Abstract

We propose a new mechanism for the stabilization of confined modes in lasers and semiconductor microcavities holding exciton-polariton condensates, with spatially uniform linear gain, cubic loss, and cubic self-focusing or defocusing nonlinearity. We demonstrated that the commonly known background instability driven by the linear gain can be suppressed by a combination of a harmonic-oscillator trapping potential and effective diffusion. Systematic numerical analysis of one- and two-dimensional (1D and 2D) versions of the model reveals a variety of stable modes, including stationary ones, breathers, and quasi-regular patterns filling the trapping area in the 1D case. In 2D, the analysis produces stationary modes, breathers, axisymmetric and rotating crescent-shaped vortices, stably rotating complexes built of up to 8 individual vortices, and, in addition, patterns featuring vortex turbulence. Existence boundaries for both 1D and 2D stationary modes are found in an exact analytical form, and an analytical approximation is developed for the full stationary states.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  24. D. A. Zezyulin, G. L. Alfimov, and V. V. Konotop, “Nonlinear modes in a complex parabolic potential,” Phys. Rev. A 81, 013606 (2010).
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  26. N. A. Veretenov, N. N. Rosanov, and S. V. Fedorov, “Motion of complexes of 3D-laser solitons,” Opt. Quant. Elect. 40, 253–262 (2008).
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    [Crossref]
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    [Crossref]
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    [Crossref]
  41. V. Skarka, N. B. Aleksić, H. Leblond, B. A. Malomed, and D. Mihalache, “Varieties of stable vortical solitons in Ginzburg-Landau media with radially inhomogeneous losses,” Phys. Rev. Lett. 105, 213901 (2010).
    [Crossref]
  42. V. Skarka, N. B. Aleksić, M. Lekić, B. N. Aleksić, B. A. Malomed, D. Mihalache, and H. Leblond, “Formation of complex two-dimensional dissipative solitons via spontaneous symmetry breaking,” Phys. Rev. A 90, 023845 (2014).
    [Crossref]

2017 (1)

H. Sakaguchi, B. A. Malomed, and D. V. Skryabin, “Spin-orbit coupling and nonlinear modes of the polariton condensate in a harmonic trap,” New J. Phys. 19, 08503 (2017).
[Crossref]

2015 (3)

J. Schachenmayer, C. Genes, E. Tignone, and G. Pupillo, “Cavity-enhanced transport of excitons,” Phys. Rev. Lett. 114, 196403 (2015).
[Crossref] [PubMed]

V. Shahnazaryan, O. Kyriienko, and I. A. Shelykh, “Adiabatic preparation of a cold exciton condensate,” Phys. Rev. B 91, 085302 (2015).
[Crossref]

N. Bobrovska and M. Matuszewski, “Adiabatic approximation and fluctuations in exciton-polariton condensates,” Phys. Rev. B 92, 035311 (2015).
[Crossref]

2014 (2)

B. A. Malomed, “Spatial solitons supported by localized gain [Invited],” J. Opt. Soc. Am. B 31, 2460–2475 (2014).
[Crossref]

V. Skarka, N. B. Aleksić, M. Lekić, B. N. Aleksić, B. A. Malomed, D. Mihalache, and H. Leblond, “Formation of complex two-dimensional dissipative solitons via spontaneous symmetry breaking,” Phys. Rev. A 90, 023845 (2014).
[Crossref]

2013 (2)

A. Askitopoulos, H. Ohadi, A. V. Kavokin, Z. Hatzopoulos, P. G. Savvidis, and P. G. Lagoudakis, “Polariton condensation in an optically induced two-dimensional potential,” Phys. Rev. B 88, 041308 (2013).
[Crossref]

V. Ardizzone, P. Lewandowski, M. H. Luk, Y. C. Tse, N. H. Kwong, A. Lucke, M. Abbarchi, E. Baudin, E. Galopin, J. Bloch, A. Lemaitre, P. T. Leung, P. Roussignol, R. Binder, J. Tignon, and S. Schumacher, “Formation and control of Turing patterns in a coherent quantum fluid,” Sci. Rep. 3, 3016 (2013).
[Crossref] [PubMed]

2012 (1)

O. V. Borovkova, Y. V. Kartashov, V. A. Vysloukh, V. E. Lobanov, B. A. Malomed, and L. Torner, “Solitons supported by spatially inhomogeneous nonlinear losses,” Opt. Exp. 20, 2657–2667 (2012).
[Crossref]

2011 (1)

2010 (5)

W. J. Firth and P. V. Paulau, “Soliton lasers stabilized by coupling to a resonant linear system,” Eur. Phys. J. D 59, 13–21 (2010).
[Crossref]

P. V. Paulau, D. Gomila, P. Colet, N. A. Loiko, N. N. Rosanov, T. Ackemann, and W. J. Firth, “Vortex solitons in lasers with feedback,” Opt. Express 18, 8859–8866 (2010).
[Crossref] [PubMed]

D. Mihalache, D. Mazilu, V. Skarka, B. A. Malomed, H. Leblond, N. B. Aleksić, and F. Lederer, “Stable topological modes in two-dimensional Ginzburg-Landau models with trapping potentials,” Phys. Rev. A 82, 023813 (2010).
[Crossref]

D. A. Zezyulin, G. L. Alfimov, and V. V. Konotop, “Nonlinear modes in a complex parabolic potential,” Phys. Rev. A 81, 013606 (2010).
[Crossref]

V. Skarka, N. B. Aleksić, H. Leblond, B. A. Malomed, and D. Mihalache, “Varieties of stable vortical solitons in Ginzburg-Landau media with radially inhomogeneous losses,” Phys. Rev. Lett. 105, 213901 (2010).
[Crossref]

2009 (1)

T. Ackemann, W. J. Firth, and G.-L. Oppo, “Fundamentals and applications of spatial dissipative solitons in photonic devices,” Adv. At. Mol. Opt. Phys. 57, 323–421 (2009).
[Crossref]

2008 (2)

W. Renninger, A. Chong, and E. Wise, “Dissipative solitons in normal-dispersion fiber lasers,” Phys. Rev. A 77, 023814 (2008).
[Crossref]

N. A. Veretenov, N. N. Rosanov, and S. V. Fedorov, “Motion of complexes of 3D-laser solitons,” Opt. Quant. Elect. 40, 253–262 (2008).
[Crossref]

2005 (1)

A. Komarov, H. Leblond, and F. Sanchez, “Quintic complex Ginzburg-Landau model for ring fiber lasers,” Phys. Rev. E 72, 025604 (2005).
[Crossref]

2004 (1)

S. Skupin, L. Bergé, U. Peschel, F. Lederer, G. Méjean, J. Yu, J. Kasparian, E. Salmon, J. P. Wolf, M. Rodriguez, L. Wöste, R. Bourayou, and R. Sauerbrey, “Filamentation of femtosecond light pulses in the air: Turbulent cells versus long-range clusters,” Phys. Rev. E 70, 046602 (2004).
[Crossref]

2002 (1)

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

2001 (2)

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]

V. Dunjko, V. Lorent, and M. Olshanii, “Bosons in cigar-shaped traps: Thomas-Fermi regime,” Tonks-Girardeau regime, and in between, Phys. Rev. Lett. 86, 5413–5416 (2001).
[Crossref]

2000 (1)

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

1998 (1)

J. Atai and B. A. Malomed, “Exact stable pulses in asymmetric linearly coupled Ginzburg-Landau equations,” Phys. Lett. A 246, 412–422 (1998).
[Crossref]

1997 (1)

D. Hochheiser, J. V. Moloney, and J. Lega, “Controlling optical turbulence,” Phys. Rev. A 55, R4011–R4014 (1997).
[Crossref]

1996 (1)

B. A. Malomed and H. G. Winful, “Stable solitons in two-component active systems,” Phys. Rev. E 53, 5365–5368 (1996).
[Crossref]

1994 (2)

J. Lega, J. V. Moloney, and A. C. Newell, “Swift-Hohenberg equation for lasers,” Phys. Rev. Lett. 73, 2978–2981 (1994).
[Crossref] [PubMed]

J. Lega, J. V. Moloney, and A. C. Newell, “Swift-Hohenberg equation for lasers,” Phys. Rev. Lett. 73, 2978–2981 (1994).
[Crossref] [PubMed]

1993 (1)

H. Sakaguchi, “Phase dynamics and localized solutions to the Ginzburg-Landau type amplitude equations,” Prog. Theor. Phys. 89, 1123–1146 (1993).
[Crossref]

1992 (1)

W. van Saarloos and P. C. Hohenberg, “Fronts, pulses, sources and sinks in generalized complex Ginzburg-Landau equation,” Physica D 56, 303–367 (1992).
[Crossref]

1989 (1)

P. Coullet, L. Gil, and F. Rocca, “Optical vortices,” Opt. Commun. 73, 403–408 (1989).
[Crossref]

1987 (1)

L. A. Lugiato and R. Lefever, “Spatial dissipative structures in passive optical systems,” Phys. Rev. Lett. 58, 2209–2211 (1987).
[Crossref] [PubMed]

1984 (1)

V. I. Petviashvili and A. M. Sergeev, “Spiral solitons in active media with excitation thresholds,” Dokl. AN SSSR (Sov. Phys. Dokl.) 276, 1380–1384 (1984).

1977 (1)

N. R. Pereira and L. Stenflo, “Nonlinear Schrödinger equation including growth and damping,” Phys. Fluids 20, 1733–1734 (1977).
[Crossref]

1972 (1)

L. M. Hocking and K. Stewartson, “Nonlinear response of a marginally unstable plane parallel flow to a 2-dimensional disturbance,” Proc. R. Soc. A 326, 289–313 (1972).
[Crossref]

Abbarchi, M.

V. Ardizzone, P. Lewandowski, M. H. Luk, Y. C. Tse, N. H. Kwong, A. Lucke, M. Abbarchi, E. Baudin, E. Galopin, J. Bloch, A. Lemaitre, P. T. Leung, P. Roussignol, R. Binder, J. Tignon, and S. Schumacher, “Formation and control of Turing patterns in a coherent quantum fluid,” Sci. Rep. 3, 3016 (2013).
[Crossref] [PubMed]

Ackemann, T.

P. V. Paulau, D. Gomila, P. Colet, N. A. Loiko, N. N. Rosanov, T. Ackemann, and W. J. Firth, “Vortex solitons in lasers with feedback,” Opt. Express 18, 8859–8866 (2010).
[Crossref] [PubMed]

T. Ackemann, W. J. Firth, and G.-L. Oppo, “Fundamentals and applications of spatial dissipative solitons in photonic devices,” Adv. At. Mol. Opt. Phys. 57, 323–421 (2009).
[Crossref]

Agrawal, G. P.

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

Aleksic, B. N.

V. Skarka, N. B. Aleksić, M. Lekić, B. N. Aleksić, B. A. Malomed, D. Mihalache, and H. Leblond, “Formation of complex two-dimensional dissipative solitons via spontaneous symmetry breaking,” Phys. Rev. A 90, 023845 (2014).
[Crossref]

Aleksic, N. B.

V. Skarka, N. B. Aleksić, M. Lekić, B. N. Aleksić, B. A. Malomed, D. Mihalache, and H. Leblond, “Formation of complex two-dimensional dissipative solitons via spontaneous symmetry breaking,” Phys. Rev. A 90, 023845 (2014).
[Crossref]

V. Skarka, N. B. Aleksić, H. Leblond, B. A. Malomed, and D. Mihalache, “Varieties of stable vortical solitons in Ginzburg-Landau media with radially inhomogeneous losses,” Phys. Rev. Lett. 105, 213901 (2010).
[Crossref]

D. Mihalache, D. Mazilu, V. Skarka, B. A. Malomed, H. Leblond, N. B. Aleksić, and F. Lederer, “Stable topological modes in two-dimensional Ginzburg-Landau models with trapping potentials,” Phys. Rev. A 82, 023813 (2010).
[Crossref]

Alfimov, G. L.

D. A. Zezyulin, G. L. Alfimov, and V. V. Konotop, “Nonlinear modes in a complex parabolic potential,” Phys. Rev. A 81, 013606 (2010).
[Crossref]

Aranson, I. S.

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

Ardizzone, V.

V. Ardizzone, P. Lewandowski, M. H. Luk, Y. C. Tse, N. H. Kwong, A. Lucke, M. Abbarchi, E. Baudin, E. Galopin, J. Bloch, A. Lemaitre, P. T. Leung, P. Roussignol, R. Binder, J. Tignon, and S. Schumacher, “Formation and control of Turing patterns in a coherent quantum fluid,” Sci. Rep. 3, 3016 (2013).
[Crossref] [PubMed]

Askitopoulos, A.

A. Askitopoulos, H. Ohadi, A. V. Kavokin, Z. Hatzopoulos, P. G. Savvidis, and P. G. Lagoudakis, “Polariton condensation in an optically induced two-dimensional potential,” Phys. Rev. B 88, 041308 (2013).
[Crossref]

Atai, J.

J. Atai and B. A. Malomed, “Exact stable pulses in asymmetric linearly coupled Ginzburg-Landau equations,” Phys. Lett. A 246, 412–422 (1998).
[Crossref]

Baudin, E.

V. Ardizzone, P. Lewandowski, M. H. Luk, Y. C. Tse, N. H. Kwong, A. Lucke, M. Abbarchi, E. Baudin, E. Galopin, J. Bloch, A. Lemaitre, P. T. Leung, P. Roussignol, R. Binder, J. Tignon, and S. Schumacher, “Formation and control of Turing patterns in a coherent quantum fluid,” Sci. Rep. 3, 3016 (2013).
[Crossref] [PubMed]

Bergé, L.

S. Skupin, L. Bergé, U. Peschel, F. Lederer, G. Méjean, J. Yu, J. Kasparian, E. Salmon, J. P. Wolf, M. Rodriguez, L. Wöste, R. Bourayou, and R. Sauerbrey, “Filamentation of femtosecond light pulses in the air: Turbulent cells versus long-range clusters,” Phys. Rev. E 70, 046602 (2004).
[Crossref]

Binder, R.

V. Ardizzone, P. Lewandowski, M. H. Luk, Y. C. Tse, N. H. Kwong, A. Lucke, M. Abbarchi, E. Baudin, E. Galopin, J. Bloch, A. Lemaitre, P. T. Leung, P. Roussignol, R. Binder, J. Tignon, and S. Schumacher, “Formation and control of Turing patterns in a coherent quantum fluid,” Sci. Rep. 3, 3016 (2013).
[Crossref] [PubMed]

Bloch, J.

V. Ardizzone, P. Lewandowski, M. H. Luk, Y. C. Tse, N. H. Kwong, A. Lucke, M. Abbarchi, E. Baudin, E. Galopin, J. Bloch, A. Lemaitre, P. T. Leung, P. Roussignol, R. Binder, J. Tignon, and S. Schumacher, “Formation and control of Turing patterns in a coherent quantum fluid,” Sci. Rep. 3, 3016 (2013).
[Crossref] [PubMed]

Bobrovska, N.

N. Bobrovska and M. Matuszewski, “Adiabatic approximation and fluctuations in exciton-polariton condensates,” Phys. Rev. B 92, 035311 (2015).
[Crossref]

Borovkova, O. V.

O. V. Borovkova, Y. V. Kartashov, V. A. Vysloukh, V. E. Lobanov, B. A. Malomed, and L. Torner, “Solitons supported by spatially inhomogeneous nonlinear losses,” Opt. Exp. 20, 2657–2667 (2012).
[Crossref]

Bourayou, R.

S. Skupin, L. Bergé, U. Peschel, F. Lederer, G. Méjean, J. Yu, J. Kasparian, E. Salmon, J. P. Wolf, M. Rodriguez, L. Wöste, R. Bourayou, and R. Sauerbrey, “Filamentation of femtosecond light pulses in the air: Turbulent cells versus long-range clusters,” Phys. Rev. E 70, 046602 (2004).
[Crossref]

Chong, A.

W. Renninger, A. Chong, and E. Wise, “Dissipative solitons in normal-dispersion fiber lasers,” Phys. Rev. A 77, 023814 (2008).
[Crossref]

Colet, P.

Coullet, P.

P. Coullet, L. Gil, and F. Rocca, “Optical vortices,” Opt. Commun. 73, 403–408 (1989).
[Crossref]

Crasovan, L.-C.

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]

Dunjko, V.

V. Dunjko, V. Lorent, and M. Olshanii, “Bosons in cigar-shaped traps: Thomas-Fermi regime,” Tonks-Girardeau regime, and in between, Phys. Rev. Lett. 86, 5413–5416 (2001).
[Crossref]

Fedorov, S. V.

N. A. Veretenov, N. N. Rosanov, and S. V. Fedorov, “Motion of complexes of 3D-laser solitons,” Opt. Quant. Elect. 40, 253–262 (2008).
[Crossref]

Firth, W. J.

P. V. Paulau, D. Gomila, P. Colet, N. A. Loiko, N. N. Rosanov, T. Ackemann, and W. J. Firth, “Vortex solitons in lasers with feedback,” Opt. Express 18, 8859–8866 (2010).
[Crossref] [PubMed]

W. J. Firth and P. V. Paulau, “Soliton lasers stabilized by coupling to a resonant linear system,” Eur. Phys. J. D 59, 13–21 (2010).
[Crossref]

T. Ackemann, W. J. Firth, and G.-L. Oppo, “Fundamentals and applications of spatial dissipative solitons in photonic devices,” Adv. At. Mol. Opt. Phys. 57, 323–421 (2009).
[Crossref]

Galopin, E.

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S. Skupin, L. Bergé, U. Peschel, F. Lederer, G. Méjean, J. Yu, J. Kasparian, E. Salmon, J. P. Wolf, M. Rodriguez, L. Wöste, R. Bourayou, and R. Sauerbrey, “Filamentation of femtosecond light pulses in the air: Turbulent cells versus long-range clusters,” Phys. Rev. E 70, 046602 (2004).
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V. Skarka, N. B. Aleksić, H. Leblond, B. A. Malomed, and D. Mihalache, “Varieties of stable vortical solitons in Ginzburg-Landau media with radially inhomogeneous losses,” Phys. Rev. Lett. 105, 213901 (2010).
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H. Sakaguchi, B. A. Malomed, and D. V. Skryabin, “Spin-orbit coupling and nonlinear modes of the polariton condensate in a harmonic trap,” New J. Phys. 19, 08503 (2017).
[Crossref]

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L. M. Hocking and K. Stewartson, “Nonlinear response of a marginally unstable plane parallel flow to a 2-dimensional disturbance,” Proc. R. Soc. A 326, 289–313 (1972).
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Figures (14)

Fig. 1
Fig. 1 The integral power (norm) of 1D ground states in the conservative limit, γ = Γ = η = 0, vs. the chemical potential, for the self-focusing (σ = +1) and defocusing (σ = −1) signs of the cubic term, with the strength of the HO potential Ω2 = 2.
Fig. 2
Fig. 2 The unstable evolution of a numerically constructed stationary trapped 1D mode for γ = Γ = 1.0, η = 0 (no diffusivity), Ω2 = 2, and σ = +1 (the self-focusing cubic term).
Fig. 3
Fig. 3 Typical examples of robust dynamical regimes in the 1D model. (a) Real and imaginary parts of a typical stable stationary ground-state mode, found at η = 0.8, Ω2 = 2. The chemical potential and integral power (norm) of this mode are µ = 0.1714 and N = 1.4120. (b) A persistent breather spontaneously generated by an unstable stationary mode, at η = 0.3. In both cases, γ = Γ = 1, Ω2 = 2, and σ = +1 (self-focusing).
Fig. 4
Fig. 4 Robust quasi-regular oscillatory states with seven (a), five (b), three (d), and single (d) peaks, developing from unstable stationary modes at η = 0.05, 0.10, 0.15, and 0.2, respectively. Other parameters are the same as in Fig. 3.
Fig. 5
Fig. 5 (a) The integral power (norm) of the numerically generated stationary 1D modes vs. diffusivity η, for σ = +1 (the self-focusing cubic term), Ω2 = 2, and γ = Γ = 0.2, 1.0, and 3.0 (solid, dashed, and dotted lines, respectively). The stability is identified by colors: red corresponds to unstable stationary modes generating quasi-regular states; blue corresponds to unstable modes developing into breathers; and black represents stable stationary modes. (b) The stability map for different robust stationary and dynamical states, labeled in the plane of (γ = Γ,η), for Ω2 = 2 and σ = +1. The dashed line is the existence boundary for the stationary states, as predicted by Eq. (18). It completely coincides with the numerically found counterpart.
Fig. 6
Fig. 6 The same as in Fig. 5, but for the 1D model with σ = 0 and Ω2 = 6. The magenta solid line in (a) additionally displays the analytical approximation (26) for γ = Γ = 0.5.
Fig. 7
Fig. 7 A stable 2D GS, numerically found for σ = 0, Γ = γ = 3.0, η = 1.5, and Ω2 = 2. The inserted panel shows radial profiles, along y = 0, of the amplitude |u(x)| (blue), as well as real (magenta) and imaginary (red) parts of u(x).
Fig. 8
Fig. 8 A persistent isotropic breather generated by an unstable stationary mode at σ = +1, Γ = γ = 2.3, η = 0.6, and Ω2 = 2.
Fig. 9
Fig. 9 (a) The local-intensity [|u (x, y)|2] and phase structure of a stable axisymmetric vortex spontaneously generated by an unstable stationary mode, at σ = 0, Γ = γ = 2.5, η = 0.6, and Ω2 = 2. (b) The evolution of the local intensity in the x and y cross-sections, illustrating the spontaneous transformation of the unstable stationary mode into the isotropic vortex.
Fig. 10
Fig. 10 A sequence of stably rotating states produced by the simulations of the 2D model for σ = +1, Γ = γ = 2.5, and Ω2 = 2, at decreasing values of diffusivity η (which are indicated in the panels): a crescent vortex (S = 1), and multi-vortex complexes with S = 2, 3, 4, 6, and 7. At η > 0.5, the model supports a a stable axisymmetric vortex with S = 1, while at η < 0.22 a transition to vortex turbulence occurs. In all panels, spatial scales are the same as in Fig. 9(a).
Fig. 11
Fig. 11 An example of the vortex-turbulent state, found for σ = 0, γ = Γ = 2, η = 0.1, and Ω2 = 2. (a) Snapshots of the local-intensity, |ψ (x, y)|2, and phase patterns (left and right panels, respectively), produced, at t = 1000, by simulations initiated with an unstable stationary state. (b) The respective evolution of the local intensity, shown in the x- and y-cross sections.
Fig. 12
Fig. 12 The local-intensity and phase structure of a stably rotating 5-vortex bound state found for σ = −1, Γ = γ = 2.0, η = 0.23, and Ω2 = 2.
Fig. 13
Fig. 13 (a) The integral power (norm) of numerically found 2D stationary modes, given by Eq. (12), vs. diffusivity η for σ = 0, Ω2 = 2, and γ = Γ = 0.5, 1.5, and 3.0 (solid, dashed, and dotted lines, respectively). The stability is identified by colors: red, blue, and magenta imply, severally, the transformation into vortex tubulence, stable axisymmetric vortices, and breathers, while black curves represent families of stable stationary modes. The green solid line displays the analytical approximation (29) for γ = Γ = 0.5. (b) Stability borders in the plane of (γ = Γ, η) for σ = 0 and Ω2 = 1. The analytical existence boundary, η thr ( 2 D ), predicted by Eq. (23), and its numerically identified counterpart are shown, respectively, by red and black dashed lines, which completely overlap. Digits in subregions represent the number of individual vortices in rotating complexes populating these subregions. In particular, 1 implies the presence of the single (S = 1) rotating crescent-shaped vortex, different from the axisymmetric vortex in the area of “Stable vortex”.
Fig. 14
Fig. 14 The same as in Fig. 13, but for σ = +1.

Tables (1)

Tables Icon

Table 1 Characteristics of rotating complexes built of S individual vortices shown in Fig. 10, i.e., with σ = +1, Γ = γ = 2.5, and Ω2 = 2 (S = 1 corresponds to the rotating crescent-shaped vortex).

Equations (30)

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i ψ t = 1 2 ( 1 i η ) 2 ψ x 2 σ | ψ | 2 ψ + i ( γ Γ | ψ | 2 ) + 1 2 Ω 2 x 2 ψ .
σ = + 1 , 0 , 1 ,
γ = Γ .
ψ ( x , t ) = e i μ t u ( x ) ,
μ u = 1 2 ( 1 i η ) d 2 u d x 2 σ | u | 2 u + i ( γ Γ | u | 2 ) u + 1 2 Ω 2 x 2 u .
N = + | u ( x ) 2 | d x ,
ψ ( x , t ) = e i μ t { u ( x ) + ε [ ν ( x ) exp ( λ t ) + w * ( x ) exp ( λ * t ) ] } ,
( μ + i λ i γ ) v = 1 2 ( 1 i η ) d 2 v d x 2 + 1 2 Ω 2 x 2 v 2 ( σ + i Γ ) | u ( x ) | 2 v ( σ + i Γ ) ( u ( x ) ) 2 w , ( μ i λ + i γ ) w = 1 2 ( 1 + i η ) d 2 w d x 2 + 1 2 Ω 2 x 2 w 2 ( σ i Γ ) | u ( x ) | 2 w ( σ i Γ ) ( u * ( x ) ) 2 v .
i ψ t = 1 2 ( 1 i η ) ( 2 r 2 + 1 r r + 1 r 2 2 θ 2 ) ψ σ | ψ | 2 ψ + i ( γ Γ | ψ | 2 ) ψ + 1 2 Ω r 2 ψ ,
ψ ( x , y , t ) = e i μ t + i S θ u ( r ) ,
μ u = 1 2 ( 1 i η ) ( d 2 d r 2 + 1 r d d r S 2 r 2 ) u σ | u | 2 u + i ( γ Γ | u | 2 ) u + 1 2 Ω r 2 u ,
N = 2 π 0 | u ( r ) | 2 r d r .
ψ ( x , t ) = { e i μ t u ( r ) + ε [ v ( x r ) exp ( λ t + i m θ ) + w * ( r ) exp ( λ * t i m θ ) ] } ,
( μ + i λ i γ ) v = 1 2 ( 1 i η ) [ d 2 d r 2 + 1 r d d r ( S + m ) 2 r 2 ] v + 1 2 Ω 2 r 2 v 2 ( σ + i Γ ) | u ( x ) | 2 v ( σ + i Γ ) ( ( u ( x ) ) 2 w , ( μ i λ + i γ ) w = 1 2 ( 1 + i η ) [ d 2 d r 2 + 1 r d d r ( S m ) 2 r 2 ] w + 1 2 Ω 2 r 2 w 2 ( σ i Γ ) | u ( x ) | 2 w ( σ i Γ ) ( u * ( x ) ) 2 v .
μ u = 1 2 ( 1 i η ) d 2 u d x 2 + i γ u + 1 2 Ω 2 x 2 u .
u lin ( 1 D ) ( x ) = A 0 exp ( Ω 2 1 i η 2 x 2 ) ,
μ lin ( 1 D ) = Ω η 2 2 ( 1 + η 2 1 ) .
γ thr ( 1 D ) = Ω 2 2 1 + η 2 1 .
ψ lin ( 1 D ) ( x ) = A 0 exp [ ( γ γ thr ( 1 D ) ) t ] × exp ( Ω 2 1 i η x 2 ) .
μ u = 1 2 ( 1 i η ) ( d 2 d r 2 + 1 r d d r S 2 r 2 ) u + i γ u + 1 2 Ω r 2 u ,
u lin ( 2 D ) ( r ) = A 0 r S exp ( Ω 2 1 i η r 2 ) ,
μ lin ( 2 D ) = ( 1 + S ) Ω η 2 ( 1 + η 2 1 ) ,
γ thr ( 2 D ) = ( 1 + S ) Ω 1 2 ( 1 + η 2 1 ) .
ψ lin ( 2 D ) ( r ) = A 0 exp [ ( γ γ thr ( 2 D ) ) t ] × r S exp ( Ω 2 1 i η r 2 ) .
+ ( γ | u | 2 Γ | u | 4 1 2 η | d u d x | 2 ) d x = 0 .
( A 0 ( 1 D ) ) 2 = 4 γ Ω η 2 2 Γ .
N = π ( 4 γ Ω η ) 2 2 Ω Γ .
0 [ γ | u | 2 Γ | u | 4 1 2 η ( | d u d r | 2 + S 2 r 2 | u | ) ] r d r = 0 .
( A 0 ( 2 D ) ) 2 = 2 1 + 2 S S ! ( 2 S ) ! Ω S Γ ( γ 1 + S 2 Ω η ) ,
N = 2 1 + 2 S π S ! ( 2 S ) ! ( Ω Γ ) 1 ( γ 1 + S 2 Ω η ) ,

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