Abstract

We introduce a model for spatiotemporal modelocking in multimode fiber lasers, which is based on the (3+1)-dimensional cubic-quintic complex Ginzburg-Landau equation (cGLE) with conservative and dissipative nonlinearities and a 2-dimensional transverse trapping potential. Systematic numerical analysis reveals a variety of stable nonlinear modes, including stable fundamental solitons and breathers, as well as solitary vortices with winding number n = 1, while vortices with n = 2 are unstable, splitting into persistently rotating bound states of two unitary vortices. A characteristic feature of the system is bistability between the fundamental and vortex spatiotemporal solitons.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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    [Crossref]
  7. K. Krupa, A. Tonello, B. M. Shalaby, M. Fabert, A. Barthé lémy, G. Millot, S. Wabnitz, and V. Couderc, “Spatial beam self-cleaning in multimode fibres,” Nat. Photonics 11(4), 237–241 (2017).
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  27. A. M. Dunlop, W. J. Firth, and E. M. Wright, “Master equation for spatiotemporal beam propagation and Kerr lens mode-locking,” Opt. Commun. 138(1-3), 211–226 (1997).
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  30. N. Akhmediev, J. M. Soto-Crespo, and P. Grelu, “Spatiotemporal optical solitons in nonlinear dissipative media: From stationary light bullets to pulsating complexes,” Chaos 17(3), 037112 (2007).
    [Crossref]
  31. N. B. Aleksić, V. Skarka, D. V. Timotijević, and D. Gauthier, “Self-stabilized spatiotemporal dynamics of dissipative light bullets generated from inputs without spherical symmetry in three-dimensional Ginzburg-Landau systems,” Phys. Rev. A 75(6), 061802 (2007).
    [Crossref]
  32. S. Chen, “Analytical spinless light-bullet solutions as attractive fixed points in the three-dimensional cubic-quintic complex Ginzburg-Landau equation,” Phys. Rev. A 86(3), 033829 (2012).
    [Crossref]
  33. D. Mihalache, D. Mazilu, F. Lederer, and H. Leblond, “Stability limits for three-dimensional vortex solitons in the Ginzburg-Landau equation with the cubic-quintic nonlinearity,” Phys. Rev. A 76(4), 045803 (2007).
    [Crossref]
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    [Crossref]
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    [Crossref]
  38. T. Mayteevarunyoo, B. A. Malomed, and D. V. Skryabin, “One- and two-dimensional modes in the complex Ginzburg-Landau equation with a trapping potential,” Opt. Express 26(7), 8849–8865 (2018).
    [Crossref]
  39. T. Mayteevarunyoo, B. A. Malomed, and D. Skryabin, “Vortex modes supported by spin-orbit coupling in a laser with saturable absorption,” New J. Phys. 20(11), 113019 (2018).
    [Crossref]
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    [Crossref]

2019 (3)

Y. Ding, X. Xiao, P. Wang, and C. Yang, “Multiple-soliton in spatiotemporal mode-locked multimode fiber lasers,” Opt. Express 27(8), 11435–11446 (2019).
[Crossref]

D. Mao, M. Li, Z. He, X. Cui, H. Lu, W. Zhang, H. Zhang, and J. Zhao, “Optical vortex fiber laser based on modulation of transverse modes in two mode fiber,” APL Photonics 4(6), 060801 (2019).
[Crossref]

B. A. Malomed, “(INVITED)Vortex solitons: Old results and new perspectives,” Phys. D (Amsterdam, Neth.) 399, 108–137 (2019).
[Crossref]

2018 (8)

T. Mayteevarunyoo, B. A. Malomed, and D. V. Skryabin, “One- and two-dimensional modes in the complex Ginzburg-Landau equation with a trapping potential,” Opt. Express 26(7), 8849–8865 (2018).
[Crossref]

T. Mayteevarunyoo, B. A. Malomed, and D. Skryabin, “Vortex modes supported by spin-orbit coupling in a laser with saturable absorption,” New J. Phys. 20(11), 113019 (2018).
[Crossref]

H. Qin, X. Xiao, P. Wang, and C. Yang, “Observation of soliton molecules in a spatiotemporal mode-locked multimode fiber laser,” Opt. Lett. 43(9), 1982–1985 (2018).
[Crossref]

T. Chen, Q. Zhang, Y. Zhang, and X. Li, “All-fiber passively mode-locked laser using nonlinear multimode interference of step-index multimode fiber,” Photonics Res. 6(11), 1033–1039 (2018).
[Crossref]

O. V. Shtyrina, M. P. Fedoruk, Y. S. Kivshar, and S. K. Turitsyn, “Coexistence of collapse and stable spatiotemporal solitons in multimode fibers,” Phys. Rev. A 97(1), 013841 (2018).
[Crossref]

W. Fu, L. G. Wright, P. Sidorenko, S. Backus, and F. W. Wise, “Several new directions for ultrafast fiber lasers [Invited],” Opt. Express 26(8), 9432–9463 (2018).
[Crossref]

L. G. Wright, Z. M. Ziegler, P. M. Lushnikov, and Z. Zhu, “Multimode Nonlinear Fiber Optics: Massively Parallel Numerical Solver, Tutorial, and Outlook,” IEEE J. Sel. Top. Quantum Electron. 24(3), 1–16 (2018).
[Crossref]

A. S. Ahsan and G. P. Agrawal, “Graded-index solitons in multimode fibers,” Opt. Lett. 43(14), 3345–3348 (2018).
[Crossref]

2017 (3)

K. Krupa, A. Tonello, B. M. Shalaby, M. Fabert, A. Barthé lémy, G. Millot, S. Wabnitz, and V. Couderc, “Spatial beam self-cleaning in multimode fibres,” Nat. Photonics 11(4), 237–241 (2017).
[Crossref]

L. G. Wright, D. N. Christodoulides, and F. W. Wise, “Spatiotemporal mode-locking in multimode fiber lasers,” Science 358(6359), 94–97 (2017).
[Crossref]

N. A. Veretenov, S. V. Fedorov, and N. N. Rosanov, “Topological Vortex and Knotted Dissipative Optical 3D Solitons Generated by 2D Vortex Solitons,” Phys. Rev. Lett. 119(26), 263901 (2017).
[Crossref]

2016 (4)

K. Krupa, A. Tonello, A. Barthélémy, V. Couderc, B. M. Shalaby, A. Bendahmane, G. Millot, and S. Wabnitz, “Observation of geometric parametric instability induced by the periodic spatial self-imaging of multimode waves,” Phys. Rev. Lett. 116(18), 183901 (2016).
[Crossref]

Z. Zhu, L. G. Wright, D. N. Christodoulides, and F. W. Wise, “Observation of multimode solitons in few-mode fiber,” Opt. Lett. 41(20), 4819–4822 (2016).
[Crossref]

Z. Liu, L. G. Wright, D. N. Christodoulides, and F. W. Wise, “Kerr self-cleaning of femtosecond-pulsed beams in graded-index multimode fiber,” Opt. Lett. 41(16), 3675–3678 (2016).
[Crossref]

I. S. Chekhovskoy, A. M. Rubenchik, O. V. Shtyrina, and M. P. Fedoruk, “Nonlinear combining and compression in multicore fibers,” Phys. Rev. A 94(4), 043848 (2016).
[Crossref]

2015 (2)

L. G. Wright, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Spatiotemporal dynamics of multimode optical solitons,” Opt. Express 23(3), 3492–3506 (2015).
[Crossref]

L. G. Wright, D. N. Christodoulides, and F. W. Wise, “Controllable spatiotemporal nonlinear effects in multimode fibres,” Nat. Photonics 9(5), 306–310 (2015).
[Crossref]

2014 (1)

2013 (4)

F. Eilenberger, K. Prater, S. Minardi, R. Geiss, U. Röpke, J. Kobelke, K. Schuster, H. Bartelt, S. Nolte, A. Tünnermann, and T. Pertsch, “Observation of discrete, vortex light bullets,” Phys. Rev. X 3(4), 041031 (2013).
[Crossref]

W. H. Renninger and F. W. Wise, “Optical solitons in graded-index multimode fibres,” Nat. Commun. 4(1), 1719 (2013).
[Crossref]

C. Jauregui, J. Limpert, and A. Tünnermann, “High-power fibre lasers,” Nat. Photonics 7(11), 861–867 (2013).
[Crossref]

D. Richardson, J. Fini, and L. Nelson, “Space-division multiplexing in optical fibres,” Nat. Photonics 7(5), 354–362 (2013).
[Crossref]

2012 (3)

P. Grelu and N. Akhmediev, “Dissipative solitons for mode-locked lasers,” Nat. Photonics 6(2), 84–92 (2012).
[Crossref]

T. F. S. Büttner, D. D. Hudson, E. C. Mägi, A. C. Bedoya, T. Taunay, and B. J. Eggleton, “Multicore, tapered optical fiber for nonlinear pulse reshaping and saturable absorption,” Opt. Lett. 37(13), 2469–2471 (2012).
[Crossref]

S. Chen, “Analytical spinless light-bullet solutions as attractive fixed points in the three-dimensional cubic-quintic complex Ginzburg-Landau equation,” Phys. Rev. A 86(3), 033829 (2012).
[Crossref]

2010 (1)

S. Minardi, F. Eilenberger, and Y. V. Kartashov, “Three-Dimensional Light Bullets in Arrays of Waveguides,” Phys. Rev. Lett. 105(26), 263901 (2010).
[Crossref]

2008 (1)

D. Mihalache, D. Mazilu, F. Lederer, and H. Leblond, “Collisions between coaxial vortex solitons in the three-dimensional cubic-quintic complex Ginzburg-Landau equation,” Phys. Rev. A 77(3), 033817 (2008).
[Crossref]

2007 (3)

D. Mihalache, D. Mazilu, F. Lederer, and H. Leblond, “Stability limits for three-dimensional vortex solitons in the Ginzburg-Landau equation with the cubic-quintic nonlinearity,” Phys. Rev. A 76(4), 045803 (2007).
[Crossref]

N. Akhmediev, J. M. Soto-Crespo, and P. Grelu, “Spatiotemporal optical solitons in nonlinear dissipative media: From stationary light bullets to pulsating complexes,” Chaos 17(3), 037112 (2007).
[Crossref]

N. B. Aleksić, V. Skarka, D. V. Timotijević, and D. Gauthier, “Self-stabilized spatiotemporal dynamics of dissipative light bullets generated from inputs without spherical symmetry in three-dimensional Ginzburg-Landau systems,” Phys. Rev. A 75(6), 061802 (2007).
[Crossref]

2005 (1)

2000 (1)

S. Raghavan and G. P. Agrawal, “Spatiotemporal solitons in inhomogeneous nonlinear media,” Opt. Commun. 180(4-6), 377–382 (2000).
[Crossref]

1997 (1)

A. M. Dunlop, W. J. Firth, and E. M. Wright, “Master equation for spatiotemporal beam propagation and Kerr lens mode-locking,” Opt. Commun. 138(1-3), 211–226 (1997).
[Crossref]

1992 (1)

1968 (2)

D. Auston, “Transverse mode locking,” IEEE J. Quantum Electron. 4(6), 420–422 (1968).
[Crossref]

P. W. Smith, “Simultaneous phase-locking of longitudinal and transverse laser modes,” Appl. Phys. Lett. 13(7), 235–237 (1968).
[Crossref]

Agrawal, G. P.

A. S. Ahsan and G. P. Agrawal, “Graded-index solitons in multimode fibers,” Opt. Lett. 43(14), 3345–3348 (2018).
[Crossref]

S. Raghavan and G. P. Agrawal, “Spatiotemporal solitons in inhomogeneous nonlinear media,” Opt. Commun. 180(4-6), 377–382 (2000).
[Crossref]

Ahsan, A. S.

Akhmediev, N.

P. Grelu and N. Akhmediev, “Dissipative solitons for mode-locked lasers,” Nat. Photonics 6(2), 84–92 (2012).
[Crossref]

N. Akhmediev, J. M. Soto-Crespo, and P. Grelu, “Spatiotemporal optical solitons in nonlinear dissipative media: From stationary light bullets to pulsating complexes,” Chaos 17(3), 037112 (2007).
[Crossref]

Aleksic, N. B.

N. B. Aleksić, V. Skarka, D. V. Timotijević, and D. Gauthier, “Self-stabilized spatiotemporal dynamics of dissipative light bullets generated from inputs without spherical symmetry in three-dimensional Ginzburg-Landau systems,” Phys. Rev. A 75(6), 061802 (2007).
[Crossref]

Auston, D.

D. Auston, “Transverse mode locking,” IEEE J. Quantum Electron. 4(6), 420–422 (1968).
[Crossref]

Backus, S.

Bartelt, H.

F. Eilenberger, K. Prater, S. Minardi, R. Geiss, U. Röpke, J. Kobelke, K. Schuster, H. Bartelt, S. Nolte, A. Tünnermann, and T. Pertsch, “Observation of discrete, vortex light bullets,” Phys. Rev. X 3(4), 041031 (2013).
[Crossref]

Barthé lémy, A.

K. Krupa, A. Tonello, B. M. Shalaby, M. Fabert, A. Barthé lémy, G. Millot, S. Wabnitz, and V. Couderc, “Spatial beam self-cleaning in multimode fibres,” Nat. Photonics 11(4), 237–241 (2017).
[Crossref]

Barthélémy, A.

K. Krupa, A. Tonello, A. Barthélémy, V. Couderc, B. M. Shalaby, A. Bendahmane, G. Millot, and S. Wabnitz, “Observation of geometric parametric instability induced by the periodic spatial self-imaging of multimode waves,” Phys. Rev. Lett. 116(18), 183901 (2016).
[Crossref]

Bedoya, A. C.

Bendahmane, A.

K. Krupa, A. Tonello, A. Barthélémy, V. Couderc, B. M. Shalaby, A. Bendahmane, G. Millot, and S. Wabnitz, “Observation of geometric parametric instability induced by the periodic spatial self-imaging of multimode waves,” Phys. Rev. Lett. 116(18), 183901 (2016).
[Crossref]

Büttner, T. F. S.

Chekhovskoy, I. S.

I. S. Chekhovskoy, A. M. Rubenchik, O. V. Shtyrina, and M. P. Fedoruk, “Nonlinear combining and compression in multicore fibers,” Phys. Rev. A 94(4), 043848 (2016).
[Crossref]

Chen, S.

S. Chen, “Analytical spinless light-bullet solutions as attractive fixed points in the three-dimensional cubic-quintic complex Ginzburg-Landau equation,” Phys. Rev. A 86(3), 033829 (2012).
[Crossref]

Chen, T.

T. Chen, Q. Zhang, Y. Zhang, and X. Li, “All-fiber passively mode-locked laser using nonlinear multimode interference of step-index multimode fiber,” Photonics Res. 6(11), 1033–1039 (2018).
[Crossref]

Christodoulides, D. N.

Couderc, V.

K. Krupa, A. Tonello, B. M. Shalaby, M. Fabert, A. Barthé lémy, G. Millot, S. Wabnitz, and V. Couderc, “Spatial beam self-cleaning in multimode fibres,” Nat. Photonics 11(4), 237–241 (2017).
[Crossref]

K. Krupa, A. Tonello, A. Barthélémy, V. Couderc, B. M. Shalaby, A. Bendahmane, G. Millot, and S. Wabnitz, “Observation of geometric parametric instability induced by the periodic spatial self-imaging of multimode waves,” Phys. Rev. Lett. 116(18), 183901 (2016).
[Crossref]

Cui, X.

D. Mao, M. Li, Z. He, X. Cui, H. Lu, W. Zhang, H. Zhang, and J. Zhao, “Optical vortex fiber laser based on modulation of transverse modes in two mode fiber,” APL Photonics 4(6), 060801 (2019).
[Crossref]

Ding, Y.

Dunlop, A. M.

A. M. Dunlop, W. J. Firth, and E. M. Wright, “Master equation for spatiotemporal beam propagation and Kerr lens mode-locking,” Opt. Commun. 138(1-3), 211–226 (1997).
[Crossref]

Eggleton, B. J.

Eilenberger, F.

F. Eilenberger, K. Prater, S. Minardi, R. Geiss, U. Röpke, J. Kobelke, K. Schuster, H. Bartelt, S. Nolte, A. Tünnermann, and T. Pertsch, “Observation of discrete, vortex light bullets,” Phys. Rev. X 3(4), 041031 (2013).
[Crossref]

S. Minardi, F. Eilenberger, and Y. V. Kartashov, “Three-Dimensional Light Bullets in Arrays of Waveguides,” Phys. Rev. Lett. 105(26), 263901 (2010).
[Crossref]

Fabert, M.

K. Krupa, A. Tonello, B. M. Shalaby, M. Fabert, A. Barthé lémy, G. Millot, S. Wabnitz, and V. Couderc, “Spatial beam self-cleaning in multimode fibres,” Nat. Photonics 11(4), 237–241 (2017).
[Crossref]

Fedorov, S. V.

N. A. Veretenov, S. V. Fedorov, and N. N. Rosanov, “Topological Vortex and Knotted Dissipative Optical 3D Solitons Generated by 2D Vortex Solitons,” Phys. Rev. Lett. 119(26), 263901 (2017).
[Crossref]

Fedoruk, M. P.

O. V. Shtyrina, M. P. Fedoruk, Y. S. Kivshar, and S. K. Turitsyn, “Coexistence of collapse and stable spatiotemporal solitons in multimode fibers,” Phys. Rev. A 97(1), 013841 (2018).
[Crossref]

I. S. Chekhovskoy, A. M. Rubenchik, O. V. Shtyrina, and M. P. Fedoruk, “Nonlinear combining and compression in multicore fibers,” Phys. Rev. A 94(4), 043848 (2016).
[Crossref]

Fini, J.

D. Richardson, J. Fini, and L. Nelson, “Space-division multiplexing in optical fibres,” Nat. Photonics 7(5), 354–362 (2013).
[Crossref]

Firth, W. J.

A. M. Dunlop, W. J. Firth, and E. M. Wright, “Master equation for spatiotemporal beam propagation and Kerr lens mode-locking,” Opt. Commun. 138(1-3), 211–226 (1997).
[Crossref]

Fu, W.

Gauthier, D.

N. B. Aleksić, V. Skarka, D. V. Timotijević, and D. Gauthier, “Self-stabilized spatiotemporal dynamics of dissipative light bullets generated from inputs without spherical symmetry in three-dimensional Ginzburg-Landau systems,” Phys. Rev. A 75(6), 061802 (2007).
[Crossref]

Geiss, R.

F. Eilenberger, K. Prater, S. Minardi, R. Geiss, U. Röpke, J. Kobelke, K. Schuster, H. Bartelt, S. Nolte, A. Tünnermann, and T. Pertsch, “Observation of discrete, vortex light bullets,” Phys. Rev. X 3(4), 041031 (2013).
[Crossref]

Grelu, P.

P. Grelu and N. Akhmediev, “Dissipative solitons for mode-locked lasers,” Nat. Photonics 6(2), 84–92 (2012).
[Crossref]

N. Akhmediev, J. M. Soto-Crespo, and P. Grelu, “Spatiotemporal optical solitons in nonlinear dissipative media: From stationary light bullets to pulsating complexes,” Chaos 17(3), 037112 (2007).
[Crossref]

He, X.-D.

He, Z.

D. Mao, M. Li, Z. He, X. Cui, H. Lu, W. Zhang, H. Zhang, and J. Zhao, “Optical vortex fiber laser based on modulation of transverse modes in two mode fiber,” APL Photonics 4(6), 060801 (2019).
[Crossref]

Hudson, D. D.

Jauregui, C.

C. Jauregui, J. Limpert, and A. Tünnermann, “High-power fibre lasers,” Nat. Photonics 7(11), 861–867 (2013).
[Crossref]

Kartashov, Y. V.

S. Minardi, F. Eilenberger, and Y. V. Kartashov, “Three-Dimensional Light Bullets in Arrays of Waveguides,” Phys. Rev. Lett. 105(26), 263901 (2010).
[Crossref]

Kivshar, Y. S.

O. V. Shtyrina, M. P. Fedoruk, Y. S. Kivshar, and S. K. Turitsyn, “Coexistence of collapse and stable spatiotemporal solitons in multimode fibers,” Phys. Rev. A 97(1), 013841 (2018).
[Crossref]

Kobelke, J.

F. Eilenberger, K. Prater, S. Minardi, R. Geiss, U. Röpke, J. Kobelke, K. Schuster, H. Bartelt, S. Nolte, A. Tünnermann, and T. Pertsch, “Observation of discrete, vortex light bullets,” Phys. Rev. X 3(4), 041031 (2013).
[Crossref]

Krupa, K.

K. Krupa, A. Tonello, B. M. Shalaby, M. Fabert, A. Barthé lémy, G. Millot, S. Wabnitz, and V. Couderc, “Spatial beam self-cleaning in multimode fibres,” Nat. Photonics 11(4), 237–241 (2017).
[Crossref]

K. Krupa, A. Tonello, A. Barthélémy, V. Couderc, B. M. Shalaby, A. Bendahmane, G. Millot, and S. Wabnitz, “Observation of geometric parametric instability induced by the periodic spatial self-imaging of multimode waves,” Phys. Rev. Lett. 116(18), 183901 (2016).
[Crossref]

Kutz, J. N.

Leblond, H.

D. Mihalache, D. Mazilu, F. Lederer, and H. Leblond, “Collisions between coaxial vortex solitons in the three-dimensional cubic-quintic complex Ginzburg-Landau equation,” Phys. Rev. A 77(3), 033817 (2008).
[Crossref]

D. Mihalache, D. Mazilu, F. Lederer, and H. Leblond, “Stability limits for three-dimensional vortex solitons in the Ginzburg-Landau equation with the cubic-quintic nonlinearity,” Phys. Rev. A 76(4), 045803 (2007).
[Crossref]

Lederer, F.

D. Mihalache, D. Mazilu, F. Lederer, and H. Leblond, “Collisions between coaxial vortex solitons in the three-dimensional cubic-quintic complex Ginzburg-Landau equation,” Phys. Rev. A 77(3), 033817 (2008).
[Crossref]

D. Mihalache, D. Mazilu, F. Lederer, and H. Leblond, “Stability limits for three-dimensional vortex solitons in the Ginzburg-Landau equation with the cubic-quintic nonlinearity,” Phys. Rev. A 76(4), 045803 (2007).
[Crossref]

Li, M.

D. Mao, M. Li, Z. He, X. Cui, H. Lu, W. Zhang, H. Zhang, and J. Zhao, “Optical vortex fiber laser based on modulation of transverse modes in two mode fiber,” APL Photonics 4(6), 060801 (2019).
[Crossref]

Li, X.

T. Chen, Q. Zhang, Y. Zhang, and X. Li, “All-fiber passively mode-locked laser using nonlinear multimode interference of step-index multimode fiber,” Photonics Res. 6(11), 1033–1039 (2018).
[Crossref]

Limpert, J.

C. Jauregui, J. Limpert, and A. Tünnermann, “High-power fibre lasers,” Nat. Photonics 7(11), 861–867 (2013).
[Crossref]

Liu, B.

Liu, Y.-F.

Liu, Z.

Lu, H.

D. Mao, M. Li, Z. He, X. Cui, H. Lu, W. Zhang, H. Zhang, and J. Zhao, “Optical vortex fiber laser based on modulation of transverse modes in two mode fiber,” APL Photonics 4(6), 060801 (2019).
[Crossref]

Lushnikov, P. M.

L. G. Wright, Z. M. Ziegler, P. M. Lushnikov, and Z. Zhu, “Multimode Nonlinear Fiber Optics: Massively Parallel Numerical Solver, Tutorial, and Outlook,” IEEE J. Sel. Top. Quantum Electron. 24(3), 1–16 (2018).
[Crossref]

Mägi, E. C.

Malomed, B. A.

B. A. Malomed, “(INVITED)Vortex solitons: Old results and new perspectives,” Phys. D (Amsterdam, Neth.) 399, 108–137 (2019).
[Crossref]

T. Mayteevarunyoo, B. A. Malomed, and D. V. Skryabin, “One- and two-dimensional modes in the complex Ginzburg-Landau equation with a trapping potential,” Opt. Express 26(7), 8849–8865 (2018).
[Crossref]

T. Mayteevarunyoo, B. A. Malomed, and D. Skryabin, “Vortex modes supported by spin-orbit coupling in a laser with saturable absorption,” New J. Phys. 20(11), 113019 (2018).
[Crossref]

Mao, D.

D. Mao, M. Li, Z. He, X. Cui, H. Lu, W. Zhang, H. Zhang, and J. Zhao, “Optical vortex fiber laser based on modulation of transverse modes in two mode fiber,” APL Photonics 4(6), 060801 (2019).
[Crossref]

Mayteevarunyoo, T.

T. Mayteevarunyoo, B. A. Malomed, and D. Skryabin, “Vortex modes supported by spin-orbit coupling in a laser with saturable absorption,” New J. Phys. 20(11), 113019 (2018).
[Crossref]

T. Mayteevarunyoo, B. A. Malomed, and D. V. Skryabin, “One- and two-dimensional modes in the complex Ginzburg-Landau equation with a trapping potential,” Opt. Express 26(7), 8849–8865 (2018).
[Crossref]

Mazilu, D.

D. Mihalache, D. Mazilu, F. Lederer, and H. Leblond, “Collisions between coaxial vortex solitons in the three-dimensional cubic-quintic complex Ginzburg-Landau equation,” Phys. Rev. A 77(3), 033817 (2008).
[Crossref]

D. Mihalache, D. Mazilu, F. Lederer, and H. Leblond, “Stability limits for three-dimensional vortex solitons in the Ginzburg-Landau equation with the cubic-quintic nonlinearity,” Phys. Rev. A 76(4), 045803 (2007).
[Crossref]

Mihalache, D.

D. Mihalache, D. Mazilu, F. Lederer, and H. Leblond, “Collisions between coaxial vortex solitons in the three-dimensional cubic-quintic complex Ginzburg-Landau equation,” Phys. Rev. A 77(3), 033817 (2008).
[Crossref]

D. Mihalache, D. Mazilu, F. Lederer, and H. Leblond, “Stability limits for three-dimensional vortex solitons in the Ginzburg-Landau equation with the cubic-quintic nonlinearity,” Phys. Rev. A 76(4), 045803 (2007).
[Crossref]

Millot, G.

K. Krupa, A. Tonello, B. M. Shalaby, M. Fabert, A. Barthé lémy, G. Millot, S. Wabnitz, and V. Couderc, “Spatial beam self-cleaning in multimode fibres,” Nat. Photonics 11(4), 237–241 (2017).
[Crossref]

K. Krupa, A. Tonello, A. Barthélémy, V. Couderc, B. M. Shalaby, A. Bendahmane, G. Millot, and S. Wabnitz, “Observation of geometric parametric instability induced by the periodic spatial self-imaging of multimode waves,” Phys. Rev. Lett. 116(18), 183901 (2016).
[Crossref]

Minardi, S.

F. Eilenberger, K. Prater, S. Minardi, R. Geiss, U. Röpke, J. Kobelke, K. Schuster, H. Bartelt, S. Nolte, A. Tünnermann, and T. Pertsch, “Observation of discrete, vortex light bullets,” Phys. Rev. X 3(4), 041031 (2013).
[Crossref]

S. Minardi, F. Eilenberger, and Y. V. Kartashov, “Three-Dimensional Light Bullets in Arrays of Waveguides,” Phys. Rev. Lett. 105(26), 263901 (2010).
[Crossref]

Nelson, L.

D. Richardson, J. Fini, and L. Nelson, “Space-division multiplexing in optical fibres,” Nat. Photonics 7(5), 354–362 (2013).
[Crossref]

Nolte, S.

F. Eilenberger, K. Prater, S. Minardi, R. Geiss, U. Röpke, J. Kobelke, K. Schuster, H. Bartelt, S. Nolte, A. Tünnermann, and T. Pertsch, “Observation of discrete, vortex light bullets,” Phys. Rev. X 3(4), 041031 (2013).
[Crossref]

Pertsch, T.

F. Eilenberger, K. Prater, S. Minardi, R. Geiss, U. Röpke, J. Kobelke, K. Schuster, H. Bartelt, S. Nolte, A. Tünnermann, and T. Pertsch, “Observation of discrete, vortex light bullets,” Phys. Rev. X 3(4), 041031 (2013).
[Crossref]

Prater, K.

F. Eilenberger, K. Prater, S. Minardi, R. Geiss, U. Röpke, J. Kobelke, K. Schuster, H. Bartelt, S. Nolte, A. Tünnermann, and T. Pertsch, “Observation of discrete, vortex light bullets,” Phys. Rev. X 3(4), 041031 (2013).
[Crossref]

Proctor, J.

Qin, H.

Raghavan, S.

S. Raghavan and G. P. Agrawal, “Spatiotemporal solitons in inhomogeneous nonlinear media,” Opt. Commun. 180(4-6), 377–382 (2000).
[Crossref]

Renninger, W. H.

Richardson, D.

D. Richardson, J. Fini, and L. Nelson, “Space-division multiplexing in optical fibres,” Nat. Photonics 7(5), 354–362 (2013).
[Crossref]

Röpke, U.

F. Eilenberger, K. Prater, S. Minardi, R. Geiss, U. Röpke, J. Kobelke, K. Schuster, H. Bartelt, S. Nolte, A. Tünnermann, and T. Pertsch, “Observation of discrete, vortex light bullets,” Phys. Rev. X 3(4), 041031 (2013).
[Crossref]

Rosanov, N. N.

N. A. Veretenov, S. V. Fedorov, and N. N. Rosanov, “Topological Vortex and Knotted Dissipative Optical 3D Solitons Generated by 2D Vortex Solitons,” Phys. Rev. Lett. 119(26), 263901 (2017).
[Crossref]

Rubenchik, A. M.

I. S. Chekhovskoy, A. M. Rubenchik, O. V. Shtyrina, and M. P. Fedoruk, “Nonlinear combining and compression in multicore fibers,” Phys. Rev. A 94(4), 043848 (2016).
[Crossref]

Schuster, K.

F. Eilenberger, K. Prater, S. Minardi, R. Geiss, U. Röpke, J. Kobelke, K. Schuster, H. Bartelt, S. Nolte, A. Tünnermann, and T. Pertsch, “Observation of discrete, vortex light bullets,” Phys. Rev. X 3(4), 041031 (2013).
[Crossref]

Shalaby, B. M.

K. Krupa, A. Tonello, B. M. Shalaby, M. Fabert, A. Barthé lémy, G. Millot, S. Wabnitz, and V. Couderc, “Spatial beam self-cleaning in multimode fibres,” Nat. Photonics 11(4), 237–241 (2017).
[Crossref]

K. Krupa, A. Tonello, A. Barthélémy, V. Couderc, B. M. Shalaby, A. Bendahmane, G. Millot, and S. Wabnitz, “Observation of geometric parametric instability induced by the periodic spatial self-imaging of multimode waves,” Phys. Rev. Lett. 116(18), 183901 (2016).
[Crossref]

Shtyrina, O. V.

O. V. Shtyrina, M. P. Fedoruk, Y. S. Kivshar, and S. K. Turitsyn, “Coexistence of collapse and stable spatiotemporal solitons in multimode fibers,” Phys. Rev. A 97(1), 013841 (2018).
[Crossref]

I. S. Chekhovskoy, A. M. Rubenchik, O. V. Shtyrina, and M. P. Fedoruk, “Nonlinear combining and compression in multicore fibers,” Phys. Rev. A 94(4), 043848 (2016).
[Crossref]

Sidorenko, P.

Skarka, V.

N. B. Aleksić, V. Skarka, D. V. Timotijević, and D. Gauthier, “Self-stabilized spatiotemporal dynamics of dissipative light bullets generated from inputs without spherical symmetry in three-dimensional Ginzburg-Landau systems,” Phys. Rev. A 75(6), 061802 (2007).
[Crossref]

Skryabin, D.

T. Mayteevarunyoo, B. A. Malomed, and D. Skryabin, “Vortex modes supported by spin-orbit coupling in a laser with saturable absorption,” New J. Phys. 20(11), 113019 (2018).
[Crossref]

Skryabin, D. V.

Smith, P. W.

P. W. Smith, “Simultaneous phase-locking of longitudinal and transverse laser modes,” Appl. Phys. Lett. 13(7), 235–237 (1968).
[Crossref]

Soto-Crespo, J. M.

N. Akhmediev, J. M. Soto-Crespo, and P. Grelu, “Spatiotemporal optical solitons in nonlinear dissipative media: From stationary light bullets to pulsating complexes,” Chaos 17(3), 037112 (2007).
[Crossref]

Taunay, T.

Timotijevic, D. V.

N. B. Aleksić, V. Skarka, D. V. Timotijević, and D. Gauthier, “Self-stabilized spatiotemporal dynamics of dissipative light bullets generated from inputs without spherical symmetry in three-dimensional Ginzburg-Landau systems,” Phys. Rev. A 75(6), 061802 (2007).
[Crossref]

Tonello, A.

K. Krupa, A. Tonello, B. M. Shalaby, M. Fabert, A. Barthé lémy, G. Millot, S. Wabnitz, and V. Couderc, “Spatial beam self-cleaning in multimode fibres,” Nat. Photonics 11(4), 237–241 (2017).
[Crossref]

K. Krupa, A. Tonello, A. Barthélémy, V. Couderc, B. M. Shalaby, A. Bendahmane, G. Millot, and S. Wabnitz, “Observation of geometric parametric instability induced by the periodic spatial self-imaging of multimode waves,” Phys. Rev. Lett. 116(18), 183901 (2016).
[Crossref]

Tünnermann, A.

C. Jauregui, J. Limpert, and A. Tünnermann, “High-power fibre lasers,” Nat. Photonics 7(11), 861–867 (2013).
[Crossref]

F. Eilenberger, K. Prater, S. Minardi, R. Geiss, U. Röpke, J. Kobelke, K. Schuster, H. Bartelt, S. Nolte, A. Tünnermann, and T. Pertsch, “Observation of discrete, vortex light bullets,” Phys. Rev. X 3(4), 041031 (2013).
[Crossref]

Turitsyn, S. K.

O. V. Shtyrina, M. P. Fedoruk, Y. S. Kivshar, and S. K. Turitsyn, “Coexistence of collapse and stable spatiotemporal solitons in multimode fibers,” Phys. Rev. A 97(1), 013841 (2018).
[Crossref]

Veretenov, N. A.

N. A. Veretenov, S. V. Fedorov, and N. N. Rosanov, “Topological Vortex and Knotted Dissipative Optical 3D Solitons Generated by 2D Vortex Solitons,” Phys. Rev. Lett. 119(26), 263901 (2017).
[Crossref]

Wabnitz, S.

K. Krupa, A. Tonello, B. M. Shalaby, M. Fabert, A. Barthé lémy, G. Millot, S. Wabnitz, and V. Couderc, “Spatial beam self-cleaning in multimode fibres,” Nat. Photonics 11(4), 237–241 (2017).
[Crossref]

K. Krupa, A. Tonello, A. Barthélémy, V. Couderc, B. M. Shalaby, A. Bendahmane, G. Millot, and S. Wabnitz, “Observation of geometric parametric instability induced by the periodic spatial self-imaging of multimode waves,” Phys. Rev. Lett. 116(18), 183901 (2016).
[Crossref]

Walton, D. T.

Wang, P.

Winful, H. G.

Wise, F. W.

Wright, E. M.

A. M. Dunlop, W. J. Firth, and E. M. Wright, “Master equation for spatiotemporal beam propagation and Kerr lens mode-locking,” Opt. Commun. 138(1-3), 211–226 (1997).
[Crossref]

Wright, L. G.

L. G. Wright, Z. M. Ziegler, P. M. Lushnikov, and Z. Zhu, “Multimode Nonlinear Fiber Optics: Massively Parallel Numerical Solver, Tutorial, and Outlook,” IEEE J. Sel. Top. Quantum Electron. 24(3), 1–16 (2018).
[Crossref]

W. Fu, L. G. Wright, P. Sidorenko, S. Backus, and F. W. Wise, “Several new directions for ultrafast fiber lasers [Invited],” Opt. Express 26(8), 9432–9463 (2018).
[Crossref]

L. G. Wright, D. N. Christodoulides, and F. W. Wise, “Spatiotemporal mode-locking in multimode fiber lasers,” Science 358(6359), 94–97 (2017).
[Crossref]

Z. Liu, L. G. Wright, D. N. Christodoulides, and F. W. Wise, “Kerr self-cleaning of femtosecond-pulsed beams in graded-index multimode fiber,” Opt. Lett. 41(16), 3675–3678 (2016).
[Crossref]

Z. Zhu, L. G. Wright, D. N. Christodoulides, and F. W. Wise, “Observation of multimode solitons in few-mode fiber,” Opt. Lett. 41(20), 4819–4822 (2016).
[Crossref]

L. G. Wright, D. N. Christodoulides, and F. W. Wise, “Controllable spatiotemporal nonlinear effects in multimode fibres,” Nat. Photonics 9(5), 306–310 (2015).
[Crossref]

L. G. Wright, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Spatiotemporal dynamics of multimode optical solitons,” Opt. Express 23(3), 3492–3506 (2015).
[Crossref]

Xiao, X.

Yang, C.

Yang, J.

J. Yang, Nonlinear Waves in Integrable and Nonintegrable Systems (SIAM: Philadelphia, 2010).

Zhang, H.

D. Mao, M. Li, Z. He, X. Cui, H. Lu, W. Zhang, H. Zhang, and J. Zhao, “Optical vortex fiber laser based on modulation of transverse modes in two mode fiber,” APL Photonics 4(6), 060801 (2019).
[Crossref]

Zhang, Q.

T. Chen, Q. Zhang, Y. Zhang, and X. Li, “All-fiber passively mode-locked laser using nonlinear multimode interference of step-index multimode fiber,” Photonics Res. 6(11), 1033–1039 (2018).
[Crossref]

Zhang, W.

D. Mao, M. Li, Z. He, X. Cui, H. Lu, W. Zhang, H. Zhang, and J. Zhao, “Optical vortex fiber laser based on modulation of transverse modes in two mode fiber,” APL Photonics 4(6), 060801 (2019).
[Crossref]

Zhang, Y.

T. Chen, Q. Zhang, Y. Zhang, and X. Li, “All-fiber passively mode-locked laser using nonlinear multimode interference of step-index multimode fiber,” Photonics Res. 6(11), 1033–1039 (2018).
[Crossref]

Zhao, J.

D. Mao, M. Li, Z. He, X. Cui, H. Lu, W. Zhang, H. Zhang, and J. Zhao, “Optical vortex fiber laser based on modulation of transverse modes in two mode fiber,” APL Photonics 4(6), 060801 (2019).
[Crossref]

Zhu, Z.

L. G. Wright, Z. M. Ziegler, P. M. Lushnikov, and Z. Zhu, “Multimode Nonlinear Fiber Optics: Massively Parallel Numerical Solver, Tutorial, and Outlook,” IEEE J. Sel. Top. Quantum Electron. 24(3), 1–16 (2018).
[Crossref]

Z. Zhu, L. G. Wright, D. N. Christodoulides, and F. W. Wise, “Observation of multimode solitons in few-mode fiber,” Opt. Lett. 41(20), 4819–4822 (2016).
[Crossref]

Ziegler, Z. M.

L. G. Wright, Z. M. Ziegler, P. M. Lushnikov, and Z. Zhu, “Multimode Nonlinear Fiber Optics: Massively Parallel Numerical Solver, Tutorial, and Outlook,” IEEE J. Sel. Top. Quantum Electron. 24(3), 1–16 (2018).
[Crossref]

APL Photonics (1)

D. Mao, M. Li, Z. He, X. Cui, H. Lu, W. Zhang, H. Zhang, and J. Zhao, “Optical vortex fiber laser based on modulation of transverse modes in two mode fiber,” APL Photonics 4(6), 060801 (2019).
[Crossref]

Appl. Phys. Lett. (1)

P. W. Smith, “Simultaneous phase-locking of longitudinal and transverse laser modes,” Appl. Phys. Lett. 13(7), 235–237 (1968).
[Crossref]

Chaos (1)

N. Akhmediev, J. M. Soto-Crespo, and P. Grelu, “Spatiotemporal optical solitons in nonlinear dissipative media: From stationary light bullets to pulsating complexes,” Chaos 17(3), 037112 (2007).
[Crossref]

IEEE J. Quantum Electron. (1)

D. Auston, “Transverse mode locking,” IEEE J. Quantum Electron. 4(6), 420–422 (1968).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (1)

L. G. Wright, Z. M. Ziegler, P. M. Lushnikov, and Z. Zhu, “Multimode Nonlinear Fiber Optics: Massively Parallel Numerical Solver, Tutorial, and Outlook,” IEEE J. Sel. Top. Quantum Electron. 24(3), 1–16 (2018).
[Crossref]

Nat. Commun. (1)

W. H. Renninger and F. W. Wise, “Optical solitons in graded-index multimode fibres,” Nat. Commun. 4(1), 1719 (2013).
[Crossref]

Nat. Photonics (5)

L. G. Wright, D. N. Christodoulides, and F. W. Wise, “Controllable spatiotemporal nonlinear effects in multimode fibres,” Nat. Photonics 9(5), 306–310 (2015).
[Crossref]

C. Jauregui, J. Limpert, and A. Tünnermann, “High-power fibre lasers,” Nat. Photonics 7(11), 861–867 (2013).
[Crossref]

D. Richardson, J. Fini, and L. Nelson, “Space-division multiplexing in optical fibres,” Nat. Photonics 7(5), 354–362 (2013).
[Crossref]

P. Grelu and N. Akhmediev, “Dissipative solitons for mode-locked lasers,” Nat. Photonics 6(2), 84–92 (2012).
[Crossref]

K. Krupa, A. Tonello, B. M. Shalaby, M. Fabert, A. Barthé lémy, G. Millot, S. Wabnitz, and V. Couderc, “Spatial beam self-cleaning in multimode fibres,” Nat. Photonics 11(4), 237–241 (2017).
[Crossref]

New J. Phys. (1)

T. Mayteevarunyoo, B. A. Malomed, and D. Skryabin, “Vortex modes supported by spin-orbit coupling in a laser with saturable absorption,” New J. Phys. 20(11), 113019 (2018).
[Crossref]

Opt. Commun. (2)

S. Raghavan and G. P. Agrawal, “Spatiotemporal solitons in inhomogeneous nonlinear media,” Opt. Commun. 180(4-6), 377–382 (2000).
[Crossref]

A. M. Dunlop, W. J. Firth, and E. M. Wright, “Master equation for spatiotemporal beam propagation and Kerr lens mode-locking,” Opt. Commun. 138(1-3), 211–226 (1997).
[Crossref]

Opt. Express (6)

Opt. Lett. (6)

Photonics Res. (1)

T. Chen, Q. Zhang, Y. Zhang, and X. Li, “All-fiber passively mode-locked laser using nonlinear multimode interference of step-index multimode fiber,” Photonics Res. 6(11), 1033–1039 (2018).
[Crossref]

Phys. D (Amsterdam, Neth.) (1)

B. A. Malomed, “(INVITED)Vortex solitons: Old results and new perspectives,” Phys. D (Amsterdam, Neth.) 399, 108–137 (2019).
[Crossref]

Phys. Rev. A (6)

O. V. Shtyrina, M. P. Fedoruk, Y. S. Kivshar, and S. K. Turitsyn, “Coexistence of collapse and stable spatiotemporal solitons in multimode fibers,” Phys. Rev. A 97(1), 013841 (2018).
[Crossref]

I. S. Chekhovskoy, A. M. Rubenchik, O. V. Shtyrina, and M. P. Fedoruk, “Nonlinear combining and compression in multicore fibers,” Phys. Rev. A 94(4), 043848 (2016).
[Crossref]

N. B. Aleksić, V. Skarka, D. V. Timotijević, and D. Gauthier, “Self-stabilized spatiotemporal dynamics of dissipative light bullets generated from inputs without spherical symmetry in three-dimensional Ginzburg-Landau systems,” Phys. Rev. A 75(6), 061802 (2007).
[Crossref]

S. Chen, “Analytical spinless light-bullet solutions as attractive fixed points in the three-dimensional cubic-quintic complex Ginzburg-Landau equation,” Phys. Rev. A 86(3), 033829 (2012).
[Crossref]

D. Mihalache, D. Mazilu, F. Lederer, and H. Leblond, “Stability limits for three-dimensional vortex solitons in the Ginzburg-Landau equation with the cubic-quintic nonlinearity,” Phys. Rev. A 76(4), 045803 (2007).
[Crossref]

D. Mihalache, D. Mazilu, F. Lederer, and H. Leblond, “Collisions between coaxial vortex solitons in the three-dimensional cubic-quintic complex Ginzburg-Landau equation,” Phys. Rev. A 77(3), 033817 (2008).
[Crossref]

Phys. Rev. Lett. (3)

S. Minardi, F. Eilenberger, and Y. V. Kartashov, “Three-Dimensional Light Bullets in Arrays of Waveguides,” Phys. Rev. Lett. 105(26), 263901 (2010).
[Crossref]

K. Krupa, A. Tonello, A. Barthélémy, V. Couderc, B. M. Shalaby, A. Bendahmane, G. Millot, and S. Wabnitz, “Observation of geometric parametric instability induced by the periodic spatial self-imaging of multimode waves,” Phys. Rev. Lett. 116(18), 183901 (2016).
[Crossref]

N. A. Veretenov, S. V. Fedorov, and N. N. Rosanov, “Topological Vortex and Knotted Dissipative Optical 3D Solitons Generated by 2D Vortex Solitons,” Phys. Rev. Lett. 119(26), 263901 (2017).
[Crossref]

Phys. Rev. X (1)

F. Eilenberger, K. Prater, S. Minardi, R. Geiss, U. Röpke, J. Kobelke, K. Schuster, H. Bartelt, S. Nolte, A. Tünnermann, and T. Pertsch, “Observation of discrete, vortex light bullets,” Phys. Rev. X 3(4), 041031 (2013).
[Crossref]

Science (1)

L. G. Wright, D. N. Christodoulides, and F. W. Wise, “Spatiotemporal mode-locking in multimode fiber lasers,” Science 358(6359), 94–97 (2017).
[Crossref]

Other (2)

J. Yang, Nonlinear Waves in Integrable and Nonintegrable Systems (SIAM: Philadelphia, 2010).

N. Akhmediev and A. Ankiewicz, eds. Dissipative solitons, Vol. 661 of Lecture Notes in Physics (Springer, 2005).

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Figures (13)
Fig. 1.
Fig. 1. (a) Numerically computed propagation constant $q$ and integral power $P$ [see Eq. (5)] of fundamental three-dimensional solitons versus $ \varepsilon$, for fixed $ \mu =1$ in Eq. (3). Red, blue and black segments correspond to stable fundamental solitons, moving breathers, and eventually destructed modes, respectively. (b) The stability chart for the stationary fundamental solitons and modes into which unstable solitons are spontaneously transformed. Indicated inside the stability area of solitons is the bistability region, in which solitary vortices with winding numbers are stable too, cf. Figure 5(b).

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Fig. 2.
Fig. 2. (a) The evolution of a stable fundamental soliton, for $ \mu =1, \varepsilon =0.5$, with $q=-0.8635,P=6.9898$, is displayed by means of isosurfaces of $\left \vert \psi \left ( x,y,z,t\right ) \right \vert$. (b) Left and right panels display spatial and temporal cross sections of the soliton’s shape, $| \psi (x,0,z,0)|$ and $| \psi (0,0,z,t)|$, respectively.

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Fig. 3.
Fig. 3. (a) The evolution of a robust quiescent breather, into which an unstable fundamental soliton transforms for $ \mu =0.3, \varepsilon =2.8$, which corresponding to $q=-2.0446,P=5.3081$, shown by means of the isosurface of $\left \vert \psi \left ( x,y,z,t\right ) \right \vert$. (b) Left and right panels show the evolution in the spatial and temporal of cross sections, $| \psi (x,0,z,0)|$ and $| \psi (0,0,z,t)|$, respectively.

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Fig. 4.
Fig. 4. A robust moving breather for $ \mu =1.0, \varepsilon =1.8$, into which the respective unstable soliton, with $q=-1.7927,P=4.5920$, spontaneously transforms. (a) The evolution in the spatial and temporal cross sections, $| \psi (x,0,z,0)|$ (left) and $| \psi (0,0,z,t)|$ (right). (b) The spatiotemporal dynamics with temporal motion.

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Fig. 5.
Fig. 5. (a) The numerically found values of $q$ (top panel) and $P$ (bottom panel) for stationary vortices with winding number $n=1$ versus $ \varepsilon$ for fixed $ \mu =1.0$. Red, blue, magenta, yellow and black segments correspond to stable vortices, splitting into fundamental solitons, chaotically oscillating modes, moving vortex breathers, and destruction of unstable vortices, respectively. (b) The stability chart for vortices with $n=1$ in the plane of $\left ( \mu , \varepsilon \right )$. The central shaded area in (b) is populated by chaotically oscillating localized modes, see an example in Fig. 8. Note that the stability area of vortices is completely covered by the region in which the fundamental solitons are stable [cf. Fig. 1(b)], i.e., it is a bistability area.

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Fig. 6.
Fig. 6. (a) The isosurface evolution of a stable vortex found at $ \mu =1.0, \varepsilon =0.2$, which corresponding to $q=-2.3130,P=15.9142$. (b) Its amplitude and phase patterns (c) Left and right panels display the evolution of the spatial and temporal cross sections $| \psi (x,0,z,0)|$ and $| \psi (0,0,z,t)|$, respectively.

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Fig. 7.
Fig. 7. The spontaneous splitting of an unstable vortex with $n=1$ into two fundamental solitons, at $ \mu =1.0, \varepsilon =0.6$. (a) The evolution of $| \psi |$ is shown in the $t$- and $x$-cross sections (top and bottom panels, respectively). (b) The temporal shape of the moving fundamental solitons, produced by the splitting, at $z=400$ (the blue line), fitted to the standard soliton’s shape, $\mathrm {sech}(t)$.

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Fig. 8.
Fig. 8. (a) The evolution of $| \psi \left ( 0,0,z,t\right ) |$ and $| \psi \left ( x,0,z,0\right ) |$ in the $t$- and $x$- cross-sections, illustrating the spontaneous transformation of an unstable stationary vortex with $n=1$ into a chaotically oscillating mode at $ \mu =1.0, \varepsilon =0.9$. (b) The respective amplitude and phase patterns at $z=1000$ (eventual values of the amplitude are small, as a result of the evolution).

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Fig. 9.
Fig. 9. A moving breather, into which an unstable stationary vortex, with $q=-2.9276,P=9.9546$, is spontaneously transformed (keeping its vorticity) at $ \mu =1.0, \varepsilon =1.2$. The evolution of cross sections $| \psi \left ( x,0,z,0\right ) |$ and $| \psi \left ( 0,0,z,t\right ) |$ is displayed in the left and right panels, respectively.

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Fig. 10.
Fig. 10. The stability chart in the plane of $\left ( \mu , \varepsilon \right )$ for vortices with $n=2$. The shaded area corresponds to the “ wobbling breathers” .

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Fig. 11.
Fig. 11. Spontaneous splitting of an unstable vortex with $n=2$ into a stably rotating bound state of two vortices with $n=1$ at $ \mu =0.4$, $ \varepsilon =0.1$ (a) The evolution of the field in the temporal and spatial cross sections, $| \psi (0,0,z,t)|$ and $| \psi (x,0,z,0)|$. (b) Amplitude and phase patterns of $ \psi (x,y)$ in the plane of $t=0$ and (bottom) the temporal profile at the central point, $x=y=0$, plotted at $z=500$ (c) The top, left and right panels zoom in on the evolution of the absolute value of the field, and spatial and temporal cross sections, respectively, in the interval of $300<z<320$. The established evolution (rotation) period is $T_{z}=2.2$.

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Fig. 12.
Fig. 12. The “ wobbling breather” at $ \mu =0.4$, $ \varepsilon =0.6$ (a) The evolution of the field in the temporal and spatial cross sections, $| \psi (0,0,z,t)|$ and $| \psi (x,0,z,0)|$. (b) The top, left and right panels zoom in on the evolution of the absolute value of the field, and spatial and temporal cross sections in the interval of $z=200-220$, respectively. The established evolution period is $T_{z}=4.5$.

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Fig. 13.
Fig. 13. (a) Splitting of an unstable breather with $n=2$ into $2$, $3$ and $4$ secondary breathers with zero intrinsic vorticity at $ \mu =0.4, \varepsilon =0.9$ (top), $ \mu =0.4, \varepsilon =1.4$ (middle), and $ \mu =0.2, \varepsilon =3.0$ (bottom), displayed by means of temporal cross sections, $| \psi \left ( 0,0,z,t\right ) |$. Panels in the right column display the respective evolution in the spatial cross section, $| \psi \left ( x,0,z,0\right ) |$. (b) The same as (a) but illustrating the isosurface evolution in 3D.

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Equations (5)

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i ∂ ψ ∂ z + L 2 τ 2 k ′ ′ ( 1 − i β ) ∂ 2 ψ ∂ t 2 + L 2 k 0 w 2 ( ∂ 2 ∂ x 2 + ∂ 2 ∂ y 2 ) ψ − k 0 g w 2 L 2 ( x 2 + y 2 ) ψ + i L ε ′ ψ + L ( η ′ − i α ′ ) | ψ | 2 ψ + L ( ν ′ + i μ ′ ) | ψ | 4 ψ = 0.
i ∂ ψ ∂ z + 1 2 ( 1 − i β ) ∂ 2 ψ ∂ t 2 + 1 2 ( ∂ 2 ∂ x 2 + ∂ 2 ∂ y 2 ) ψ − ( x 2 + y 2 − i ε ) ψ + ( η − i α ) | ψ | 2 ψ + ( ν + i μ ) | ψ | 4 ψ = 0.
α = A α t h r ,   α t h r ≡ ( 2 ε μ )
− q ϕ + 1 2 ( ∂ 2 ∂ t 2 + ∂ 2 ∂ x 2 + ∂ 2 ∂ y 2 ) ϕ − ( x 2 + y 2 − i ε ) ϕ + ( η − i α ) | ϕ | 2 ϕ + i μ | ϕ | 4 ϕ = 0.
P = ∫ ∫ ∫ | ψ ( x , y , t ) | 2 d x d y d t .

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