Abstract

We consider two-dimensional (2D) localized modes in the second-harmonic-generating (χ(2)) system with the harmonic-oscillator (HO) trapping potential. In addition to its realization in optics, the system describes the mean-field dynamics of mixed atomic-molecular Bose–Einstein condensates (BECs). The existence and stability of various modes is determined by their total power, N, topological charge, m/2 [m is the intrinsic vorticity of the second-harmonic (SH) field], and χ(2) mismatch, q. The analysis is carried out in a numerical form and, in parallel, by means of the variational approximation (VA), which produces results that agree well with numerical findings. Below a certain power threshold, NNc(m)(q), all trapped modes are of the single-color type, represented by the SH component only, while the fundamental frequency (FF) one is absent. In contrast with the usual situation, where such modes are always unstable, we demonstrate that they are stable, for m=0, 1, 2 (the mode with m=1 may be formally considered as a semivortex with topological charge m/2=1/2), at NNc(m)(q), and unstable above this threshold. On the other hand, Nc(m)(q)0 at qqmax (in our notation, qmax=1); hence the single-color modes are unstable in the latter case. At N=Nc(m), the modes with m=0 and m=2 undergo a pitchfork bifurcation, which gives rise to two-color states, which remain completely stable for m=0. The two-color vortices with m=2 (topological charge 1) have an upper stability border, N=Nc2(q). Above the border, they exhibit periodic splittings and recombinations, while keeping their vorticity. The semivortex does not bifurcate; at N=Nc(m=1), it exhibits quasi-chaotic oscillations and a rotating “groove” resembling a screw-edge dislocation induced by the semi-integer vorticity.

© 2012 Optical Society of America

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  1. G. I. Stegeman, D. J. Hagan, and L. Torner, “χ(2) cascading phenomena and their applications to all-optical signal processing, mode-locking, pulse compression and solitons,” Opt. Quantum Electron. 28, 1691–1740 (1996).
    [CrossRef]
  2. C. Etrich, F. Lederer, B. A. Malomed, T. Peschel, and U. Peschel, “Optical solitons in media with a quadratic nonlinearity,” Prog. Opt. 41, 483–568 (2000).
    [CrossRef]
  3. A. V. Buryak, P. Di Trapani, D. V. Skryabin, and S. Trillo, “Optical solitons due to quadratic nonlinearities: from basic physics to futuristic applications,” Phys. Rep. 370, 63–235 (2002).
    [CrossRef]
  4. B. A. Malomed, D. Mihalache, F. Wise, and L. Torner, “Spatiotemporal optical solitons,” J. Opt. B 7, R53–R72 (2005).
    [CrossRef]
  5. Y. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic, 2003).
  6. A. A. Kanashov and A. M. Rubenchik, “On diffraction and dispersion effect on three wave interaction,” Physica D 4, 122–134 (1981).
    [CrossRef]
  7. W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. VanStryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, “Observation of two-dimensional spatial solitary waves in a quadratic medium,” Phys. Rev. Lett. 74, 5036–5039 (1995).
    [CrossRef]
  8. 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]
  9. X. Liu, L. J. Qian, and F. W. Wise, “Generation of optical spatiotemporal solitons,” Phys. Rev. Lett. 82, 4631–4634 (1999).
    [CrossRef]
  10. X. Liu, K. Beckwitt, and F. Wise, “Two-dimensional optical spatiotemporal solitons in quadratic media,” Phys. Rev. E 62, 1328–1340 (2000).
    [CrossRef]
  11. F. A. Bovino, M. Braccini, and C. Sibilia, “Orbital angular momentum in noncollinear second-harmonic generation by off-axis vortex beams,” J. Opt. Soc. Am. B 28, 2806–2811 (2011).
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  13. L. Torner and D. V. Petrov, “Azimuthal instabilities and self-breaking of beams into sets of solitons in bulk second-harmonic generation,” Electron. Lett. 33, 608–610 (1997).
    [CrossRef]
  14. D. V. Skryabin and W. J. Firth, “Instabilities of higher-order parametric solitons: Filamentation versus coalescence,” Phys. Rev. E 58, R1252–R1255 (1998).
    [CrossRef]
  15. J. P. Torres, J. M. Soto-Crespo, L. Torner, and D. V. Petrov, “Solitary-wave vortices in quadratic nonlinear media,” J. Opt. Soc. Am. B 15, 625–627 (1998).
    [CrossRef]
  16. D. V. Petrov, L. Torner, J. Martorell, R. Vilaseca, J. P. Torres, and C. Cojocaru, “Observation of azimuthal modulational instability and formation of patterns of optical solitons in a quadratic nonlinear crystal,” Opt. Lett. 23, 1444–1446 (1998).
    [CrossRef]
  17. J. P. Torres, J. M. Soto-Crespo, L. Torner, and D. V. Petrov, “Solitary-wave vortices in type II second-harmonic generation,” Opt. Commun. 149, 77–83 (1998).
    [CrossRef]
  18. G. Molina-Terriza, E. M. Wright, and L. Torner, “Propagation and control of noncanonical optical vortices,” Opt. Lett. 26, 163–165 (2001).
    [CrossRef]
  19. V. I. Kruglov, Y. A. Logvin, and V. M. Volkov, “The theory of spiral laser-beams in nonlinear media,” J. Mod. Opt. 39, 2277–2291 (1992).
    [CrossRef]
  20. C. J. Pethick and H. Smith, Bose-Einstein Condensate in Dilute Gas (Cambridge University, 2008).
  21. F. Dalfovo and S. Stringari, “Bosons in anisotropic traps: ground state and vortices,” Phys. Rev. A 53, 2477–2485 (1996).
    [CrossRef]
  22. R. J. Dodd, “Approximate solutions of the nonlinear Schrödinger equation for ground and excited states of Bose-Einstein condensates,” J. Res. Natl. Inst. Stand. Technol. 101, 545–552 (1996).
    [CrossRef]
  23. T. J. Alexander and L. Bergé, “Ground states and vortices of matter-wave condensates and optical guided waves,” Phys. Rev. E 65, 026611 (2002).
    [CrossRef]
  24. L. D. Carr and C. W. Clark, “Vortices in attractive Bose-Einstein condensates in two dimensions,” Phys. Rev. Lett. 97, 010403 (2006).
    [CrossRef]
  25. D. Mihalache, D. Mazilu, B. A. Malomed, and F. Lederer, “Vortex stability in nearly-two-dimensional Bose-Einstein condensates with attraction,” Phys. Rev. A 73, 043615 (2006).
    [CrossRef]
  26. L. D. Carr and C. W. Clark, “Vortices and ring solitons in Bose-Einstein condensates,” Phys. Rev. A 74, 043613 (2006).
    [CrossRef]
  27. G. Herring, L. D. Carr, R. Carretero-González, P. G. Kevrekidis, and D. J. Frantzeskakis, “Radially symmetric nonlinear states of harmonically trapped Bose-Einstein condensates,” Phys. Rev. A 77, 043607 (2008).
    [CrossRef]
  28. F. Du, Y. W. Lu, and S. T. Wu, “Electrically tunable liquid-crystal photonic crystal fiber,” Appl. Phys. Lett. 85, 2181–2183(2004).
    [CrossRef]
  29. F. Luan, A. K. George, T. D. Hedeley, G. J. Pearce, D. M. Bird, J. C. Knight, and P. S. J. Russell, “All-solid photonic bandgap fiber,” Opt. Lett. 29, 2369–2371 (2004).
    [CrossRef]
  30. P. D. Drummond, K. V. Kheruntsyan, and H. He, “Coherent molecular solitons in Bose-Einstein condensates,” Phys. Rev. Lett. 81, 3055–3058 (1998).
    [CrossRef]
  31. D. J. Heinzen, R. Wynar, P. D. Drummond, and K. V. Kheruntsyan, “Superchemistry: Dynamics of coupled atomic and molecular Bose-Einstein condensates,” Phys. Rev. Lett. 84, 5029–5033 (2000).
    [CrossRef]
  32. J. J. Hope and M. K. Olsen, “Quantum superchemistry: Dynamical quantum effects in coupled atomic and molecular Bose-Einstein condensates,” Phys. Rev. Lett. 86, 3220–3223 (2001).
    [CrossRef]
  33. T. Hornung, S. Gordienko, R. de Vivie-Riedle, and B. J. Verhaar, “Optimal conversion of an atomic to a molecular Bose-Einstein condensate,” Phys. Rev. A 66, 043607 (2002).
    [CrossRef]
  34. Z. Y. Xu, Y. V. Kartashov, L. C. Crasovan, D. Mihalache, and L. Torner, “Multicolor vortex solitons in two-dimensional photonic lattices,” Phys. Rev. E 71, 016616 (2005).
    [CrossRef]
  35. M. J. Werner and P. D. Drummond, “Strongly coupled nonlinear parametric solitary waves,” Opt. Lett. 19, 613–615 (1994).
    [CrossRef]
  36. D. Mihalache, D. Mazilu, L. C. Crasovan, I. Towers, B. A. Malomed, A. V. Buryak, L. Torner, and F. Lederer, “Stable three-dimensional spinning optical solitons supported by competing quadratic and cubic nonlinearities,” Phys. Rev. E 66, 016613 (2002).
    [CrossRef]

2011 (1)

2008 (1)

G. Herring, L. D. Carr, R. Carretero-González, P. G. Kevrekidis, and D. J. Frantzeskakis, “Radially symmetric nonlinear states of harmonically trapped Bose-Einstein condensates,” Phys. Rev. A 77, 043607 (2008).
[CrossRef]

2006 (3)

L. D. Carr and C. W. Clark, “Vortices in attractive Bose-Einstein condensates in two dimensions,” Phys. Rev. Lett. 97, 010403 (2006).
[CrossRef]

D. Mihalache, D. Mazilu, B. A. Malomed, and F. Lederer, “Vortex stability in nearly-two-dimensional Bose-Einstein condensates with attraction,” Phys. Rev. A 73, 043615 (2006).
[CrossRef]

L. D. Carr and C. W. Clark, “Vortices and ring solitons in Bose-Einstein condensates,” Phys. Rev. A 74, 043613 (2006).
[CrossRef]

2005 (2)

Z. Y. Xu, Y. V. Kartashov, L. C. Crasovan, D. Mihalache, and L. Torner, “Multicolor vortex solitons in two-dimensional photonic lattices,” Phys. Rev. E 71, 016616 (2005).
[CrossRef]

B. A. Malomed, D. Mihalache, F. Wise, and L. Torner, “Spatiotemporal optical solitons,” J. Opt. B 7, R53–R72 (2005).
[CrossRef]

2004 (2)

F. Du, Y. W. Lu, and S. T. Wu, “Electrically tunable liquid-crystal photonic crystal fiber,” Appl. Phys. Lett. 85, 2181–2183(2004).
[CrossRef]

F. Luan, A. K. George, T. D. Hedeley, G. J. Pearce, D. M. Bird, J. C. Knight, and P. S. J. Russell, “All-solid photonic bandgap fiber,” Opt. Lett. 29, 2369–2371 (2004).
[CrossRef]

2002 (4)

T. J. Alexander and L. Bergé, “Ground states and vortices of matter-wave condensates and optical guided waves,” Phys. Rev. E 65, 026611 (2002).
[CrossRef]

T. Hornung, S. Gordienko, R. de Vivie-Riedle, and B. J. Verhaar, “Optimal conversion of an atomic to a molecular Bose-Einstein condensate,” Phys. Rev. A 66, 043607 (2002).
[CrossRef]

D. Mihalache, D. Mazilu, L. C. Crasovan, I. Towers, B. A. Malomed, A. V. Buryak, L. Torner, and F. Lederer, “Stable three-dimensional spinning optical solitons supported by competing quadratic and cubic nonlinearities,” Phys. Rev. E 66, 016613 (2002).
[CrossRef]

A. V. Buryak, P. Di Trapani, D. V. Skryabin, and S. Trillo, “Optical solitons due to quadratic nonlinearities: from basic physics to futuristic applications,” Phys. Rep. 370, 63–235 (2002).
[CrossRef]

2001 (2)

G. Molina-Terriza, E. M. Wright, and L. Torner, “Propagation and control of noncanonical optical vortices,” Opt. Lett. 26, 163–165 (2001).
[CrossRef]

J. J. Hope and M. K. Olsen, “Quantum superchemistry: Dynamical quantum effects in coupled atomic and molecular Bose-Einstein condensates,” Phys. Rev. Lett. 86, 3220–3223 (2001).
[CrossRef]

2000 (3)

D. J. Heinzen, R. Wynar, P. D. Drummond, and K. V. Kheruntsyan, “Superchemistry: Dynamics of coupled atomic and molecular Bose-Einstein condensates,” Phys. Rev. Lett. 84, 5029–5033 (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]

C. Etrich, F. Lederer, B. A. Malomed, T. Peschel, and U. Peschel, “Optical solitons in media with a quadratic nonlinearity,” Prog. Opt. 41, 483–568 (2000).
[CrossRef]

1999 (1)

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

1998 (5)

D. V. Skryabin and W. J. Firth, “Instabilities of higher-order parametric solitons: Filamentation versus coalescence,” Phys. Rev. E 58, R1252–R1255 (1998).
[CrossRef]

J. P. Torres, J. M. Soto-Crespo, L. Torner, and D. V. Petrov, “Solitary-wave vortices in quadratic nonlinear media,” J. Opt. Soc. Am. B 15, 625–627 (1998).
[CrossRef]

D. V. Petrov, L. Torner, J. Martorell, R. Vilaseca, J. P. Torres, and C. Cojocaru, “Observation of azimuthal modulational instability and formation of patterns of optical solitons in a quadratic nonlinear crystal,” Opt. Lett. 23, 1444–1446 (1998).
[CrossRef]

J. P. Torres, J. M. Soto-Crespo, L. Torner, and D. V. Petrov, “Solitary-wave vortices in type II second-harmonic generation,” Opt. Commun. 149, 77–83 (1998).
[CrossRef]

P. D. Drummond, K. V. Kheruntsyan, and H. He, “Coherent molecular solitons in Bose-Einstein condensates,” Phys. Rev. Lett. 81, 3055–3058 (1998).
[CrossRef]

1997 (3)

W. J. Firth and D. V. Skryabin, “Optical solitons carrying orbital angular momentum,” Phys. Rev. Lett. 79, 2450–2453 (1997).
[CrossRef]

L. Torner and D. V. Petrov, “Azimuthal instabilities and self-breaking of beams into sets of solitons in bulk second-harmonic generation,” Electron. Lett. 33, 608–610 (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]

1996 (3)

G. I. Stegeman, D. J. Hagan, and L. Torner, “χ(2) cascading phenomena and their applications to all-optical signal processing, mode-locking, pulse compression and solitons,” Opt. Quantum Electron. 28, 1691–1740 (1996).
[CrossRef]

F. Dalfovo and S. Stringari, “Bosons in anisotropic traps: ground state and vortices,” Phys. Rev. A 53, 2477–2485 (1996).
[CrossRef]

R. J. Dodd, “Approximate solutions of the nonlinear Schrödinger equation for ground and excited states of Bose-Einstein condensates,” J. Res. Natl. Inst. Stand. Technol. 101, 545–552 (1996).
[CrossRef]

1995 (1)

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. VanStryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, “Observation of two-dimensional spatial solitary waves in a quadratic medium,” Phys. Rev. Lett. 74, 5036–5039 (1995).
[CrossRef]

1994 (1)

1992 (1)

V. I. Kruglov, Y. A. Logvin, and V. M. Volkov, “The theory of spiral laser-beams in nonlinear media,” J. Mod. Opt. 39, 2277–2291 (1992).
[CrossRef]

1981 (1)

A. A. Kanashov and A. M. Rubenchik, “On diffraction and dispersion effect on three wave interaction,” Physica D 4, 122–134 (1981).
[CrossRef]

Agrawal, G. P.

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

Alexander, T. J.

T. J. Alexander and L. Bergé, “Ground states and vortices of matter-wave condensates and optical guided waves,” Phys. Rev. E 65, 026611 (2002).
[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]

Beckwitt, K.

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

Bergé, L.

T. J. Alexander and L. Bergé, “Ground states and vortices of matter-wave condensates and optical guided waves,” Phys. Rev. E 65, 026611 (2002).
[CrossRef]

Berntson, A.

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]

Bird, D. M.

Bovino, F. A.

Braccini, M.

Buryak, A. V.

A. V. Buryak, P. Di Trapani, D. V. Skryabin, and S. Trillo, “Optical solitons due to quadratic nonlinearities: from basic physics to futuristic applications,” Phys. Rep. 370, 63–235 (2002).
[CrossRef]

D. Mihalache, D. Mazilu, L. C. Crasovan, I. Towers, B. A. Malomed, A. V. Buryak, L. Torner, and F. Lederer, “Stable three-dimensional spinning optical solitons supported by competing quadratic and cubic nonlinearities,” Phys. Rev. E 66, 016613 (2002).
[CrossRef]

Carr, L. D.

G. Herring, L. D. Carr, R. Carretero-González, P. G. Kevrekidis, and D. J. Frantzeskakis, “Radially symmetric nonlinear states of harmonically trapped Bose-Einstein condensates,” Phys. Rev. A 77, 043607 (2008).
[CrossRef]

L. D. Carr and C. W. Clark, “Vortices and ring solitons in Bose-Einstein condensates,” Phys. Rev. A 74, 043613 (2006).
[CrossRef]

L. D. Carr and C. W. Clark, “Vortices in attractive Bose-Einstein condensates in two dimensions,” Phys. Rev. Lett. 97, 010403 (2006).
[CrossRef]

Carretero-González, R.

G. Herring, L. D. Carr, R. Carretero-González, P. G. Kevrekidis, and D. J. Frantzeskakis, “Radially symmetric nonlinear states of harmonically trapped Bose-Einstein condensates,” Phys. Rev. A 77, 043607 (2008).
[CrossRef]

Clark, C. W.

L. D. Carr and C. W. Clark, “Vortices and ring solitons in Bose-Einstein condensates,” Phys. Rev. A 74, 043613 (2006).
[CrossRef]

L. D. Carr and C. W. Clark, “Vortices in attractive Bose-Einstein condensates in two dimensions,” Phys. Rev. Lett. 97, 010403 (2006).
[CrossRef]

Cojocaru, C.

Crasovan, L. C.

Z. Y. Xu, Y. V. Kartashov, L. C. Crasovan, D. Mihalache, and L. Torner, “Multicolor vortex solitons in two-dimensional photonic lattices,” Phys. Rev. E 71, 016616 (2005).
[CrossRef]

D. Mihalache, D. Mazilu, L. C. Crasovan, I. Towers, B. A. Malomed, A. V. Buryak, L. Torner, and F. Lederer, “Stable three-dimensional spinning optical solitons supported by competing quadratic and cubic nonlinearities,” Phys. Rev. E 66, 016613 (2002).
[CrossRef]

Dalfovo, F.

F. Dalfovo and S. Stringari, “Bosons in anisotropic traps: ground state and vortices,” Phys. Rev. A 53, 2477–2485 (1996).
[CrossRef]

de Vivie-Riedle, R.

T. Hornung, S. Gordienko, R. de Vivie-Riedle, and B. J. Verhaar, “Optimal conversion of an atomic to a molecular Bose-Einstein condensate,” Phys. Rev. A 66, 043607 (2002).
[CrossRef]

Di Trapani, P.

A. V. Buryak, P. Di Trapani, D. V. Skryabin, and S. Trillo, “Optical solitons due to quadratic nonlinearities: from basic physics to futuristic applications,” Phys. Rep. 370, 63–235 (2002).
[CrossRef]

Dodd, R. J.

R. J. Dodd, “Approximate solutions of the nonlinear Schrödinger equation for ground and excited states of Bose-Einstein condensates,” J. Res. Natl. Inst. Stand. Technol. 101, 545–552 (1996).
[CrossRef]

Drummond, P.

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]

Drummond, P. D.

D. J. Heinzen, R. Wynar, P. D. Drummond, and K. V. Kheruntsyan, “Superchemistry: Dynamics of coupled atomic and molecular Bose-Einstein condensates,” Phys. Rev. Lett. 84, 5029–5033 (2000).
[CrossRef]

P. D. Drummond, K. V. Kheruntsyan, and H. He, “Coherent molecular solitons in Bose-Einstein condensates,” Phys. Rev. Lett. 81, 3055–3058 (1998).
[CrossRef]

M. J. Werner and P. D. Drummond, “Strongly coupled nonlinear parametric solitary waves,” Opt. Lett. 19, 613–615 (1994).
[CrossRef]

Du, F.

F. Du, Y. W. Lu, and S. T. Wu, “Electrically tunable liquid-crystal photonic crystal fiber,” Appl. Phys. Lett. 85, 2181–2183(2004).
[CrossRef]

Etrich, C.

C. Etrich, F. Lederer, B. A. Malomed, T. Peschel, and U. Peschel, “Optical solitons in media with a quadratic nonlinearity,” Prog. Opt. 41, 483–568 (2000).
[CrossRef]

Firth, W. J.

D. V. Skryabin and W. J. Firth, “Instabilities of higher-order parametric solitons: Filamentation versus coalescence,” Phys. Rev. E 58, R1252–R1255 (1998).
[CrossRef]

W. J. Firth and D. V. Skryabin, “Optical solitons carrying orbital angular momentum,” Phys. Rev. Lett. 79, 2450–2453 (1997).
[CrossRef]

Frantzeskakis, D. J.

G. Herring, L. D. Carr, R. Carretero-González, P. G. Kevrekidis, and D. J. Frantzeskakis, “Radially symmetric nonlinear states of harmonically trapped Bose-Einstein condensates,” Phys. Rev. A 77, 043607 (2008).
[CrossRef]

George, A. K.

Gordienko, S.

T. Hornung, S. Gordienko, R. de Vivie-Riedle, and B. J. Verhaar, “Optimal conversion of an atomic to a molecular Bose-Einstein condensate,” Phys. Rev. A 66, 043607 (2002).
[CrossRef]

Hagan, D. J.

G. I. Stegeman, D. J. Hagan, and L. Torner, “χ(2) cascading phenomena and their applications to all-optical signal processing, mode-locking, pulse compression and solitons,” Opt. Quantum Electron. 28, 1691–1740 (1996).
[CrossRef]

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. VanStryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, “Observation of two-dimensional spatial solitary waves in a quadratic medium,” Phys. Rev. Lett. 74, 5036–5039 (1995).
[CrossRef]

He, H.

P. D. Drummond, K. V. Kheruntsyan, and H. He, “Coherent molecular solitons in Bose-Einstein condensates,” Phys. Rev. Lett. 81, 3055–3058 (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]

Hedeley, T. D.

Heinzen, D. J.

D. J. Heinzen, R. Wynar, P. D. Drummond, and K. V. Kheruntsyan, “Superchemistry: Dynamics of coupled atomic and molecular Bose-Einstein condensates,” Phys. Rev. Lett. 84, 5029–5033 (2000).
[CrossRef]

Herring, G.

G. Herring, L. D. Carr, R. Carretero-González, P. G. Kevrekidis, and D. J. Frantzeskakis, “Radially symmetric nonlinear states of harmonically trapped Bose-Einstein condensates,” Phys. Rev. A 77, 043607 (2008).
[CrossRef]

Hope, J. J.

J. J. Hope and M. K. Olsen, “Quantum superchemistry: Dynamical quantum effects in coupled atomic and molecular Bose-Einstein condensates,” Phys. Rev. Lett. 86, 3220–3223 (2001).
[CrossRef]

Hornung, T.

T. Hornung, S. Gordienko, R. de Vivie-Riedle, and B. J. Verhaar, “Optimal conversion of an atomic to a molecular Bose-Einstein condensate,” Phys. Rev. A 66, 043607 (2002).
[CrossRef]

Kanashov, A. A.

A. A. Kanashov and A. M. Rubenchik, “On diffraction and dispersion effect on three wave interaction,” Physica D 4, 122–134 (1981).
[CrossRef]

Kartashov, Y. V.

Z. Y. Xu, Y. V. Kartashov, L. C. Crasovan, D. Mihalache, and L. Torner, “Multicolor vortex solitons in two-dimensional photonic lattices,” Phys. Rev. E 71, 016616 (2005).
[CrossRef]

Kevrekidis, P. G.

G. Herring, L. D. Carr, R. Carretero-González, P. G. Kevrekidis, and D. J. Frantzeskakis, “Radially symmetric nonlinear states of harmonically trapped Bose-Einstein condensates,” Phys. Rev. A 77, 043607 (2008).
[CrossRef]

Kheruntsyan, K. V.

D. J. Heinzen, R. Wynar, P. D. Drummond, and K. V. Kheruntsyan, “Superchemistry: Dynamics of coupled atomic and molecular Bose-Einstein condensates,” Phys. Rev. Lett. 84, 5029–5033 (2000).
[CrossRef]

P. D. Drummond, K. V. Kheruntsyan, and H. He, “Coherent molecular solitons in Bose-Einstein condensates,” Phys. Rev. Lett. 81, 3055–3058 (1998).
[CrossRef]

Kivshar, Y. S.

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

Knight, J. C.

Kruglov, V. I.

V. I. Kruglov, Y. A. Logvin, and V. M. Volkov, “The theory of spiral laser-beams in nonlinear media,” J. Mod. Opt. 39, 2277–2291 (1992).
[CrossRef]

Lederer, F.

D. Mihalache, D. Mazilu, B. A. Malomed, and F. Lederer, “Vortex stability in nearly-two-dimensional Bose-Einstein condensates with attraction,” Phys. Rev. A 73, 043615 (2006).
[CrossRef]

D. Mihalache, D. Mazilu, L. C. Crasovan, I. Towers, B. A. Malomed, A. V. Buryak, L. Torner, and F. Lederer, “Stable three-dimensional spinning optical solitons supported by competing quadratic and cubic nonlinearities,” Phys. Rev. E 66, 016613 (2002).
[CrossRef]

C. Etrich, F. Lederer, B. A. Malomed, T. Peschel, and U. Peschel, “Optical solitons in media with a quadratic nonlinearity,” Prog. Opt. 41, 483–568 (2000).
[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]

Logvin, Y. A.

V. I. Kruglov, Y. A. Logvin, and V. M. Volkov, “The theory of spiral laser-beams in nonlinear media,” J. Mod. Opt. 39, 2277–2291 (1992).
[CrossRef]

Lu, Y. W.

F. Du, Y. W. Lu, and S. T. Wu, “Electrically tunable liquid-crystal photonic crystal fiber,” Appl. Phys. Lett. 85, 2181–2183(2004).
[CrossRef]

Luan, F.

Malomed, B. A.

D. Mihalache, D. Mazilu, B. A. Malomed, and F. Lederer, “Vortex stability in nearly-two-dimensional Bose-Einstein condensates with attraction,” Phys. Rev. A 73, 043615 (2006).
[CrossRef]

B. A. Malomed, D. Mihalache, F. Wise, and L. Torner, “Spatiotemporal optical solitons,” J. Opt. B 7, R53–R72 (2005).
[CrossRef]

D. Mihalache, D. Mazilu, L. C. Crasovan, I. Towers, B. A. Malomed, A. V. Buryak, L. Torner, and F. Lederer, “Stable three-dimensional spinning optical solitons supported by competing quadratic and cubic nonlinearities,” Phys. Rev. E 66, 016613 (2002).
[CrossRef]

C. Etrich, F. Lederer, B. A. Malomed, T. Peschel, and U. Peschel, “Optical solitons in media with a quadratic nonlinearity,” Prog. Opt. 41, 483–568 (2000).
[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]

Martorell, J.

Mazilu, D.

D. Mihalache, D. Mazilu, B. A. Malomed, and F. Lederer, “Vortex stability in nearly-two-dimensional Bose-Einstein condensates with attraction,” Phys. Rev. A 73, 043615 (2006).
[CrossRef]

D. Mihalache, D. Mazilu, L. C. Crasovan, I. Towers, B. A. Malomed, A. V. Buryak, L. Torner, and F. Lederer, “Stable three-dimensional spinning optical solitons supported by competing quadratic and cubic nonlinearities,” Phys. Rev. E 66, 016613 (2002).
[CrossRef]

Menyuk, C. R.

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. VanStryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, “Observation of two-dimensional spatial solitary waves in a quadratic medium,” Phys. Rev. Lett. 74, 5036–5039 (1995).
[CrossRef]

Mihalache, D.

D. Mihalache, D. Mazilu, B. A. Malomed, and F. Lederer, “Vortex stability in nearly-two-dimensional Bose-Einstein condensates with attraction,” Phys. Rev. A 73, 043615 (2006).
[CrossRef]

Z. Y. Xu, Y. V. Kartashov, L. C. Crasovan, D. Mihalache, and L. Torner, “Multicolor vortex solitons in two-dimensional photonic lattices,” Phys. Rev. E 71, 016616 (2005).
[CrossRef]

B. A. Malomed, D. Mihalache, F. Wise, and L. Torner, “Spatiotemporal optical solitons,” J. Opt. B 7, R53–R72 (2005).
[CrossRef]

D. Mihalache, D. Mazilu, L. C. Crasovan, I. Towers, B. A. Malomed, A. V. Buryak, L. Torner, and F. Lederer, “Stable three-dimensional spinning optical solitons supported by competing quadratic and cubic nonlinearities,” Phys. Rev. E 66, 016613 (2002).
[CrossRef]

Molina-Terriza, G.

Olsen, M. K.

J. J. Hope and M. K. Olsen, “Quantum superchemistry: Dynamical quantum effects in coupled atomic and molecular Bose-Einstein condensates,” Phys. Rev. Lett. 86, 3220–3223 (2001).
[CrossRef]

Pearce, G. J.

Peschel, T.

C. Etrich, F. Lederer, B. A. Malomed, T. Peschel, and U. Peschel, “Optical solitons in media with a quadratic nonlinearity,” Prog. Opt. 41, 483–568 (2000).
[CrossRef]

Peschel, U.

C. Etrich, F. Lederer, B. A. Malomed, T. Peschel, and U. Peschel, “Optical solitons in media with a quadratic nonlinearity,” Prog. Opt. 41, 483–568 (2000).
[CrossRef]

Pethick, C. J.

C. J. Pethick and H. Smith, Bose-Einstein Condensate in Dilute Gas (Cambridge University, 2008).

Petrov, D. V.

J. P. Torres, J. M. Soto-Crespo, L. Torner, and D. V. Petrov, “Solitary-wave vortices in type II second-harmonic generation,” Opt. Commun. 149, 77–83 (1998).
[CrossRef]

J. P. Torres, J. M. Soto-Crespo, L. Torner, and D. V. Petrov, “Solitary-wave vortices in quadratic nonlinear media,” J. Opt. Soc. Am. B 15, 625–627 (1998).
[CrossRef]

D. V. Petrov, L. Torner, J. Martorell, R. Vilaseca, J. P. Torres, and C. Cojocaru, “Observation of azimuthal modulational instability and formation of patterns of optical solitons in a quadratic nonlinear crystal,” Opt. Lett. 23, 1444–1446 (1998).
[CrossRef]

L. Torner and D. V. Petrov, “Azimuthal instabilities and self-breaking of beams into sets of solitons in bulk second-harmonic generation,” Electron. Lett. 33, 608–610 (1997).
[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]

Rubenchik, A. M.

A. A. Kanashov and A. M. Rubenchik, “On diffraction and dispersion effect on three wave interaction,” Physica D 4, 122–134 (1981).
[CrossRef]

Russell, P. S. J.

Sibilia, C.

Skryabin, D. V.

A. V. Buryak, P. Di Trapani, D. V. Skryabin, and S. Trillo, “Optical solitons due to quadratic nonlinearities: from basic physics to futuristic applications,” Phys. Rep. 370, 63–235 (2002).
[CrossRef]

D. V. Skryabin and W. J. Firth, “Instabilities of higher-order parametric solitons: Filamentation versus coalescence,” Phys. Rev. E 58, R1252–R1255 (1998).
[CrossRef]

W. J. Firth and D. V. Skryabin, “Optical solitons carrying orbital angular momentum,” Phys. Rev. Lett. 79, 2450–2453 (1997).
[CrossRef]

Smith, H.

C. J. Pethick and H. Smith, Bose-Einstein Condensate in Dilute Gas (Cambridge University, 2008).

Soto-Crespo, J. M.

J. P. Torres, J. M. Soto-Crespo, L. Torner, and D. V. Petrov, “Solitary-wave vortices in type II second-harmonic generation,” Opt. Commun. 149, 77–83 (1998).
[CrossRef]

J. P. Torres, J. M. Soto-Crespo, L. Torner, and D. V. Petrov, “Solitary-wave vortices in quadratic nonlinear media,” J. Opt. Soc. Am. B 15, 625–627 (1998).
[CrossRef]

Stegeman, G. I.

G. I. Stegeman, D. J. Hagan, and L. Torner, “χ(2) cascading phenomena and their applications to all-optical signal processing, mode-locking, pulse compression and solitons,” Opt. Quantum Electron. 28, 1691–1740 (1996).
[CrossRef]

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. VanStryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, “Observation of two-dimensional spatial solitary waves in a quadratic medium,” Phys. Rev. Lett. 74, 5036–5039 (1995).
[CrossRef]

Stringari, S.

F. Dalfovo and S. Stringari, “Bosons in anisotropic traps: ground state and vortices,” Phys. Rev. A 53, 2477–2485 (1996).
[CrossRef]

Torner, L.

B. A. Malomed, D. Mihalache, F. Wise, and L. Torner, “Spatiotemporal optical solitons,” J. Opt. B 7, R53–R72 (2005).
[CrossRef]

Z. Y. Xu, Y. V. Kartashov, L. C. Crasovan, D. Mihalache, and L. Torner, “Multicolor vortex solitons in two-dimensional photonic lattices,” Phys. Rev. E 71, 016616 (2005).
[CrossRef]

D. Mihalache, D. Mazilu, L. C. Crasovan, I. Towers, B. A. Malomed, A. V. Buryak, L. Torner, and F. Lederer, “Stable three-dimensional spinning optical solitons supported by competing quadratic and cubic nonlinearities,” Phys. Rev. E 66, 016613 (2002).
[CrossRef]

G. Molina-Terriza, E. M. Wright, and L. Torner, “Propagation and control of noncanonical optical vortices,” Opt. Lett. 26, 163–165 (2001).
[CrossRef]

J. P. Torres, J. M. Soto-Crespo, L. Torner, and D. V. Petrov, “Solitary-wave vortices in type II second-harmonic generation,” Opt. Commun. 149, 77–83 (1998).
[CrossRef]

J. P. Torres, J. M. Soto-Crespo, L. Torner, and D. V. Petrov, “Solitary-wave vortices in quadratic nonlinear media,” J. Opt. Soc. Am. B 15, 625–627 (1998).
[CrossRef]

D. V. Petrov, L. Torner, J. Martorell, R. Vilaseca, J. P. Torres, and C. Cojocaru, “Observation of azimuthal modulational instability and formation of patterns of optical solitons in a quadratic nonlinear crystal,” Opt. Lett. 23, 1444–1446 (1998).
[CrossRef]

L. Torner and D. V. Petrov, “Azimuthal instabilities and self-breaking of beams into sets of solitons in bulk second-harmonic generation,” Electron. Lett. 33, 608–610 (1997).
[CrossRef]

G. I. Stegeman, D. J. Hagan, and L. Torner, “χ(2) cascading phenomena and their applications to all-optical signal processing, mode-locking, pulse compression and solitons,” Opt. Quantum Electron. 28, 1691–1740 (1996).
[CrossRef]

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. VanStryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, “Observation of two-dimensional spatial solitary waves in a quadratic medium,” Phys. Rev. Lett. 74, 5036–5039 (1995).
[CrossRef]

Torres, J. P.

Torruellas, W. E.

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. VanStryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, “Observation of two-dimensional spatial solitary waves in a quadratic medium,” Phys. Rev. Lett. 74, 5036–5039 (1995).
[CrossRef]

Towers, I.

D. Mihalache, D. Mazilu, L. C. Crasovan, I. Towers, B. A. Malomed, A. V. Buryak, L. Torner, and F. Lederer, “Stable three-dimensional spinning optical solitons supported by competing quadratic and cubic nonlinearities,” Phys. Rev. E 66, 016613 (2002).
[CrossRef]

Trillo, S.

A. V. Buryak, P. Di Trapani, D. V. Skryabin, and S. Trillo, “Optical solitons due to quadratic nonlinearities: from basic physics to futuristic applications,” Phys. Rep. 370, 63–235 (2002).
[CrossRef]

VanStryland, E. W.

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. VanStryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, “Observation of two-dimensional spatial solitary waves in a quadratic medium,” Phys. Rev. Lett. 74, 5036–5039 (1995).
[CrossRef]

Verhaar, B. J.

T. Hornung, S. Gordienko, R. de Vivie-Riedle, and B. J. Verhaar, “Optimal conversion of an atomic to a molecular Bose-Einstein condensate,” Phys. Rev. A 66, 043607 (2002).
[CrossRef]

Vilaseca, R.

Volkov, V. M.

V. I. Kruglov, Y. A. Logvin, and V. M. Volkov, “The theory of spiral laser-beams in nonlinear media,” J. Mod. Opt. 39, 2277–2291 (1992).
[CrossRef]

Wang, Z.

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. VanStryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, “Observation of two-dimensional spatial solitary waves in a quadratic medium,” Phys. Rev. Lett. 74, 5036–5039 (1995).
[CrossRef]

Werner, M. J.

Wise, F.

B. A. Malomed, D. Mihalache, F. Wise, and L. Torner, “Spatiotemporal optical solitons,” J. Opt. B 7, R53–R72 (2005).
[CrossRef]

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.

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

Wright, E. M.

Wu, S. T.

F. Du, Y. W. Lu, and S. T. Wu, “Electrically tunable liquid-crystal photonic crystal fiber,” Appl. Phys. Lett. 85, 2181–2183(2004).
[CrossRef]

Wynar, R.

D. J. Heinzen, R. Wynar, P. D. Drummond, and K. V. Kheruntsyan, “Superchemistry: Dynamics of coupled atomic and molecular Bose-Einstein condensates,” Phys. Rev. Lett. 84, 5029–5033 (2000).
[CrossRef]

Xu, Z. Y.

Z. Y. Xu, Y. V. Kartashov, L. C. Crasovan, D. Mihalache, and L. Torner, “Multicolor vortex solitons in two-dimensional photonic lattices,” Phys. Rev. E 71, 016616 (2005).
[CrossRef]

Appl. Phys. Lett. (1)

F. Du, Y. W. Lu, and S. T. Wu, “Electrically tunable liquid-crystal photonic crystal fiber,” Appl. Phys. Lett. 85, 2181–2183(2004).
[CrossRef]

Electron. Lett. (1)

L. Torner and D. V. Petrov, “Azimuthal instabilities and self-breaking of beams into sets of solitons in bulk second-harmonic generation,” Electron. Lett. 33, 608–610 (1997).
[CrossRef]

J. Mod. Opt. (1)

V. I. Kruglov, Y. A. Logvin, and V. M. Volkov, “The theory of spiral laser-beams in nonlinear media,” J. Mod. Opt. 39, 2277–2291 (1992).
[CrossRef]

J. Opt. B (1)

B. A. Malomed, D. Mihalache, F. Wise, and L. Torner, “Spatiotemporal optical solitons,” J. Opt. B 7, R53–R72 (2005).
[CrossRef]

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

J. Res. Natl. Inst. Stand. Technol. (1)

R. J. Dodd, “Approximate solutions of the nonlinear Schrödinger equation for ground and excited states of Bose-Einstein condensates,” J. Res. Natl. Inst. Stand. Technol. 101, 545–552 (1996).
[CrossRef]

Opt. Commun. (1)

J. P. Torres, J. M. Soto-Crespo, L. Torner, and D. V. Petrov, “Solitary-wave vortices in type II second-harmonic generation,” Opt. Commun. 149, 77–83 (1998).
[CrossRef]

Opt. Lett. (4)

Opt. Quantum Electron. (1)

G. I. Stegeman, D. J. Hagan, and L. Torner, “χ(2) cascading phenomena and their applications to all-optical signal processing, mode-locking, pulse compression and solitons,” Opt. Quantum Electron. 28, 1691–1740 (1996).
[CrossRef]

Phys. Rep. (1)

A. V. Buryak, P. Di Trapani, D. V. Skryabin, and S. Trillo, “Optical solitons due to quadratic nonlinearities: from basic physics to futuristic applications,” Phys. Rep. 370, 63–235 (2002).
[CrossRef]

Phys. Rev. A (5)

T. Hornung, S. Gordienko, R. de Vivie-Riedle, and B. J. Verhaar, “Optimal conversion of an atomic to a molecular Bose-Einstein condensate,” Phys. Rev. A 66, 043607 (2002).
[CrossRef]

D. Mihalache, D. Mazilu, B. A. Malomed, and F. Lederer, “Vortex stability in nearly-two-dimensional Bose-Einstein condensates with attraction,” Phys. Rev. A 73, 043615 (2006).
[CrossRef]

L. D. Carr and C. W. Clark, “Vortices and ring solitons in Bose-Einstein condensates,” Phys. Rev. A 74, 043613 (2006).
[CrossRef]

G. Herring, L. D. Carr, R. Carretero-González, P. G. Kevrekidis, and D. J. Frantzeskakis, “Radially symmetric nonlinear states of harmonically trapped Bose-Einstein condensates,” Phys. Rev. A 77, 043607 (2008).
[CrossRef]

F. Dalfovo and S. Stringari, “Bosons in anisotropic traps: ground state and vortices,” Phys. Rev. A 53, 2477–2485 (1996).
[CrossRef]

Phys. Rev. E (6)

T. J. Alexander and L. Bergé, “Ground states and vortices of matter-wave condensates and optical guided waves,” Phys. Rev. E 65, 026611 (2002).
[CrossRef]

Z. Y. Xu, Y. V. Kartashov, L. C. Crasovan, D. Mihalache, and L. Torner, “Multicolor vortex solitons in two-dimensional photonic lattices,” Phys. Rev. E 71, 016616 (2005).
[CrossRef]

D. Mihalache, D. Mazilu, L. C. Crasovan, I. Towers, B. A. Malomed, A. V. Buryak, L. Torner, and F. Lederer, “Stable three-dimensional spinning optical solitons supported by competing quadratic and cubic nonlinearities,” Phys. Rev. E 66, 016613 (2002).
[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]

D. V. Skryabin and W. J. Firth, “Instabilities of higher-order parametric solitons: Filamentation versus coalescence,” Phys. Rev. E 58, R1252–R1255 (1998).
[CrossRef]

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

Phys. Rev. Lett. (7)

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. VanStryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, “Observation of two-dimensional spatial solitary waves in a quadratic medium,” Phys. Rev. Lett. 74, 5036–5039 (1995).
[CrossRef]

W. J. Firth and D. V. Skryabin, “Optical solitons carrying orbital angular momentum,” Phys. Rev. Lett. 79, 2450–2453 (1997).
[CrossRef]

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

P. D. Drummond, K. V. Kheruntsyan, and H. He, “Coherent molecular solitons in Bose-Einstein condensates,” Phys. Rev. Lett. 81, 3055–3058 (1998).
[CrossRef]

D. J. Heinzen, R. Wynar, P. D. Drummond, and K. V. Kheruntsyan, “Superchemistry: Dynamics of coupled atomic and molecular Bose-Einstein condensates,” Phys. Rev. Lett. 84, 5029–5033 (2000).
[CrossRef]

J. J. Hope and M. K. Olsen, “Quantum superchemistry: Dynamical quantum effects in coupled atomic and molecular Bose-Einstein condensates,” Phys. Rev. Lett. 86, 3220–3223 (2001).
[CrossRef]

L. D. Carr and C. W. Clark, “Vortices in attractive Bose-Einstein condensates in two dimensions,” Phys. Rev. Lett. 97, 010403 (2006).
[CrossRef]

Physica D (1)

A. A. Kanashov and A. M. Rubenchik, “On diffraction and dispersion effect on three wave interaction,” Physica D 4, 122–134 (1981).
[CrossRef]

Prog. Opt. (1)

C. Etrich, F. Lederer, B. A. Malomed, T. Peschel, and U. Peschel, “Optical solitons in media with a quadratic nonlinearity,” Prog. Opt. 41, 483–568 (2000).
[CrossRef]

Other (2)

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

C. J. Pethick and H. Smith, Bose-Einstein Condensate in Dilute Gas (Cambridge University, 2008).

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Figures (7)

Fig. 1.
Fig. 1.

(a) Critical power for the onset of the parametric instability of the single-color beams with topological charges 0 (m=0) and 1 (m=2), the beams being unstable at N>Nc. Chains of rhombuses and dashed and dotted curves show, respectively, numerical results and the prediction of the VA produced by Eqs. (10), (13), and (14). Examples of periodic oscillations of perturbed solutions at N>Nc are shown for m=0, q=0, and N=3 in (b), and for m=2, q=0, and N=6.5 in (c). Amplitudes Su0,v0(z) and Su1,v2 are defined by Eqs. (15) and (16), respectively.

Fig. 2.
Fig. 2.

(a) Critical power for the onset of the parametric instability of the single-color semivortex with topological charge 1/2 (m=1). Chains of rhombuses and the dashed and dotted curves show, respectively, numerical results and the prediction of the VA produced by Eqs. (20) and (21). (b) An example of quasi-chaotic oscillations of an unstable perturbed solution is shown for m=1, q=0, and N=5.8. (c) 3D profile of the FF component of the same solution at z=1000.

Fig. 3.
Fig. 3.

Contour plots of the FF component of the same solution that is displayed in Figs. 2(b) and 2(c) at z=1000 (a), z=1002 (b), and z=1004 (c).

Fig. 4.
Fig. 4.

Amplitude umax of the FF field (a) and propagation constant (b) of the solitary-vortex beams with m=2 at q=0, of both the single- and two-color types, at N<Nc(m=2) and N>Nc(m=2), respectively. The pitchfork bifurcation occurs at N=Nc(m=2). Chains of rhombuses and the dashed curve show, respectively, numerical results and the prediction of the VA based on ansatz (22).

Fig. 5.
Fig. 5.

(a) The stability area of the two-color vortices with topological charge 1 (m=2), in the plane of the mismatch (q) and total power (N), is Nc<N<Nc2. The numerically found stability borders and their counterparts predicted by the VA are displayed by chains of symbols and dashed lines, respectively [in fact, Nc(q) is the same border as one labeled by m=2 in Fig. 1(a)]. (b) Peak value of the integral amplitude of the zeroth angular harmonic of the SH field, Sv0(z) [defined as per Eq. (15)], as found from the simulations of the oscillatory solutions at N>N2c27.5 for q=0.

Fig. 6.
Fig. 6.

Periodic evolution of the integrally defined amplitudes of the first angular harmonic in the FF field (a), and the zeroth harmonic in the SH field (b).

Fig. 7.
Fig. 7.

The regime of periodic splittings and recoveries of an unstable two-color vortex (the same one whose evolution is presented in Fig. 6) is illustrated by a sequence of profiles of |u(x,y,z)| at z=90 (a), 140 (b), and 190 (c).

Equations (27)

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

iuz+12(2x2+2y2)u+u*vU(x,y)u=0,2ivz+12(2x2+2y2)vqv+12u24U(x,y)v=0,
L={[iuzu*+2ivzv*12(|ux|2+|uy|2+|vx|2+|vy|2)]U(r)(|u|2+4|v|2)q|v|2+12(u2v*+u2*v)}dxdy.
2ivz+12(2x2+2y2)vqv4U(x,y)v=0.
v=vm0exp(i(μz+mθ))rmexp(r2/2),
Nm=4πm!vm02.
u=u0(z)exp(αr2),v=v0(z)exp(r2/2),
idu0dz=(α+116α)u04α4α+1u0*v0,
idv0dz=(12+q2)v024α+1u02.
idu0dz=(α+116α14q4)u04α4α+1v00(u0)*.
Nc(m=0)=4π(α+116α14q4)2(4α+14α)2.
u=u1(z)rexp[αr2+iθ(i/4)(3+q)z],
idu1dz=(2α+18α34q4)u164α2(4α+1)3v20(u1)*.
Nc(m=2)=π(4α+1)6512α4(2α+18α34q4)2.
Nc(m=0,2)0atq1,Nc(m=0,2)π[1+(m/2)](1q)2at0<1q1.
Su0,v0(z)|{u(x,y),v(x,y)}dxdy|.
Su1,v2(z)|{u(x,y)eiθ,v(x,y)e2iθ}dxdy|.
γ+γ=1+q/2.
γu0=(α+116α)u04α2(α+α)+1v10(u1)*,γu1=(2α+18α)u116α2[2(α+α)+1]2v10(u0)*.
128αα2v102[2(α+α)+1]4=(α+116αγ)(2α+18αγ).
Nc(m=1)=π(α+α+1/2)42αα2×[(α+116α12q4α2132α)2+(α+116α)(2α+18α1q2)].
Nc(m=1)0atq1,Nc(m=1)2π(1q)2at0<1q1;
u(x,y,z)=u10rexp(iμz+iθαr2),v(x,y,z)=v20r2exp(2iμz+2iθβr2),
N=π[(2α)2u102+β3v202],
u(x,y,z)=u1(z)rexp(iθαr2)+u1(z)rexp(iθα1r2)+u3(z)r3exp(3iθα3r2),v(x,y,z)=v2(z)r2exp(2iθβr2)+v0(z)exp(β0r2)+v4(z)r4exp(4iθβ4r2).
idu1dz=(2α1+18α1)u14α1u1*v0(α+α1+β0)224α12u3*v2(α1+α3+β)4,idu3dz=(4α3+14α3)u316α34u1*v2(α1+α3+β)464α34u1*v4(α+α3+β4)4,idv0dz=[β0+12(4β0+q)]v0β0u1u1(α+α1+β0)2,idv4dz=[5β42+12(4β4+q)]v416β45u1u3(α+α3+β4)5.
γ0=μ+γ1,γ3=2μγ1,γ4=3μγ1.
6α12α34(8v20)2(α1+α3+β)8=[2α1+18α1γ132α12β0u102(α+α1+β0)4(4β0+β01+4q8γ0)]×[4α3+14α3γ32α34β45(64u10)2(α+α3+β4)10(20β4+5β41+4q8γ4)].

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