Abstract

We study the stability of three-dimensional, propagation-invariant wave packets in self-focusing media with anomalous dispersion at intensities where nonlinear absorption is significant. For light bullets with spherical symmetry in the two transversal dimensions and time, we show that nonlinear losses (NLLs) act as a strongly stabilizing mechanism for light bullet propagation, and, in practical settings, that longer-lived light bullet propagation requires, paradoxically, higher NLLs during the propagation, at the expense of higher input energy.

© 2013 Optical Society of America

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References

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  1. M. A. Porras, A. Parola, D. Faccio, A. Dubietis, and P. Di Trapani, “Nonlinear unbalanced Bessel beams: stationary conical waves supported by nonlinear losses,” Phys. Rev. Lett. 93, 153902 (2004).
    [CrossRef]
  2. M. A. Porras, A. Parola, and P. Di Trapani, “Nonlinear unbalanced O-waves: nonsolitary, conical light bullets in nonlinear dissipative media,” J. Opt. Soc. Am. B 22, 1406–1413 (2005).
    [CrossRef]
  3. A. Alexandrescu and V. M. Pérez-García, “Matter-wave solitons supported by dissipation,” Phys. Rev. A 73, 053610 (2006).
    [CrossRef]
  4. W. Liu, J. F. Gravel, F. Théberge, A. Becker, and S. L. Chin, “Background reservoir: its crucial role for long-distance propagation of femtosecond laser pulses in air,” Appl. Phys. B 80, 857–860 (2005).
    [CrossRef]
  5. M. A. Porras, “Nonlinear light bullets in purely lossy, self-focusing media,” Appl. Phys. B 103, 591–596 (2011).
    [CrossRef]
  6. P. Polesana, A. Couairon, D. Faccio, A. Parola, M. A. Porras, A. Dubietis, A. Piskarskas, and P. Di Trapani, “Observation of conical waves in focusing, dispersive, and dissipative Kerr media,” Phys. Rev. Lett. 99, 223902 (2007).
    [CrossRef]
  7. Y. Silberberg, “Collapse of optical pulses,” Opt. Lett. 15, 1282–1284 (1990).
    [CrossRef]
  8. S. Malaguti, G. Bellanca, and S. Trillo, “Two-dimensional envelope localized waves in the anomalous dispersion regime,” Opt. Lett. 33, 1117–1119 (2008).
    [CrossRef]
  9. M. A. Porras, “A dissipative attractor in the spatiotemporal collapse of ultrashort light pulses,” Opt. Express 18, 7376–7383 (2010).
    [CrossRef]
  10. L. Bergé and S. Skupin, “Self-channeling of ultrashort laser pulses in materials with anomalous dispersion,” Phys. Rev. E 71, 065601 (2005).
    [CrossRef]
  11. D. Moll and A. L. Gaeta, “Role of dispersion in multiple-collapse dynamics,” Opt. Lett. 29, 995–997 (2004).
    [CrossRef]
  12. L. W. Liou, X. D. Cao, C. McKinstrie, and G. P. Agrawal, “Spatio-temporal instabilities in dispersive nonlinear media,” Phys. Rev. A 46, 4202–4208 (1992).
    [CrossRef]
  13. V. E. Zakharov and A. M. Rubenchik, “Instability of waveguides and solitons in nonlinear media,” Sov. Phys. JETP 38, 494–500 (1974).
  14. Y. R. Shen, The Principles of Nonlinear Optics (Wiley-Interscience, 1984).
  15. M. A. Porras, A. Parola, D. Faccio, A. Couairon, and P. Di Trapani, “Light-filament dynamics and the spatiotemporal instability of the Townes profile,” Phys. Rev. A 76, 011803(R) (2007).
    [CrossRef]
  16. A. De Rossi, C. Conti, and S. Trillo, “Stability, multistability, and wobbling of optical gap solitons,” Phys. Rev. Lett. 81, 85–88 (1998).
    [CrossRef]
  17. I. V. Barashenkov, D. E. Pelinovsky, and E. V. Zemlyanaya, “Vibrations and oscillatory instabilities of gap solitons,” Phys. Rev. Lett. 80, 5117–5120 (1998).
    [CrossRef]

2011

M. A. Porras, “Nonlinear light bullets in purely lossy, self-focusing media,” Appl. Phys. B 103, 591–596 (2011).
[CrossRef]

2010

2008

2007

P. Polesana, A. Couairon, D. Faccio, A. Parola, M. A. Porras, A. Dubietis, A. Piskarskas, and P. Di Trapani, “Observation of conical waves in focusing, dispersive, and dissipative Kerr media,” Phys. Rev. Lett. 99, 223902 (2007).
[CrossRef]

M. A. Porras, A. Parola, D. Faccio, A. Couairon, and P. Di Trapani, “Light-filament dynamics and the spatiotemporal instability of the Townes profile,” Phys. Rev. A 76, 011803(R) (2007).
[CrossRef]

2006

A. Alexandrescu and V. M. Pérez-García, “Matter-wave solitons supported by dissipation,” Phys. Rev. A 73, 053610 (2006).
[CrossRef]

2005

W. Liu, J. F. Gravel, F. Théberge, A. Becker, and S. L. Chin, “Background reservoir: its crucial role for long-distance propagation of femtosecond laser pulses in air,” Appl. Phys. B 80, 857–860 (2005).
[CrossRef]

M. A. Porras, A. Parola, and P. Di Trapani, “Nonlinear unbalanced O-waves: nonsolitary, conical light bullets in nonlinear dissipative media,” J. Opt. Soc. Am. B 22, 1406–1413 (2005).
[CrossRef]

L. Bergé and S. Skupin, “Self-channeling of ultrashort laser pulses in materials with anomalous dispersion,” Phys. Rev. E 71, 065601 (2005).
[CrossRef]

2004

D. Moll and A. L. Gaeta, “Role of dispersion in multiple-collapse dynamics,” Opt. Lett. 29, 995–997 (2004).
[CrossRef]

M. A. Porras, A. Parola, D. Faccio, A. Dubietis, and P. Di Trapani, “Nonlinear unbalanced Bessel beams: stationary conical waves supported by nonlinear losses,” Phys. Rev. Lett. 93, 153902 (2004).
[CrossRef]

1998

A. De Rossi, C. Conti, and S. Trillo, “Stability, multistability, and wobbling of optical gap solitons,” Phys. Rev. Lett. 81, 85–88 (1998).
[CrossRef]

I. V. Barashenkov, D. E. Pelinovsky, and E. V. Zemlyanaya, “Vibrations and oscillatory instabilities of gap solitons,” Phys. Rev. Lett. 80, 5117–5120 (1998).
[CrossRef]

1992

L. W. Liou, X. D. Cao, C. McKinstrie, and G. P. Agrawal, “Spatio-temporal instabilities in dispersive nonlinear media,” Phys. Rev. A 46, 4202–4208 (1992).
[CrossRef]

1990

1974

V. E. Zakharov and A. M. Rubenchik, “Instability of waveguides and solitons in nonlinear media,” Sov. Phys. JETP 38, 494–500 (1974).

Agrawal, G. P.

L. W. Liou, X. D. Cao, C. McKinstrie, and G. P. Agrawal, “Spatio-temporal instabilities in dispersive nonlinear media,” Phys. Rev. A 46, 4202–4208 (1992).
[CrossRef]

Alexandrescu, A.

A. Alexandrescu and V. M. Pérez-García, “Matter-wave solitons supported by dissipation,” Phys. Rev. A 73, 053610 (2006).
[CrossRef]

Barashenkov, I. V.

I. V. Barashenkov, D. E. Pelinovsky, and E. V. Zemlyanaya, “Vibrations and oscillatory instabilities of gap solitons,” Phys. Rev. Lett. 80, 5117–5120 (1998).
[CrossRef]

Becker, A.

W. Liu, J. F. Gravel, F. Théberge, A. Becker, and S. L. Chin, “Background reservoir: its crucial role for long-distance propagation of femtosecond laser pulses in air,” Appl. Phys. B 80, 857–860 (2005).
[CrossRef]

Bellanca, G.

Bergé, L.

L. Bergé and S. Skupin, “Self-channeling of ultrashort laser pulses in materials with anomalous dispersion,” Phys. Rev. E 71, 065601 (2005).
[CrossRef]

Cao, X. D.

L. W. Liou, X. D. Cao, C. McKinstrie, and G. P. Agrawal, “Spatio-temporal instabilities in dispersive nonlinear media,” Phys. Rev. A 46, 4202–4208 (1992).
[CrossRef]

Chin, S. L.

W. Liu, J. F. Gravel, F. Théberge, A. Becker, and S. L. Chin, “Background reservoir: its crucial role for long-distance propagation of femtosecond laser pulses in air,” Appl. Phys. B 80, 857–860 (2005).
[CrossRef]

Conti, C.

A. De Rossi, C. Conti, and S. Trillo, “Stability, multistability, and wobbling of optical gap solitons,” Phys. Rev. Lett. 81, 85–88 (1998).
[CrossRef]

Couairon, A.

M. A. Porras, A. Parola, D. Faccio, A. Couairon, and P. Di Trapani, “Light-filament dynamics and the spatiotemporal instability of the Townes profile,” Phys. Rev. A 76, 011803(R) (2007).
[CrossRef]

P. Polesana, A. Couairon, D. Faccio, A. Parola, M. A. Porras, A. Dubietis, A. Piskarskas, and P. Di Trapani, “Observation of conical waves in focusing, dispersive, and dissipative Kerr media,” Phys. Rev. Lett. 99, 223902 (2007).
[CrossRef]

De Rossi, A.

A. De Rossi, C. Conti, and S. Trillo, “Stability, multistability, and wobbling of optical gap solitons,” Phys. Rev. Lett. 81, 85–88 (1998).
[CrossRef]

Di Trapani, P.

M. A. Porras, A. Parola, D. Faccio, A. Couairon, and P. Di Trapani, “Light-filament dynamics and the spatiotemporal instability of the Townes profile,” Phys. Rev. A 76, 011803(R) (2007).
[CrossRef]

P. Polesana, A. Couairon, D. Faccio, A. Parola, M. A. Porras, A. Dubietis, A. Piskarskas, and P. Di Trapani, “Observation of conical waves in focusing, dispersive, and dissipative Kerr media,” Phys. Rev. Lett. 99, 223902 (2007).
[CrossRef]

M. A. Porras, A. Parola, and P. Di Trapani, “Nonlinear unbalanced O-waves: nonsolitary, conical light bullets in nonlinear dissipative media,” J. Opt. Soc. Am. B 22, 1406–1413 (2005).
[CrossRef]

M. A. Porras, A. Parola, D. Faccio, A. Dubietis, and P. Di Trapani, “Nonlinear unbalanced Bessel beams: stationary conical waves supported by nonlinear losses,” Phys. Rev. Lett. 93, 153902 (2004).
[CrossRef]

Dubietis, A.

P. Polesana, A. Couairon, D. Faccio, A. Parola, M. A. Porras, A. Dubietis, A. Piskarskas, and P. Di Trapani, “Observation of conical waves in focusing, dispersive, and dissipative Kerr media,” Phys. Rev. Lett. 99, 223902 (2007).
[CrossRef]

M. A. Porras, A. Parola, D. Faccio, A. Dubietis, and P. Di Trapani, “Nonlinear unbalanced Bessel beams: stationary conical waves supported by nonlinear losses,” Phys. Rev. Lett. 93, 153902 (2004).
[CrossRef]

Faccio, D.

P. Polesana, A. Couairon, D. Faccio, A. Parola, M. A. Porras, A. Dubietis, A. Piskarskas, and P. Di Trapani, “Observation of conical waves in focusing, dispersive, and dissipative Kerr media,” Phys. Rev. Lett. 99, 223902 (2007).
[CrossRef]

M. A. Porras, A. Parola, D. Faccio, A. Couairon, and P. Di Trapani, “Light-filament dynamics and the spatiotemporal instability of the Townes profile,” Phys. Rev. A 76, 011803(R) (2007).
[CrossRef]

M. A. Porras, A. Parola, D. Faccio, A. Dubietis, and P. Di Trapani, “Nonlinear unbalanced Bessel beams: stationary conical waves supported by nonlinear losses,” Phys. Rev. Lett. 93, 153902 (2004).
[CrossRef]

Gaeta, A. L.

Gravel, J. F.

W. Liu, J. F. Gravel, F. Théberge, A. Becker, and S. L. Chin, “Background reservoir: its crucial role for long-distance propagation of femtosecond laser pulses in air,” Appl. Phys. B 80, 857–860 (2005).
[CrossRef]

Liou, L. W.

L. W. Liou, X. D. Cao, C. McKinstrie, and G. P. Agrawal, “Spatio-temporal instabilities in dispersive nonlinear media,” Phys. Rev. A 46, 4202–4208 (1992).
[CrossRef]

Liu, W.

W. Liu, J. F. Gravel, F. Théberge, A. Becker, and S. L. Chin, “Background reservoir: its crucial role for long-distance propagation of femtosecond laser pulses in air,” Appl. Phys. B 80, 857–860 (2005).
[CrossRef]

Malaguti, S.

McKinstrie, C.

L. W. Liou, X. D. Cao, C. McKinstrie, and G. P. Agrawal, “Spatio-temporal instabilities in dispersive nonlinear media,” Phys. Rev. A 46, 4202–4208 (1992).
[CrossRef]

Moll, D.

Parola, A.

M. A. Porras, A. Parola, D. Faccio, A. Couairon, and P. Di Trapani, “Light-filament dynamics and the spatiotemporal instability of the Townes profile,” Phys. Rev. A 76, 011803(R) (2007).
[CrossRef]

P. Polesana, A. Couairon, D. Faccio, A. Parola, M. A. Porras, A. Dubietis, A. Piskarskas, and P. Di Trapani, “Observation of conical waves in focusing, dispersive, and dissipative Kerr media,” Phys. Rev. Lett. 99, 223902 (2007).
[CrossRef]

M. A. Porras, A. Parola, and P. Di Trapani, “Nonlinear unbalanced O-waves: nonsolitary, conical light bullets in nonlinear dissipative media,” J. Opt. Soc. Am. B 22, 1406–1413 (2005).
[CrossRef]

M. A. Porras, A. Parola, D. Faccio, A. Dubietis, and P. Di Trapani, “Nonlinear unbalanced Bessel beams: stationary conical waves supported by nonlinear losses,” Phys. Rev. Lett. 93, 153902 (2004).
[CrossRef]

Pelinovsky, D. E.

I. V. Barashenkov, D. E. Pelinovsky, and E. V. Zemlyanaya, “Vibrations and oscillatory instabilities of gap solitons,” Phys. Rev. Lett. 80, 5117–5120 (1998).
[CrossRef]

Pérez-García, V. M.

A. Alexandrescu and V. M. Pérez-García, “Matter-wave solitons supported by dissipation,” Phys. Rev. A 73, 053610 (2006).
[CrossRef]

Piskarskas, A.

P. Polesana, A. Couairon, D. Faccio, A. Parola, M. A. Porras, A. Dubietis, A. Piskarskas, and P. Di Trapani, “Observation of conical waves in focusing, dispersive, and dissipative Kerr media,” Phys. Rev. Lett. 99, 223902 (2007).
[CrossRef]

Polesana, P.

P. Polesana, A. Couairon, D. Faccio, A. Parola, M. A. Porras, A. Dubietis, A. Piskarskas, and P. Di Trapani, “Observation of conical waves in focusing, dispersive, and dissipative Kerr media,” Phys. Rev. Lett. 99, 223902 (2007).
[CrossRef]

Porras, M. A.

M. A. Porras, “Nonlinear light bullets in purely lossy, self-focusing media,” Appl. Phys. B 103, 591–596 (2011).
[CrossRef]

M. A. Porras, “A dissipative attractor in the spatiotemporal collapse of ultrashort light pulses,” Opt. Express 18, 7376–7383 (2010).
[CrossRef]

P. Polesana, A. Couairon, D. Faccio, A. Parola, M. A. Porras, A. Dubietis, A. Piskarskas, and P. Di Trapani, “Observation of conical waves in focusing, dispersive, and dissipative Kerr media,” Phys. Rev. Lett. 99, 223902 (2007).
[CrossRef]

M. A. Porras, A. Parola, D. Faccio, A. Couairon, and P. Di Trapani, “Light-filament dynamics and the spatiotemporal instability of the Townes profile,” Phys. Rev. A 76, 011803(R) (2007).
[CrossRef]

M. A. Porras, A. Parola, and P. Di Trapani, “Nonlinear unbalanced O-waves: nonsolitary, conical light bullets in nonlinear dissipative media,” J. Opt. Soc. Am. B 22, 1406–1413 (2005).
[CrossRef]

M. A. Porras, A. Parola, D. Faccio, A. Dubietis, and P. Di Trapani, “Nonlinear unbalanced Bessel beams: stationary conical waves supported by nonlinear losses,” Phys. Rev. Lett. 93, 153902 (2004).
[CrossRef]

Rubenchik, A. M.

V. E. Zakharov and A. M. Rubenchik, “Instability of waveguides and solitons in nonlinear media,” Sov. Phys. JETP 38, 494–500 (1974).

Shen, Y. R.

Y. R. Shen, The Principles of Nonlinear Optics (Wiley-Interscience, 1984).

Silberberg, Y.

Skupin, S.

L. Bergé and S. Skupin, “Self-channeling of ultrashort laser pulses in materials with anomalous dispersion,” Phys. Rev. E 71, 065601 (2005).
[CrossRef]

Théberge, F.

W. Liu, J. F. Gravel, F. Théberge, A. Becker, and S. L. Chin, “Background reservoir: its crucial role for long-distance propagation of femtosecond laser pulses in air,” Appl. Phys. B 80, 857–860 (2005).
[CrossRef]

Trillo, S.

S. Malaguti, G. Bellanca, and S. Trillo, “Two-dimensional envelope localized waves in the anomalous dispersion regime,” Opt. Lett. 33, 1117–1119 (2008).
[CrossRef]

A. De Rossi, C. Conti, and S. Trillo, “Stability, multistability, and wobbling of optical gap solitons,” Phys. Rev. Lett. 81, 85–88 (1998).
[CrossRef]

Zakharov, V. E.

V. E. Zakharov and A. M. Rubenchik, “Instability of waveguides and solitons in nonlinear media,” Sov. Phys. JETP 38, 494–500 (1974).

Zemlyanaya, E. V.

I. V. Barashenkov, D. E. Pelinovsky, and E. V. Zemlyanaya, “Vibrations and oscillatory instabilities of gap solitons,” Phys. Rev. Lett. 80, 5117–5120 (1998).
[CrossRef]

Appl. Phys. B

W. Liu, J. F. Gravel, F. Théberge, A. Becker, and S. L. Chin, “Background reservoir: its crucial role for long-distance propagation of femtosecond laser pulses in air,” Appl. Phys. B 80, 857–860 (2005).
[CrossRef]

M. A. Porras, “Nonlinear light bullets in purely lossy, self-focusing media,” Appl. Phys. B 103, 591–596 (2011).
[CrossRef]

J. Opt. Soc. Am. B

Opt. Express

Opt. Lett.

Phys. Rev. A

L. W. Liou, X. D. Cao, C. McKinstrie, and G. P. Agrawal, “Spatio-temporal instabilities in dispersive nonlinear media,” Phys. Rev. A 46, 4202–4208 (1992).
[CrossRef]

M. A. Porras, A. Parola, D. Faccio, A. Couairon, and P. Di Trapani, “Light-filament dynamics and the spatiotemporal instability of the Townes profile,” Phys. Rev. A 76, 011803(R) (2007).
[CrossRef]

A. Alexandrescu and V. M. Pérez-García, “Matter-wave solitons supported by dissipation,” Phys. Rev. A 73, 053610 (2006).
[CrossRef]

Phys. Rev. E

L. Bergé and S. Skupin, “Self-channeling of ultrashort laser pulses in materials with anomalous dispersion,” Phys. Rev. E 71, 065601 (2005).
[CrossRef]

Phys. Rev. Lett.

P. Polesana, A. Couairon, D. Faccio, A. Parola, M. A. Porras, A. Dubietis, A. Piskarskas, and P. Di Trapani, “Observation of conical waves in focusing, dispersive, and dissipative Kerr media,” Phys. Rev. Lett. 99, 223902 (2007).
[CrossRef]

A. De Rossi, C. Conti, and S. Trillo, “Stability, multistability, and wobbling of optical gap solitons,” Phys. Rev. Lett. 81, 85–88 (1998).
[CrossRef]

I. V. Barashenkov, D. E. Pelinovsky, and E. V. Zemlyanaya, “Vibrations and oscillatory instabilities of gap solitons,” Phys. Rev. Lett. 80, 5117–5120 (1998).
[CrossRef]

M. A. Porras, A. Parola, D. Faccio, A. Dubietis, and P. Di Trapani, “Nonlinear unbalanced Bessel beams: stationary conical waves supported by nonlinear losses,” Phys. Rev. Lett. 93, 153902 (2004).
[CrossRef]

Sov. Phys. JETP

V. E. Zakharov and A. M. Rubenchik, “Instability of waveguides and solitons in nonlinear media,” Sov. Phys. JETP 38, 494–500 (1974).

Other

Y. R. Shen, The Principles of Nonlinear Optics (Wiley-Interscience, 1984).

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Figures (10)

Fig. 1.
Fig. 1.

Open circles: for several values of M, numerically obtained values of γmax determining the maximum intensity I0,max of LLBs and NLO waves as a function of the normalized axial wave-vector shift α=δ/kNL, with kNL=k0n2I0,max/n0. Dashed lines: for large M, the values of γmax of NLO waves follow the approximate linear variation [2] γmax=γmax,LLB(1+|α|), where γmax,LLB is the value of γmax for LLBs (α=0).

Fig. 2.
Fig. 2.

Amplitude profiles (sections y=0 or x=0) of LLBs at carrier frequency ω0=1.21525fs1 (carrier wavelength 1550 nm) in fused silica (n given by Sellmeier relation, and assuming n2=2.2×1016cm2/W, M=10, and β(M)=5.113×10116cm17/W9) with peak intensities I0=10.43, 14.62488, 14.716059, and I0,max=14.716061TW/cm2 from top to bottom.

Fig. 3.
Fig. 3.

Energy losses per unit propagation length, β(M)(k0|k0|)1/20dr4πr2a2M, in J/cm, of LLBs in fused silica at 1550 nm as a function of their peak intensity I0. Losses grow without bound in the limit of I0=I0,max.

Fig. 4.
Fig. 4.

(a) Variation with propagation distance z of the peak intensity of spatiotemporal symmetric, input Gaussian wave packets at 1550 nm of envelope A=Iexp(r2/s2) of width s=80μm (duration t=k0|k0|s=32.36fs) of increasing intensities I=2.32×1012 (light gray curve), 3.46×1012 (dark gray curve), and 4.594×1012W/cm2 (black curve) (peak powers equal to 20, 30, and 40 critical powers for self-focusing) launched in fused silica, evaluated numerically from Eq. (2). (b) Gaussian radial intensity profiles at the entrance of the medium z=0, and at z=0.32cm in the collapse region for the input intensities in (a) (light gray, dark gray, and black curves). The dashed curve represents the intensity radial profile of the LLB at I0,max and infinite NLLs in fused silica at 1550 nm.

Fig. 5.
Fig. 5.

(a) Axial evolution of the normalized axial intensity for perturbed LLBs with γ=0.1 (gray curve) and γ=1.5 (black curve) in media with M=10. (b) Axial evolution of the intensity of the growing perturbation in logarithmic scale in order to better appreciate its exponentially growing oscillations.

Fig. 6.
Fig. 6.

Variation of the exponential gain and axial oscillation frequency of the most unstable eigenmode of LLB with increasing NLLs parameter γ, from γ0 to γmax=3.15294301 in the case of M=10 and α=0.

Fig. 7.
Fig. 7.

For M=10, radial profiles of the dominant unstable mode (u,v) of the LLB with γ=0.1 [(a) and (b)], and for the LLB with γ=1.5 [(c) and (d)].

Fig. 8.
Fig. 8.

Axial intensity (black) and energy (gray) along propagation distance z of initially truncated LLBs in fused silica at 1550 nm with the indicated intensities (γ=0.100, 0.200, 2.0, 3.15, and 3.15294). The initial LLBs are truncated by multiplying their profiles by the super-Gaussian exp(r4/rt4), with rt=270μm.

Fig. 9.
Fig. 9.

Axial intensity of initial LLBs propagating in fused silica at 1550 nm with the indicated intensity (γ=3.15294) close to Imax, truncated initially at increasing radii rt=270, 360, 450, and 540μm.

Fig. 10.
Fig. 10.

For M=10 and α=0.264, variation of the exponential gain and axial oscillation frequency of the most unstable eigenmode of NLO waves when increasing the NLL parameter γ from 0 to γmax=4.03984401 for M=10 and α=0.264. The gray curves represent the case where M=10 and α=0 (LLBs) for comparison.

Equations (12)

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

zA=i2k0ΔA+i|k0|2t2A+ik0n2n0|A|2Aβ(M)2|A|2M2A,
zA=i2k0ΔA+ik0n2n0|A|2Aβ(M)2|A|2M2A,
dadr+2rdadr(dφdr)2a2k0δa+2k02n2n0a3=0,
(dφdr+2rdφdr)a2+dφdrda2dr+k0β(M)a2M=0,
ζA˜=i2ΔA˜+i|A˜|2A˜2γ|A˜|2M2A˜,
γ=n0β(M)I0M24k0n2.
d2a˜dρ2+2ρda˜dρ(dφdρ)2a˜2αa˜+2a˜3=0,
(d2φdρ2+2ρdφdρ)a˜2+dφdρda˜2dρ+4γa˜2M=0,
A˜(ρ,ζ)={a˜(ρ)eiφ(ρ)+ϵ[u(ρ)eiκζ+v(ρ)eiκζ]}eiαζ
(HffH)(uv)=κ(uv),
ϵ[u(ρ)eiκRζi+v(ρ)eiκRζi]=f(ρ,ζi)eκIζi,
ϵu(ρ)=f(ρ,ζ1)eκIζeiκR(ζ1ζ2)f(ρ,ζ2)eκIζ2eiκRζ2[1e2iκR(ζ1ζ2)],ϵv(ρ)=f(ρ,ζ1)eκIζ1f(ρ,ζ2)eκIζ2eiκR(ζ1ζ2)eiκRζ1[1e2iκR(ζ1ζ2)],

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