Abstract

We demonstrate that necklace-shaped arrays of localized spatial beams can merge into stable fundamental or vortex solitons in a generic model of laser cavities, based on the two-dimensional complex Ginzburg-Landau equation with the cubic-quintic nonlinearity. The outcome of the fusion is controlled by the number of “beads” in the initial necklace, 2N, and its topological charge, M. We predict and confirm by systematic simulations that the vorticity of the emerging soliton is |N-M|. Threshold characteristics of the fusion are found and explained too. If the initial radius of the array (R0) is too large, it simply keeps the necklace shape (if R 0 is somewhat smaller, the necklace features a partial fusion), while, if R 0 is too small, the array disappears.

© 2007 Optical Society of America

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    [CrossRef] [PubMed]
  8. Y. V. Kartashov, L.-C. Crasovan, D. Mihalache, and L. Torner, "Robust propagation of two-color soliton clusters supported by competing nonlinearities." Phys. Rev. Lett. 89, 273902 (2002).
    [CrossRef]
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  13. Y. J. He and H. Z. Wang, "(1+1)-dimensional dipole solitons supported by optical lattice," Opt. Express 14, 9832-9837 (2006).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  17. L.-C. Crasovan, B. A. Malomed, and D. Mihalache, "Stable vortex solitons in the two-dimensional Ginzburg-Landau equation," Phys. Rev. E 63, 016605 (2001).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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  23. F. Smektala, C. Quemard, V. Couderc, and A. Barthelemy, "Non-linear optical properties of chalcogenide glasses measured by Z-scan," J. Non-Cryst. Solids 274, 232-237 (2000).
    [CrossRef]
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    [CrossRef]
  25. C. Zhan, D. Zhang, D. Zhu, D. Wang, Y. Li, D. Li, Z. Lu, L. Zhao, and Y. Nie, "Third- and fifth-order optical nonlinearities in a new stilbazolium derivative," J. Opt. Soc. Am. B 19, 369-375 (2002).
    [CrossRef]
  26. N. N. Rosanov, S. V. Fedorov, and A. N. Shatsev, "Motion of clusters of weakly coupled two-dimensional cavity solitons," JETP 102, 547-555 (2006); "Curvilinear motion of multivortex laser-soliton complexes with strong and weak coupling," Phys. Rev. Lett. 95, 053903 (2005).
    [CrossRef]
  27. S. V. Fedorov, N. N. Rosanov, and A. N. Shatsev, "Two-dimensional solitons in B-class lasers with saturable absorption," Opt. Spectra. 102, 449-455 (2007).
    [CrossRef]
  28. B. A. Malomed, "Potential of interaction between two- and three-dimensional solitons," Phys. Rev. E 58, 7928-7933 (1998).
    [CrossRef]
  29. B. Kneer, T. Wong, K. Vogel, W. P. Schleich, and D. F. Walls, "Generic model of an atom laser," Phys. Rev. A 58, 4841 (1998).
    [CrossRef]
  30. H. J. Miesner, D. M. Stamper-Kurn, M. R. D. S. Durfee, S. Inouye, and W. Ketterle, "Bosonic stimulation in the formation of a Bose-Einstein condensate," Science 279, 1005-1007 (1998).
    [CrossRef] [PubMed]
  31. E. W. Hagley, L. Deng, M. Kozuma, J. Wen, K. Helmerson, S. L. Rolston, and W. D. Phillips, "A well-collimated quasi-continuous atom laser," Science 283, 1706-1709 (1999).
  32. P. D. Drummond, K. V. Kheruntsyan and H. He, "Novel solitons in parametric amplifiers and atom lasers," J. Opt. B: Quantum Semiclassical Opt. 1, 387-395 (1999).
    [CrossRef]
  33. D. Schneble, Y. Torii, M. Boyd, E. W. Streed, D. E. Pritchard, and W. Ketterle, "The onset of matter-wave amplification in a superradiant Bose-Einstein condensate," Science 300, 475-478 (2003).
    [CrossRef] [PubMed]
  34. L. D. Carr and J. Brand, "Pulsed atomic soliton laser," Phys. Rev. A 70, 033607 (2004).
    [CrossRef]
  35. M. I. Rodas-Verde, H. Michinel, and V. M. Pérez-García, "Controllable soliton emission from a Bose-Einstein condensate," Phys. Rev. Lett. 95, 153903 (2005).
    [CrossRef] [PubMed]
  36. P. Y. P. Chen and B. A. Malomed, "A model of a dual-core matter-wave soliton laser," J. Phys. B 38, 4221-4234 (2005); "Stable circulation modes in a dual-core matter-wave soliton laser," ibid.39, 2803-2813 (2006).
    [CrossRef]
  37. A. V. Carpentier, H. Michinel, M. I. Rodas-Verde, and V. M. Perez-Garcia, "Analysis of an atom laser based on the spatial control of the scattering length," Phys. Rev. A 74, 013619 (2006).
    [CrossRef]

2007 (1)

S. V. Fedorov, N. N. Rosanov, and A. N. Shatsev, "Two-dimensional solitons in B-class lasers with saturable absorption," Opt. Spectra. 102, 449-455 (2007).
[CrossRef]

2006 (5)

D. Mihalache, D. Mazilu, F. Lederer, Y. V. Kartashov, L.-C. Crasovan, L. Torner, and B. A. Malomed, "Stable vortex tori in the three-dimensional cubic-quintic Ginzburg-Landau equation," Phys. Rev. Lett. 97, 073904 (2006).
[CrossRef] [PubMed]

Y. J. He, H. H. Fan, J. W. Dong, and H. Z. Wang, "Self-trapped spatiotemporal necklace-ring solitons in the Ginzburg-Landau equation," Phys. Rev. E 74, 016611 (2006).
[CrossRef]

N. N. Rosanov, S. V. Fedorov, and A. N. Shatsev, "Motion of clusters of weakly coupled two-dimensional cavity solitons," JETP 102, 547-555 (2006); "Curvilinear motion of multivortex laser-soliton complexes with strong and weak coupling," Phys. Rev. Lett. 95, 053903 (2005).
[CrossRef]

A. V. Carpentier, H. Michinel, M. I. Rodas-Verde, and V. M. Perez-Garcia, "Analysis of an atom laser based on the spatial control of the scattering length," Phys. Rev. A 74, 013619 (2006).
[CrossRef]

Y. J. He and H. Z. Wang, "(1+1)-dimensional dipole solitons supported by optical lattice," Opt. Express 14, 9832-9837 (2006).
[CrossRef] [PubMed]

2005 (4)

Z. Chen, H. Martin, E. D. Eugenieva, J. Xu, and J. Yang, "Formation of discrete solitons in light-induced photonic lattices," Opt. Express 13, 1816 (2005).
[CrossRef] [PubMed]

Y. V. Kartashov, R. Carretero-González, B. A. Malomed, V. A. Vysloukh, and L. Torner, "Multipole-mode solitons in Bessel optical lattices," Opt. Express 13, 10703 (2005).
[CrossRef] [PubMed]

M. I. Rodas-Verde, H. Michinel, and V. M. Pérez-García, "Controllable soliton emission from a Bose-Einstein condensate," Phys. Rev. Lett. 95, 153903 (2005).
[CrossRef] [PubMed]

J. Yang, I. Makasyuk, P. G. Kevrekidis, H. Martin, B. A. Malomed, D. J. Frantzeskakis, and Z. Chen, "Necklace-like solitons in optically induced photonic lattices," Phys. Rev. Lett. 94, 113902 (2005).
[CrossRef] [PubMed]

2004 (1)

L. D. Carr and J. Brand, "Pulsed atomic soliton laser," Phys. Rev. A 70, 033607 (2004).
[CrossRef]

2003 (3)

D. Schneble, Y. Torii, M. Boyd, E. W. Streed, D. E. Pritchard, and W. Ketterle, "The onset of matter-wave amplification in a superradiant Bose-Einstein condensate," Science 300, 475-478 (2003).
[CrossRef] [PubMed]

D. Mihalache, D. Mazilu, L.-C. Crasovan, B. A. Malomed, F. Lederer, and L. Torner, "Robust soliton clusters in media with competing cubic and quintic nonlinearities," Phys. Rev. E 68, 046612 (2003).
[CrossRef]

G. Boudebs, S. Cherukulappurath, H. Leblond, J. Troles, F. Smektala, and F. Sanchez, "Experimental and theoretical study of higher-order nonlinearities in chalcogenide glasses," Opt. Commun. 219, 427-433 (2003).
[CrossRef]

2002 (5)

D. V. Skryabin and A. G. Vladimirov, "Vortex-induced rotation of clusters of localized states in the complex Ginzburg-Landau equation," Phys. Rev. Lett. 89, 044101 (2002).
[CrossRef] [PubMed]

A. S. Desyatnikov and Yu. S. Kivshar, "Rotating optical soliton clusters," Phys. Rev. Lett. 88, 053901 (2002).
[CrossRef] [PubMed]

Y. V. Kartashov, L.-C. Crasovan, D. Mihalache, and L. Torner, "Robust propagation of two-color soliton clusters supported by competing nonlinearities." Phys. Rev. Lett. 89, 273902 (2002).
[CrossRef]

C. Zhan, D. Zhang, D. Zhu, D. Wang, Y. Li, D. Li, Z. Lu, L. Zhao, and Y. Nie, "Third- and fifth-order optical nonlinearities in a new stilbazolium derivative," J. Opt. Soc. Am. B 19, 369-375 (2002).
[CrossRef]

I. S. Aranson and L. Kramer, "The world of the complex Ginzburg-Landau equation", Rev. Mod. Phys. 74, 99-143 (2002).
[CrossRef]

2001 (3)

D. Neshev, W. Krolikowski, D. E. Pelinovsky, G. McCarthy, and Y. S. Kivshar, "Transverse instability of vector solitons and generation of dipole arrays," Phys. Rev. Lett. 87, 103903 (2001).
[CrossRef] [PubMed]

T. Carmon, R. Uzdin, C. Pigier, Z. H. Musslimani, M. Segev, and A. Nepomnyashchy, "Rotating propeller solitons," Phys. Rev. Lett. 87, 143901 (2001).
[CrossRef] [PubMed]

L.-C. Crasovan, B. A. Malomed, and D. Mihalache, "Stable vortex solitons in the two-dimensional Ginzburg-Landau equation," Phys. Rev. E 63, 016605 (2001).
[CrossRef]

2000 (2)

F. Smektala, C. Quemard, V. Couderc, and A. Barthelemy, "Non-linear optical properties of chalcogenide glasses measured by Z-scan," J. Non-Cryst. Solids 274, 232-237 (2000).
[CrossRef]

J. J. Garcia-Ripoll, V. M. Perez-Garcia, E. A. Ostrovskaya, and Y. S. Kivshar, "Dipole-mode vector solitons," Phys. Rev. Lett. 85, 82-85 (2000).
[CrossRef] [PubMed]

1999 (2)

E. W. Hagley, L. Deng, M. Kozuma, J. Wen, K. Helmerson, S. L. Rolston, and W. D. Phillips, "A well-collimated quasi-continuous atom laser," Science 283, 1706-1709 (1999).

P. D. Drummond, K. V. Kheruntsyan and H. He, "Novel solitons in parametric amplifiers and atom lasers," J. Opt. B: Quantum Semiclassical Opt. 1, 387-395 (1999).
[CrossRef]

1998 (4)

B. A. Malomed, "Potential of interaction between two- and three-dimensional solitons," Phys. Rev. E 58, 7928-7933 (1998).
[CrossRef]

B. Kneer, T. Wong, K. Vogel, W. P. Schleich, and D. F. Walls, "Generic model of an atom laser," Phys. Rev. A 58, 4841 (1998).
[CrossRef]

H. J. Miesner, D. M. Stamper-Kurn, M. R. D. S. Durfee, S. Inouye, and W. Ketterle, "Bosonic stimulation in the formation of a Bose-Einstein condensate," Science 279, 1005-1007 (1998).
[CrossRef] [PubMed]

M. Solj?ci?, S. Sears, and M. Segev, "Self-trapping of ‘necklace’ beams in self-focusing Kerr media," Phys. Rev. Lett. 81, 4851 (1998); "Integer and fractional angular momentum borne on self-trapped necklace-ring beams," ibid.86, 420 (2001).
[CrossRef]

1994 (1)

J. Lega, J. V. Moloney, and A. C. Newell, "Swift-Hohenberg equation for lasers," Phys. Rev. Lett. 73, 2978 (1994).
[CrossRef] [PubMed]

1991 (1)

B. A. Malomed, "Bound solitons in the nonlinear Schrödinger-Ginzburg-Landau equation," Phys. Rev. A 44, 6954 (1991).
[CrossRef] [PubMed]

Aranson, I. S.

I. S. Aranson and L. Kramer, "The world of the complex Ginzburg-Landau equation", Rev. Mod. Phys. 74, 99-143 (2002).
[CrossRef]

Barthelemy, A.

F. Smektala, C. Quemard, V. Couderc, and A. Barthelemy, "Non-linear optical properties of chalcogenide glasses measured by Z-scan," J. Non-Cryst. Solids 274, 232-237 (2000).
[CrossRef]

Boudebs, G.

G. Boudebs, S. Cherukulappurath, H. Leblond, J. Troles, F. Smektala, and F. Sanchez, "Experimental and theoretical study of higher-order nonlinearities in chalcogenide glasses," Opt. Commun. 219, 427-433 (2003).
[CrossRef]

Boyd, M.

D. Schneble, Y. Torii, M. Boyd, E. W. Streed, D. E. Pritchard, and W. Ketterle, "The onset of matter-wave amplification in a superradiant Bose-Einstein condensate," Science 300, 475-478 (2003).
[CrossRef] [PubMed]

Brand, J.

L. D. Carr and J. Brand, "Pulsed atomic soliton laser," Phys. Rev. A 70, 033607 (2004).
[CrossRef]

Carmon, T.

T. Carmon, R. Uzdin, C. Pigier, Z. H. Musslimani, M. Segev, and A. Nepomnyashchy, "Rotating propeller solitons," Phys. Rev. Lett. 87, 143901 (2001).
[CrossRef] [PubMed]

Carpentier, A. V.

A. V. Carpentier, H. Michinel, M. I. Rodas-Verde, and V. M. Perez-Garcia, "Analysis of an atom laser based on the spatial control of the scattering length," Phys. Rev. A 74, 013619 (2006).
[CrossRef]

Carr, L. D.

L. D. Carr and J. Brand, "Pulsed atomic soliton laser," Phys. Rev. A 70, 033607 (2004).
[CrossRef]

Carretero-González, R.

Chen, Z.

Z. Chen, H. Martin, E. D. Eugenieva, J. Xu, and J. Yang, "Formation of discrete solitons in light-induced photonic lattices," Opt. Express 13, 1816 (2005).
[CrossRef] [PubMed]

J. Yang, I. Makasyuk, P. G. Kevrekidis, H. Martin, B. A. Malomed, D. J. Frantzeskakis, and Z. Chen, "Necklace-like solitons in optically induced photonic lattices," Phys. Rev. Lett. 94, 113902 (2005).
[CrossRef] [PubMed]

Cherukulappurath, S.

G. Boudebs, S. Cherukulappurath, H. Leblond, J. Troles, F. Smektala, and F. Sanchez, "Experimental and theoretical study of higher-order nonlinearities in chalcogenide glasses," Opt. Commun. 219, 427-433 (2003).
[CrossRef]

Couderc, V.

F. Smektala, C. Quemard, V. Couderc, and A. Barthelemy, "Non-linear optical properties of chalcogenide glasses measured by Z-scan," J. Non-Cryst. Solids 274, 232-237 (2000).
[CrossRef]

Crasovan, L.-C.

D. Mihalache, D. Mazilu, F. Lederer, Y. V. Kartashov, L.-C. Crasovan, L. Torner, and B. A. Malomed, "Stable vortex tori in the three-dimensional cubic-quintic Ginzburg-Landau equation," Phys. Rev. Lett. 97, 073904 (2006).
[CrossRef] [PubMed]

D. Mihalache, D. Mazilu, L.-C. Crasovan, B. A. Malomed, F. Lederer, and L. Torner, "Robust soliton clusters in media with competing cubic and quintic nonlinearities," Phys. Rev. E 68, 046612 (2003).
[CrossRef]

Y. V. Kartashov, L.-C. Crasovan, D. Mihalache, and L. Torner, "Robust propagation of two-color soliton clusters supported by competing nonlinearities." Phys. Rev. Lett. 89, 273902 (2002).
[CrossRef]

L.-C. Crasovan, B. A. Malomed, and D. Mihalache, "Stable vortex solitons in the two-dimensional Ginzburg-Landau equation," Phys. Rev. E 63, 016605 (2001).
[CrossRef]

Deng, L.

E. W. Hagley, L. Deng, M. Kozuma, J. Wen, K. Helmerson, S. L. Rolston, and W. D. Phillips, "A well-collimated quasi-continuous atom laser," Science 283, 1706-1709 (1999).

Desyatnikov, A. S.

A. S. Desyatnikov and Yu. S. Kivshar, "Rotating optical soliton clusters," Phys. Rev. Lett. 88, 053901 (2002).
[CrossRef] [PubMed]

Dong, J. W.

Y. J. He, H. H. Fan, J. W. Dong, and H. Z. Wang, "Self-trapped spatiotemporal necklace-ring solitons in the Ginzburg-Landau equation," Phys. Rev. E 74, 016611 (2006).
[CrossRef]

Drummond, P. D.

P. D. Drummond, K. V. Kheruntsyan and H. He, "Novel solitons in parametric amplifiers and atom lasers," J. Opt. B: Quantum Semiclassical Opt. 1, 387-395 (1999).
[CrossRef]

Durfee, M. R. D. S.

H. J. Miesner, D. M. Stamper-Kurn, M. R. D. S. Durfee, S. Inouye, and W. Ketterle, "Bosonic stimulation in the formation of a Bose-Einstein condensate," Science 279, 1005-1007 (1998).
[CrossRef] [PubMed]

Eugenieva, E. D.

Fan, H. H.

Y. J. He, H. H. Fan, J. W. Dong, and H. Z. Wang, "Self-trapped spatiotemporal necklace-ring solitons in the Ginzburg-Landau equation," Phys. Rev. E 74, 016611 (2006).
[CrossRef]

Fedorov, S. V.

S. V. Fedorov, N. N. Rosanov, and A. N. Shatsev, "Two-dimensional solitons in B-class lasers with saturable absorption," Opt. Spectra. 102, 449-455 (2007).
[CrossRef]

N. N. Rosanov, S. V. Fedorov, and A. N. Shatsev, "Motion of clusters of weakly coupled two-dimensional cavity solitons," JETP 102, 547-555 (2006); "Curvilinear motion of multivortex laser-soliton complexes with strong and weak coupling," Phys. Rev. Lett. 95, 053903 (2005).
[CrossRef]

Frantzeskakis, D. J.

J. Yang, I. Makasyuk, P. G. Kevrekidis, H. Martin, B. A. Malomed, D. J. Frantzeskakis, and Z. Chen, "Necklace-like solitons in optically induced photonic lattices," Phys. Rev. Lett. 94, 113902 (2005).
[CrossRef] [PubMed]

Garcia-Ripoll, J. J.

J. J. Garcia-Ripoll, V. M. Perez-Garcia, E. A. Ostrovskaya, and Y. S. Kivshar, "Dipole-mode vector solitons," Phys. Rev. Lett. 85, 82-85 (2000).
[CrossRef] [PubMed]

Hagley, E. W.

E. W. Hagley, L. Deng, M. Kozuma, J. Wen, K. Helmerson, S. L. Rolston, and W. D. Phillips, "A well-collimated quasi-continuous atom laser," Science 283, 1706-1709 (1999).

He, H.

P. D. Drummond, K. V. Kheruntsyan and H. He, "Novel solitons in parametric amplifiers and atom lasers," J. Opt. B: Quantum Semiclassical Opt. 1, 387-395 (1999).
[CrossRef]

He, Y. J.

Y. J. He, H. H. Fan, J. W. Dong, and H. Z. Wang, "Self-trapped spatiotemporal necklace-ring solitons in the Ginzburg-Landau equation," Phys. Rev. E 74, 016611 (2006).
[CrossRef]

Y. J. He and H. Z. Wang, "(1+1)-dimensional dipole solitons supported by optical lattice," Opt. Express 14, 9832-9837 (2006).
[CrossRef] [PubMed]

Helmerson, K.

E. W. Hagley, L. Deng, M. Kozuma, J. Wen, K. Helmerson, S. L. Rolston, and W. D. Phillips, "A well-collimated quasi-continuous atom laser," Science 283, 1706-1709 (1999).

Inouye, S.

H. J. Miesner, D. M. Stamper-Kurn, M. R. D. S. Durfee, S. Inouye, and W. Ketterle, "Bosonic stimulation in the formation of a Bose-Einstein condensate," Science 279, 1005-1007 (1998).
[CrossRef] [PubMed]

Kartashov, Y. V.

D. Mihalache, D. Mazilu, F. Lederer, Y. V. Kartashov, L.-C. Crasovan, L. Torner, and B. A. Malomed, "Stable vortex tori in the three-dimensional cubic-quintic Ginzburg-Landau equation," Phys. Rev. Lett. 97, 073904 (2006).
[CrossRef] [PubMed]

Y. V. Kartashov, R. Carretero-González, B. A. Malomed, V. A. Vysloukh, and L. Torner, "Multipole-mode solitons in Bessel optical lattices," Opt. Express 13, 10703 (2005).
[CrossRef] [PubMed]

Y. V. Kartashov, L.-C. Crasovan, D. Mihalache, and L. Torner, "Robust propagation of two-color soliton clusters supported by competing nonlinearities." Phys. Rev. Lett. 89, 273902 (2002).
[CrossRef]

Ketterle, W.

D. Schneble, Y. Torii, M. Boyd, E. W. Streed, D. E. Pritchard, and W. Ketterle, "The onset of matter-wave amplification in a superradiant Bose-Einstein condensate," Science 300, 475-478 (2003).
[CrossRef] [PubMed]

H. J. Miesner, D. M. Stamper-Kurn, M. R. D. S. Durfee, S. Inouye, and W. Ketterle, "Bosonic stimulation in the formation of a Bose-Einstein condensate," Science 279, 1005-1007 (1998).
[CrossRef] [PubMed]

Kevrekidis, P. G.

J. Yang, I. Makasyuk, P. G. Kevrekidis, H. Martin, B. A. Malomed, D. J. Frantzeskakis, and Z. Chen, "Necklace-like solitons in optically induced photonic lattices," Phys. Rev. Lett. 94, 113902 (2005).
[CrossRef] [PubMed]

Kheruntsyan, K. V.

P. D. Drummond, K. V. Kheruntsyan and H. He, "Novel solitons in parametric amplifiers and atom lasers," J. Opt. B: Quantum Semiclassical Opt. 1, 387-395 (1999).
[CrossRef]

Kivshar, Y. S.

D. Neshev, W. Krolikowski, D. E. Pelinovsky, G. McCarthy, and Y. S. Kivshar, "Transverse instability of vector solitons and generation of dipole arrays," Phys. Rev. Lett. 87, 103903 (2001).
[CrossRef] [PubMed]

J. J. Garcia-Ripoll, V. M. Perez-Garcia, E. A. Ostrovskaya, and Y. S. Kivshar, "Dipole-mode vector solitons," Phys. Rev. Lett. 85, 82-85 (2000).
[CrossRef] [PubMed]

Kivshar, Yu. S.

A. S. Desyatnikov and Yu. S. Kivshar, "Rotating optical soliton clusters," Phys. Rev. Lett. 88, 053901 (2002).
[CrossRef] [PubMed]

Kneer, B.

B. Kneer, T. Wong, K. Vogel, W. P. Schleich, and D. F. Walls, "Generic model of an atom laser," Phys. Rev. A 58, 4841 (1998).
[CrossRef]

Kozuma, M.

E. W. Hagley, L. Deng, M. Kozuma, J. Wen, K. Helmerson, S. L. Rolston, and W. D. Phillips, "A well-collimated quasi-continuous atom laser," Science 283, 1706-1709 (1999).

Kramer, L.

I. S. Aranson and L. Kramer, "The world of the complex Ginzburg-Landau equation", Rev. Mod. Phys. 74, 99-143 (2002).
[CrossRef]

Krolikowski, W.

D. Neshev, W. Krolikowski, D. E. Pelinovsky, G. McCarthy, and Y. S. Kivshar, "Transverse instability of vector solitons and generation of dipole arrays," Phys. Rev. Lett. 87, 103903 (2001).
[CrossRef] [PubMed]

Leblond, H.

G. Boudebs, S. Cherukulappurath, H. Leblond, J. Troles, F. Smektala, and F. Sanchez, "Experimental and theoretical study of higher-order nonlinearities in chalcogenide glasses," Opt. Commun. 219, 427-433 (2003).
[CrossRef]

Lederer, F.

D. Mihalache, D. Mazilu, F. Lederer, Y. V. Kartashov, L.-C. Crasovan, L. Torner, and B. A. Malomed, "Stable vortex tori in the three-dimensional cubic-quintic Ginzburg-Landau equation," Phys. Rev. Lett. 97, 073904 (2006).
[CrossRef] [PubMed]

D. Mihalache, D. Mazilu, L.-C. Crasovan, B. A. Malomed, F. Lederer, and L. Torner, "Robust soliton clusters in media with competing cubic and quintic nonlinearities," Phys. Rev. E 68, 046612 (2003).
[CrossRef]

Lega, J.

J. Lega, J. V. Moloney, and A. C. Newell, "Swift-Hohenberg equation for lasers," Phys. Rev. Lett. 73, 2978 (1994).
[CrossRef] [PubMed]

Li, D.

Li, Y.

Lu, Z.

Makasyuk, I.

J. Yang, I. Makasyuk, P. G. Kevrekidis, H. Martin, B. A. Malomed, D. J. Frantzeskakis, and Z. Chen, "Necklace-like solitons in optically induced photonic lattices," Phys. Rev. Lett. 94, 113902 (2005).
[CrossRef] [PubMed]

Malomed, B. A.

D. Mihalache, D. Mazilu, F. Lederer, Y. V. Kartashov, L.-C. Crasovan, L. Torner, and B. A. Malomed, "Stable vortex tori in the three-dimensional cubic-quintic Ginzburg-Landau equation," Phys. Rev. Lett. 97, 073904 (2006).
[CrossRef] [PubMed]

J. Yang, I. Makasyuk, P. G. Kevrekidis, H. Martin, B. A. Malomed, D. J. Frantzeskakis, and Z. Chen, "Necklace-like solitons in optically induced photonic lattices," Phys. Rev. Lett. 94, 113902 (2005).
[CrossRef] [PubMed]

Y. V. Kartashov, R. Carretero-González, B. A. Malomed, V. A. Vysloukh, and L. Torner, "Multipole-mode solitons in Bessel optical lattices," Opt. Express 13, 10703 (2005).
[CrossRef] [PubMed]

D. Mihalache, D. Mazilu, L.-C. Crasovan, B. A. Malomed, F. Lederer, and L. Torner, "Robust soliton clusters in media with competing cubic and quintic nonlinearities," Phys. Rev. E 68, 046612 (2003).
[CrossRef]

L.-C. Crasovan, B. A. Malomed, and D. Mihalache, "Stable vortex solitons in the two-dimensional Ginzburg-Landau equation," Phys. Rev. E 63, 016605 (2001).
[CrossRef]

B. A. Malomed, "Potential of interaction between two- and three-dimensional solitons," Phys. Rev. E 58, 7928-7933 (1998).
[CrossRef]

B. A. Malomed, "Bound solitons in the nonlinear Schrödinger-Ginzburg-Landau equation," Phys. Rev. A 44, 6954 (1991).
[CrossRef] [PubMed]

Martin, H.

J. Yang, I. Makasyuk, P. G. Kevrekidis, H. Martin, B. A. Malomed, D. J. Frantzeskakis, and Z. Chen, "Necklace-like solitons in optically induced photonic lattices," Phys. Rev. Lett. 94, 113902 (2005).
[CrossRef] [PubMed]

Z. Chen, H. Martin, E. D. Eugenieva, J. Xu, and J. Yang, "Formation of discrete solitons in light-induced photonic lattices," Opt. Express 13, 1816 (2005).
[CrossRef] [PubMed]

Mazilu, D.

D. Mihalache, D. Mazilu, F. Lederer, Y. V. Kartashov, L.-C. Crasovan, L. Torner, and B. A. Malomed, "Stable vortex tori in the three-dimensional cubic-quintic Ginzburg-Landau equation," Phys. Rev. Lett. 97, 073904 (2006).
[CrossRef] [PubMed]

D. Mihalache, D. Mazilu, L.-C. Crasovan, B. A. Malomed, F. Lederer, and L. Torner, "Robust soliton clusters in media with competing cubic and quintic nonlinearities," Phys. Rev. E 68, 046612 (2003).
[CrossRef]

McCarthy, G.

D. Neshev, W. Krolikowski, D. E. Pelinovsky, G. McCarthy, and Y. S. Kivshar, "Transverse instability of vector solitons and generation of dipole arrays," Phys. Rev. Lett. 87, 103903 (2001).
[CrossRef] [PubMed]

Michinel, H.

A. V. Carpentier, H. Michinel, M. I. Rodas-Verde, and V. M. Perez-Garcia, "Analysis of an atom laser based on the spatial control of the scattering length," Phys. Rev. A 74, 013619 (2006).
[CrossRef]

M. I. Rodas-Verde, H. Michinel, and V. M. Pérez-García, "Controllable soliton emission from a Bose-Einstein condensate," Phys. Rev. Lett. 95, 153903 (2005).
[CrossRef] [PubMed]

Miesner, H. J.

H. J. Miesner, D. M. Stamper-Kurn, M. R. D. S. Durfee, S. Inouye, and W. Ketterle, "Bosonic stimulation in the formation of a Bose-Einstein condensate," Science 279, 1005-1007 (1998).
[CrossRef] [PubMed]

Mihalache, D.

D. Mihalache, D. Mazilu, F. Lederer, Y. V. Kartashov, L.-C. Crasovan, L. Torner, and B. A. Malomed, "Stable vortex tori in the three-dimensional cubic-quintic Ginzburg-Landau equation," Phys. Rev. Lett. 97, 073904 (2006).
[CrossRef] [PubMed]

D. Mihalache, D. Mazilu, L.-C. Crasovan, B. A. Malomed, F. Lederer, and L. Torner, "Robust soliton clusters in media with competing cubic and quintic nonlinearities," Phys. Rev. E 68, 046612 (2003).
[CrossRef]

Y. V. Kartashov, L.-C. Crasovan, D. Mihalache, and L. Torner, "Robust propagation of two-color soliton clusters supported by competing nonlinearities." Phys. Rev. Lett. 89, 273902 (2002).
[CrossRef]

L.-C. Crasovan, B. A. Malomed, and D. Mihalache, "Stable vortex solitons in the two-dimensional Ginzburg-Landau equation," Phys. Rev. E 63, 016605 (2001).
[CrossRef]

Moloney, J. V.

J. Lega, J. V. Moloney, and A. C. Newell, "Swift-Hohenberg equation for lasers," Phys. Rev. Lett. 73, 2978 (1994).
[CrossRef] [PubMed]

Musslimani, Z. H.

T. Carmon, R. Uzdin, C. Pigier, Z. H. Musslimani, M. Segev, and A. Nepomnyashchy, "Rotating propeller solitons," Phys. Rev. Lett. 87, 143901 (2001).
[CrossRef] [PubMed]

Nepomnyashchy, A.

T. Carmon, R. Uzdin, C. Pigier, Z. H. Musslimani, M. Segev, and A. Nepomnyashchy, "Rotating propeller solitons," Phys. Rev. Lett. 87, 143901 (2001).
[CrossRef] [PubMed]

Neshev, D.

D. Neshev, W. Krolikowski, D. E. Pelinovsky, G. McCarthy, and Y. S. Kivshar, "Transverse instability of vector solitons and generation of dipole arrays," Phys. Rev. Lett. 87, 103903 (2001).
[CrossRef] [PubMed]

Newell, A. C.

J. Lega, J. V. Moloney, and A. C. Newell, "Swift-Hohenberg equation for lasers," Phys. Rev. Lett. 73, 2978 (1994).
[CrossRef] [PubMed]

Nie, Y.

Ostrovskaya, E. A.

J. J. Garcia-Ripoll, V. M. Perez-Garcia, E. A. Ostrovskaya, and Y. S. Kivshar, "Dipole-mode vector solitons," Phys. Rev. Lett. 85, 82-85 (2000).
[CrossRef] [PubMed]

Pelinovsky, D. E.

D. Neshev, W. Krolikowski, D. E. Pelinovsky, G. McCarthy, and Y. S. Kivshar, "Transverse instability of vector solitons and generation of dipole arrays," Phys. Rev. Lett. 87, 103903 (2001).
[CrossRef] [PubMed]

Perez-Garcia, V. M.

A. V. Carpentier, H. Michinel, M. I. Rodas-Verde, and V. M. Perez-Garcia, "Analysis of an atom laser based on the spatial control of the scattering length," Phys. Rev. A 74, 013619 (2006).
[CrossRef]

J. J. Garcia-Ripoll, V. M. Perez-Garcia, E. A. Ostrovskaya, and Y. S. Kivshar, "Dipole-mode vector solitons," Phys. Rev. Lett. 85, 82-85 (2000).
[CrossRef] [PubMed]

Pérez-García, V. M.

M. I. Rodas-Verde, H. Michinel, and V. M. Pérez-García, "Controllable soliton emission from a Bose-Einstein condensate," Phys. Rev. Lett. 95, 153903 (2005).
[CrossRef] [PubMed]

Phillips, W. D.

E. W. Hagley, L. Deng, M. Kozuma, J. Wen, K. Helmerson, S. L. Rolston, and W. D. Phillips, "A well-collimated quasi-continuous atom laser," Science 283, 1706-1709 (1999).

Pigier, C.

T. Carmon, R. Uzdin, C. Pigier, Z. H. Musslimani, M. Segev, and A. Nepomnyashchy, "Rotating propeller solitons," Phys. Rev. Lett. 87, 143901 (2001).
[CrossRef] [PubMed]

Pritchard, D. E.

D. Schneble, Y. Torii, M. Boyd, E. W. Streed, D. E. Pritchard, and W. Ketterle, "The onset of matter-wave amplification in a superradiant Bose-Einstein condensate," Science 300, 475-478 (2003).
[CrossRef] [PubMed]

Quemard, C.

F. Smektala, C. Quemard, V. Couderc, and A. Barthelemy, "Non-linear optical properties of chalcogenide glasses measured by Z-scan," J. Non-Cryst. Solids 274, 232-237 (2000).
[CrossRef]

Rodas-Verde, M. I.

A. V. Carpentier, H. Michinel, M. I. Rodas-Verde, and V. M. Perez-Garcia, "Analysis of an atom laser based on the spatial control of the scattering length," Phys. Rev. A 74, 013619 (2006).
[CrossRef]

M. I. Rodas-Verde, H. Michinel, and V. M. Pérez-García, "Controllable soliton emission from a Bose-Einstein condensate," Phys. Rev. Lett. 95, 153903 (2005).
[CrossRef] [PubMed]

Rolston, S. L.

E. W. Hagley, L. Deng, M. Kozuma, J. Wen, K. Helmerson, S. L. Rolston, and W. D. Phillips, "A well-collimated quasi-continuous atom laser," Science 283, 1706-1709 (1999).

Rosanov, N. N.

S. V. Fedorov, N. N. Rosanov, and A. N. Shatsev, "Two-dimensional solitons in B-class lasers with saturable absorption," Opt. Spectra. 102, 449-455 (2007).
[CrossRef]

N. N. Rosanov, S. V. Fedorov, and A. N. Shatsev, "Motion of clusters of weakly coupled two-dimensional cavity solitons," JETP 102, 547-555 (2006); "Curvilinear motion of multivortex laser-soliton complexes with strong and weak coupling," Phys. Rev. Lett. 95, 053903 (2005).
[CrossRef]

Sanchez, F.

G. Boudebs, S. Cherukulappurath, H. Leblond, J. Troles, F. Smektala, and F. Sanchez, "Experimental and theoretical study of higher-order nonlinearities in chalcogenide glasses," Opt. Commun. 219, 427-433 (2003).
[CrossRef]

Schleich, W. P.

B. Kneer, T. Wong, K. Vogel, W. P. Schleich, and D. F. Walls, "Generic model of an atom laser," Phys. Rev. A 58, 4841 (1998).
[CrossRef]

Schneble, D.

D. Schneble, Y. Torii, M. Boyd, E. W. Streed, D. E. Pritchard, and W. Ketterle, "The onset of matter-wave amplification in a superradiant Bose-Einstein condensate," Science 300, 475-478 (2003).
[CrossRef] [PubMed]

Sears, S.

M. Solj?ci?, S. Sears, and M. Segev, "Self-trapping of ‘necklace’ beams in self-focusing Kerr media," Phys. Rev. Lett. 81, 4851 (1998); "Integer and fractional angular momentum borne on self-trapped necklace-ring beams," ibid.86, 420 (2001).
[CrossRef]

Segev, M.

T. Carmon, R. Uzdin, C. Pigier, Z. H. Musslimani, M. Segev, and A. Nepomnyashchy, "Rotating propeller solitons," Phys. Rev. Lett. 87, 143901 (2001).
[CrossRef] [PubMed]

M. Solj?ci?, S. Sears, and M. Segev, "Self-trapping of ‘necklace’ beams in self-focusing Kerr media," Phys. Rev. Lett. 81, 4851 (1998); "Integer and fractional angular momentum borne on self-trapped necklace-ring beams," ibid.86, 420 (2001).
[CrossRef]

Shatsev, A. N.

S. V. Fedorov, N. N. Rosanov, and A. N. Shatsev, "Two-dimensional solitons in B-class lasers with saturable absorption," Opt. Spectra. 102, 449-455 (2007).
[CrossRef]

N. N. Rosanov, S. V. Fedorov, and A. N. Shatsev, "Motion of clusters of weakly coupled two-dimensional cavity solitons," JETP 102, 547-555 (2006); "Curvilinear motion of multivortex laser-soliton complexes with strong and weak coupling," Phys. Rev. Lett. 95, 053903 (2005).
[CrossRef]

Skryabin, D. V.

D. V. Skryabin and A. G. Vladimirov, "Vortex-induced rotation of clusters of localized states in the complex Ginzburg-Landau equation," Phys. Rev. Lett. 89, 044101 (2002).
[CrossRef] [PubMed]

Smektala, F.

G. Boudebs, S. Cherukulappurath, H. Leblond, J. Troles, F. Smektala, and F. Sanchez, "Experimental and theoretical study of higher-order nonlinearities in chalcogenide glasses," Opt. Commun. 219, 427-433 (2003).
[CrossRef]

F. Smektala, C. Quemard, V. Couderc, and A. Barthelemy, "Non-linear optical properties of chalcogenide glasses measured by Z-scan," J. Non-Cryst. Solids 274, 232-237 (2000).
[CrossRef]

Soljacic, M.

M. Solj?ci?, S. Sears, and M. Segev, "Self-trapping of ‘necklace’ beams in self-focusing Kerr media," Phys. Rev. Lett. 81, 4851 (1998); "Integer and fractional angular momentum borne on self-trapped necklace-ring beams," ibid.86, 420 (2001).
[CrossRef]

Stamper-Kurn, D. M.

H. J. Miesner, D. M. Stamper-Kurn, M. R. D. S. Durfee, S. Inouye, and W. Ketterle, "Bosonic stimulation in the formation of a Bose-Einstein condensate," Science 279, 1005-1007 (1998).
[CrossRef] [PubMed]

Streed, E. W.

D. Schneble, Y. Torii, M. Boyd, E. W. Streed, D. E. Pritchard, and W. Ketterle, "The onset of matter-wave amplification in a superradiant Bose-Einstein condensate," Science 300, 475-478 (2003).
[CrossRef] [PubMed]

Torii, Y.

D. Schneble, Y. Torii, M. Boyd, E. W. Streed, D. E. Pritchard, and W. Ketterle, "The onset of matter-wave amplification in a superradiant Bose-Einstein condensate," Science 300, 475-478 (2003).
[CrossRef] [PubMed]

Torner, L.

D. Mihalache, D. Mazilu, F. Lederer, Y. V. Kartashov, L.-C. Crasovan, L. Torner, and B. A. Malomed, "Stable vortex tori in the three-dimensional cubic-quintic Ginzburg-Landau equation," Phys. Rev. Lett. 97, 073904 (2006).
[CrossRef] [PubMed]

Y. V. Kartashov, R. Carretero-González, B. A. Malomed, V. A. Vysloukh, and L. Torner, "Multipole-mode solitons in Bessel optical lattices," Opt. Express 13, 10703 (2005).
[CrossRef] [PubMed]

D. Mihalache, D. Mazilu, L.-C. Crasovan, B. A. Malomed, F. Lederer, and L. Torner, "Robust soliton clusters in media with competing cubic and quintic nonlinearities," Phys. Rev. E 68, 046612 (2003).
[CrossRef]

Y. V. Kartashov, L.-C. Crasovan, D. Mihalache, and L. Torner, "Robust propagation of two-color soliton clusters supported by competing nonlinearities." Phys. Rev. Lett. 89, 273902 (2002).
[CrossRef]

Troles, J.

G. Boudebs, S. Cherukulappurath, H. Leblond, J. Troles, F. Smektala, and F. Sanchez, "Experimental and theoretical study of higher-order nonlinearities in chalcogenide glasses," Opt. Commun. 219, 427-433 (2003).
[CrossRef]

Uzdin, R.

T. Carmon, R. Uzdin, C. Pigier, Z. H. Musslimani, M. Segev, and A. Nepomnyashchy, "Rotating propeller solitons," Phys. Rev. Lett. 87, 143901 (2001).
[CrossRef] [PubMed]

Vladimirov, A. G.

D. V. Skryabin and A. G. Vladimirov, "Vortex-induced rotation of clusters of localized states in the complex Ginzburg-Landau equation," Phys. Rev. Lett. 89, 044101 (2002).
[CrossRef] [PubMed]

Vogel, K.

B. Kneer, T. Wong, K. Vogel, W. P. Schleich, and D. F. Walls, "Generic model of an atom laser," Phys. Rev. A 58, 4841 (1998).
[CrossRef]

Vysloukh, V. A.

Walls, D. F.

B. Kneer, T. Wong, K. Vogel, W. P. Schleich, and D. F. Walls, "Generic model of an atom laser," Phys. Rev. A 58, 4841 (1998).
[CrossRef]

Wang, D.

Wang, H. Z.

Y. J. He, H. H. Fan, J. W. Dong, and H. Z. Wang, "Self-trapped spatiotemporal necklace-ring solitons in the Ginzburg-Landau equation," Phys. Rev. E 74, 016611 (2006).
[CrossRef]

Y. J. He and H. Z. Wang, "(1+1)-dimensional dipole solitons supported by optical lattice," Opt. Express 14, 9832-9837 (2006).
[CrossRef] [PubMed]

Wen, J.

E. W. Hagley, L. Deng, M. Kozuma, J. Wen, K. Helmerson, S. L. Rolston, and W. D. Phillips, "A well-collimated quasi-continuous atom laser," Science 283, 1706-1709 (1999).

Wong, T.

B. Kneer, T. Wong, K. Vogel, W. P. Schleich, and D. F. Walls, "Generic model of an atom laser," Phys. Rev. A 58, 4841 (1998).
[CrossRef]

Xu, J.

Yang, J.

Z. Chen, H. Martin, E. D. Eugenieva, J. Xu, and J. Yang, "Formation of discrete solitons in light-induced photonic lattices," Opt. Express 13, 1816 (2005).
[CrossRef] [PubMed]

J. Yang, I. Makasyuk, P. G. Kevrekidis, H. Martin, B. A. Malomed, D. J. Frantzeskakis, and Z. Chen, "Necklace-like solitons in optically induced photonic lattices," Phys. Rev. Lett. 94, 113902 (2005).
[CrossRef] [PubMed]

Zhan, C.

Zhang, D.

Zhao, L.

Zhu, D.

J. Non-Cryst. Solids (1)

F. Smektala, C. Quemard, V. Couderc, and A. Barthelemy, "Non-linear optical properties of chalcogenide glasses measured by Z-scan," J. Non-Cryst. Solids 274, 232-237 (2000).
[CrossRef]

J. Opt. B: Quantum Semiclassical Opt. (1)

P. D. Drummond, K. V. Kheruntsyan and H. He, "Novel solitons in parametric amplifiers and atom lasers," J. Opt. B: Quantum Semiclassical Opt. 1, 387-395 (1999).
[CrossRef]

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

JETP (1)

N. N. Rosanov, S. V. Fedorov, and A. N. Shatsev, "Motion of clusters of weakly coupled two-dimensional cavity solitons," JETP 102, 547-555 (2006); "Curvilinear motion of multivortex laser-soliton complexes with strong and weak coupling," Phys. Rev. Lett. 95, 053903 (2005).
[CrossRef]

Opt. Commun. (1)

G. Boudebs, S. Cherukulappurath, H. Leblond, J. Troles, F. Smektala, and F. Sanchez, "Experimental and theoretical study of higher-order nonlinearities in chalcogenide glasses," Opt. Commun. 219, 427-433 (2003).
[CrossRef]

Opt. Express (3)

Opt. Spectra. (1)

S. V. Fedorov, N. N. Rosanov, and A. N. Shatsev, "Two-dimensional solitons in B-class lasers with saturable absorption," Opt. Spectra. 102, 449-455 (2007).
[CrossRef]

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[CrossRef]

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[CrossRef] [PubMed]

Phys. Rev. E (4)

Y. J. He, H. H. Fan, J. W. Dong, and H. Z. Wang, "Self-trapped spatiotemporal necklace-ring solitons in the Ginzburg-Landau equation," Phys. Rev. E 74, 016611 (2006).
[CrossRef]

D. Mihalache, D. Mazilu, L.-C. Crasovan, B. A. Malomed, F. Lederer, and L. Torner, "Robust soliton clusters in media with competing cubic and quintic nonlinearities," Phys. Rev. E 68, 046612 (2003).
[CrossRef]

L.-C. Crasovan, B. A. Malomed, and D. Mihalache, "Stable vortex solitons in the two-dimensional Ginzburg-Landau equation," Phys. Rev. E 63, 016605 (2001).
[CrossRef]

B. A. Malomed, "Potential of interaction between two- and three-dimensional solitons," Phys. Rev. E 58, 7928-7933 (1998).
[CrossRef]

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D. Mihalache, D. Mazilu, F. Lederer, Y. V. Kartashov, L.-C. Crasovan, L. Torner, and B. A. Malomed, "Stable vortex tori in the three-dimensional cubic-quintic Ginzburg-Landau equation," Phys. Rev. Lett. 97, 073904 (2006).
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Other (5)

P. Y. P. Chen and B. A. Malomed, "A model of a dual-core matter-wave soliton laser," J. Phys. B 38, 4221-4234 (2005); "Stable circulation modes in a dual-core matter-wave soliton laser," ibid.39, 2803-2813 (2006).
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N. N. Rosanov, Spatial Hysteresis and Optical Patterns (Springer-Verlag, 2002).

B. A. Malomed, "Complex Ginzburg-Landau equation," in Encyclopedia of Nonlinear Science, A. Scott, ed., (Routledge, New York, 2005) pp. 157-160; "Solitary pulses in linearly coupled Ginzburg-Landau equations," Chaos 17, 037117 (2007).

N. Akhmediev and A. Ankiewicz, "Dissipative Solitons," Lecture Notes in Physics (Springer, Berlin, 2005) Vol. 661.
[CrossRef]

D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed, "Stability of dissipative optical solitons in the three-dimensional cubic-quintic Ginzburg-Landau equation," Phys. Rev. A 75, 033811 (2007); "Stability limits for three-dimensional vortex solitons in the Ginzburg-Landau equation with the cubic-quintic nonlinearity," ibid. A 76, 045803 (2007).
[CrossRef]

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Figures (4)

Fig. 1.
Fig. 1.

The fusion into a fundamental soliton of the necklace array whose initial radius is not too large, R 0RFmax , and the topological charge is equal to half the number of “beads” in the array, M=N. (a): The largest radius, admitting the fusion, versus N. (b, c): Examples of the fusion for R 0=4 and M=N=5 (b) or M=N=3 (c) [the examples are shown by means of contour plots of the local power, |u(x,y)|2].

Fig. 2.
Fig. 2.

The fusion into a stable vortex soliton of the necklace whose initial radius and topological charge satisfy constraints RV minR 0≤1.8N, and either MVmin≤M<N or N<M≤MVmax≡N-M. (a): The minimum initial radius admitting the fusion versus N; (b): the minimum topological charge versus the initial radius for different fixed values of modulation number N. (c-e): Examples of the formation of the vortex soliton with N=5 and R 0=5 for M=4 (c), M=3 (d), and M=2 (e). Additionally, panel (f) displays an example of the decay of the necklace cluster with N=5 and M=4, in the case of R 0<RV min (here, R 0=1.9 and RV min =2).

Fig. 3.
Fig. 3.

(a). Slow expansion of the necklace array with N=5, M=0 and R 0=11, in the case when its initial radius slightly exceeds the maximum value 1.8N, see Eq. (3). (b): In the same case, but with M=3, ten initial “beads” fuse into eight and, eventually, into six individual elements.

Fig. 4.
Fig. 4.

Formation of “frozen” patterns which look like stable soliton necklaces, with the initial necklace radius exceeding RNmin and an arbitrary value of the topological charge (M). (a): RNmin as a function of modulation number N (b, c): Examples of the formation of stable necklace rings with N=5 and R 0=12 for M=2 (b) and M=0 (c).

Equations (3)

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i u z + i α u + ( 1 2 i β ) ( u XX + u YY ) + ( 1 i ε ) u 2 u ( ν i μ ) u 2 u = 0 ,
u ( Z = 0 , r , θ ) = A sech ( ( r R 0 ) w ) cos ( N θ ) exp ( i M θ ) .
R m in V R 0 1.8 N ,

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