Abstract

We develop a theory of soliton interactions in a bulk χ(2) medium. In particular, we derive a general analytical model based on the effective particle approach to predict the result of the soliton interactions and also verify the validity of the model by direct numerical simulations. Among the major phenomena investigated are soliton scattering, spiraling, and fusion and also power exchange between the colliding solitons.

© 1999 Optical Society of America

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  1. M. Segev and G. I. Stegeman, Phys. Today 51, 42 (1998), and references therein.
    [CrossRef]
  2. K. A. Gorshkov and L. A. Ostrovsky, Physica D 3, 428 (1981).
    [CrossRef]
  3. V. I. Karpman and V. V. Solov’ev, Physica D 3, 487 (1981).
    [CrossRef]
  4. V. V. Steblina, Yu. S. Kivshar, and A. V. Buryak, Opt. Lett. 23, 156 (1998).
    [CrossRef]
  5. W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. Van Stryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, Phys. Rev. Lett. 74, 5036 (1995).
    [CrossRef] [PubMed]
  6. G. C. Duree, J. L. Shultz, G. J. Salamo, M. Segev, A. Yariv, B. Crosignani, P. Di Porto, E. J. Sharp, and R. R. Neurgaonkar, Phys. Rev. Lett. 71, 553 (1993).
    [CrossRef]
  7. V. Tikhonenko, J. Christou, and B. Luther-Davies, J. Opt. Soc. Am. B 12, 2046 (1995).
    [CrossRef]
  8. V. Tikhonenko, J. Christou, and B. Luther-Davies, Phys. Rev. Lett. 76, 2698 (1996).
    [CrossRef] [PubMed]
  9. M. Shih, M. Segev, and G. Salamo, Phys. Rev. Lett. 78, 2551 (1997); A. V. Buryak, Yu. S. Kivshar, M. Shih, and M. Segev, Phys. Rev. Lett. (to be published).
    [CrossRef]
  10. L. Poladian, A. W. Snyder, and D. J. Mitchell, Opt. Commun. 85, 59 (1991).
    [CrossRef]
  11. A. V. Buryak, Yu. S. Kivshar, and V. V. Steblina, Phys. Rev. A 52, 1670 (1995).
    [CrossRef] [PubMed]
  12. A. V. Buryak, Yu. S. Kivshar, and S. Trillo, Phys. Rev. Lett. 77, 5210 (1996).
    [CrossRef] [PubMed]
  13. A. V. Buryak and Yu. S. Kivshar, Phys. Rev. Lett. 78, 3286 (1997).
    [CrossRef]
  14. M. T. G. Canva, R. A. Fuerst, D. Baboiu, G. I. Stegeman, and G. Assanto, Opt. Lett. 22, 1683 (1997).
    [CrossRef]
  15. R. A. Fuerst, M. T. G. Canva, D. Baboiu, and G. I. Stegeman, Opt. Lett. 22, 1748 (1997).
    [CrossRef]
  16. G. I. Stegeman, D. J. Hagan, and L. Torner, Opt. Quantum Electron. 28, 1691 (1996).
    [CrossRef]
  17. M. J. Werner and P. D. Drummond, J. Opt. Soc. Am. B 10, 2390 (1993).
    [CrossRef]
  18. D. M. Baboiu, G. I. Stegeman, and L. Torner, Opt. Lett. 20, 2282 (1995).
    [CrossRef]
  19. C. Etrich, U. Peschel, F. Lederer, and B. Malomed, Phys. Rev. A 52, R3444 (1995).
    [CrossRef]
  20. M. Haelterman, S. Trillo, and P. Ferro, Opt. Lett. 22, 84 (1997).
    [CrossRef] [PubMed]
  21. D. M. Baboiu and G. I. Stegeman, J. Opt. Soc. Am. B 14, 3143 (1997).
    [CrossRef]
  22. G. Leo, G. Assanto, and W. E. Torruellas, Opt. Lett. 22, 7 (1997).
    [CrossRef] [PubMed]
  23. G. Leo, G. Assanto, and W. E. Torruellas, Opt. Commun. 134, 223 (1997).
    [CrossRef]
  24. C. Etrich, U. Peschel, F. Lederer, B. Malomed, and Yu. S. Kivshar, Phys. Rev. E 54, 4321 (1996); D. E. Pelenovsky, Yu. S. Kivshar, and V. V. Afanasjev, Physica D 116, 121 (1998); Yu. S. Kivshar, D. E. Pelenovsky, T. Cretegny, and M. Peyrard, Phys. Rev. Lett. PRLTAO 80, 5032 (1998).
    [CrossRef]
  25. Y. Baek, R. Schiek, G. I. Stegeman, I. Baumann, and W. Sohler, Opt. Lett. 22, 1550 (1997).
    [CrossRef]
  26. B. Costantini, C. De Angelis, A. Barthelemy, B. Bourliaguest, and V. Kermene, Opt. Lett. 23, 424 (1998).
    [CrossRef]
  27. C. R. Menyuk, R. Schiek, and L. Torner, J. Opt. Soc. Am. B 11, 2434 (1994).
    [CrossRef]
  28. A. V. Buryak and Yu. S. Kivshar, Phys. Lett. A 197, 407 (1995).
    [CrossRef]
  29. A. V. Buryak, “Solitons due to second harmonic generation,” Ph.D. dissertation (Australian National University, Canberra, 1996).
  30. V. V. Steblina, Yu. S. Kivshar, M. Lisak, and B. A. Malomed, Opt. Commun. 118, 345 (1995).
    [CrossRef]
  31. D. E. Pelinovsky, A. V. Buryak, and Yu. S. Kivshar, Phys. Rev. Lett. 75, 591 (1995).
    [CrossRef] [PubMed]
  32. T. P. Taha and M. J. Ablowitz, J. Comput. Phys. 55, 203 (1984).
    [CrossRef]
  33. H. Goldstein, Classical Mechanics (Addison-Wesley, Reading, Mass., 1980), Chap. 3.
  34. A. W. Snyder, A. V. Buryak, and D. J. Mitchell, Opt. Lett. 23, 4 (1998).
    [CrossRef]
  35. A. V. Buryak, Yu. S. Kivshar, and S. Trillo, J. Opt. Soc. Am. B 14, 3110 (1997).
    [CrossRef]
  36. G. D. Miller, R. G. Batchko, W. M. Tulloch, D. R. Weise, M. M. Fejer, and R. L. Byer, Opt. Lett. 22, 1834 (1997).
    [CrossRef]
  37. D. E. Edmundson and R. H. Enns, Opt. Lett. 18, 1609 (1993).
    [CrossRef] [PubMed]
  38. D. E. Edmundson and R. H. Enns, Phys. Rev. A 51, 2491 (1995).
    [CrossRef] [PubMed]
  39. B. Malomed, P. Drummond, H. He, A. Berntson, D. Anderson, and M. Lisak, Phys. Rev. E 56, 4725 (1997).
    [CrossRef]

1998 (4)

1997 (11)

1996 (3)

G. I. Stegeman, D. J. Hagan, and L. Torner, Opt. Quantum Electron. 28, 1691 (1996).
[CrossRef]

A. V. Buryak, Yu. S. Kivshar, and S. Trillo, Phys. Rev. Lett. 77, 5210 (1996).
[CrossRef] [PubMed]

V. Tikhonenko, J. Christou, and B. Luther-Davies, Phys. Rev. Lett. 76, 2698 (1996).
[CrossRef] [PubMed]

1995 (9)

A. V. Buryak, Yu. S. Kivshar, and V. V. Steblina, Phys. Rev. A 52, 1670 (1995).
[CrossRef] [PubMed]

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. Van Stryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, Phys. Rev. Lett. 74, 5036 (1995).
[CrossRef] [PubMed]

C. Etrich, U. Peschel, F. Lederer, and B. Malomed, Phys. Rev. A 52, R3444 (1995).
[CrossRef]

A. V. Buryak and Yu. S. Kivshar, Phys. Lett. A 197, 407 (1995).
[CrossRef]

V. V. Steblina, Yu. S. Kivshar, M. Lisak, and B. A. Malomed, Opt. Commun. 118, 345 (1995).
[CrossRef]

D. E. Pelinovsky, A. V. Buryak, and Yu. S. Kivshar, Phys. Rev. Lett. 75, 591 (1995).
[CrossRef] [PubMed]

V. Tikhonenko, J. Christou, and B. Luther-Davies, J. Opt. Soc. Am. B 12, 2046 (1995).
[CrossRef]

D. M. Baboiu, G. I. Stegeman, and L. Torner, Opt. Lett. 20, 2282 (1995).
[CrossRef]

D. E. Edmundson and R. H. Enns, Phys. Rev. A 51, 2491 (1995).
[CrossRef] [PubMed]

1994 (1)

1993 (3)

D. E. Edmundson and R. H. Enns, Opt. Lett. 18, 1609 (1993).
[CrossRef] [PubMed]

G. C. Duree, J. L. Shultz, G. J. Salamo, M. Segev, A. Yariv, B. Crosignani, P. Di Porto, E. J. Sharp, and R. R. Neurgaonkar, Phys. Rev. Lett. 71, 553 (1993).
[CrossRef]

M. J. Werner and P. D. Drummond, J. Opt. Soc. Am. B 10, 2390 (1993).
[CrossRef]

1991 (1)

L. Poladian, A. W. Snyder, and D. J. Mitchell, Opt. Commun. 85, 59 (1991).
[CrossRef]

1984 (1)

T. P. Taha and M. J. Ablowitz, J. Comput. Phys. 55, 203 (1984).
[CrossRef]

1981 (2)

K. A. Gorshkov and L. A. Ostrovsky, Physica D 3, 428 (1981).
[CrossRef]

V. I. Karpman and V. V. Solov’ev, Physica D 3, 487 (1981).
[CrossRef]

Ablowitz, M. J.

T. P. Taha and M. J. Ablowitz, J. Comput. Phys. 55, 203 (1984).
[CrossRef]

Anderson, D.

B. Malomed, P. Drummond, H. He, A. Berntson, D. Anderson, and M. Lisak, Phys. Rev. E 56, 4725 (1997).
[CrossRef]

Assanto, G.

Baboiu, D.

Baboiu, D. M.

Baek, Y.

Barthelemy, A.

Batchko, R. G.

Baumann, I.

Berntson, A.

B. Malomed, P. Drummond, H. He, A. Berntson, D. Anderson, and M. Lisak, Phys. Rev. E 56, 4725 (1997).
[CrossRef]

Bourliaguest, B.

Buryak, A. V.

A. W. Snyder, A. V. Buryak, and D. J. Mitchell, Opt. Lett. 23, 4 (1998).
[CrossRef]

V. V. Steblina, Yu. S. Kivshar, and A. V. Buryak, Opt. Lett. 23, 156 (1998).
[CrossRef]

A. V. Buryak, Yu. S. Kivshar, and S. Trillo, J. Opt. Soc. Am. B 14, 3110 (1997).
[CrossRef]

A. V. Buryak and Yu. S. Kivshar, Phys. Rev. Lett. 78, 3286 (1997).
[CrossRef]

A. V. Buryak, Yu. S. Kivshar, and S. Trillo, Phys. Rev. Lett. 77, 5210 (1996).
[CrossRef] [PubMed]

D. E. Pelinovsky, A. V. Buryak, and Yu. S. Kivshar, Phys. Rev. Lett. 75, 591 (1995).
[CrossRef] [PubMed]

A. V. Buryak, Yu. S. Kivshar, and V. V. Steblina, Phys. Rev. A 52, 1670 (1995).
[CrossRef] [PubMed]

A. V. Buryak and Yu. S. Kivshar, Phys. Lett. A 197, 407 (1995).
[CrossRef]

Byer, R. L.

Canva, M. T. G.

Christou, J.

V. Tikhonenko, J. Christou, and B. Luther-Davies, Phys. Rev. Lett. 76, 2698 (1996).
[CrossRef] [PubMed]

V. Tikhonenko, J. Christou, and B. Luther-Davies, J. Opt. Soc. Am. B 12, 2046 (1995).
[CrossRef]

Costantini, B.

Crosignani, B.

G. C. Duree, J. L. Shultz, G. J. Salamo, M. Segev, A. Yariv, B. Crosignani, P. Di Porto, E. J. Sharp, and R. R. Neurgaonkar, Phys. Rev. Lett. 71, 553 (1993).
[CrossRef]

De Angelis, C.

Di Porto, P.

G. C. Duree, J. L. Shultz, G. J. Salamo, M. Segev, A. Yariv, B. Crosignani, P. Di Porto, E. J. Sharp, and R. R. Neurgaonkar, Phys. Rev. Lett. 71, 553 (1993).
[CrossRef]

Drummond, P.

B. Malomed, P. Drummond, H. He, A. Berntson, D. Anderson, and M. Lisak, Phys. Rev. E 56, 4725 (1997).
[CrossRef]

Drummond, P. D.

Duree, G. C.

G. C. Duree, J. L. Shultz, G. J. Salamo, M. Segev, A. Yariv, B. Crosignani, P. Di Porto, E. J. Sharp, and R. R. Neurgaonkar, Phys. Rev. Lett. 71, 553 (1993).
[CrossRef]

Edmundson, D. E.

Enns, R. H.

Etrich, C.

C. Etrich, U. Peschel, F. Lederer, and B. Malomed, Phys. Rev. A 52, R3444 (1995).
[CrossRef]

Fejer, M. M.

Ferro, P.

Fuerst, R. A.

Gorshkov, K. A.

K. A. Gorshkov and L. A. Ostrovsky, Physica D 3, 428 (1981).
[CrossRef]

Haelterman, M.

Hagan, D. J.

G. I. Stegeman, D. J. Hagan, and L. Torner, Opt. Quantum Electron. 28, 1691 (1996).
[CrossRef]

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. Van Stryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, Phys. Rev. Lett. 74, 5036 (1995).
[CrossRef] [PubMed]

He, H.

B. Malomed, P. Drummond, H. He, A. Berntson, D. Anderson, and M. Lisak, Phys. Rev. E 56, 4725 (1997).
[CrossRef]

Karpman, V. I.

V. I. Karpman and V. V. Solov’ev, Physica D 3, 487 (1981).
[CrossRef]

Kermene, V.

Kivshar, Yu. S.

V. V. Steblina, Yu. S. Kivshar, and A. V. Buryak, Opt. Lett. 23, 156 (1998).
[CrossRef]

A. V. Buryak, Yu. S. Kivshar, and S. Trillo, J. Opt. Soc. Am. B 14, 3110 (1997).
[CrossRef]

A. V. Buryak and Yu. S. Kivshar, Phys. Rev. Lett. 78, 3286 (1997).
[CrossRef]

A. V. Buryak, Yu. S. Kivshar, and S. Trillo, Phys. Rev. Lett. 77, 5210 (1996).
[CrossRef] [PubMed]

D. E. Pelinovsky, A. V. Buryak, and Yu. S. Kivshar, Phys. Rev. Lett. 75, 591 (1995).
[CrossRef] [PubMed]

A. V. Buryak, Yu. S. Kivshar, and V. V. Steblina, Phys. Rev. A 52, 1670 (1995).
[CrossRef] [PubMed]

V. V. Steblina, Yu. S. Kivshar, M. Lisak, and B. A. Malomed, Opt. Commun. 118, 345 (1995).
[CrossRef]

A. V. Buryak and Yu. S. Kivshar, Phys. Lett. A 197, 407 (1995).
[CrossRef]

Lederer, F.

C. Etrich, U. Peschel, F. Lederer, and B. Malomed, Phys. Rev. A 52, R3444 (1995).
[CrossRef]

Leo, G.

G. Leo, G. Assanto, and W. E. Torruellas, Opt. Commun. 134, 223 (1997).
[CrossRef]

G. Leo, G. Assanto, and W. E. Torruellas, Opt. Lett. 22, 7 (1997).
[CrossRef] [PubMed]

Lisak, M.

B. Malomed, P. Drummond, H. He, A. Berntson, D. Anderson, and M. Lisak, Phys. Rev. E 56, 4725 (1997).
[CrossRef]

V. V. Steblina, Yu. S. Kivshar, M. Lisak, and B. A. Malomed, Opt. Commun. 118, 345 (1995).
[CrossRef]

Luther-Davies, B.

V. Tikhonenko, J. Christou, and B. Luther-Davies, Phys. Rev. Lett. 76, 2698 (1996).
[CrossRef] [PubMed]

V. Tikhonenko, J. Christou, and B. Luther-Davies, J. Opt. Soc. Am. B 12, 2046 (1995).
[CrossRef]

Malomed, B.

B. Malomed, P. Drummond, H. He, A. Berntson, D. Anderson, and M. Lisak, Phys. Rev. E 56, 4725 (1997).
[CrossRef]

C. Etrich, U. Peschel, F. Lederer, and B. Malomed, Phys. Rev. A 52, R3444 (1995).
[CrossRef]

Malomed, B. A.

V. V. Steblina, Yu. S. Kivshar, M. Lisak, and B. A. Malomed, Opt. Commun. 118, 345 (1995).
[CrossRef]

Menyuk, C. R.

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. Van Stryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, Phys. Rev. Lett. 74, 5036 (1995).
[CrossRef] [PubMed]

C. R. Menyuk, R. Schiek, and L. Torner, J. Opt. Soc. Am. B 11, 2434 (1994).
[CrossRef]

Miller, G. D.

Mitchell, D. J.

A. W. Snyder, A. V. Buryak, and D. J. Mitchell, Opt. Lett. 23, 4 (1998).
[CrossRef]

L. Poladian, A. W. Snyder, and D. J. Mitchell, Opt. Commun. 85, 59 (1991).
[CrossRef]

Neurgaonkar, R. R.

G. C. Duree, J. L. Shultz, G. J. Salamo, M. Segev, A. Yariv, B. Crosignani, P. Di Porto, E. J. Sharp, and R. R. Neurgaonkar, Phys. Rev. Lett. 71, 553 (1993).
[CrossRef]

Ostrovsky, L. A.

K. A. Gorshkov and L. A. Ostrovsky, Physica D 3, 428 (1981).
[CrossRef]

Pelinovsky, D. E.

D. E. Pelinovsky, A. V. Buryak, and Yu. S. Kivshar, Phys. Rev. Lett. 75, 591 (1995).
[CrossRef] [PubMed]

Peschel, U.

C. Etrich, U. Peschel, F. Lederer, and B. Malomed, Phys. Rev. A 52, R3444 (1995).
[CrossRef]

Poladian, L.

L. Poladian, A. W. Snyder, and D. J. Mitchell, Opt. Commun. 85, 59 (1991).
[CrossRef]

Salamo, G. J.

G. C. Duree, J. L. Shultz, G. J. Salamo, M. Segev, A. Yariv, B. Crosignani, P. Di Porto, E. J. Sharp, and R. R. Neurgaonkar, Phys. Rev. Lett. 71, 553 (1993).
[CrossRef]

Schiek, R.

Segev, M.

M. Segev and G. I. Stegeman, Phys. Today 51, 42 (1998), and references therein.
[CrossRef]

G. C. Duree, J. L. Shultz, G. J. Salamo, M. Segev, A. Yariv, B. Crosignani, P. Di Porto, E. J. Sharp, and R. R. Neurgaonkar, Phys. Rev. Lett. 71, 553 (1993).
[CrossRef]

Sharp, E. J.

G. C. Duree, J. L. Shultz, G. J. Salamo, M. Segev, A. Yariv, B. Crosignani, P. Di Porto, E. J. Sharp, and R. R. Neurgaonkar, Phys. Rev. Lett. 71, 553 (1993).
[CrossRef]

Shultz, J. L.

G. C. Duree, J. L. Shultz, G. J. Salamo, M. Segev, A. Yariv, B. Crosignani, P. Di Porto, E. J. Sharp, and R. R. Neurgaonkar, Phys. Rev. Lett. 71, 553 (1993).
[CrossRef]

Snyder, A. W.

A. W. Snyder, A. V. Buryak, and D. J. Mitchell, Opt. Lett. 23, 4 (1998).
[CrossRef]

L. Poladian, A. W. Snyder, and D. J. Mitchell, Opt. Commun. 85, 59 (1991).
[CrossRef]

Sohler, W.

Solov’ev, V. V.

V. I. Karpman and V. V. Solov’ev, Physica D 3, 487 (1981).
[CrossRef]

Steblina, V. V.

V. V. Steblina, Yu. S. Kivshar, and A. V. Buryak, Opt. Lett. 23, 156 (1998).
[CrossRef]

A. V. Buryak, Yu. S. Kivshar, and V. V. Steblina, Phys. Rev. A 52, 1670 (1995).
[CrossRef] [PubMed]

V. V. Steblina, Yu. S. Kivshar, M. Lisak, and B. A. Malomed, Opt. Commun. 118, 345 (1995).
[CrossRef]

Stegeman, G. I.

Taha, T. P.

T. P. Taha and M. J. Ablowitz, J. Comput. Phys. 55, 203 (1984).
[CrossRef]

Tikhonenko, V.

V. Tikhonenko, J. Christou, and B. Luther-Davies, Phys. Rev. Lett. 76, 2698 (1996).
[CrossRef] [PubMed]

V. Tikhonenko, J. Christou, and B. Luther-Davies, J. Opt. Soc. Am. B 12, 2046 (1995).
[CrossRef]

Torner, L.

G. I. Stegeman, D. J. Hagan, and L. Torner, Opt. Quantum Electron. 28, 1691 (1996).
[CrossRef]

D. M. Baboiu, G. I. Stegeman, and L. Torner, Opt. Lett. 20, 2282 (1995).
[CrossRef]

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. Van Stryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, Phys. Rev. Lett. 74, 5036 (1995).
[CrossRef] [PubMed]

C. R. Menyuk, R. Schiek, and L. Torner, J. Opt. Soc. Am. B 11, 2434 (1994).
[CrossRef]

Torruellas, W. E.

G. Leo, G. Assanto, and W. E. Torruellas, Opt. Commun. 134, 223 (1997).
[CrossRef]

G. Leo, G. Assanto, and W. E. Torruellas, Opt. Lett. 22, 7 (1997).
[CrossRef] [PubMed]

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. Van Stryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, Phys. Rev. Lett. 74, 5036 (1995).
[CrossRef] [PubMed]

Trillo, S.

Tulloch, W. M.

Van Stryland, E. W.

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. Van Stryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, Phys. Rev. Lett. 74, 5036 (1995).
[CrossRef] [PubMed]

Wang, Z.

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. Van Stryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, Phys. Rev. Lett. 74, 5036 (1995).
[CrossRef] [PubMed]

Weise, D. R.

Werner, M. J.

Yariv, A.

G. C. Duree, J. L. Shultz, G. J. Salamo, M. Segev, A. Yariv, B. Crosignani, P. Di Porto, E. J. Sharp, and R. R. Neurgaonkar, Phys. Rev. Lett. 71, 553 (1993).
[CrossRef]

J. Comput. Phys. (1)

T. P. Taha and M. J. Ablowitz, J. Comput. Phys. 55, 203 (1984).
[CrossRef]

J. Opt. Soc. Am. B (5)

Opt. Commun. (3)

L. Poladian, A. W. Snyder, and D. J. Mitchell, Opt. Commun. 85, 59 (1991).
[CrossRef]

G. Leo, G. Assanto, and W. E. Torruellas, Opt. Commun. 134, 223 (1997).
[CrossRef]

V. V. Steblina, Yu. S. Kivshar, M. Lisak, and B. A. Malomed, Opt. Commun. 118, 345 (1995).
[CrossRef]

Opt. Lett. (11)

Opt. Quantum Electron. (1)

G. I. Stegeman, D. J. Hagan, and L. Torner, Opt. Quantum Electron. 28, 1691 (1996).
[CrossRef]

Phys. Lett. A (1)

A. V. Buryak and Yu. S. Kivshar, Phys. Lett. A 197, 407 (1995).
[CrossRef]

Phys. Rev. A (3)

C. Etrich, U. Peschel, F. Lederer, and B. Malomed, Phys. Rev. A 52, R3444 (1995).
[CrossRef]

A. V. Buryak, Yu. S. Kivshar, and V. V. Steblina, Phys. Rev. A 52, 1670 (1995).
[CrossRef] [PubMed]

D. E. Edmundson and R. H. Enns, Phys. Rev. A 51, 2491 (1995).
[CrossRef] [PubMed]

Phys. Rev. E (1)

B. Malomed, P. Drummond, H. He, A. Berntson, D. Anderson, and M. Lisak, Phys. Rev. E 56, 4725 (1997).
[CrossRef]

Phys. Rev. Lett. (6)

A. V. Buryak, Yu. S. Kivshar, and S. Trillo, Phys. Rev. Lett. 77, 5210 (1996).
[CrossRef] [PubMed]

A. V. Buryak and Yu. S. Kivshar, Phys. Rev. Lett. 78, 3286 (1997).
[CrossRef]

V. Tikhonenko, J. Christou, and B. Luther-Davies, Phys. Rev. Lett. 76, 2698 (1996).
[CrossRef] [PubMed]

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. Van Stryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, Phys. Rev. Lett. 74, 5036 (1995).
[CrossRef] [PubMed]

G. C. Duree, J. L. Shultz, G. J. Salamo, M. Segev, A. Yariv, B. Crosignani, P. Di Porto, E. J. Sharp, and R. R. Neurgaonkar, Phys. Rev. Lett. 71, 553 (1993).
[CrossRef]

D. E. Pelinovsky, A. V. Buryak, and Yu. S. Kivshar, Phys. Rev. Lett. 75, 591 (1995).
[CrossRef] [PubMed]

Phys. Today (1)

M. Segev and G. I. Stegeman, Phys. Today 51, 42 (1998), and references therein.
[CrossRef]

Physica D (2)

K. A. Gorshkov and L. A. Ostrovsky, Physica D 3, 428 (1981).
[CrossRef]

V. I. Karpman and V. V. Solov’ev, Physica D 3, 487 (1981).
[CrossRef]

Other (4)

M. Shih, M. Segev, and G. Salamo, Phys. Rev. Lett. 78, 2551 (1997); A. V. Buryak, Yu. S. Kivshar, M. Shih, and M. Segev, Phys. Rev. Lett. (to be published).
[CrossRef]

A. V. Buryak, “Solitons due to second harmonic generation,” Ph.D. dissertation (Australian National University, Canberra, 1996).

C. Etrich, U. Peschel, F. Lederer, B. Malomed, and Yu. S. Kivshar, Phys. Rev. E 54, 4321 (1996); D. E. Pelenovsky, Yu. S. Kivshar, and V. V. Afanasjev, Physica D 116, 121 (1998); Yu. S. Kivshar, D. E. Pelenovsky, T. Cretegny, and M. Peyrard, Phys. Rev. Lett. PRLTAO 80, 5032 (1998).
[CrossRef]

H. Goldstein, Classical Mechanics (Addison-Wesley, Reading, Mass., 1980), Chap. 3.

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Figures (7)

Fig. 1
Fig. 1

Qualitative sketch of the effective interaction potential Ueff(R, ϕ) for (a) out-of-phase interaction (ϕ=π) and (b) in-phase interaction (ϕ=0).

Fig. 2
Fig. 2

Sequences of soliton positions in the (X, Y) plane shown at different propagation distances for (a) soliton scattering (ϕ=π, S=3.6), (b) soliton reflection by spiraling (ϕ=0, S=11.6), and (c) soliton fusion (ϕ=0, S=11.0). Results were obtained for β=0.5, Δ=1.0, and C0=0.2.

Fig. 3
Fig. 3

Three-dimensional view of soliton interactions when the parameters are (a) the same as in Fig. 2(b) and (b) the same as in Fig. 2(c).

Fig. 4
Fig. 4

Soliton scattering angle θ versus impact parameter S [which is defined as the minimum distance between the trajectories of noninteracting solitons; also see Fig. 2(a)] for out-of-phase collisions (ϕ=π). Solid curve, analytical results; filled circles, results of numerical simulations. Results were obtained for β=0.5, Δ=1.0, and C0=0.2.

Fig. 5
Fig. 5

Soliton scattering angle θ versus initial relative phase ϕ0 [see the definitions in Fig. 2(a)]. Solid curve, analytical results; filled circles, results of numerical simulations. Results were obtained for β=0.5, Δ=1.0, S=11.6, and C0=0.2.

Fig. 6
Fig. 6

Sequences of soliton positions in the (X, Y) plane shown at different propagation distances for soliton scattering (ϕ=π/8, S=3.2) with significant power exchange. Results were obtained for β=0.5, Δ=1.0, and C0=0.2.

Fig. 7
Fig. 7

Normalized exchanged power between the solitons ΔQ/(2Q) versus initial relative phase ϕ0. Solid curve, analytical results; filled circles, results of numerical simulations. Results were obtained for S=3.6, β=0.5, Δ=1.0, and C0=0.2.

Equations (105)

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2ik1 E1z˜+2E1+8πω12c2χ(2)E2E1* exp(-iδkz˜)=0,
2ik2 E2z˜+2E2+16πω12c2χ(2)E12 exp(iδkz˜)=0,
E1=vc216πω12χ(2)R02,
E2=wc28πω12χ(2)R02exp(iδkRdz)
i vz+2vx2+2vy2+wv*=0,
iσ wz+2wx2+2wy2-σΔw+v22=0,
Q=--(|v|2+2σ|w|2)dxdy,
Px=i2--v* vx-v v*x+σw* wx-w w*xdxdy,
Py=i2--v* vy-v v*y+σw* wy-w w*ydxdy.
v(x, y, z)=V(x-Cxz, y-Cyz, z)exp(iβz),
w(x, y, z)=W(x-Cxz, y-Cyz, z)exp(2iβz)
2Vx2+2Vy2-iCx Vx-iCy Vy-βV+WV*=0,
2Wx2+2Wy2-iσCx Wx-iσCy Wy-σ(2β+Δ)W
+V22=0.
Vs(β, Cx, Cy; x, y)
=Vsβ-Cx24-Cy24, 0, 0; x , yexp[i(Cxx+Cyy)/2],
Ws(β, Cx, Cy; x, y)
=Wsβ-Cx24-Cy24, 0, 0; x , yexp[i(Cxx+Cyy)].
Qs(β, Cx, Cy)=Qsβ-Cx24-Cy24, 0, 0,
Psx(β, Cx, Cy)=-Cx2Qsβ-Cx24-Cy24, 0, 0,
Psy(β, Cx, Cy)=-Cy2Qsβ-Cx24-Cy24, 0, 0.
Qβ=Q˜β,
PxCx=-Q˜2+Cx24Q˜β,
PyCy=-Q˜2+Cy24Q˜β,
QCx=Pxβ=-Cx2Q˜β,
QCy=Pyβ=-Cy2Q˜β,
PxCy=PyCx=CxCy4Q˜β,
Q˜Q(β-Cx2/4-Cy2/4, 0, 0).
(Q, Px, Py)(β, Cx, Cy)detQβQCxQCyPxβPxCxPxCyPyβPyCxPyCy=0,
Q˜24Q˜β=0.
β-Cx24-Cy24>max0,-Δ2.
Q˜β>0.
-Q˜(1)β(1)ϕ¨(1)+12Uϕ(1)=0,
-Q˜(2)β(2)ϕ¨(2)+12Uϕ(2)=0,
Q˜(1)2x¨(1)+12Ux(1)=0,
Q˜(2)2x¨(2)+12Ux(2)=0,
Q˜(1)2y¨(1)+12Uy(1)=0,
Q˜(2)2y¨(2)+12Uy(2)=0,
Mϕϕ¨+Uϕ=0,
MXX¨+UX=0,
MYY¨+UY=0,
Mϕϕ¨+Ueffϕ=0,MRR¨+UeffR=0,
L=1/2MRR˙2+1/2Mϕϕ˙2-Ueff(R, ϕ).
MRπ0[|Vs(r)|2+2σ|Ws(r)|2]rdr,
Mϕ-2MR/β,
Ueff(R, ϕ)=MRS2C022R2+U1(R)cos(ϕ)+U2(R)cos(2ϕ),
U1(R)-2--[Ws(2)Vs(2)Vs(1)+Ws(1)Vs(1)Vs(2)]dxdy,
U2(R)---[Vs(2)2Ws(1)+Vs(1)2Ws(2)]dxdy,
U1(R)-K1A2β4 exp(-βR)/R,
U2(R)-K2B2[2(Δ+2β)]1/4×exp[-[2(Δ+2β)]1/2R]/R,
U1(R)=-2330.0 exp(-0.7071 R)/R,
U2(R)=-8216.0 exp(-1.4142 R),
Q(1)=Q-12Qβ0Δβ,Q(2)=Q+12Qβ0Δβ,
Mϕ=-Qβ0=const.,
MX=MY=Q/2-Qβ02(Δβ)2/8.
ΔQ=Qβ0Δβ,
Hin=1/2MRR˙2+1/2Mϕϕ˙2.
Imin=Gk1ω15[χ(2)]2R04,
vv(1)+v(2)=V(1)[x-x(1), y-y(1), Z]exp[iϕ(1)]+V(2)[x-x(2), y-y(2), Z]exp[iϕ(2)],
ww(1)+w(2)=W(1)[x-x(1), y-y(1), Z]exp[2iϕ(1)]+W(2)[x-x(2), y-y(2), Z]exp[2iϕ(2)],
ϕ(j)(z)=0zβ(j)(Z)dz+ϕ0(j),
x(j)(z)=0zCx(j)(Z)dz+x0(j),
y(j)(z)=0zCy(j)(Z)dz+y0(j),
V(j)(x, y, z)=V0(j)[β(j)(Z), Cx(j)(Z), Cy(j)(Z); x, y]+n=1+nVn(j)(x, y, z),
W(j)(x, y, z)=W0(j)[β(j)(Z), Cx(j)(Z), Cy(j)(Z); x, y]+n=1+nWn(j)(x, y, z).
2V0(j)x2+2V0(j)y2-iCx(j) V0(j)x-iCy(j) V0(j)y-β(j)V0(j)
+W0(j)V0(j)*=0,
2W0(j)x2+2W0(j)y2-iσCx(j) W0(j)x-iσCy(j) W0(j)y
-σ[2β(j)+Δ]W0(j)+V0(j)22=0,
V0(j)=Vs[β(j),Cx(j),Cy(j); x-x(j), y-y(j)],
W0(j)=Ws[β(j),Cx(j),Cy(j); x-x(j), y-y(j)].
LˆV1(1)W1(1)V1(1)*W1(1)*=iβ˙(1)-V0β(1)(1)-σW0β(1)(1)V0β(1)(1)*σW0β(1)(1)*+iC˙x(1)-V0Cx(1)(1)-σW0Cx(1)(1)V0Cx(1)(1)*σW0Cx(1)(1)*+iC˙y(1)-V0Cy(1)(1)-σW0Cy(1)(1)V0Cy(1)(1)*σW0Cy(1)(1)*-1W0(2)V0(2)* exp{i[ϕ(2)-ϕ(1)]}1/2V0(2)2 exp{2i[ϕ(2)-ϕ(1)]}W0(2)*V0(2) exp{-i[ϕ(2)-ϕ(1)]}1/2V0(2)*2 exp{-2i[ϕ(2)-ϕ(1)]},
Lˆ=AV0(1)*W0(1)0V0(1)B00W0(1)*0CV0(1)00V0(1)*D,
A2x2+2y2-iCx(1) x-iCy(1) y-β(1),
B2x2+2y2-iCx(1) x-iCy(1) y-σ[2β(1)+Δ],
C2x2+2y2+iσCx(1) x+iσCy(1) y-β(1),
D2x2+2y2+iσCx(1) x+iσCy(1) y-σ[2β(1)+Δ].
Vh1(1)=iV0(1),Wh1(1)=iσW0(1),
Vh1(1)*=-iV0(1)*,Wh1(1)*=-iσW0(1)*;
Vh2(1)=V0x(1),Wh2(1)=W0x(1),
Vh2(1)*=V0x(1)*,Wh2(1)*=W0x(1)*;
Vh3(1)=V0y(1),Wh3(1)=W0y(1),
Vh3(1)*=V0y(1)*,Wh3(1)*=W0y(1)*;
--[Vhq(1)*Vrhs(1)+Whq(1)*Wrhs(1)+Vhq(1)Vrhs(1)*
+Whq(1)Wrhs(1)*]dxdy=0,
β˙(1) Q(1)β(1)+C˙x(1) Px(1)β(1)+C˙y(1) Py(1)β(1)-U(1)ϕ(1)=0,
β˙(1) Q(1)Cx(1)+C˙x(1) Px(1)Cx(1)+C˙y(1) Py(1)Cx(1)-U(1)x(1)=0,
β˙(1) Q(1)Cy(1)+C˙x(1) Px(1)Cy(1)+C˙y(1) Py(1)Cy(1)-U(1)y(1)=0,
U(1)=-Re --(2W0(2)V0(2)*V0(1)*×exp{i[ϕ(2)-ϕ(1)]}+V0(2)2W0(1)*×exp{2i[ϕ(2)-ϕ(1)]})dxdy.
U=U1 cos(ϕ)+U2 cos(2ϕ),
ϕ[Cx(1)x(1)-Cx(2)x(2)+Cy(1)y(1)-Cy(2)y(2)]/2+ϕ(2)-ϕ(1),
U1-2--[W˜0(2)V˜0(2)V˜0(1)+W˜0(1)V˜0(1)V˜0(2)]dxdy,
U2---[V˜0(2)2W˜0(1)+V˜0(1)2W˜0(2)]dxdy.
V˜0(j)[β(j),Cx(j), Cy(j); x-x(j), y-y(j)]V0(j)β(j)-Cx(j)24-Cy(j)24, 0, 0; x-x(j), y-y(j),
W˜0(j)[β(j), Cx(j), Cy(j); x-x(j), y-y(j)]W0(j)β(j)-Cx(j)24-Cy(j)24, 0, 0; x-x(j), y-y(j),
ϕ¨(1) Q(1)β(1)+x¨(1) Px(1)β(1)+y¨(1) Py(1)β(1)-12Uϕ(1)=0,
ϕ¨(1) Q(1)Cx(1)+x¨(1) Px(1)Cx(1)+y¨(1) Py(1)Cx(1)-12Ux(1)=0,
ϕ¨(1) Q(1)Cy(1)+x¨(1) Px(1)Cy(1)+y¨(1) Py(1)Cy(1)-12Uy(1)=0,
ϕ¨(2) Q(2)β(2)+x¨(2) Px(2)β(2)+y¨(2) Py(2)β(2)-12Uϕ(2)=0,
ϕ¨(2) Q(2)Cx(2)+x¨(2) Px(2)Cx(2)+y¨(2) Py(2)Cx(2)-12Ux(2)=0,
ϕ¨(2) Q(2)Cy(2)+x¨(2) Px(2)Cy(2)+y¨(2) Py(2)Cy(2)-12Uy(2)=0.
ϕ¨(1) Q(1)β(1)+x¨(1) Px(1)β(1)-12Uϕ(1)=0,
ϕ¨(1) Q(1)Cx(1)+x¨(1) Px(1)Cx(1)-12Ux(1)=0,
ϕ¨(2) Q(2)β(2)+x¨(2) Px(2)β(2)-12Uϕ(2)=0,
ϕ¨(2) Q(2)Cx(2)+x¨(2) Px(2)Cx(2)-12Ux(2)=0,

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