. Lett. 92, 011101 (2008).
[CrossRef]

Genevet, P.

F. Pedaci, S. Barland, E. Gaboche, P. Genevet, M. Giudici, J. R. Tredicce, T. Ackemann, A. J. Scroggie, W. J. Firth, and G.-L. Oppo, Appl. Phys. Lett. 92, 011101 (2008).
[CrossRef]

Giudici, M.

F. Pedaci, S. Barland, E. Gaboche, P. Genevet, M. Giudici, J. R. Tredicce, T. Ackemann, A. J. Scroggie, W. J. Firth, and G.-L. Oppo, Appl. Phys. Lett. 92, 011101 (2008).
[CrossRef]

Grelu, P.

Gütlich, B.

C. Cleff, B. Gütlich, and C. Denz, Phys. Rev. Lett. 100, 233902 (2008).
[CrossRef] [PubMed]

Kartashov, Y. V.

Kivshar, Y. S.

T. S. Ku, M.-F. Shih, A. A. Sukhorukov, and Y. S. Kivshar, Phys. Rev. Lett. 94, 063904 (2005).
[CrossRef] [PubMed]

Ku, T. S.

T. S. Ku, M.-F. Shih, A. A. Sukhorukov, and Y. S. Kivshar, Phys. Rev. Lett. 94, 063904 (2005).
[CrossRef] [PubMed]

Leblond, H.

D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed, Phys. Rev. A 77, 033817 (2008).
[CrossRef]

D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed, Phys. Rev. A 75, 033811 (2007).
[CrossRef]

Lederer, F.

D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed, Phys. Rev. A 77, 033817 (2008).
[CrossRef]

D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed, Phys. Rev. A 75, 033811 (2007).
[CrossRef]

D. Mihalache, D. Mazilu, F. Lederer, Y. V. Kartashov, L.-C. Crasovan, L. Torner, and B. A. Malomed, Phys. Rev. Lett. 97, 073904 (2006).
[CrossRef] [PubMed]

Lega, J.

J. Lega, J. V. Moloney, and A. C. Newell, Phys. Rev. Lett. 73, 2978 (1994).
[CrossRef] [PubMed]

Lugiato, L. A.

L. Spinelli, G. Tissoni, M. Brambilla, F. Prati, and L. A. Lugiato, Phys. Rev. A 58, 2542 (1998).
[CrossRef]

Malomed, B. A.

D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed, Phys. Rev. A 77, 033817 (2008).
[CrossRef]

D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed, Phys. Rev. A 75, 033811 (2007).
[CrossRef]

D. Mihalache, D. Mazilu, F. Lederer, Y. V. Kartashov, L.-C. Crasovan, L. Torner, and B. A. Malomed, Phys. Rev. Lett. 97, 073904 (2006).
[CrossRef] [PubMed]

V. V. Afanasjev, B. A. Malomed, and P. L. Chu, Phys. Rev. E 56, 6020 (1997).
[CrossRef]

Mazilu, D.

D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed, Phys. Rev. A 77, 033817 (2008).
[CrossRef]

D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed, Phys. Rev. A 75, 033811 (2007).
[CrossRef]

D. Mihalache, D. Mazilu, F. Lederer, Y. V. Kartashov, L.-C. Crasovan, L. Torner, and B. A. Malomed, Phys. Rev. Lett. 97, 073904 (2006).
[CrossRef] [PubMed]

Mihalache, D.

D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed, Phys. Rev. A 77, 033817 (2008).
[CrossRef]

D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed, Phys. Rev. A 75, 033811 (2007).
[CrossRef]

D. Mihalache, D. Mazilu, F. Lederer, Y. V. Kartashov, L.-C. Crasovan, L. Torner, and B. A. Malomed, Phys. Rev. Lett. 97, 073904 (2006).
[CrossRef] [PubMed]

Moloney, J. V.

J. Lega, J. V. Moloney, and A. C. Newell, Phys. Rev. Lett. 73, 2978 (1994).
[CrossRef] [PubMed]

Montgomery, D.

Newell, A. C.

J. Lega, J. V. Moloney, and A. C. Newell, Phys. Rev. Lett. 73, 2978 (1994).
[CrossRef] [PubMed]

Oppo, G.-L.

F. Pedaci, S. Barland, E. Gaboche, P. Genevet, M. Giudici, J. R. Tredicce, T. Ackemann, A. J. Scroggie, W. J. Firth, and G.-L. Oppo, Appl. Phys. Lett. 92, 011101 (2008).
[CrossRef]

Pedaci, F.

F. Pedaci, S. Barland, E. Gaboche, P. Genevet, M. Giudici, J. R. Tredicce, T. Ackemann, A. J. Scroggie, W. J. Firth, and G.-L. Oppo, Appl. Phys. Lett. 92, 011101 (2008).
[CrossRef]

Prati, F.

L. Spinelli, G. Tissoni, M. Brambilla, F. Prati, and L. A. Lugiato, Phys. Rev. A 58, 2542 (1998).
[CrossRef]

Saffman, M.

Scroggie, A. J.

F. Pedaci, S. Barland, E. Gaboche, P. Genevet, M. Giudici, J. R. Tredicce, T. Ackemann, A. J. Scroggie, W. J. Firth, and G.-L. Oppo, Appl. Phys. Lett. 92, 011101 (2008).
[CrossRef]

W. J. Firth and A. J. Scroggie, Phys. Rev. Lett. 76, 1623 (1996).
[CrossRef] [PubMed]

Shih, M.-F.

T. S. Ku, M.-F. Shih, A. A. Sukhorukov, and Y. S. Kivshar, Phys. Rev. Lett. 94, 063904 (2005).
[CrossRef] [PubMed]

Skarka, V.

V. Skarka and N. B. Aleksić, Phys. Rev. Lett. 96, 013903 (2006).
[CrossRef] [PubMed]

Skryabin, D. V.

D. V. Skryabin and A. G. Vladimirov, Phys. Rev. Lett. 89, 044101 (2002).
[CrossRef] [PubMed]

Soto-Crespo, J. M.

Spinelli, L.

L. Spinelli, G. Tissoni, M. Brambilla, F. Prati, and L. A. Lugiato, Phys. Rev. A 58, 2542 (1998).
[CrossRef]

Sukhorukov, A. A.

T. S. Ku, M.-F. Shih, A. A. Sukhorukov, and Y. S. Kivshar, Phys. Rev. Lett. 94, 063904 (2005).
[CrossRef] [PubMed]

Tissoni, G.

L. Spinelli, G. Tissoni, M. Brambilla, F. Prati, and L. A. Lugiato, Phys. Rev. A 58, 2542 (1998).
[CrossRef]

Torner, L.

Tredicce, J. R.

F. Pedaci, S. Barland, E. Gaboche, P. Genevet, M. Giudici, J. R. Tredicce, T. Ackemann, A. J. Scroggie, W. J. Firth, and G.-L. Oppo, Appl. Phys. Lett. 92, 011101 (2008).
[CrossRef]

Vladimirov, A. G.

D. V. Skryabin and A. G. Vladimirov, Phys. Rev. Lett. 89, 044101 (2002).
[CrossRef] [PubMed]

Vysloukh, V. A.

Appl. Phys. Lett.

F. Pedaci, S. Barland, E. Gaboche, P. Genevet, M. Giudici, J. R. Tredicce, T. Ackemann, A. J. Scroggie, W. J. Firth, and G.-L. Oppo, Appl. Phys. Lett. 92, 011101 (2008).
[CrossRef]

J. Opt. Soc. Am. B

Opt. Express

Opt. Lett.

Phys. Rev. A

D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed, Phys. Rev. A 75, 033811 (2007).
[CrossRef]

L. Spinelli, G. Tissoni, M. Brambilla, F. Prati, and L. A. Lugiato, Phys. Rev. A 58, 2542 (1998).
[CrossRef]

D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed, Phys. Rev. A 77, 033817 (2008).
[CrossRef]

Phys. Rev. E

V. V. Afanasjev, B. A. Malomed, and P. L. Chu, Phys. Rev. E 56, 6020 (1997).
[CrossRef]

Phys. Rev. Lett.

D. V. Skryabin and A. G. Vladimirov, Phys. Rev. Lett. 89, 044101 (2002).
[CrossRef] [PubMed]

V. Skarka and N. B. Aleksić, Phys. Rev. Lett. 96, 013903 (2006).
[CrossRef] [PubMed]

J. Lega, J. V. Moloney, and A. C. Newell, Phys. Rev. Lett. 73, 2978 (1994).
[CrossRef] [PubMed]

D. Mihalache, D. Mazilu, F. Lederer, Y. V. Kartashov, L.-C. Crasovan, L. Torner, and B. A. Malomed, Phys. Rev. Lett. 97, 073904 (2006).
[CrossRef] [PubMed]

C. Cleff, B. Gütlich, and C. Denz, Phys. Rev. Lett. 100, 233902 (2008).
[CrossRef] [PubMed]

T. S. Ku, M.-F. Shih, A. A. Sukhorukov, and Y. S. Kivshar, Phys. Rev. Lett. 94, 063904 (2005).
[CrossRef] [PubMed]

W. J. Firth and A. J. Scroggie, Phys. Rev. Lett. 76, 1623 (1996).
[CrossRef] [PubMed]

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Figures (3)

Fig. 1
Fig. 1

(a) Profile of a single stable 1D soliton for α = 0 [for α ≠ 0, the soliton moves at a constant velocity; cf. panel (f)]. (b) Solutions of the energy-balance equation (curve E) and momentum-balance equation (other curves). (c), (d) Phase difference and separation, x 0 , between the bound solitons, versus α; results of direct simulations for (e) α = 0 and (f) α = 0.06, with the initial separation, x 0 = 7.8 , predicted by the analytical approach for the bound state.

Fig. 2
Fig. 2

Same as in Fig. 1, but for 2D solitons. In the last two panels, x 0 = 5.5 for (e) α = 0 and (f) α = 0.4.

Fig. 3
Fig. 3

Same as in Figs. 1, 2, but for 3D solitons [both panels (a) and (b) display its profile]. In the last two panels, x 0 = 8.7 , for (f) α = 0 and (g) α = 0.08.

Equations (6)

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

i u z + ( 1 2 ) Δ u + | u | 2 u ν | u | 4 u = i R [ u ] α x u ,
u = A ( z ) exp { [ x q ( z ) ] 2 + y 2 + t 2 2 w 2 ( z ) + i ψ ( z ) + i c ( z ) [ ( x q ( z ) ) 2 + y 2 + t 2 ] + i L ( z ) [ x q ( z ) ] } ,
d E d z = + + + ( u * G + u G * ) d x d y d t ,
d M d z = i + + + ( u x * G u x G * ) d x d y d t ,
u ( x , y , t ) = u 0 ( x x 0 2 , y , t ) exp ( i φ ) + u 0 ( x + x 0 2 , y , t ) ,
d E ( u 0 , x 0 , ϕ ) d z = d M ( u 0 , x 0 , ϕ ) d z = 0 .

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