Abstract

We present a new species of spatiotemporal solitons (STSs, alias light bullets) that may be formed through the interaction of two waves with different polarizations in a birefringent Kerr medium near the point at which the second-order group-velocity dispersion (GVD) vanishes in the presence of a uniform cw background in one component. The analysis is based on an asymptotic reduction of the two cross-phase, modulation-coupled (2+1)-dimensional nonlinear Schrödinger equations, incorporating the third-order GVD to the Davey–Stewartson (DS) systems of type I or II for the self-defocusing and focusing Kerr nonlinearity, respectively. The STSs are then predicted following the pattern of the dromion solutions of the DS-I system and lump solitons of the DS-II system. Strictly speaking, the solitons are unstable, but they are shown to be stable objects on experimentally relevant scales of the propagation distance.

© 2006 Optical Society of America

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  1. Y. Silberberg, "Collapse of optical pulses," Opt. Lett. 15, 1282-1284 (1990).
  2. R. H. Enns, D. E. Edmundson, S. S. Rangnekar, and A. E. Kaplan, "Optical switching between bistable soliton states--a theoretical review," Opt. Quantum Electron. 24, S1295-S1314 (1992).
    [CrossRef]
  3. R. McLeod, K. Wagner, and S. Blair, "(3+1)-dimensional optical soliton dragging logic," Phys. Rev. A 52, 3254-3278 (1995).
    [CrossRef]
  4. D. E. Edmundson and R. H. Enns, "Particlelike nature of colliding three-dimensional optical solitons," Phys. Rev. A 51, 2491-2498 (1995).
    [CrossRef]
  5. D. E. Edmundson, "Unstable higher modes of a three-dimensional nonlinear Schrodinger equation," Phys. Rev. E 55, 7636-7644 (1997).
    [CrossRef]
  6. K. Hayata and M. Koshiba, "Multidimensional solitons in quadratic nonlinear media," Phys. Rev. Lett. 71, 3275-3278 (1993).
    [CrossRef]
  7. B. A. Malomed, P. Drummond, H. He, A. Berntson, D. Anderson, and M. Lisak, "Spatiotemporal solitons in multidimensional optical media with a quadratic nonlinearity," Phys. Rev. E 56, 4725-4735 (1997).
    [CrossRef]
  8. D. V. Skryabin and W. J. Firth, "Generation and stability of optical bullets in quadratic nonlinear media," Opt. Commun. 148, 79-84 (1998).
    [CrossRef]
  9. D. Mihalache, D. Mazilu, B. A. Malomed, and L. Torner, "Asymmetric spatio-temporal optical solitons in media with quadratic nonlinearity," Opt. Commun. 152, 365-370 (1998).
    [CrossRef]
  10. I. N. Towers, B. A. Malomed, and F. W. Wise, "Light bullets in quadratic media with normal dispersion at the second harmonic," Phys. Rev. Lett. 90, 123902 (2003).
    [CrossRef]
  11. H. Leblond, "Bidimensional optical solitons in a quadratic medium," J. Phys. A 31, 5129-5143 (1998).
    [CrossRef]
  12. M. L. Quiroga-Teixeiro, A. Berntson, and H. Michinel, "Internal dynamics of nonlinear beams in their ground states: short- and long-lived excitation," J. Opt. Soc. Am. B 16, 1697-1704 (1999).
  13. D. Mihalache, D. Mazilu, L.-C. Crasovan, I. Towers, A. V. Buryak, B. A. Malomed, L. Torner, J. P. Torres, and F. Lederer, "Stable spinning optical solitons in three dimensions," Phys. Rev. Lett. 88, 073902 (2002).
    [CrossRef]
  14. D. Mihalache, D. Mazilu, I. Towers, B. A. Malomed, and F. Lederer, "Stable spatiotemporal spinning solitons in a bimodal cubic-quintic medium," Phys. Rev. E 67, 056608 (2003).
    [CrossRef]
  15. A. B. Blagoeva, S. G. Dinev, A. A. Dreischuh, and A. Naidenov, "Light bullets formation in a bulk media," IEEE J. Quantum Electron. 27, 2060-2065 (1991).
    [CrossRef]
  16. Y. Chen and J. Atai, "Dark optical bullets in light self-trapping," Opt. Lett. 20, 133-135 (1995).
  17. D. J. Frantzeskakis, K. Hizanidis, B. A. Malomed, and C. Polymilis, "Stable anti-dark light bullets supported by the third-order dispersion," Phys. Lett. A 248, 203-207 (1998).
    [CrossRef]
  18. H. E. Nistazakis, D. J. Frantzeskakis, and B. A. Malomed, "Collisions between spatiotemporal solitons of different dimensionality in a planar waveguide," Phys. Rev. E 64, 026604 (2001).
    [CrossRef]
  19. M. Blaauboer, B. A. Malomed, and G. Kurizki, "Spatiotemporally localized multidimensional solitons in self-induced transparency media," Phys. Rev. Lett. 84, 1906-1909 (2000).
    [CrossRef]
  20. M. Blaauboer, G. Kurizki, and B. A. Malomed, "Spatiotemporally localized solitons in resonantly absorbing Bragg reflectors," Phys. Rev. E 62, R57-R59 (2000).
    [CrossRef]
  21. I. Towers and B. A. Malomed, "Stable (2+1)-dimensional solitons in a layered medium with sign-alternating Kerr nonlinearity," J. Opt. Soc. Am. B 19, 537-543 (2002).
  22. M. Matuszewski, M. Trippenbach, B. A. Malomed, E. Infeld, and M. Skorupski, "Two-dimensional dispersion-managed light bullets in Kerr media," Phys. Rev. E 70, 016603 (2004).
    [CrossRef]
  23. X. Liu, L. J. Qian, and F. W. Wise, "Generation of optical spatiotemporal solitons," Phys. Rev. Lett. 82, 4631-4634 (1999).
    [CrossRef]
  24. X. Liu, K. Beckwitt, and F. Wise, "Two-dimensional optical spatiotemporal solitons in quadratic media," Phys. Rev. E 62, 1328-1340 (2000).
    [CrossRef]
  25. H. S. Eisenberg, R. Morandotti, Y. Silberberg, S. Bar-Ad, D. Ross, and J. S. Aitchison, "Kerr spatiotemporal self-focusing in a planar glass waveguide," Phys. Rev. Lett. 87, 043902 (2001).
    [CrossRef]
  26. G. Gottwald, R. Grimshaw, and B. A. Malomed, "Stable two-dimensional parametric solitons in fluid systems," Phys. Lett. A 248, 208-218 (1998).
    [CrossRef]
  27. H. Saito and M. Ueda, "Dynamically stabilized bright solitons in a two-dimensional Bose-Einstein condensate," Phys. Rev. Lett. 90, 040403 (2003).
    [CrossRef]
  28. F. Kh. Abdullaev, J. G. Caputo, R. A. Kraenkel, and B. A. Malomed, "Controlling collapse in Bose-Einstein condensates by temporal modulation of the scattering length," Phys. Rev. A 67, 013605 (2003).
    [CrossRef]
  29. G. D. Montesinos, V. M. Pérez-García, and P. J. Torres, "Stabilization of solitons of the multidimensional nonlinear Schrodinger equation: matter-wave breathers," Physica D 191, 193-210 (2004).
    [CrossRef]
  30. B. B. Baizakov, B. A. Malomed, and M. Salerno, "Multidimensional solitons in a low-dimensional periodic potential," Phys. Rev. A 70, 053613 (2004).
    [CrossRef]
  31. D. Mihalache, D. Mazilu, F. Lederer, Y. V. Kartashov, L. C. Crasovan, and L. Torner, "Stable three-dimensional spatiotemporal solitons in a two-dimensional photonic lattice," Phys. Rev. E 70, 055603 (2004).
    [CrossRef]
  32. P. G. Kevrekidis, G. Theocharis, D. J. Frantzeskakis, and B. A. Malomed, "Feshbach resonance management for Bose-Einstein condensates," Phys. Rev. Lett. 90, 230401 (2003).
    [CrossRef]
  33. G. D. Montesinos, V. M. Pérez-García, and H. Michinel, Stabilized two-dimensional vector solitons," Phys. Rev. Lett. 92, 133901 (2004).
    [CrossRef]
  34. M. Trippenbach, M. Matuszewski, and B. A. Malomed, Stabilization of three-dimensional matter-waves solitons in an optical lattice," Europhys. Lett. 70, 8-14 (2005).
    [CrossRef]
  35. M. Matuszewski, E. Infeld, B. A. Malomed, and M. Trippenbach, Fully three dimensional breather solitons can be created using Feshbach resonances," Phys. Rev. Lett. 95, 050403 (2005).
    [CrossRef]
  36. B. A. Malomed, D. Mihalache, F. Wise, and L. Torner, Spatiotemporal optical solitons," J. Opt. 7, R53-R72 (2005).
  37. Yu. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic, 2003).
  38. B. A. Malomed, Soliton Management in Periodic Systems (Springer, 2006).
  39. G. P. Agrawal, Nonlinear Fiber Optics (Academic,1995).
  40. A. Davey and K. Stewartson, "3-dimensional packets of surface-waves," Proc. R. Soc. London, Ser. A 338, 101-110 (1974).
  41. M. J. Ablowitz and P. A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering (Cambridge U. Press, 1991).
  42. M. Boiti, J. J. Leon, L. Martina, and F. Pempinelli, "Scattering of localized solitons in the plane," Phys. Lett. A 132, 432-439 (1988).
    [CrossRef]
  43. A. S. Fokas and P. M. Santini, "Coherent structures in multidimensions," Phys. Rev. Lett. 63, 1329-1333 (1989).
    [CrossRef]
  44. A. S. Fokas and P. M. Santini, "Dromions and a boundary-value problem for the Davey-Stewartson-1 equation," Physica D 44, 99-130 (1990).
    [CrossRef]
  45. V. A. Arkadiev, A. K. Pogrebkov, and M. C. Polivanov, "Inverse scattering transform method and soliton-solutions for Davey-Stewartson-II equation," Physica D 36, 189-197 (1989).
    [CrossRef]
  46. P. Di Trapani, W. Chinaglia, S. Minardi, A. Piskarkas, and G. Valiulis, "Observation of quadratic optical vortex solitons," Phys. Rev. Lett. 84, 3843-3846 (2000).
    [CrossRef]
  47. K. Beckwitt, Y. F. Chen, F. W. Wise, and B. A. Malomed, "Temporal solitons in quadratic nonlinear media with opposite group-velocity dispersions at the fundamental and second harmonics," Phys. Rev. E 68, 057601 (2003).
    [CrossRef]
  48. V. Djordjevic and L. Redekopp, "2-dimensional packets of capillary-gravity waves," J. Fluid Mech. 79, 703-714 (1977).
    [CrossRef]
  49. Yu. S. Kivshar, "Stable vector solitons composed of bright and dark pulses," Opt. Lett. 17, 1322-1325 (1992).
  50. R. E. Kates and D. J. Kaup, "2-dimensional nonlinear Schrodinger-equations and their properties," Physica D 75, 458-470 (1994).
  51. D. J. Benney and G. J. Roskes, "Wave instabilities," Stud. Appl. Math. 48, 377-385 (1969).
  52. M. J. Ablowitz, G. Biondini, and S. Blair, "Nonlinear Schrodinger equations with mean terms in nonresonant multidimensional quadratic materials," Phys. Rev. E 63, 046605 (2001).
    [CrossRef]
  53. G. Huang, L. Deng, and C. Hang, "Davey-Stewartson description of two-dimensional nonlinear excitations in Bose-Einstein condensates," Phys. Rev. E 72, 036621 (2005).
    [CrossRef]
  54. D. B. Khismatullin and I. Sh. Akhatov, "Sound-ultrasound interaction in bubbly fluids: theory and possible applications," Phys. Fluids 13, 3582-3598 (2001).
    [CrossRef]
  55. H. Leblond, "Electromagnetic waves in ferromagnets: a Davey-Stewartson-type model," J. Phys. A 32, 7907-7932 (1999).
    [CrossRef]
  56. G. Huang, V. V. Konotop, H. W. Tam, and B. Hu, "Nonlinear modulation of multidimensional lattice waves," Phys. Rev. E 64, 056619 (2001).
    [CrossRef]
  57. D. Anker and N. C. Freeman, "Soliton solutions of Davey-Stewartson equation for long waves," Proc. R. Soc. London, Ser. A 360, 529-540 (1978).
  58. J. Satsuma and M. J. Ablowitz, "2-dimensional lumps in non-linear dispersive systems," J. Math. Phys. 20, 1496-1503 (1979).
    [CrossRef]
  59. A. Nakamura, "Explode-decay mode lump solitons of a two-dimensional non-linear Schrodinger-equation," Phys. Lett. A 88, 55-56 (1982).
    [CrossRef]

2005 (4)

M. Trippenbach, M. Matuszewski, and B. A. Malomed, Stabilization of three-dimensional matter-waves solitons in an optical lattice," Europhys. Lett. 70, 8-14 (2005).
[CrossRef]

M. Matuszewski, E. Infeld, B. A. Malomed, and M. Trippenbach, Fully three dimensional breather solitons can be created using Feshbach resonances," Phys. Rev. Lett. 95, 050403 (2005).
[CrossRef]

B. A. Malomed, D. Mihalache, F. Wise, and L. Torner, Spatiotemporal optical solitons," J. Opt. 7, R53-R72 (2005).

G. Huang, L. Deng, and C. Hang, "Davey-Stewartson description of two-dimensional nonlinear excitations in Bose-Einstein condensates," Phys. Rev. E 72, 036621 (2005).
[CrossRef]

2004 (5)

M. Matuszewski, M. Trippenbach, B. A. Malomed, E. Infeld, and M. Skorupski, "Two-dimensional dispersion-managed light bullets in Kerr media," Phys. Rev. E 70, 016603 (2004).
[CrossRef]

G. D. Montesinos, V. M. Pérez-García, and P. J. Torres, "Stabilization of solitons of the multidimensional nonlinear Schrodinger equation: matter-wave breathers," Physica D 191, 193-210 (2004).
[CrossRef]

B. B. Baizakov, B. A. Malomed, and M. Salerno, "Multidimensional solitons in a low-dimensional periodic potential," Phys. Rev. A 70, 053613 (2004).
[CrossRef]

D. Mihalache, D. Mazilu, F. Lederer, Y. V. Kartashov, L. C. Crasovan, and L. Torner, "Stable three-dimensional spatiotemporal solitons in a two-dimensional photonic lattice," Phys. Rev. E 70, 055603 (2004).
[CrossRef]

G. D. Montesinos, V. M. Pérez-García, and H. Michinel, Stabilized two-dimensional vector solitons," Phys. Rev. Lett. 92, 133901 (2004).
[CrossRef]

2003 (6)

H. Saito and M. Ueda, "Dynamically stabilized bright solitons in a two-dimensional Bose-Einstein condensate," Phys. Rev. Lett. 90, 040403 (2003).
[CrossRef]

F. Kh. Abdullaev, J. G. Caputo, R. A. Kraenkel, and B. A. Malomed, "Controlling collapse in Bose-Einstein condensates by temporal modulation of the scattering length," Phys. Rev. A 67, 013605 (2003).
[CrossRef]

P. G. Kevrekidis, G. Theocharis, D. J. Frantzeskakis, and B. A. Malomed, "Feshbach resonance management for Bose-Einstein condensates," Phys. Rev. Lett. 90, 230401 (2003).
[CrossRef]

I. N. Towers, B. A. Malomed, and F. W. Wise, "Light bullets in quadratic media with normal dispersion at the second harmonic," Phys. Rev. Lett. 90, 123902 (2003).
[CrossRef]

D. Mihalache, D. Mazilu, I. Towers, B. A. Malomed, and F. Lederer, "Stable spatiotemporal spinning solitons in a bimodal cubic-quintic medium," Phys. Rev. E 67, 056608 (2003).
[CrossRef]

K. Beckwitt, Y. F. Chen, F. W. Wise, and B. A. Malomed, "Temporal solitons in quadratic nonlinear media with opposite group-velocity dispersions at the fundamental and second harmonics," Phys. Rev. E 68, 057601 (2003).
[CrossRef]

2002 (2)

D. Mihalache, D. Mazilu, L.-C. Crasovan, I. Towers, A. V. Buryak, B. A. Malomed, L. Torner, J. P. Torres, and F. Lederer, "Stable spinning optical solitons in three dimensions," Phys. Rev. Lett. 88, 073902 (2002).
[CrossRef]

I. Towers and B. A. Malomed, "Stable (2+1)-dimensional solitons in a layered medium with sign-alternating Kerr nonlinearity," J. Opt. Soc. Am. B 19, 537-543 (2002).

2001 (5)

H. E. Nistazakis, D. J. Frantzeskakis, and B. A. Malomed, "Collisions between spatiotemporal solitons of different dimensionality in a planar waveguide," Phys. Rev. E 64, 026604 (2001).
[CrossRef]

H. S. Eisenberg, R. Morandotti, Y. Silberberg, S. Bar-Ad, D. Ross, and J. S. Aitchison, "Kerr spatiotemporal self-focusing in a planar glass waveguide," Phys. Rev. Lett. 87, 043902 (2001).
[CrossRef]

D. B. Khismatullin and I. Sh. Akhatov, "Sound-ultrasound interaction in bubbly fluids: theory and possible applications," Phys. Fluids 13, 3582-3598 (2001).
[CrossRef]

G. Huang, V. V. Konotop, H. W. Tam, and B. Hu, "Nonlinear modulation of multidimensional lattice waves," Phys. Rev. E 64, 056619 (2001).
[CrossRef]

M. J. Ablowitz, G. Biondini, and S. Blair, "Nonlinear Schrodinger equations with mean terms in nonresonant multidimensional quadratic materials," Phys. Rev. E 63, 046605 (2001).
[CrossRef]

2000 (4)

P. Di Trapani, W. Chinaglia, S. Minardi, A. Piskarkas, and G. Valiulis, "Observation of quadratic optical vortex solitons," Phys. Rev. Lett. 84, 3843-3846 (2000).
[CrossRef]

M. Blaauboer, B. A. Malomed, and G. Kurizki, "Spatiotemporally localized multidimensional solitons in self-induced transparency media," Phys. Rev. Lett. 84, 1906-1909 (2000).
[CrossRef]

M. Blaauboer, G. Kurizki, and B. A. Malomed, "Spatiotemporally localized solitons in resonantly absorbing Bragg reflectors," Phys. Rev. E 62, R57-R59 (2000).
[CrossRef]

X. Liu, K. Beckwitt, and F. Wise, "Two-dimensional optical spatiotemporal solitons in quadratic media," Phys. Rev. E 62, 1328-1340 (2000).
[CrossRef]

1999 (3)

X. Liu, L. J. Qian, and F. W. Wise, "Generation of optical spatiotemporal solitons," Phys. Rev. Lett. 82, 4631-4634 (1999).
[CrossRef]

M. L. Quiroga-Teixeiro, A. Berntson, and H. Michinel, "Internal dynamics of nonlinear beams in their ground states: short- and long-lived excitation," J. Opt. Soc. Am. B 16, 1697-1704 (1999).

H. Leblond, "Electromagnetic waves in ferromagnets: a Davey-Stewartson-type model," J. Phys. A 32, 7907-7932 (1999).
[CrossRef]

1998 (5)

H. Leblond, "Bidimensional optical solitons in a quadratic medium," J. Phys. A 31, 5129-5143 (1998).
[CrossRef]

D. V. Skryabin and W. J. Firth, "Generation and stability of optical bullets in quadratic nonlinear media," Opt. Commun. 148, 79-84 (1998).
[CrossRef]

D. Mihalache, D. Mazilu, B. A. Malomed, and L. Torner, "Asymmetric spatio-temporal optical solitons in media with quadratic nonlinearity," Opt. Commun. 152, 365-370 (1998).
[CrossRef]

G. Gottwald, R. Grimshaw, and B. A. Malomed, "Stable two-dimensional parametric solitons in fluid systems," Phys. Lett. A 248, 208-218 (1998).
[CrossRef]

D. J. Frantzeskakis, K. Hizanidis, B. A. Malomed, and C. Polymilis, "Stable anti-dark light bullets supported by the third-order dispersion," Phys. Lett. A 248, 203-207 (1998).
[CrossRef]

1997 (2)

D. E. Edmundson, "Unstable higher modes of a three-dimensional nonlinear Schrodinger equation," Phys. Rev. E 55, 7636-7644 (1997).
[CrossRef]

B. A. Malomed, P. Drummond, H. He, A. Berntson, D. Anderson, and M. Lisak, "Spatiotemporal solitons in multidimensional optical media with a quadratic nonlinearity," Phys. Rev. E 56, 4725-4735 (1997).
[CrossRef]

1995 (3)

Y. Chen and J. Atai, "Dark optical bullets in light self-trapping," Opt. Lett. 20, 133-135 (1995).

R. McLeod, K. Wagner, and S. Blair, "(3+1)-dimensional optical soliton dragging logic," Phys. Rev. A 52, 3254-3278 (1995).
[CrossRef]

D. E. Edmundson and R. H. Enns, "Particlelike nature of colliding three-dimensional optical solitons," Phys. Rev. A 51, 2491-2498 (1995).
[CrossRef]

1994 (1)

R. E. Kates and D. J. Kaup, "2-dimensional nonlinear Schrodinger-equations and their properties," Physica D 75, 458-470 (1994).

1993 (1)

K. Hayata and M. Koshiba, "Multidimensional solitons in quadratic nonlinear media," Phys. Rev. Lett. 71, 3275-3278 (1993).
[CrossRef]

1992 (2)

R. H. Enns, D. E. Edmundson, S. S. Rangnekar, and A. E. Kaplan, "Optical switching between bistable soliton states--a theoretical review," Opt. Quantum Electron. 24, S1295-S1314 (1992).
[CrossRef]

Yu. S. Kivshar, "Stable vector solitons composed of bright and dark pulses," Opt. Lett. 17, 1322-1325 (1992).

1991 (1)

A. B. Blagoeva, S. G. Dinev, A. A. Dreischuh, and A. Naidenov, "Light bullets formation in a bulk media," IEEE J. Quantum Electron. 27, 2060-2065 (1991).
[CrossRef]

1990 (2)

Y. Silberberg, "Collapse of optical pulses," Opt. Lett. 15, 1282-1284 (1990).

A. S. Fokas and P. M. Santini, "Dromions and a boundary-value problem for the Davey-Stewartson-1 equation," Physica D 44, 99-130 (1990).
[CrossRef]

1989 (2)

V. A. Arkadiev, A. K. Pogrebkov, and M. C. Polivanov, "Inverse scattering transform method and soliton-solutions for Davey-Stewartson-II equation," Physica D 36, 189-197 (1989).
[CrossRef]

A. S. Fokas and P. M. Santini, "Coherent structures in multidimensions," Phys. Rev. Lett. 63, 1329-1333 (1989).
[CrossRef]

1988 (1)

M. Boiti, J. J. Leon, L. Martina, and F. Pempinelli, "Scattering of localized solitons in the plane," Phys. Lett. A 132, 432-439 (1988).
[CrossRef]

1982 (1)

A. Nakamura, "Explode-decay mode lump solitons of a two-dimensional non-linear Schrodinger-equation," Phys. Lett. A 88, 55-56 (1982).
[CrossRef]

1979 (1)

J. Satsuma and M. J. Ablowitz, "2-dimensional lumps in non-linear dispersive systems," J. Math. Phys. 20, 1496-1503 (1979).
[CrossRef]

1978 (1)

D. Anker and N. C. Freeman, "Soliton solutions of Davey-Stewartson equation for long waves," Proc. R. Soc. London, Ser. A 360, 529-540 (1978).

1977 (1)

V. Djordjevic and L. Redekopp, "2-dimensional packets of capillary-gravity waves," J. Fluid Mech. 79, 703-714 (1977).
[CrossRef]

1974 (1)

A. Davey and K. Stewartson, "3-dimensional packets of surface-waves," Proc. R. Soc. London, Ser. A 338, 101-110 (1974).

1969 (1)

D. J. Benney and G. J. Roskes, "Wave instabilities," Stud. Appl. Math. 48, 377-385 (1969).

Abdullaev, F. Kh.

F. Kh. Abdullaev, J. G. Caputo, R. A. Kraenkel, and B. A. Malomed, "Controlling collapse in Bose-Einstein condensates by temporal modulation of the scattering length," Phys. Rev. A 67, 013605 (2003).
[CrossRef]

Ablowitz, M. J.

M. J. Ablowitz, G. Biondini, and S. Blair, "Nonlinear Schrodinger equations with mean terms in nonresonant multidimensional quadratic materials," Phys. Rev. E 63, 046605 (2001).
[CrossRef]

J. Satsuma and M. J. Ablowitz, "2-dimensional lumps in non-linear dispersive systems," J. Math. Phys. 20, 1496-1503 (1979).
[CrossRef]

M. J. Ablowitz and P. A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering (Cambridge U. Press, 1991).

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics (Academic,1995).

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

Aitchison, J. S.

H. S. Eisenberg, R. Morandotti, Y. Silberberg, S. Bar-Ad, D. Ross, and J. S. Aitchison, "Kerr spatiotemporal self-focusing in a planar glass waveguide," Phys. Rev. Lett. 87, 043902 (2001).
[CrossRef]

Akhatov, I. Sh.

D. B. Khismatullin and I. Sh. Akhatov, "Sound-ultrasound interaction in bubbly fluids: theory and possible applications," Phys. Fluids 13, 3582-3598 (2001).
[CrossRef]

Anderson, D.

B. A. Malomed, P. Drummond, H. He, A. Berntson, D. Anderson, and M. Lisak, "Spatiotemporal solitons in multidimensional optical media with a quadratic nonlinearity," Phys. Rev. E 56, 4725-4735 (1997).
[CrossRef]

Anker, D.

D. Anker and N. C. Freeman, "Soliton solutions of Davey-Stewartson equation for long waves," Proc. R. Soc. London, Ser. A 360, 529-540 (1978).

Arkadiev, V. A.

V. A. Arkadiev, A. K. Pogrebkov, and M. C. Polivanov, "Inverse scattering transform method and soliton-solutions for Davey-Stewartson-II equation," Physica D 36, 189-197 (1989).
[CrossRef]

Atai, J.

Baizakov, B. B.

B. B. Baizakov, B. A. Malomed, and M. Salerno, "Multidimensional solitons in a low-dimensional periodic potential," Phys. Rev. A 70, 053613 (2004).
[CrossRef]

Bar-Ad, S.

H. S. Eisenberg, R. Morandotti, Y. Silberberg, S. Bar-Ad, D. Ross, and J. S. Aitchison, "Kerr spatiotemporal self-focusing in a planar glass waveguide," Phys. Rev. Lett. 87, 043902 (2001).
[CrossRef]

Beckwitt, K.

K. Beckwitt, Y. F. Chen, F. W. Wise, and B. A. Malomed, "Temporal solitons in quadratic nonlinear media with opposite group-velocity dispersions at the fundamental and second harmonics," Phys. Rev. E 68, 057601 (2003).
[CrossRef]

X. Liu, K. Beckwitt, and F. Wise, "Two-dimensional optical spatiotemporal solitons in quadratic media," Phys. Rev. E 62, 1328-1340 (2000).
[CrossRef]

Benney, D. J.

D. J. Benney and G. J. Roskes, "Wave instabilities," Stud. Appl. Math. 48, 377-385 (1969).

Berntson, A.

M. L. Quiroga-Teixeiro, A. Berntson, and H. Michinel, "Internal dynamics of nonlinear beams in their ground states: short- and long-lived excitation," J. Opt. Soc. Am. B 16, 1697-1704 (1999).

B. A. Malomed, P. Drummond, H. He, A. Berntson, D. Anderson, and M. Lisak, "Spatiotemporal solitons in multidimensional optical media with a quadratic nonlinearity," Phys. Rev. E 56, 4725-4735 (1997).
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M. Blaauboer, B. A. Malomed, and G. Kurizki, "Spatiotemporally localized multidimensional solitons in self-induced transparency media," Phys. Rev. Lett. 84, 1906-1909 (2000).
[CrossRef]

M. Blaauboer, G. Kurizki, and B. A. Malomed, "Spatiotemporally localized solitons in resonantly absorbing Bragg reflectors," Phys. Rev. E 62, R57-R59 (2000).
[CrossRef]

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D. Mihalache, D. Mazilu, L.-C. Crasovan, I. Towers, A. V. Buryak, B. A. Malomed, L. Torner, J. P. Torres, and F. Lederer, "Stable spinning optical solitons in three dimensions," Phys. Rev. Lett. 88, 073902 (2002).
[CrossRef]

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F. Kh. Abdullaev, J. G. Caputo, R. A. Kraenkel, and B. A. Malomed, "Controlling collapse in Bose-Einstein condensates by temporal modulation of the scattering length," Phys. Rev. A 67, 013605 (2003).
[CrossRef]

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Chen, Y. F.

K. Beckwitt, Y. F. Chen, F. W. Wise, and B. A. Malomed, "Temporal solitons in quadratic nonlinear media with opposite group-velocity dispersions at the fundamental and second harmonics," Phys. Rev. E 68, 057601 (2003).
[CrossRef]

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P. Di Trapani, W. Chinaglia, S. Minardi, A. Piskarkas, and G. Valiulis, "Observation of quadratic optical vortex solitons," Phys. Rev. Lett. 84, 3843-3846 (2000).
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D. Mihalache, D. Mazilu, L.-C. Crasovan, I. Towers, A. V. Buryak, B. A. Malomed, L. Torner, J. P. Torres, and F. Lederer, "Stable spinning optical solitons in three dimensions," Phys. Rev. Lett. 88, 073902 (2002).
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B. A. Malomed, P. Drummond, H. He, A. Berntson, D. Anderson, and M. Lisak, "Spatiotemporal solitons in multidimensional optical media with a quadratic nonlinearity," Phys. Rev. E 56, 4725-4735 (1997).
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P. G. Kevrekidis, G. Theocharis, D. J. Frantzeskakis, and B. A. Malomed, "Feshbach resonance management for Bose-Einstein condensates," Phys. Rev. Lett. 90, 230401 (2003).
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H. E. Nistazakis, D. J. Frantzeskakis, and B. A. Malomed, "Collisions between spatiotemporal solitons of different dimensionality in a planar waveguide," Phys. Rev. E 64, 026604 (2001).
[CrossRef]

D. J. Frantzeskakis, K. Hizanidis, B. A. Malomed, and C. Polymilis, "Stable anti-dark light bullets supported by the third-order dispersion," Phys. Lett. A 248, 203-207 (1998).
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G. Gottwald, R. Grimshaw, and B. A. Malomed, "Stable two-dimensional parametric solitons in fluid systems," Phys. Lett. A 248, 208-218 (1998).
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B. A. Malomed, P. Drummond, H. He, A. Berntson, D. Anderson, and M. Lisak, "Spatiotemporal solitons in multidimensional optical media with a quadratic nonlinearity," Phys. Rev. E 56, 4725-4735 (1997).
[CrossRef]

Hizanidis, K.

D. J. Frantzeskakis, K. Hizanidis, B. A. Malomed, and C. Polymilis, "Stable anti-dark light bullets supported by the third-order dispersion," Phys. Lett. A 248, 203-207 (1998).
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G. Huang, L. Deng, and C. Hang, "Davey-Stewartson description of two-dimensional nonlinear excitations in Bose-Einstein condensates," Phys. Rev. E 72, 036621 (2005).
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G. Huang, V. V. Konotop, H. W. Tam, and B. Hu, "Nonlinear modulation of multidimensional lattice waves," Phys. Rev. E 64, 056619 (2001).
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M. Matuszewski, E. Infeld, B. A. Malomed, and M. Trippenbach, Fully three dimensional breather solitons can be created using Feshbach resonances," Phys. Rev. Lett. 95, 050403 (2005).
[CrossRef]

M. Matuszewski, M. Trippenbach, B. A. Malomed, E. Infeld, and M. Skorupski, "Two-dimensional dispersion-managed light bullets in Kerr media," Phys. Rev. E 70, 016603 (2004).
[CrossRef]

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R. H. Enns, D. E. Edmundson, S. S. Rangnekar, and A. E. Kaplan, "Optical switching between bistable soliton states--a theoretical review," Opt. Quantum Electron. 24, S1295-S1314 (1992).
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D. Mihalache, D. Mazilu, F. Lederer, Y. V. Kartashov, L. C. Crasovan, and L. Torner, "Stable three-dimensional spatiotemporal solitons in a two-dimensional photonic lattice," Phys. Rev. E 70, 055603 (2004).
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P. G. Kevrekidis, G. Theocharis, D. J. Frantzeskakis, and B. A. Malomed, "Feshbach resonance management for Bose-Einstein condensates," Phys. Rev. Lett. 90, 230401 (2003).
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G. Huang, V. V. Konotop, H. W. Tam, and B. Hu, "Nonlinear modulation of multidimensional lattice waves," Phys. Rev. E 64, 056619 (2001).
[CrossRef]

Koshiba, M.

K. Hayata and M. Koshiba, "Multidimensional solitons in quadratic nonlinear media," Phys. Rev. Lett. 71, 3275-3278 (1993).
[CrossRef]

Kraenkel, R. A.

F. Kh. Abdullaev, J. G. Caputo, R. A. Kraenkel, and B. A. Malomed, "Controlling collapse in Bose-Einstein condensates by temporal modulation of the scattering length," Phys. Rev. A 67, 013605 (2003).
[CrossRef]

Kurizki, G.

M. Blaauboer, G. Kurizki, and B. A. Malomed, "Spatiotemporally localized solitons in resonantly absorbing Bragg reflectors," Phys. Rev. E 62, R57-R59 (2000).
[CrossRef]

M. Blaauboer, B. A. Malomed, and G. Kurizki, "Spatiotemporally localized multidimensional solitons in self-induced transparency media," Phys. Rev. Lett. 84, 1906-1909 (2000).
[CrossRef]

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D. Mihalache, D. Mazilu, F. Lederer, Y. V. Kartashov, L. C. Crasovan, and L. Torner, "Stable three-dimensional spatiotemporal solitons in a two-dimensional photonic lattice," Phys. Rev. E 70, 055603 (2004).
[CrossRef]

D. Mihalache, D. Mazilu, I. Towers, B. A. Malomed, and F. Lederer, "Stable spatiotemporal spinning solitons in a bimodal cubic-quintic medium," Phys. Rev. E 67, 056608 (2003).
[CrossRef]

D. Mihalache, D. Mazilu, L.-C. Crasovan, I. Towers, A. V. Buryak, B. A. Malomed, L. Torner, J. P. Torres, and F. Lederer, "Stable spinning optical solitons in three dimensions," Phys. Rev. Lett. 88, 073902 (2002).
[CrossRef]

Leon, J. J.

M. Boiti, J. J. Leon, L. Martina, and F. Pempinelli, "Scattering of localized solitons in the plane," Phys. Lett. A 132, 432-439 (1988).
[CrossRef]

Lisak, M.

B. A. Malomed, P. Drummond, H. He, A. Berntson, D. Anderson, and M. Lisak, "Spatiotemporal solitons in multidimensional optical media with a quadratic nonlinearity," Phys. Rev. E 56, 4725-4735 (1997).
[CrossRef]

Liu, X.

X. Liu, K. Beckwitt, and F. Wise, "Two-dimensional optical spatiotemporal solitons in quadratic media," Phys. Rev. E 62, 1328-1340 (2000).
[CrossRef]

X. Liu, L. J. Qian, and F. W. Wise, "Generation of optical spatiotemporal solitons," Phys. Rev. Lett. 82, 4631-4634 (1999).
[CrossRef]

Malomed, B. A.

M. Matuszewski, E. Infeld, B. A. Malomed, and M. Trippenbach, Fully three dimensional breather solitons can be created using Feshbach resonances," Phys. Rev. Lett. 95, 050403 (2005).
[CrossRef]

B. A. Malomed, D. Mihalache, F. Wise, and L. Torner, Spatiotemporal optical solitons," J. Opt. 7, R53-R72 (2005).

M. Trippenbach, M. Matuszewski, and B. A. Malomed, Stabilization of three-dimensional matter-waves solitons in an optical lattice," Europhys. Lett. 70, 8-14 (2005).
[CrossRef]

M. Matuszewski, M. Trippenbach, B. A. Malomed, E. Infeld, and M. Skorupski, "Two-dimensional dispersion-managed light bullets in Kerr media," Phys. Rev. E 70, 016603 (2004).
[CrossRef]

B. B. Baizakov, B. A. Malomed, and M. Salerno, "Multidimensional solitons in a low-dimensional periodic potential," Phys. Rev. A 70, 053613 (2004).
[CrossRef]

P. G. Kevrekidis, G. Theocharis, D. J. Frantzeskakis, and B. A. Malomed, "Feshbach resonance management for Bose-Einstein condensates," Phys. Rev. Lett. 90, 230401 (2003).
[CrossRef]

F. Kh. Abdullaev, J. G. Caputo, R. A. Kraenkel, and B. A. Malomed, "Controlling collapse in Bose-Einstein condensates by temporal modulation of the scattering length," Phys. Rev. A 67, 013605 (2003).
[CrossRef]

I. N. Towers, B. A. Malomed, and F. W. Wise, "Light bullets in quadratic media with normal dispersion at the second harmonic," Phys. Rev. Lett. 90, 123902 (2003).
[CrossRef]

D. Mihalache, D. Mazilu, I. Towers, B. A. Malomed, and F. Lederer, "Stable spatiotemporal spinning solitons in a bimodal cubic-quintic medium," Phys. Rev. E 67, 056608 (2003).
[CrossRef]

K. Beckwitt, Y. F. Chen, F. W. Wise, and B. A. Malomed, "Temporal solitons in quadratic nonlinear media with opposite group-velocity dispersions at the fundamental and second harmonics," Phys. Rev. E 68, 057601 (2003).
[CrossRef]

I. Towers and B. A. Malomed, "Stable (2+1)-dimensional solitons in a layered medium with sign-alternating Kerr nonlinearity," J. Opt. Soc. Am. B 19, 537-543 (2002).

D. Mihalache, D. Mazilu, L.-C. Crasovan, I. Towers, A. V. Buryak, B. A. Malomed, L. Torner, J. P. Torres, and F. Lederer, "Stable spinning optical solitons in three dimensions," Phys. Rev. Lett. 88, 073902 (2002).
[CrossRef]

H. E. Nistazakis, D. J. Frantzeskakis, and B. A. Malomed, "Collisions between spatiotemporal solitons of different dimensionality in a planar waveguide," Phys. Rev. E 64, 026604 (2001).
[CrossRef]

M. Blaauboer, B. A. Malomed, and G. Kurizki, "Spatiotemporally localized multidimensional solitons in self-induced transparency media," Phys. Rev. Lett. 84, 1906-1909 (2000).
[CrossRef]

M. Blaauboer, G. Kurizki, and B. A. Malomed, "Spatiotemporally localized solitons in resonantly absorbing Bragg reflectors," Phys. Rev. E 62, R57-R59 (2000).
[CrossRef]

D. J. Frantzeskakis, K. Hizanidis, B. A. Malomed, and C. Polymilis, "Stable anti-dark light bullets supported by the third-order dispersion," Phys. Lett. A 248, 203-207 (1998).
[CrossRef]

D. Mihalache, D. Mazilu, B. A. Malomed, and L. Torner, "Asymmetric spatio-temporal optical solitons in media with quadratic nonlinearity," Opt. Commun. 152, 365-370 (1998).
[CrossRef]

G. Gottwald, R. Grimshaw, and B. A. Malomed, "Stable two-dimensional parametric solitons in fluid systems," Phys. Lett. A 248, 208-218 (1998).
[CrossRef]

B. A. Malomed, P. Drummond, H. He, A. Berntson, D. Anderson, and M. Lisak, "Spatiotemporal solitons in multidimensional optical media with a quadratic nonlinearity," Phys. Rev. E 56, 4725-4735 (1997).
[CrossRef]

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

Martina, L.

M. Boiti, J. J. Leon, L. Martina, and F. Pempinelli, "Scattering of localized solitons in the plane," Phys. Lett. A 132, 432-439 (1988).
[CrossRef]

Matuszewski, M.

M. Trippenbach, M. Matuszewski, and B. A. Malomed, Stabilization of three-dimensional matter-waves solitons in an optical lattice," Europhys. Lett. 70, 8-14 (2005).
[CrossRef]

M. Matuszewski, E. Infeld, B. A. Malomed, and M. Trippenbach, Fully three dimensional breather solitons can be created using Feshbach resonances," Phys. Rev. Lett. 95, 050403 (2005).
[CrossRef]

M. Matuszewski, M. Trippenbach, B. A. Malomed, E. Infeld, and M. Skorupski, "Two-dimensional dispersion-managed light bullets in Kerr media," Phys. Rev. E 70, 016603 (2004).
[CrossRef]

Mazilu, D.

D. Mihalache, D. Mazilu, F. Lederer, Y. V. Kartashov, L. C. Crasovan, and L. Torner, "Stable three-dimensional spatiotemporal solitons in a two-dimensional photonic lattice," Phys. Rev. E 70, 055603 (2004).
[CrossRef]

D. Mihalache, D. Mazilu, I. Towers, B. A. Malomed, and F. Lederer, "Stable spatiotemporal spinning solitons in a bimodal cubic-quintic medium," Phys. Rev. E 67, 056608 (2003).
[CrossRef]

D. Mihalache, D. Mazilu, L.-C. Crasovan, I. Towers, A. V. Buryak, B. A. Malomed, L. Torner, J. P. Torres, and F. Lederer, "Stable spinning optical solitons in three dimensions," Phys. Rev. Lett. 88, 073902 (2002).
[CrossRef]

D. Mihalache, D. Mazilu, B. A. Malomed, and L. Torner, "Asymmetric spatio-temporal optical solitons in media with quadratic nonlinearity," Opt. Commun. 152, 365-370 (1998).
[CrossRef]

McLeod, R.

R. McLeod, K. Wagner, and S. Blair, "(3+1)-dimensional optical soliton dragging logic," Phys. Rev. A 52, 3254-3278 (1995).
[CrossRef]

Michinel, H.

G. D. Montesinos, V. M. Pérez-García, and H. Michinel, Stabilized two-dimensional vector solitons," Phys. Rev. Lett. 92, 133901 (2004).
[CrossRef]

M. L. Quiroga-Teixeiro, A. Berntson, and H. Michinel, "Internal dynamics of nonlinear beams in their ground states: short- and long-lived excitation," J. Opt. Soc. Am. B 16, 1697-1704 (1999).

Mihalache, D.

B. A. Malomed, D. Mihalache, F. Wise, and L. Torner, Spatiotemporal optical solitons," J. Opt. 7, R53-R72 (2005).

D. Mihalache, D. Mazilu, F. Lederer, Y. V. Kartashov, L. C. Crasovan, and L. Torner, "Stable three-dimensional spatiotemporal solitons in a two-dimensional photonic lattice," Phys. Rev. E 70, 055603 (2004).
[CrossRef]

D. Mihalache, D. Mazilu, I. Towers, B. A. Malomed, and F. Lederer, "Stable spatiotemporal spinning solitons in a bimodal cubic-quintic medium," Phys. Rev. E 67, 056608 (2003).
[CrossRef]

D. Mihalache, D. Mazilu, L.-C. Crasovan, I. Towers, A. V. Buryak, B. A. Malomed, L. Torner, J. P. Torres, and F. Lederer, "Stable spinning optical solitons in three dimensions," Phys. Rev. Lett. 88, 073902 (2002).
[CrossRef]

D. Mihalache, D. Mazilu, B. A. Malomed, and L. Torner, "Asymmetric spatio-temporal optical solitons in media with quadratic nonlinearity," Opt. Commun. 152, 365-370 (1998).
[CrossRef]

Minardi, S.

P. Di Trapani, W. Chinaglia, S. Minardi, A. Piskarkas, and G. Valiulis, "Observation of quadratic optical vortex solitons," Phys. Rev. Lett. 84, 3843-3846 (2000).
[CrossRef]

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G. D. Montesinos, V. M. Pérez-García, and H. Michinel, Stabilized two-dimensional vector solitons," Phys. Rev. Lett. 92, 133901 (2004).
[CrossRef]

G. D. Montesinos, V. M. Pérez-García, and P. J. Torres, "Stabilization of solitons of the multidimensional nonlinear Schrodinger equation: matter-wave breathers," Physica D 191, 193-210 (2004).
[CrossRef]

Morandotti, R.

H. S. Eisenberg, R. Morandotti, Y. Silberberg, S. Bar-Ad, D. Ross, and J. S. Aitchison, "Kerr spatiotemporal self-focusing in a planar glass waveguide," Phys. Rev. Lett. 87, 043902 (2001).
[CrossRef]

Naidenov, A.

A. B. Blagoeva, S. G. Dinev, A. A. Dreischuh, and A. Naidenov, "Light bullets formation in a bulk media," IEEE J. Quantum Electron. 27, 2060-2065 (1991).
[CrossRef]

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Nistazakis, H. E.

H. E. Nistazakis, D. J. Frantzeskakis, and B. A. Malomed, "Collisions between spatiotemporal solitons of different dimensionality in a planar waveguide," Phys. Rev. E 64, 026604 (2001).
[CrossRef]

Pempinelli, F.

M. Boiti, J. J. Leon, L. Martina, and F. Pempinelli, "Scattering of localized solitons in the plane," Phys. Lett. A 132, 432-439 (1988).
[CrossRef]

Pérez-García, V. M.

G. D. Montesinos, V. M. Pérez-García, and P. J. Torres, "Stabilization of solitons of the multidimensional nonlinear Schrodinger equation: matter-wave breathers," Physica D 191, 193-210 (2004).
[CrossRef]

G. D. Montesinos, V. M. Pérez-García, and H. Michinel, Stabilized two-dimensional vector solitons," Phys. Rev. Lett. 92, 133901 (2004).
[CrossRef]

Piskarkas, A.

P. Di Trapani, W. Chinaglia, S. Minardi, A. Piskarkas, and G. Valiulis, "Observation of quadratic optical vortex solitons," Phys. Rev. Lett. 84, 3843-3846 (2000).
[CrossRef]

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[CrossRef]

Polymilis, C.

D. J. Frantzeskakis, K. Hizanidis, B. A. Malomed, and C. Polymilis, "Stable anti-dark light bullets supported by the third-order dispersion," Phys. Lett. A 248, 203-207 (1998).
[CrossRef]

Qian, L. J.

X. Liu, L. J. Qian, and F. W. Wise, "Generation of optical spatiotemporal solitons," Phys. Rev. Lett. 82, 4631-4634 (1999).
[CrossRef]

Quiroga-Teixeiro, M. L.

Rangnekar, S. S.

R. H. Enns, D. E. Edmundson, S. S. Rangnekar, and A. E. Kaplan, "Optical switching between bistable soliton states--a theoretical review," Opt. Quantum Electron. 24, S1295-S1314 (1992).
[CrossRef]

Redekopp, L.

V. Djordjevic and L. Redekopp, "2-dimensional packets of capillary-gravity waves," J. Fluid Mech. 79, 703-714 (1977).
[CrossRef]

Roskes, G. J.

D. J. Benney and G. J. Roskes, "Wave instabilities," Stud. Appl. Math. 48, 377-385 (1969).

Ross, D.

H. S. Eisenberg, R. Morandotti, Y. Silberberg, S. Bar-Ad, D. Ross, and J. S. Aitchison, "Kerr spatiotemporal self-focusing in a planar glass waveguide," Phys. Rev. Lett. 87, 043902 (2001).
[CrossRef]

Saito, H.

H. Saito and M. Ueda, "Dynamically stabilized bright solitons in a two-dimensional Bose-Einstein condensate," Phys. Rev. Lett. 90, 040403 (2003).
[CrossRef]

Salerno, M.

B. B. Baizakov, B. A. Malomed, and M. Salerno, "Multidimensional solitons in a low-dimensional periodic potential," Phys. Rev. A 70, 053613 (2004).
[CrossRef]

Santini, P. M.

A. S. Fokas and P. M. Santini, "Dromions and a boundary-value problem for the Davey-Stewartson-1 equation," Physica D 44, 99-130 (1990).
[CrossRef]

A. S. Fokas and P. M. Santini, "Coherent structures in multidimensions," Phys. Rev. Lett. 63, 1329-1333 (1989).
[CrossRef]

Satsuma, J.

J. Satsuma and M. J. Ablowitz, "2-dimensional lumps in non-linear dispersive systems," J. Math. Phys. 20, 1496-1503 (1979).
[CrossRef]

Silberberg, Y.

H. S. Eisenberg, R. Morandotti, Y. Silberberg, S. Bar-Ad, D. Ross, and J. S. Aitchison, "Kerr spatiotemporal self-focusing in a planar glass waveguide," Phys. Rev. Lett. 87, 043902 (2001).
[CrossRef]

Y. Silberberg, "Collapse of optical pulses," Opt. Lett. 15, 1282-1284 (1990).

Skorupski, M.

M. Matuszewski, M. Trippenbach, B. A. Malomed, E. Infeld, and M. Skorupski, "Two-dimensional dispersion-managed light bullets in Kerr media," Phys. Rev. E 70, 016603 (2004).
[CrossRef]

Skryabin, D. V.

D. V. Skryabin and W. J. Firth, "Generation and stability of optical bullets in quadratic nonlinear media," Opt. Commun. 148, 79-84 (1998).
[CrossRef]

Stewartson, K.

A. Davey and K. Stewartson, "3-dimensional packets of surface-waves," Proc. R. Soc. London, Ser. A 338, 101-110 (1974).

Tam, H. W.

G. Huang, V. V. Konotop, H. W. Tam, and B. Hu, "Nonlinear modulation of multidimensional lattice waves," Phys. Rev. E 64, 056619 (2001).
[CrossRef]

Theocharis, G.

P. G. Kevrekidis, G. Theocharis, D. J. Frantzeskakis, and B. A. Malomed, "Feshbach resonance management for Bose-Einstein condensates," Phys. Rev. Lett. 90, 230401 (2003).
[CrossRef]

Torner, L.

B. A. Malomed, D. Mihalache, F. Wise, and L. Torner, Spatiotemporal optical solitons," J. Opt. 7, R53-R72 (2005).

D. Mihalache, D. Mazilu, F. Lederer, Y. V. Kartashov, L. C. Crasovan, and L. Torner, "Stable three-dimensional spatiotemporal solitons in a two-dimensional photonic lattice," Phys. Rev. E 70, 055603 (2004).
[CrossRef]

D. Mihalache, D. Mazilu, L.-C. Crasovan, I. Towers, A. V. Buryak, B. A. Malomed, L. Torner, J. P. Torres, and F. Lederer, "Stable spinning optical solitons in three dimensions," Phys. Rev. Lett. 88, 073902 (2002).
[CrossRef]

D. Mihalache, D. Mazilu, B. A. Malomed, and L. Torner, "Asymmetric spatio-temporal optical solitons in media with quadratic nonlinearity," Opt. Commun. 152, 365-370 (1998).
[CrossRef]

Torres, J. P.

D. Mihalache, D. Mazilu, L.-C. Crasovan, I. Towers, A. V. Buryak, B. A. Malomed, L. Torner, J. P. Torres, and F. Lederer, "Stable spinning optical solitons in three dimensions," Phys. Rev. Lett. 88, 073902 (2002).
[CrossRef]

Torres, P. J.

G. D. Montesinos, V. M. Pérez-García, and P. J. Torres, "Stabilization of solitons of the multidimensional nonlinear Schrodinger equation: matter-wave breathers," Physica D 191, 193-210 (2004).
[CrossRef]

Towers, I.

D. Mihalache, D. Mazilu, I. Towers, B. A. Malomed, and F. Lederer, "Stable spatiotemporal spinning solitons in a bimodal cubic-quintic medium," Phys. Rev. E 67, 056608 (2003).
[CrossRef]

D. Mihalache, D. Mazilu, L.-C. Crasovan, I. Towers, A. V. Buryak, B. A. Malomed, L. Torner, J. P. Torres, and F. Lederer, "Stable spinning optical solitons in three dimensions," Phys. Rev. Lett. 88, 073902 (2002).
[CrossRef]

I. Towers and B. A. Malomed, "Stable (2+1)-dimensional solitons in a layered medium with sign-alternating Kerr nonlinearity," J. Opt. Soc. Am. B 19, 537-543 (2002).

Towers, I. N.

I. N. Towers, B. A. Malomed, and F. W. Wise, "Light bullets in quadratic media with normal dispersion at the second harmonic," Phys. Rev. Lett. 90, 123902 (2003).
[CrossRef]

Trapani, P. Di

P. Di Trapani, W. Chinaglia, S. Minardi, A. Piskarkas, and G. Valiulis, "Observation of quadratic optical vortex solitons," Phys. Rev. Lett. 84, 3843-3846 (2000).
[CrossRef]

Trippenbach, M.

M. Matuszewski, E. Infeld, B. A. Malomed, and M. Trippenbach, Fully three dimensional breather solitons can be created using Feshbach resonances," Phys. Rev. Lett. 95, 050403 (2005).
[CrossRef]

M. Trippenbach, M. Matuszewski, and B. A. Malomed, Stabilization of three-dimensional matter-waves solitons in an optical lattice," Europhys. Lett. 70, 8-14 (2005).
[CrossRef]

M. Matuszewski, M. Trippenbach, B. A. Malomed, E. Infeld, and M. Skorupski, "Two-dimensional dispersion-managed light bullets in Kerr media," Phys. Rev. E 70, 016603 (2004).
[CrossRef]

Ueda, M.

H. Saito and M. Ueda, "Dynamically stabilized bright solitons in a two-dimensional Bose-Einstein condensate," Phys. Rev. Lett. 90, 040403 (2003).
[CrossRef]

Valiulis, G.

P. Di Trapani, W. Chinaglia, S. Minardi, A. Piskarkas, and G. Valiulis, "Observation of quadratic optical vortex solitons," Phys. Rev. Lett. 84, 3843-3846 (2000).
[CrossRef]

Wagner, K.

R. McLeod, K. Wagner, and S. Blair, "(3+1)-dimensional optical soliton dragging logic," Phys. Rev. A 52, 3254-3278 (1995).
[CrossRef]

Wise, F.

B. A. Malomed, D. Mihalache, F. Wise, and L. Torner, Spatiotemporal optical solitons," J. Opt. 7, R53-R72 (2005).

X. Liu, K. Beckwitt, and F. Wise, "Two-dimensional optical spatiotemporal solitons in quadratic media," Phys. Rev. E 62, 1328-1340 (2000).
[CrossRef]

Wise, F. W.

I. N. Towers, B. A. Malomed, and F. W. Wise, "Light bullets in quadratic media with normal dispersion at the second harmonic," Phys. Rev. Lett. 90, 123902 (2003).
[CrossRef]

K. Beckwitt, Y. F. Chen, F. W. Wise, and B. A. Malomed, "Temporal solitons in quadratic nonlinear media with opposite group-velocity dispersions at the fundamental and second harmonics," Phys. Rev. E 68, 057601 (2003).
[CrossRef]

X. Liu, L. J. Qian, and F. W. Wise, "Generation of optical spatiotemporal solitons," Phys. Rev. Lett. 82, 4631-4634 (1999).
[CrossRef]

Europhys. Lett. (1)

M. Trippenbach, M. Matuszewski, and B. A. Malomed, Stabilization of three-dimensional matter-waves solitons in an optical lattice," Europhys. Lett. 70, 8-14 (2005).
[CrossRef]

IEEE J. Quantum Electron. (1)

A. B. Blagoeva, S. G. Dinev, A. A. Dreischuh, and A. Naidenov, "Light bullets formation in a bulk media," IEEE J. Quantum Electron. 27, 2060-2065 (1991).
[CrossRef]

J. Fluid Mech. (1)

V. Djordjevic and L. Redekopp, "2-dimensional packets of capillary-gravity waves," J. Fluid Mech. 79, 703-714 (1977).
[CrossRef]

J. Math. Phys. (1)

J. Satsuma and M. J. Ablowitz, "2-dimensional lumps in non-linear dispersive systems," J. Math. Phys. 20, 1496-1503 (1979).
[CrossRef]

J. Opt. (1)

B. A. Malomed, D. Mihalache, F. Wise, and L. Torner, Spatiotemporal optical solitons," J. Opt. 7, R53-R72 (2005).

J. Opt. Soc. Am. B (2)

J. Phys. A (2)

H. Leblond, "Bidimensional optical solitons in a quadratic medium," J. Phys. A 31, 5129-5143 (1998).
[CrossRef]

H. Leblond, "Electromagnetic waves in ferromagnets: a Davey-Stewartson-type model," J. Phys. A 32, 7907-7932 (1999).
[CrossRef]

Opt. Commun. (2)

D. V. Skryabin and W. J. Firth, "Generation and stability of optical bullets in quadratic nonlinear media," Opt. Commun. 148, 79-84 (1998).
[CrossRef]

D. Mihalache, D. Mazilu, B. A. Malomed, and L. Torner, "Asymmetric spatio-temporal optical solitons in media with quadratic nonlinearity," Opt. Commun. 152, 365-370 (1998).
[CrossRef]

Opt. Lett. (3)

Opt. Quantum Electron. (1)

R. H. Enns, D. E. Edmundson, S. S. Rangnekar, and A. E. Kaplan, "Optical switching between bistable soliton states--a theoretical review," Opt. Quantum Electron. 24, S1295-S1314 (1992).
[CrossRef]

Phys. Fluids (1)

D. B. Khismatullin and I. Sh. Akhatov, "Sound-ultrasound interaction in bubbly fluids: theory and possible applications," Phys. Fluids 13, 3582-3598 (2001).
[CrossRef]

Phys. Lett. A (4)

A. Nakamura, "Explode-decay mode lump solitons of a two-dimensional non-linear Schrodinger-equation," Phys. Lett. A 88, 55-56 (1982).
[CrossRef]

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[CrossRef]

D. J. Frantzeskakis, K. Hizanidis, B. A. Malomed, and C. Polymilis, "Stable anti-dark light bullets supported by the third-order dispersion," Phys. Lett. A 248, 203-207 (1998).
[CrossRef]

G. Gottwald, R. Grimshaw, and B. A. Malomed, "Stable two-dimensional parametric solitons in fluid systems," Phys. Lett. A 248, 208-218 (1998).
[CrossRef]

Phys. Rev. A (4)

F. Kh. Abdullaev, J. G. Caputo, R. A. Kraenkel, and B. A. Malomed, "Controlling collapse in Bose-Einstein condensates by temporal modulation of the scattering length," Phys. Rev. A 67, 013605 (2003).
[CrossRef]

B. B. Baizakov, B. A. Malomed, and M. Salerno, "Multidimensional solitons in a low-dimensional periodic potential," Phys. Rev. A 70, 053613 (2004).
[CrossRef]

R. McLeod, K. Wagner, and S. Blair, "(3+1)-dimensional optical soliton dragging logic," Phys. Rev. A 52, 3254-3278 (1995).
[CrossRef]

D. E. Edmundson and R. H. Enns, "Particlelike nature of colliding three-dimensional optical solitons," Phys. Rev. A 51, 2491-2498 (1995).
[CrossRef]

Phys. Rev. E (12)

D. E. Edmundson, "Unstable higher modes of a three-dimensional nonlinear Schrodinger equation," Phys. Rev. E 55, 7636-7644 (1997).
[CrossRef]

B. A. Malomed, P. Drummond, H. He, A. Berntson, D. Anderson, and M. Lisak, "Spatiotemporal solitons in multidimensional optical media with a quadratic nonlinearity," Phys. Rev. E 56, 4725-4735 (1997).
[CrossRef]

H. E. Nistazakis, D. J. Frantzeskakis, and B. A. Malomed, "Collisions between spatiotemporal solitons of different dimensionality in a planar waveguide," Phys. Rev. E 64, 026604 (2001).
[CrossRef]

D. Mihalache, D. Mazilu, I. Towers, B. A. Malomed, and F. Lederer, "Stable spatiotemporal spinning solitons in a bimodal cubic-quintic medium," Phys. Rev. E 67, 056608 (2003).
[CrossRef]

M. Blaauboer, G. Kurizki, and B. A. Malomed, "Spatiotemporally localized solitons in resonantly absorbing Bragg reflectors," Phys. Rev. E 62, R57-R59 (2000).
[CrossRef]

D. Mihalache, D. Mazilu, F. Lederer, Y. V. Kartashov, L. C. Crasovan, and L. Torner, "Stable three-dimensional spatiotemporal solitons in a two-dimensional photonic lattice," Phys. Rev. E 70, 055603 (2004).
[CrossRef]

M. Matuszewski, M. Trippenbach, B. A. Malomed, E. Infeld, and M. Skorupski, "Two-dimensional dispersion-managed light bullets in Kerr media," Phys. Rev. E 70, 016603 (2004).
[CrossRef]

X. Liu, K. Beckwitt, and F. Wise, "Two-dimensional optical spatiotemporal solitons in quadratic media," Phys. Rev. E 62, 1328-1340 (2000).
[CrossRef]

K. Beckwitt, Y. F. Chen, F. W. Wise, and B. A. Malomed, "Temporal solitons in quadratic nonlinear media with opposite group-velocity dispersions at the fundamental and second harmonics," Phys. Rev. E 68, 057601 (2003).
[CrossRef]

M. J. Ablowitz, G. Biondini, and S. Blair, "Nonlinear Schrodinger equations with mean terms in nonresonant multidimensional quadratic materials," Phys. Rev. E 63, 046605 (2001).
[CrossRef]

G. Huang, L. Deng, and C. Hang, "Davey-Stewartson description of two-dimensional nonlinear excitations in Bose-Einstein condensates," Phys. Rev. E 72, 036621 (2005).
[CrossRef]

G. Huang, V. V. Konotop, H. W. Tam, and B. Hu, "Nonlinear modulation of multidimensional lattice waves," Phys. Rev. E 64, 056619 (2001).
[CrossRef]

Phys. Rev. Lett. (12)

P. Di Trapani, W. Chinaglia, S. Minardi, A. Piskarkas, and G. Valiulis, "Observation of quadratic optical vortex solitons," Phys. Rev. Lett. 84, 3843-3846 (2000).
[CrossRef]

A. S. Fokas and P. M. Santini, "Coherent structures in multidimensions," Phys. Rev. Lett. 63, 1329-1333 (1989).
[CrossRef]

M. Matuszewski, E. Infeld, B. A. Malomed, and M. Trippenbach, Fully three dimensional breather solitons can be created using Feshbach resonances," Phys. Rev. Lett. 95, 050403 (2005).
[CrossRef]

H. S. Eisenberg, R. Morandotti, Y. Silberberg, S. Bar-Ad, D. Ross, and J. S. Aitchison, "Kerr spatiotemporal self-focusing in a planar glass waveguide," Phys. Rev. Lett. 87, 043902 (2001).
[CrossRef]

X. Liu, L. J. Qian, and F. W. Wise, "Generation of optical spatiotemporal solitons," Phys. Rev. Lett. 82, 4631-4634 (1999).
[CrossRef]

H. Saito and M. Ueda, "Dynamically stabilized bright solitons in a two-dimensional Bose-Einstein condensate," Phys. Rev. Lett. 90, 040403 (2003).
[CrossRef]

P. G. Kevrekidis, G. Theocharis, D. J. Frantzeskakis, and B. A. Malomed, "Feshbach resonance management for Bose-Einstein condensates," Phys. Rev. Lett. 90, 230401 (2003).
[CrossRef]

G. D. Montesinos, V. M. Pérez-García, and H. Michinel, Stabilized two-dimensional vector solitons," Phys. Rev. Lett. 92, 133901 (2004).
[CrossRef]

D. Mihalache, D. Mazilu, L.-C. Crasovan, I. Towers, A. V. Buryak, B. A. Malomed, L. Torner, J. P. Torres, and F. Lederer, "Stable spinning optical solitons in three dimensions," Phys. Rev. Lett. 88, 073902 (2002).
[CrossRef]

M. Blaauboer, B. A. Malomed, and G. Kurizki, "Spatiotemporally localized multidimensional solitons in self-induced transparency media," Phys. Rev. Lett. 84, 1906-1909 (2000).
[CrossRef]

I. N. Towers, B. A. Malomed, and F. W. Wise, "Light bullets in quadratic media with normal dispersion at the second harmonic," Phys. Rev. Lett. 90, 123902 (2003).
[CrossRef]

K. Hayata and M. Koshiba, "Multidimensional solitons in quadratic nonlinear media," Phys. Rev. Lett. 71, 3275-3278 (1993).
[CrossRef]

Physica D (4)

G. D. Montesinos, V. M. Pérez-García, and P. J. Torres, "Stabilization of solitons of the multidimensional nonlinear Schrodinger equation: matter-wave breathers," Physica D 191, 193-210 (2004).
[CrossRef]

A. S. Fokas and P. M. Santini, "Dromions and a boundary-value problem for the Davey-Stewartson-1 equation," Physica D 44, 99-130 (1990).
[CrossRef]

V. A. Arkadiev, A. K. Pogrebkov, and M. C. Polivanov, "Inverse scattering transform method and soliton-solutions for Davey-Stewartson-II equation," Physica D 36, 189-197 (1989).
[CrossRef]

R. E. Kates and D. J. Kaup, "2-dimensional nonlinear Schrodinger-equations and their properties," Physica D 75, 458-470 (1994).

Proc. R. Soc. London, Ser. A (2)

A. Davey and K. Stewartson, "3-dimensional packets of surface-waves," Proc. R. Soc. London, Ser. A 338, 101-110 (1974).

D. Anker and N. C. Freeman, "Soliton solutions of Davey-Stewartson equation for long waves," Proc. R. Soc. London, Ser. A 360, 529-540 (1978).

Stud. Appl. Math. (1)

D. J. Benney and G. J. Roskes, "Wave instabilities," Stud. Appl. Math. 48, 377-385 (1969).

Other (4)

M. J. Ablowitz and P. A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering (Cambridge U. Press, 1991).

Yu. 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).

G. P. Agrawal, Nonlinear Fiber Optics (Academic,1995).

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Figures (4)

Fig. 1
Fig. 1

Initial condition (at z = 0 ) for the case of the defocusing Kerr nonlinearity ( s = 1 ) . Parameters in the NLS model are ϵ 1 = ϵ 2 = ϵ = 0.1 , η = 0.9 , σ = 2 , δ = 0.1 , and β = 1.01 , whereas the dromion parameters are ν R = μ R = 1 , ν I = 1 , μ I = 1 , Q 0 = i , A = 0.5 , v 0 = 1 , and Ω = 1.56 . Top panel: a 3D plot of the dromion in the u component. Middle panel: a spatiotemporal (in the x t plane) contour plot of the dromion profile. Bottom panel: contour plot of the v component for a background of finite extent (super-Gaussian) with v 0 = 1 . In physical units, the spatiotemporal soliton (i.e., the dromion) in the u component has the initial peak power of 6.3 W and spatial and temporal widths (full width at half-maximum, FWHM) of 1.77 mm and 0.33 ps , respectively.

Fig. 2
Fig. 2

Same as in Fig. 1, but at z = 15 (in physical units, it corresponds to z = 4 m ). Note that, after having passed this distance, the dromion has lost only 23 % of its initial power, and it propagates without major changes in its shape. On the other hand, each of the dark stripes in the v component has split into two. The stripes generated by the splitting travel along the + z direction and drag the dromion’s component in the u field.

Fig. 3
Fig. 3

Initial condition for the case of the focusing Kerr nonlinearity ( s = + 1 ) . Parameters in the NLS equations are the same as in the case of defocusing nonlinearity (see, e.g., the caption of Fig. 1), while the lump’s parameters are κ = 1 , r = 0 , p = 1 , A = 0.2 , and Ω = 1.56 . Top panel: a 3D plot of the lump soliton in the u component. Middle panel: spatiotemporal (in the x t plane) contour plot of the lump’s profile. Bottom panel: contour plot of the v component for the background of a finite extent (super-Gaussian) with v 0 = 1 . In physical units, this spatiotemporal soliton (i.e., the lump) in the u component has an initial peak power of 5.9 W and spatial and temporal widths (FWHM) of 4.96 mm and 0.54 ps , respectively.

Fig. 4
Fig. 4

Same as in Fig. 3, but at z = 15 (in physical units, it corresponds to 4 m ). Note that, up to z = 13 ( 3.5 m ) , the lump soliton propagates practically undistorted. Nevertheless, at z = 15 the modulational instability has already set in, leading to the destruction of the background in the v component. Note that the characteristic modulational instability peaks (see also discussion in the text) are presented by the dark regions at the edges of the figure. At the same time, the lump, as well as the localized structure on top of the background in the v component, starts collapsing.

Equations (36)

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i ( E 1 , 2 z + k 1 , 2 E 1 , 2 t ) 1 2 k 1 , 2 2 E 1 , 2 t 2 + 1 2 k 1 , 2 2 E 1 , 2 x 2
+ R 1 , 2 ( E 1 , 2 2 + σ E 2 , 1 2 ) E 1 , 2 = i 6 k 1 , 2 3 E 1 , 2 t 3 ,
σ = 2 2 cos 2 θ 2 + cos 2 θ .
i u z + 1 2 ( u x x ϵ 1 u t t ) + s ( η u 2 + σ v 2 ) u i 6 u t t t = 0 ,
i ( v z δ v t ) + 1 2 ( γ v x x ϵ 2 v t t ) + s ( 1 η v 2 + σ u 2 ) v i 6 β v t t t = 0 ,
s = sign ( n 2 ( l ) ) , η = n 2 ( 1 ) n 2 ( 2 ) , ϵ 1 , 2 = t 0 k 1 , 2 k 1 ,
δ = t 0 2 ( k 1 k 2 ) k 1 , γ = k 1 k 2 , β = k 2 k 1 .
4 ( k z + ω δ β 6 ω 3 ) 2 = ( γ k x 2 + ϵ ω 2 ) ( γ k x 2 + ϵ ω 2 4 s η ) .
u = q exp ( i s v 0 2 σ z ) , v = ( v 0 + ρ ) exp ( i s v 0 2 η 1 z + i ϕ ) ,
i q z + 1 2 ( q x x ϵ q t t ) + s ( η q 2 + 2 σ v 0 ρ ) q = i 6 q t t t ,
ρ z δ ρ t + 1 2 v 0 ( γ ϕ x x ϵ ϕ t t ) = β 6 ρ t t t ,
v 0 ( ϕ z δ ϕ t ) 1 2 ( γ ρ x x ϵ ρ t t )
s v 0 ( 2 v 0 η ρ + σ q 2 ) = β 6 v 0 ϕ t t t .
Z ϵ z , X ϵ x , T ϵ ( t C z ) ,
ρ ϵ V , ϕ ϵ Φ , q ϵ 1 2 Q exp [ i ( K + Ω 2 2 ) z i Ω t ] ,
K = 1 6 Ω 3 , C = 1 2 Ω 2 ,
i Q Z + 1 2 ( Q X X Ω Q T T ) + s ( η Q 2 + 2 σ v 0 V ) Q = 0 .
1 2 v 0 γ Φ X X ( C + δ ) V T = 0 ,
v 0 ( C + δ ) Φ T + s v 0 ( 2 v 0 η V + σ Q 2 ) = 0 .
s γ v 0 2 η V X X + ( C + δ ) 2 V T T + 1 2 s γ σ v 0 ( Q 2 ) X X = 0 .
X = A η X , Z = η A 2 Z , T = A η Ω T ,
Q = A Q , V = ( η A 2 2 σ v 0 ) V ,
( C + δ ) 2 ( 1 2 Ω 2 + δ ) 2 = ( γ v 0 2 η ) Ω
i Q Z + 1 2 ( Q X X λ Q T T ) + s ( Q 2 + V ) Q = 0 ,
V X X + s V T T + σ 2 ( Q 2 ) X X = 0 ,
Q ( ξ , ζ , Z ) = 4 Q 0 ( μ R ν R ) 1 2 F ( ξ , ζ , Z ) G ( ξ , ζ , Z ) ,
F ( ξ , ζ , Z ) = exp [ ν R ( ξ + ξ 0 ) μ R ( ζ + ζ 0 ) ] exp [ i ( ν I ξ μ I ζ + ( μ 2 + ν 2 ) Z ) ] ,
G ( ξ , ζ , Z ) = [ 1 + exp ( 2 ν R ( ξ + ξ 0 ) ) ] [ 1 + exp ( 2 μ R ( ζ + ζ 0 ) ) ] + Q 0 2 .
V 1 lim ξ V ( ξ , ζ , Z ) = 2 μ R 2 sech 2 [ μ R ( ζ ζ 0 ) ] ,
V 2 lim ζ V ( ξ , ζ , Z ) = 2 ν R 2 sech 2 [ ν R ( ξ ξ 0 ) ] .
V ( ξ , ζ , Z ) = 1 2 ( ξ ̃ ( Q 2 ) ζ ̃ d ξ ̃ + ζ ̃ ( Q 2 ) ξ ̃ d ζ ̃ ) ( Q 2 + V 1 + V 2 ) ,
u ( x , t , 0 ) = ϵ 1 2 A Q ( x , t , 0 ) exp ( i Ω t ) ,
v ( x , t , 0 ) = [ v 0 + ϵ ( η A 2 2 σ v 0 ) V ( x , t , 0 ) ] v b ,
Q ( X , T , Z ) = 2 κ ¯ exp [ p W p W ¯ i ( p 2 + p ¯ 2 ) Z ] W + r 2 i p Z 2 + κ 2 ,
V ( X , Z , T ) = g ( X , Z , T ) + g ¯ ( X , Z , T ) Q ( X , Z , T ) 2 ,
g ( X , Z , T ) = 4 ( W + r 2 i p Z ¯ ) 2 ( W + r 2 i p Z 2 + κ 2 ) 2 ,

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