Abstract

In this work, new plain and composite high-energy solitons of the cubic–quintic Swift–Hohenberg equation were numerically found. Starting from a composite pulse found by Soto-Crespo and Akhmediev and changing some parameter values allowed us to find these high energy pulses. We also found the region in the parameter space in which these solutions exist. Some pulse characteristics, namely, temporal and spectral profiles and chirp, are presented. The study of the pulse energy shows its independence of the dispersion parameter but its dependence on the nonlinear gain. An extreme amplitude pulse has also been found.

© 2016 Chinese Laser Press

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Corrections

S. C. V. Latas, "High-energy plain and composite pulses in a laser modeled by the complex Swift–Hohenberg equation: publisher’s note," Photon. Res. 4, 101-101 (2016)
https://www.osapublishing.org/prj/abstract.cfm?uri=prj-4-3-101

21 March 2016: Corrections were made to Figs. 1, 3, and 5.


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References

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  1. N. Akhmediev and A. Ankiewicz, Dissipative Solitons (Springer-Verlag, 2005).
  2. J. Lega, J. Moloney, and A. Newell, “Swift–Hohenberg equation for lasers,” Phys. Rev. Lett. 73, 2978–2981 (1994).
    [Crossref]
  3. H. Sakaguchi and H. Brand, “Localized patterns for the quintic complex Swift–Hohenberg equation,” Phys. D 117, 95–105 (1998).
    [Crossref]
  4. I. Aranson and L. Kramer, “The world of the complex Ginzburg–Landau equation,” Rev. Mod. Phys. 74, 99–143 (2002).
    [Crossref]
  5. K. Maruno, A. Ankiewicz, and N. Akhmediev, “Exact soliton solutions of the one-dimensional complex Swift–Hohenberg equation,” Phys. D 176, 44–66 (2003).
    [Crossref]
  6. J. Soto-Crespo and N. Akhmediev, “Composite solitons and two-pulse generation in passively mode-locked lasers modeled by the complex quintic Swift–Hohenberg equation,” Phys. Rev. E 66, 066610 (2002).
    [Crossref]
  7. H. Wang and L. Yanti, “An efficient numerical method for the quintic complex Swift–Hohenberg equation,” Numer. Math. Theor. Appl. 4, 237–254 (2011).
  8. V. Afanajev, N. Akhmediev, and J. Soto-Crespo, “Three forms of localized solutions of the quintic complex Ginzburg–Landau equation,” Phys. Rev. E 53, 1931–1939 (1996).
    [Crossref]
  9. N. Akhmediev, A. Rodrigues, and G. Town, “Interaction of dual frequency pulses in passively mode-locked lasers,” Opt. Commun. 187, 419–426 (2001).
    [Crossref]
  10. N. Akhmediev, J. Soto-Crespo, and P. Grelu, “Roadmap to ultra-short record high-energy pulses out of laser oscillators,” Phys. Lett. A 372, 3124–3128 (2008).
    [Crossref]
  11. W. Chang, J. Soto-Crespo, P. Vouzas, and N. Akhmediev, “Extreme amplitude spikes in a laser model described by the complex Ginzburg–Landau equation,” Opt. Lett. 40, 2949–2952 (2015).
    [Crossref]
  12. Z. Liu, S. Zhang, and F. Wise, “Rogue waves in a normal dispersion fiber laser,” Opt. Lett. 40, 1366–1369 (2015).
    [Crossref]
  13. C. Lecaplain, P. Grelu, J. M. Soto-Crespo, and N. Akhmediev, “Dissipative rogue waves generated by chaotic pulse bunching in a mode-locked laser,” Phys. Rev. Lett. 108, 233901 (2012).
    [Crossref]
  14. W. Chang, A. Ankiewicz, J. Soto-Crespo, and N. Akhmediev, “Dissipative soliton resonances in laser models with parameter management,” J. Opt. Soc. Am. B 25, 1972–1977 (2008).
    [Crossref]
  15. M. Ferreira, Nonlinear Effects in Optical Fibers (Wiley, 2011).
  16. F. If, P. Berg, P. L. Christiansen, and O. Skovgaard, “Split-step spectral method for nonlinear Schrodinger-equation with absorbing boundaries,” J. Comp. Phys. 72, 501–503 (1987).
    [Crossref]

2015 (2)

2012 (1)

C. Lecaplain, P. Grelu, J. M. Soto-Crespo, and N. Akhmediev, “Dissipative rogue waves generated by chaotic pulse bunching in a mode-locked laser,” Phys. Rev. Lett. 108, 233901 (2012).
[Crossref]

2011 (1)

H. Wang and L. Yanti, “An efficient numerical method for the quintic complex Swift–Hohenberg equation,” Numer. Math. Theor. Appl. 4, 237–254 (2011).

2008 (2)

W. Chang, A. Ankiewicz, J. Soto-Crespo, and N. Akhmediev, “Dissipative soliton resonances in laser models with parameter management,” J. Opt. Soc. Am. B 25, 1972–1977 (2008).
[Crossref]

N. Akhmediev, J. Soto-Crespo, and P. Grelu, “Roadmap to ultra-short record high-energy pulses out of laser oscillators,” Phys. Lett. A 372, 3124–3128 (2008).
[Crossref]

2003 (1)

K. Maruno, A. Ankiewicz, and N. Akhmediev, “Exact soliton solutions of the one-dimensional complex Swift–Hohenberg equation,” Phys. D 176, 44–66 (2003).
[Crossref]

2002 (2)

J. Soto-Crespo and N. Akhmediev, “Composite solitons and two-pulse generation in passively mode-locked lasers modeled by the complex quintic Swift–Hohenberg equation,” Phys. Rev. E 66, 066610 (2002).
[Crossref]

I. Aranson and L. Kramer, “The world of the complex Ginzburg–Landau equation,” Rev. Mod. Phys. 74, 99–143 (2002).
[Crossref]

2001 (1)

N. Akhmediev, A. Rodrigues, and G. Town, “Interaction of dual frequency pulses in passively mode-locked lasers,” Opt. Commun. 187, 419–426 (2001).
[Crossref]

1998 (1)

H. Sakaguchi and H. Brand, “Localized patterns for the quintic complex Swift–Hohenberg equation,” Phys. D 117, 95–105 (1998).
[Crossref]

1996 (1)

V. Afanajev, N. Akhmediev, and J. Soto-Crespo, “Three forms of localized solutions of the quintic complex Ginzburg–Landau equation,” Phys. Rev. E 53, 1931–1939 (1996).
[Crossref]

1994 (1)

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

1987 (1)

F. If, P. Berg, P. L. Christiansen, and O. Skovgaard, “Split-step spectral method for nonlinear Schrodinger-equation with absorbing boundaries,” J. Comp. Phys. 72, 501–503 (1987).
[Crossref]

Afanajev, V.

V. Afanajev, N. Akhmediev, and J. Soto-Crespo, “Three forms of localized solutions of the quintic complex Ginzburg–Landau equation,” Phys. Rev. E 53, 1931–1939 (1996).
[Crossref]

Akhmediev, N.

W. Chang, J. Soto-Crespo, P. Vouzas, and N. Akhmediev, “Extreme amplitude spikes in a laser model described by the complex Ginzburg–Landau equation,” Opt. Lett. 40, 2949–2952 (2015).
[Crossref]

C. Lecaplain, P. Grelu, J. M. Soto-Crespo, and N. Akhmediev, “Dissipative rogue waves generated by chaotic pulse bunching in a mode-locked laser,” Phys. Rev. Lett. 108, 233901 (2012).
[Crossref]

W. Chang, A. Ankiewicz, J. Soto-Crespo, and N. Akhmediev, “Dissipative soliton resonances in laser models with parameter management,” J. Opt. Soc. Am. B 25, 1972–1977 (2008).
[Crossref]

N. Akhmediev, J. Soto-Crespo, and P. Grelu, “Roadmap to ultra-short record high-energy pulses out of laser oscillators,” Phys. Lett. A 372, 3124–3128 (2008).
[Crossref]

K. Maruno, A. Ankiewicz, and N. Akhmediev, “Exact soliton solutions of the one-dimensional complex Swift–Hohenberg equation,” Phys. D 176, 44–66 (2003).
[Crossref]

J. Soto-Crespo and N. Akhmediev, “Composite solitons and two-pulse generation in passively mode-locked lasers modeled by the complex quintic Swift–Hohenberg equation,” Phys. Rev. E 66, 066610 (2002).
[Crossref]

N. Akhmediev, A. Rodrigues, and G. Town, “Interaction of dual frequency pulses in passively mode-locked lasers,” Opt. Commun. 187, 419–426 (2001).
[Crossref]

V. Afanajev, N. Akhmediev, and J. Soto-Crespo, “Three forms of localized solutions of the quintic complex Ginzburg–Landau equation,” Phys. Rev. E 53, 1931–1939 (1996).
[Crossref]

N. Akhmediev and A. Ankiewicz, Dissipative Solitons (Springer-Verlag, 2005).

Ankiewicz, A.

W. Chang, A. Ankiewicz, J. Soto-Crespo, and N. Akhmediev, “Dissipative soliton resonances in laser models with parameter management,” J. Opt. Soc. Am. B 25, 1972–1977 (2008).
[Crossref]

K. Maruno, A. Ankiewicz, and N. Akhmediev, “Exact soliton solutions of the one-dimensional complex Swift–Hohenberg equation,” Phys. D 176, 44–66 (2003).
[Crossref]

N. Akhmediev and A. Ankiewicz, Dissipative Solitons (Springer-Verlag, 2005).

Aranson, I.

I. Aranson and L. Kramer, “The world of the complex Ginzburg–Landau equation,” Rev. Mod. Phys. 74, 99–143 (2002).
[Crossref]

Berg, P.

F. If, P. Berg, P. L. Christiansen, and O. Skovgaard, “Split-step spectral method for nonlinear Schrodinger-equation with absorbing boundaries,” J. Comp. Phys. 72, 501–503 (1987).
[Crossref]

Brand, H.

H. Sakaguchi and H. Brand, “Localized patterns for the quintic complex Swift–Hohenberg equation,” Phys. D 117, 95–105 (1998).
[Crossref]

Chang, W.

Christiansen, P. L.

F. If, P. Berg, P. L. Christiansen, and O. Skovgaard, “Split-step spectral method for nonlinear Schrodinger-equation with absorbing boundaries,” J. Comp. Phys. 72, 501–503 (1987).
[Crossref]

Ferreira, M.

M. Ferreira, Nonlinear Effects in Optical Fibers (Wiley, 2011).

Grelu, P.

C. Lecaplain, P. Grelu, J. M. Soto-Crespo, and N. Akhmediev, “Dissipative rogue waves generated by chaotic pulse bunching in a mode-locked laser,” Phys. Rev. Lett. 108, 233901 (2012).
[Crossref]

N. Akhmediev, J. Soto-Crespo, and P. Grelu, “Roadmap to ultra-short record high-energy pulses out of laser oscillators,” Phys. Lett. A 372, 3124–3128 (2008).
[Crossref]

If, F.

F. If, P. Berg, P. L. Christiansen, and O. Skovgaard, “Split-step spectral method for nonlinear Schrodinger-equation with absorbing boundaries,” J. Comp. Phys. 72, 501–503 (1987).
[Crossref]

Kramer, L.

I. Aranson and L. Kramer, “The world of the complex Ginzburg–Landau equation,” Rev. Mod. Phys. 74, 99–143 (2002).
[Crossref]

Lecaplain, C.

C. Lecaplain, P. Grelu, J. M. Soto-Crespo, and N. Akhmediev, “Dissipative rogue waves generated by chaotic pulse bunching in a mode-locked laser,” Phys. Rev. Lett. 108, 233901 (2012).
[Crossref]

Lega, J.

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

Liu, Z.

Maruno, K.

K. Maruno, A. Ankiewicz, and N. Akhmediev, “Exact soliton solutions of the one-dimensional complex Swift–Hohenberg equation,” Phys. D 176, 44–66 (2003).
[Crossref]

Moloney, J.

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

Newell, A.

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

Rodrigues, A.

N. Akhmediev, A. Rodrigues, and G. Town, “Interaction of dual frequency pulses in passively mode-locked lasers,” Opt. Commun. 187, 419–426 (2001).
[Crossref]

Sakaguchi, H.

H. Sakaguchi and H. Brand, “Localized patterns for the quintic complex Swift–Hohenberg equation,” Phys. D 117, 95–105 (1998).
[Crossref]

Skovgaard, O.

F. If, P. Berg, P. L. Christiansen, and O. Skovgaard, “Split-step spectral method for nonlinear Schrodinger-equation with absorbing boundaries,” J. Comp. Phys. 72, 501–503 (1987).
[Crossref]

Soto-Crespo, J.

W. Chang, J. Soto-Crespo, P. Vouzas, and N. Akhmediev, “Extreme amplitude spikes in a laser model described by the complex Ginzburg–Landau equation,” Opt. Lett. 40, 2949–2952 (2015).
[Crossref]

N. Akhmediev, J. Soto-Crespo, and P. Grelu, “Roadmap to ultra-short record high-energy pulses out of laser oscillators,” Phys. Lett. A 372, 3124–3128 (2008).
[Crossref]

W. Chang, A. Ankiewicz, J. Soto-Crespo, and N. Akhmediev, “Dissipative soliton resonances in laser models with parameter management,” J. Opt. Soc. Am. B 25, 1972–1977 (2008).
[Crossref]

J. Soto-Crespo and N. Akhmediev, “Composite solitons and two-pulse generation in passively mode-locked lasers modeled by the complex quintic Swift–Hohenberg equation,” Phys. Rev. E 66, 066610 (2002).
[Crossref]

V. Afanajev, N. Akhmediev, and J. Soto-Crespo, “Three forms of localized solutions of the quintic complex Ginzburg–Landau equation,” Phys. Rev. E 53, 1931–1939 (1996).
[Crossref]

Soto-Crespo, J. M.

C. Lecaplain, P. Grelu, J. M. Soto-Crespo, and N. Akhmediev, “Dissipative rogue waves generated by chaotic pulse bunching in a mode-locked laser,” Phys. Rev. Lett. 108, 233901 (2012).
[Crossref]

Town, G.

N. Akhmediev, A. Rodrigues, and G. Town, “Interaction of dual frequency pulses in passively mode-locked lasers,” Opt. Commun. 187, 419–426 (2001).
[Crossref]

Vouzas, P.

Wang, H.

H. Wang and L. Yanti, “An efficient numerical method for the quintic complex Swift–Hohenberg equation,” Numer. Math. Theor. Appl. 4, 237–254 (2011).

Wise, F.

Yanti, L.

H. Wang and L. Yanti, “An efficient numerical method for the quintic complex Swift–Hohenberg equation,” Numer. Math. Theor. Appl. 4, 237–254 (2011).

Zhang, S.

J. Comp. Phys. (1)

F. If, P. Berg, P. L. Christiansen, and O. Skovgaard, “Split-step spectral method for nonlinear Schrodinger-equation with absorbing boundaries,” J. Comp. Phys. 72, 501–503 (1987).
[Crossref]

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

Numer. Math. Theor. Appl. (1)

H. Wang and L. Yanti, “An efficient numerical method for the quintic complex Swift–Hohenberg equation,” Numer. Math. Theor. Appl. 4, 237–254 (2011).

Opt. Commun. (1)

N. Akhmediev, A. Rodrigues, and G. Town, “Interaction of dual frequency pulses in passively mode-locked lasers,” Opt. Commun. 187, 419–426 (2001).
[Crossref]

Opt. Lett. (2)

Phys. D (2)

K. Maruno, A. Ankiewicz, and N. Akhmediev, “Exact soliton solutions of the one-dimensional complex Swift–Hohenberg equation,” Phys. D 176, 44–66 (2003).
[Crossref]

H. Sakaguchi and H. Brand, “Localized patterns for the quintic complex Swift–Hohenberg equation,” Phys. D 117, 95–105 (1998).
[Crossref]

Phys. Lett. A (1)

N. Akhmediev, J. Soto-Crespo, and P. Grelu, “Roadmap to ultra-short record high-energy pulses out of laser oscillators,” Phys. Lett. A 372, 3124–3128 (2008).
[Crossref]

Phys. Rev. E (2)

J. Soto-Crespo and N. Akhmediev, “Composite solitons and two-pulse generation in passively mode-locked lasers modeled by the complex quintic Swift–Hohenberg equation,” Phys. Rev. E 66, 066610 (2002).
[Crossref]

V. Afanajev, N. Akhmediev, and J. Soto-Crespo, “Three forms of localized solutions of the quintic complex Ginzburg–Landau equation,” Phys. Rev. E 53, 1931–1939 (1996).
[Crossref]

Phys. Rev. Lett. (2)

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

C. Lecaplain, P. Grelu, J. M. Soto-Crespo, and N. Akhmediev, “Dissipative rogue waves generated by chaotic pulse bunching in a mode-locked laser,” Phys. Rev. Lett. 108, 233901 (2012).
[Crossref]

Rev. Mod. Phys. (1)

I. Aranson and L. Kramer, “The world of the complex Ginzburg–Landau equation,” Rev. Mod. Phys. 74, 99–143 (2002).
[Crossref]

Other (2)

N. Akhmediev and A. Ankiewicz, Dissipative Solitons (Springer-Verlag, 2005).

M. Ferreira, Nonlinear Effects in Optical Fibers (Wiley, 2011).

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Figures (6)

Fig. 1.
Fig. 1. Region of existence of dissipative solitons (darker area), in the plane (ϵ, D), for the following parameter values: β=0.3, δ=0.5, μ=0.001, ν=0, and γ=0.05. Dissipative pulses do not exist beyond the lower and upper boundaries. Nevertheless, their region of existence extends beyond the left and right boundaries, for values of |D|>2. The marks (circles, squares, and triangles) correspond to examples of pulses presented in the following figures.
Fig. 2.
Fig. 2. (a) Amplitude and (b) spectral pulse profiles for four values of ϵ, namely, ϵ=0.4 (thick dashed curves), ϵ=1.3 (solid curves), ϵ=1.35 (dashed–dotted curves), and ϵ=1.5 (dashed curves). The curves correspond to the triangles along the vertical line D=0 in Fig. 1. (The other parameter values are β=0.3, δ=0.5, μ=0.001, ν=0, and γ=0.05.)
Fig. 3.
Fig. 3. Energy Q of dissipative pulses versus dispersion parameter, D, for three different values of ϵ, associated with PPs, NCPs, and WCPs.
Fig. 4.
Fig. 4. Pulses’ (a) amplitude, (b) chirp, and (c) spectra for four different values of D. The pulse curves correspond to the squares along the horizontal line ϵ=1.2 in Fig. 1. (The other parameter values are β=0.3, δ=0.5, μ=0.001, ν=0, and γ=0.05.)
Fig. 5.
Fig. 5. Pulse (a) amplitudes, (b) chirp, and (c) spectra for the same four different values of D as in Fig. 4. The pulses represented are WCPs for D<0 and NCPs for D>0. Similar profiles of WCPs and NCPs were obtained in both dispersion regimes. The pulse curves correspond to the circles along the horizontal line ϵ=1.4 in Fig. 1. (The other parameter values are β=0.3, δ=0.5, μ=0.001, ν=0, and γ=0.05.)
Fig. 6.
Fig. 6. Pulse (a) evolution, (b) amplitude, and (c) spectrum profiles of a plain pulse solution. A small change in some parameter values can produce a significant growth of the pulse amplitude. [The parameter values are D=0, β=0.3, δ=0.5, ϵ=0.35, μ=0, ν=0.000025, and γ=0.05, as shown in (b).]

Equations (1)

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iuZ+D2uTT+|u|2u+ν|u|4u=iδu+iβuTT+iϵ|u|2u+iμ|u|4u+iγuTTTT,

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