Abstract

We numerically investigate the dynamic evolution of pulsating solitons based on complex cubic-quintic Ginzburg-Landau equation with gain dynamics effects. We show that an additional soliton can be generated by the disturbance caused by a dispersion wave emitted by a single-period pulsating soliton and these solitons form soliton molecule. More complicated oscillating processes, such as snaking pulsation and double-periodic pulsation are actuated by periodic collision of the entangled solitons. Moreover, the dispersive wave, caused by high gain parameters and the soliton collision, appears periodically which is in sync with the pulsating process. These results are consistent with the recent experiments of soliton pulsations measured by dispersive Fourier transform techniques, and will stimulate further experimental research of the complex multi-soliton bunches in dissipative systems.

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

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References

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  1. P. Grelu and N. Akhmediev, “Dissipative solitons for mode-locked lasers,” Nat. Photonics 6, 84–92 (2012).
    [Crossref]
  2. V. S. Grigoryan and T. S. Muradyan, “Evolution of light pulses into autosolitons in nonlinear amplifying media,” J. Opt. Soc. Am. B 41, 1757–1765 (1991).
    [Crossref]
  3. M. Remoissenet and J. A. Whitehead, “Waves called solitons: Concepts and experiments,” Am. J. Phys. 63, 381–382 (1996).
    [Crossref]
  4. N. Akhmediev and A. Ankiewicz, “Dissipative Solitons: From Optics to Biology and Medicine,” Lecture Notes in Physics (Springer-Verlag, 2008).
  5. 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] [PubMed]
  6. K. Goda and B. Jalali, “Dispersive fourier transformation for fast continuous single-shot measurements,” Nat. Photonics 7, 102–112 (2013).
    [Crossref]
  7. B. H. Kolner and M. Nazarathy, “Temporal imaging with a time lens,” Opt. Lett. 14, 630 (1989).
    [Crossref] [PubMed]
  8. P. Ryczkowski, M. Narhi, C. Billet, J. M. Merolla, G. Genty, and J. M. Dudley, “Real-time full-field characterization of transient dissipative soliton dynamics in a mode-locked laser,” Nat. Photonics 12, 221–227 (2018).
    [Crossref]
  9. K. Krupa, K. Nithyanandan, U. Andral, P. Tchofo-Dinda, and P. Grelu, “Real-time observation of internal motion within ultrafast dissipative optical soliton molecules,” Phys. Rev. Lett. 118, 243901 (2017).
    [Crossref] [PubMed]
  10. X. M. Liu, X. K. Yao, and Y. D. Cui, “Real-time observation of the buildup of soliton molecules,” Phys. Rev. Lett. 121, 023905 (2018).
    [Crossref] [PubMed]
  11. A. F. J. Runge, M. Erkintalo, and N. G. R. Broderick, “Observation of soliton explosions in a passively mode-locked fiber laser,” Optica 2, 36–39 (2015).
    [Crossref]
  12. A. Klein, G. Masri, H. Duadi, K. Sulimany, O. Lib, H. Steinberg, S. A. Kolpakov, and M. Fridman, “Ultrafast rogue wave patterns in fiber lasers,” Optica 5, 774–778 (2018).
    [Crossref]
  13. Y. Jeong, L. A. Vazquez-Zuniga, S. Lee, and Y. Kwon, “On the formation of noise-like pulses in fiber ring cavity configurations,” Opt. Fiber Technol. 20, 575–592 (2014).
    [Crossref]
  14. J. M. Sotocrespo, N. Akhmediev, and A. Ankiewicz, “Pulsating, creeping, and erupting solitons in dissipative systems,” Phys. Rev. Lett. 85, 2937–2940 (2000).
    [Crossref]
  15. N. Akhmediev, J. M. Sotocrespo, and G. Town, “Pulsating solitons, chaotic solitons, period doubling, and pulse coexistence in modelocked lasers,” Phys Rev E Stat Nonlin Soft Matter Phys 63, 056602 (2001).
    [Crossref]
  16. W. Chang, A. Ankiewicz, N. Akhmediev, and J. M. Soto-Crespo, “Creeping solitons in dissipative systems and their bifurcations,” Phys. Rev. E 76, 016607 (2007).
    [Crossref]
  17. J. M. Soto-Crespo, P. Grelu, N. Akhmediev, and N. Devine, “Soliton complexes in dissipative systems: vibrating, shaking, and mixed soliton pairs,” Phys. Rev. E Stat. Nonlinear & Soft Matter Phys. 75, 016613 (2007).
    [Crossref]
  18. A. K. Komarov, “Passive mode locking of fiber lasers upon doubling the period of repetition of ultrashort pulses in the output radiation,” Opt. & Spectrosc. 102, 637–642 (2007).
    [Crossref]
  19. Z. H. Wang, Z. Zhi, Y. G. Liu, R. J. He, J. Zhao, G. D. Wang, S. C. Wang, and G. Yang, “Self-organized compound pattern and pulsation of dissipative solitons in a passively mode-locked fiber laser,” Opt. Lett. 43, 478–481 (2018).
    [Crossref] [PubMed]
  20. A. Komarov, F. Amrani, A. Dmitriev, K. Komarov, D. Meshcheriakov, and F. Sanchez, “Dispersive-wave mechanism of interaction between ultrashort pulses in passive mode-locked fiber lasers,” Phys. Rev. A 85, 281–289 (2012).
    [Crossref]
  21. O. Thual and S. Fauve, “Localized structures generated by subcritical instabilities: Counterprogating waves,” J. De Physique 49, 1829–1833 (1988).
    [Crossref]
  22. A. Niang, F. Amrani, M. Salhi, H. Leblond, and F. Sanchez, “Influence of gain dynamics on dissipative soliton interaction in the presence of a continuous wave,” Phys. Rev. A 92, 033831 (2015).
    [Crossref]
  23. S. M. J. Kelly, “Characteristic sideband instability of periodically amplified average soliton,” Electron. Lett. 28, 806–807 (1992).
    [Crossref]

2018 (4)

P. Ryczkowski, M. Narhi, C. Billet, J. M. Merolla, G. Genty, and J. M. Dudley, “Real-time full-field characterization of transient dissipative soliton dynamics in a mode-locked laser,” Nat. Photonics 12, 221–227 (2018).
[Crossref]

X. M. Liu, X. K. Yao, and Y. D. Cui, “Real-time observation of the buildup of soliton molecules,” Phys. Rev. Lett. 121, 023905 (2018).
[Crossref] [PubMed]

A. Klein, G. Masri, H. Duadi, K. Sulimany, O. Lib, H. Steinberg, S. A. Kolpakov, and M. Fridman, “Ultrafast rogue wave patterns in fiber lasers,” Optica 5, 774–778 (2018).
[Crossref]

Z. H. Wang, Z. Zhi, Y. G. Liu, R. J. He, J. Zhao, G. D. Wang, S. C. Wang, and G. Yang, “Self-organized compound pattern and pulsation of dissipative solitons in a passively mode-locked fiber laser,” Opt. Lett. 43, 478–481 (2018).
[Crossref] [PubMed]

2017 (1)

K. Krupa, K. Nithyanandan, U. Andral, P. Tchofo-Dinda, and P. Grelu, “Real-time observation of internal motion within ultrafast dissipative optical soliton molecules,” Phys. Rev. Lett. 118, 243901 (2017).
[Crossref] [PubMed]

2015 (2)

A. F. J. Runge, M. Erkintalo, and N. G. R. Broderick, “Observation of soliton explosions in a passively mode-locked fiber laser,” Optica 2, 36–39 (2015).
[Crossref]

A. Niang, F. Amrani, M. Salhi, H. Leblond, and F. Sanchez, “Influence of gain dynamics on dissipative soliton interaction in the presence of a continuous wave,” Phys. Rev. A 92, 033831 (2015).
[Crossref]

2014 (1)

Y. Jeong, L. A. Vazquez-Zuniga, S. Lee, and Y. Kwon, “On the formation of noise-like pulses in fiber ring cavity configurations,” Opt. Fiber Technol. 20, 575–592 (2014).
[Crossref]

2013 (1)

K. Goda and B. Jalali, “Dispersive fourier transformation for fast continuous single-shot measurements,” Nat. Photonics 7, 102–112 (2013).
[Crossref]

2012 (3)

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

A. Komarov, F. Amrani, A. Dmitriev, K. Komarov, D. Meshcheriakov, and F. Sanchez, “Dispersive-wave mechanism of interaction between ultrashort pulses in passive mode-locked fiber lasers,” Phys. Rev. A 85, 281–289 (2012).
[Crossref]

2007 (3)

W. Chang, A. Ankiewicz, N. Akhmediev, and J. M. Soto-Crespo, “Creeping solitons in dissipative systems and their bifurcations,” Phys. Rev. E 76, 016607 (2007).
[Crossref]

J. M. Soto-Crespo, P. Grelu, N. Akhmediev, and N. Devine, “Soliton complexes in dissipative systems: vibrating, shaking, and mixed soliton pairs,” Phys. Rev. E Stat. Nonlinear & Soft Matter Phys. 75, 016613 (2007).
[Crossref]

A. K. Komarov, “Passive mode locking of fiber lasers upon doubling the period of repetition of ultrashort pulses in the output radiation,” Opt. & Spectrosc. 102, 637–642 (2007).
[Crossref]

2001 (1)

N. Akhmediev, J. M. Sotocrespo, and G. Town, “Pulsating solitons, chaotic solitons, period doubling, and pulse coexistence in modelocked lasers,” Phys Rev E Stat Nonlin Soft Matter Phys 63, 056602 (2001).
[Crossref]

2000 (1)

J. M. Sotocrespo, N. Akhmediev, and A. Ankiewicz, “Pulsating, creeping, and erupting solitons in dissipative systems,” Phys. Rev. Lett. 85, 2937–2940 (2000).
[Crossref]

1996 (1)

M. Remoissenet and J. A. Whitehead, “Waves called solitons: Concepts and experiments,” Am. J. Phys. 63, 381–382 (1996).
[Crossref]

1992 (1)

S. M. J. Kelly, “Characteristic sideband instability of periodically amplified average soliton,” Electron. Lett. 28, 806–807 (1992).
[Crossref]

1991 (1)

V. S. Grigoryan and T. S. Muradyan, “Evolution of light pulses into autosolitons in nonlinear amplifying media,” J. Opt. Soc. Am. B 41, 1757–1765 (1991).
[Crossref]

1989 (1)

1988 (1)

O. Thual and S. Fauve, “Localized structures generated by subcritical instabilities: Counterprogating waves,” J. De Physique 49, 1829–1833 (1988).
[Crossref]

Akhmediev, N.

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

J. M. Soto-Crespo, P. Grelu, N. Akhmediev, and N. Devine, “Soliton complexes in dissipative systems: vibrating, shaking, and mixed soliton pairs,” Phys. Rev. E Stat. Nonlinear & Soft Matter Phys. 75, 016613 (2007).
[Crossref]

W. Chang, A. Ankiewicz, N. Akhmediev, and J. M. Soto-Crespo, “Creeping solitons in dissipative systems and their bifurcations,” Phys. Rev. E 76, 016607 (2007).
[Crossref]

N. Akhmediev, J. M. Sotocrespo, and G. Town, “Pulsating solitons, chaotic solitons, period doubling, and pulse coexistence in modelocked lasers,” Phys Rev E Stat Nonlin Soft Matter Phys 63, 056602 (2001).
[Crossref]

J. M. Sotocrespo, N. Akhmediev, and A. Ankiewicz, “Pulsating, creeping, and erupting solitons in dissipative systems,” Phys. Rev. Lett. 85, 2937–2940 (2000).
[Crossref]

N. Akhmediev and A. Ankiewicz, “Dissipative Solitons: From Optics to Biology and Medicine,” Lecture Notes in Physics (Springer-Verlag, 2008).

Amrani, F.

A. Niang, F. Amrani, M. Salhi, H. Leblond, and F. Sanchez, “Influence of gain dynamics on dissipative soliton interaction in the presence of a continuous wave,” Phys. Rev. A 92, 033831 (2015).
[Crossref]

A. Komarov, F. Amrani, A. Dmitriev, K. Komarov, D. Meshcheriakov, and F. Sanchez, “Dispersive-wave mechanism of interaction between ultrashort pulses in passive mode-locked fiber lasers,” Phys. Rev. A 85, 281–289 (2012).
[Crossref]

Andral, U.

K. Krupa, K. Nithyanandan, U. Andral, P. Tchofo-Dinda, and P. Grelu, “Real-time observation of internal motion within ultrafast dissipative optical soliton molecules,” Phys. Rev. Lett. 118, 243901 (2017).
[Crossref] [PubMed]

Ankiewicz, A.

W. Chang, A. Ankiewicz, N. Akhmediev, and J. M. Soto-Crespo, “Creeping solitons in dissipative systems and their bifurcations,” Phys. Rev. E 76, 016607 (2007).
[Crossref]

J. M. Sotocrespo, N. Akhmediev, and A. Ankiewicz, “Pulsating, creeping, and erupting solitons in dissipative systems,” Phys. Rev. Lett. 85, 2937–2940 (2000).
[Crossref]

N. Akhmediev and A. Ankiewicz, “Dissipative Solitons: From Optics to Biology and Medicine,” Lecture Notes in Physics (Springer-Verlag, 2008).

Billet, C.

P. Ryczkowski, M. Narhi, C. Billet, J. M. Merolla, G. Genty, and J. M. Dudley, “Real-time full-field characterization of transient dissipative soliton dynamics in a mode-locked laser,” Nat. Photonics 12, 221–227 (2018).
[Crossref]

Broderick, N. G. R.

Chang, W.

W. Chang, A. Ankiewicz, N. Akhmediev, and J. M. Soto-Crespo, “Creeping solitons in dissipative systems and their bifurcations,” Phys. Rev. E 76, 016607 (2007).
[Crossref]

Cui, Y. D.

X. M. Liu, X. K. Yao, and Y. D. Cui, “Real-time observation of the buildup of soliton molecules,” Phys. Rev. Lett. 121, 023905 (2018).
[Crossref] [PubMed]

Devine, N.

J. M. Soto-Crespo, P. Grelu, N. Akhmediev, and N. Devine, “Soliton complexes in dissipative systems: vibrating, shaking, and mixed soliton pairs,” Phys. Rev. E Stat. Nonlinear & Soft Matter Phys. 75, 016613 (2007).
[Crossref]

Dmitriev, A.

A. Komarov, F. Amrani, A. Dmitriev, K. Komarov, D. Meshcheriakov, and F. Sanchez, “Dispersive-wave mechanism of interaction between ultrashort pulses in passive mode-locked fiber lasers,” Phys. Rev. A 85, 281–289 (2012).
[Crossref]

Duadi, H.

Dudley, J. M.

P. Ryczkowski, M. Narhi, C. Billet, J. M. Merolla, G. Genty, and J. M. Dudley, “Real-time full-field characterization of transient dissipative soliton dynamics in a mode-locked laser,” Nat. Photonics 12, 221–227 (2018).
[Crossref]

Erkintalo, M.

Fauve, S.

O. Thual and S. Fauve, “Localized structures generated by subcritical instabilities: Counterprogating waves,” J. De Physique 49, 1829–1833 (1988).
[Crossref]

Fridman, M.

Genty, G.

P. Ryczkowski, M. Narhi, C. Billet, J. M. Merolla, G. Genty, and J. M. Dudley, “Real-time full-field characterization of transient dissipative soliton dynamics in a mode-locked laser,” Nat. Photonics 12, 221–227 (2018).
[Crossref]

Goda, K.

K. Goda and B. Jalali, “Dispersive fourier transformation for fast continuous single-shot measurements,” Nat. Photonics 7, 102–112 (2013).
[Crossref]

Grelu, P.

K. Krupa, K. Nithyanandan, U. Andral, P. Tchofo-Dinda, and P. Grelu, “Real-time observation of internal motion within ultrafast dissipative optical soliton molecules,” Phys. Rev. Lett. 118, 243901 (2017).
[Crossref] [PubMed]

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

J. M. Soto-Crespo, P. Grelu, N. Akhmediev, and N. Devine, “Soliton complexes in dissipative systems: vibrating, shaking, and mixed soliton pairs,” Phys. Rev. E Stat. Nonlinear & Soft Matter Phys. 75, 016613 (2007).
[Crossref]

Grigoryan, V. S.

V. S. Grigoryan and T. S. Muradyan, “Evolution of light pulses into autosolitons in nonlinear amplifying media,” J. Opt. Soc. Am. B 41, 1757–1765 (1991).
[Crossref]

He, R. J.

Jalali, B.

K. Goda and B. Jalali, “Dispersive fourier transformation for fast continuous single-shot measurements,” Nat. Photonics 7, 102–112 (2013).
[Crossref]

Jeong, Y.

Y. Jeong, L. A. Vazquez-Zuniga, S. Lee, and Y. Kwon, “On the formation of noise-like pulses in fiber ring cavity configurations,” Opt. Fiber Technol. 20, 575–592 (2014).
[Crossref]

Kelly, S. M. J.

S. M. J. Kelly, “Characteristic sideband instability of periodically amplified average soliton,” Electron. Lett. 28, 806–807 (1992).
[Crossref]

Klein, A.

Kolner, B. H.

Kolpakov, S. A.

Komarov, A.

A. Komarov, F. Amrani, A. Dmitriev, K. Komarov, D. Meshcheriakov, and F. Sanchez, “Dispersive-wave mechanism of interaction between ultrashort pulses in passive mode-locked fiber lasers,” Phys. Rev. A 85, 281–289 (2012).
[Crossref]

Komarov, A. K.

A. K. Komarov, “Passive mode locking of fiber lasers upon doubling the period of repetition of ultrashort pulses in the output radiation,” Opt. & Spectrosc. 102, 637–642 (2007).
[Crossref]

Komarov, K.

A. Komarov, F. Amrani, A. Dmitriev, K. Komarov, D. Meshcheriakov, and F. Sanchez, “Dispersive-wave mechanism of interaction between ultrashort pulses in passive mode-locked fiber lasers,” Phys. Rev. A 85, 281–289 (2012).
[Crossref]

Krupa, K.

K. Krupa, K. Nithyanandan, U. Andral, P. Tchofo-Dinda, and P. Grelu, “Real-time observation of internal motion within ultrafast dissipative optical soliton molecules,” Phys. Rev. Lett. 118, 243901 (2017).
[Crossref] [PubMed]

Kwon, Y.

Y. Jeong, L. A. Vazquez-Zuniga, S. Lee, and Y. Kwon, “On the formation of noise-like pulses in fiber ring cavity configurations,” Opt. Fiber Technol. 20, 575–592 (2014).
[Crossref]

Leblond, H.

A. Niang, F. Amrani, M. Salhi, H. Leblond, and F. Sanchez, “Influence of gain dynamics on dissipative soliton interaction in the presence of a continuous wave,” Phys. Rev. A 92, 033831 (2015).
[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] [PubMed]

Lee, S.

Y. Jeong, L. A. Vazquez-Zuniga, S. Lee, and Y. Kwon, “On the formation of noise-like pulses in fiber ring cavity configurations,” Opt. Fiber Technol. 20, 575–592 (2014).
[Crossref]

Lib, O.

Liu, X. M.

X. M. Liu, X. K. Yao, and Y. D. Cui, “Real-time observation of the buildup of soliton molecules,” Phys. Rev. Lett. 121, 023905 (2018).
[Crossref] [PubMed]

Liu, Y. G.

Masri, G.

Merolla, J. M.

P. Ryczkowski, M. Narhi, C. Billet, J. M. Merolla, G. Genty, and J. M. Dudley, “Real-time full-field characterization of transient dissipative soliton dynamics in a mode-locked laser,” Nat. Photonics 12, 221–227 (2018).
[Crossref]

Meshcheriakov, D.

A. Komarov, F. Amrani, A. Dmitriev, K. Komarov, D. Meshcheriakov, and F. Sanchez, “Dispersive-wave mechanism of interaction between ultrashort pulses in passive mode-locked fiber lasers,” Phys. Rev. A 85, 281–289 (2012).
[Crossref]

Muradyan, T. S.

V. S. Grigoryan and T. S. Muradyan, “Evolution of light pulses into autosolitons in nonlinear amplifying media,” J. Opt. Soc. Am. B 41, 1757–1765 (1991).
[Crossref]

Narhi, M.

P. Ryczkowski, M. Narhi, C. Billet, J. M. Merolla, G. Genty, and J. M. Dudley, “Real-time full-field characterization of transient dissipative soliton dynamics in a mode-locked laser,” Nat. Photonics 12, 221–227 (2018).
[Crossref]

Nazarathy, M.

Niang, A.

A. Niang, F. Amrani, M. Salhi, H. Leblond, and F. Sanchez, “Influence of gain dynamics on dissipative soliton interaction in the presence of a continuous wave,” Phys. Rev. A 92, 033831 (2015).
[Crossref]

Nithyanandan, K.

K. Krupa, K. Nithyanandan, U. Andral, P. Tchofo-Dinda, and P. Grelu, “Real-time observation of internal motion within ultrafast dissipative optical soliton molecules,” Phys. Rev. Lett. 118, 243901 (2017).
[Crossref] [PubMed]

Remoissenet, M.

M. Remoissenet and J. A. Whitehead, “Waves called solitons: Concepts and experiments,” Am. J. Phys. 63, 381–382 (1996).
[Crossref]

Runge, A. F. J.

Ryczkowski, P.

P. Ryczkowski, M. Narhi, C. Billet, J. M. Merolla, G. Genty, and J. M. Dudley, “Real-time full-field characterization of transient dissipative soliton dynamics in a mode-locked laser,” Nat. Photonics 12, 221–227 (2018).
[Crossref]

Salhi, M.

A. Niang, F. Amrani, M. Salhi, H. Leblond, and F. Sanchez, “Influence of gain dynamics on dissipative soliton interaction in the presence of a continuous wave,” Phys. Rev. A 92, 033831 (2015).
[Crossref]

Sanchez, F.

A. Niang, F. Amrani, M. Salhi, H. Leblond, and F. Sanchez, “Influence of gain dynamics on dissipative soliton interaction in the presence of a continuous wave,” Phys. Rev. A 92, 033831 (2015).
[Crossref]

A. Komarov, F. Amrani, A. Dmitriev, K. Komarov, D. Meshcheriakov, and F. Sanchez, “Dispersive-wave mechanism of interaction between ultrashort pulses in passive mode-locked fiber lasers,” Phys. Rev. A 85, 281–289 (2012).
[Crossref]

Sotocrespo, J. M.

N. Akhmediev, J. M. Sotocrespo, and G. Town, “Pulsating solitons, chaotic solitons, period doubling, and pulse coexistence in modelocked lasers,” Phys Rev E Stat Nonlin Soft Matter Phys 63, 056602 (2001).
[Crossref]

J. M. Sotocrespo, N. Akhmediev, and A. Ankiewicz, “Pulsating, creeping, and erupting solitons in dissipative systems,” Phys. Rev. Lett. 85, 2937–2940 (2000).
[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] [PubMed]

W. Chang, A. Ankiewicz, N. Akhmediev, and J. M. Soto-Crespo, “Creeping solitons in dissipative systems and their bifurcations,” Phys. Rev. E 76, 016607 (2007).
[Crossref]

J. M. Soto-Crespo, P. Grelu, N. Akhmediev, and N. Devine, “Soliton complexes in dissipative systems: vibrating, shaking, and mixed soliton pairs,” Phys. Rev. E Stat. Nonlinear & Soft Matter Phys. 75, 016613 (2007).
[Crossref]

Steinberg, H.

Sulimany, K.

Tchofo-Dinda, P.

K. Krupa, K. Nithyanandan, U. Andral, P. Tchofo-Dinda, and P. Grelu, “Real-time observation of internal motion within ultrafast dissipative optical soliton molecules,” Phys. Rev. Lett. 118, 243901 (2017).
[Crossref] [PubMed]

Thual, O.

O. Thual and S. Fauve, “Localized structures generated by subcritical instabilities: Counterprogating waves,” J. De Physique 49, 1829–1833 (1988).
[Crossref]

Town, G.

N. Akhmediev, J. M. Sotocrespo, and G. Town, “Pulsating solitons, chaotic solitons, period doubling, and pulse coexistence in modelocked lasers,” Phys Rev E Stat Nonlin Soft Matter Phys 63, 056602 (2001).
[Crossref]

Vazquez-Zuniga, L. A.

Y. Jeong, L. A. Vazquez-Zuniga, S. Lee, and Y. Kwon, “On the formation of noise-like pulses in fiber ring cavity configurations,” Opt. Fiber Technol. 20, 575–592 (2014).
[Crossref]

Wang, G. D.

Wang, S. C.

Wang, Z. H.

Whitehead, J. A.

M. Remoissenet and J. A. Whitehead, “Waves called solitons: Concepts and experiments,” Am. J. Phys. 63, 381–382 (1996).
[Crossref]

Yang, G.

Yao, X. K.

X. M. Liu, X. K. Yao, and Y. D. Cui, “Real-time observation of the buildup of soliton molecules,” Phys. Rev. Lett. 121, 023905 (2018).
[Crossref] [PubMed]

Zhao, J.

Zhi, Z.

Am. J. Phys. (1)

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Supplementary Material (2)

NameDescription
» Visualization 1       Formation of soliton molecule
» Visualization 2       Double-periodic pulsating soliton

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Figures (10)

Fig. 1
Fig. 1 A map of soliton behaviors of CQGL-type equation solutions in the parameter plane (g0, Is).
Fig. 2
Fig. 2 (a) Evolution of temporal intensity profile at point 1 in Fig. 1 with g0 = 2.4 and Is = 0.32. (b) Spectrum at z = 3000.
Fig. 3
Fig. 3 (a) Evolution of temporal intensity profile at the point 2 in Fig. 1 with g0 = 2.4 and Is = 0.324. (b) The evolution trajectories in (σF, Q) space. The upper one is for 1 < z < 1000, the lower one is for 1000 < z < 6000. (c) The enlarged diagrams of the time profiles from 3000 < z < 3030. (d) The spectrum corresponds to (c).
Fig. 4
Fig. 4 (a) Evolution of the time profiles of the soliton at the point 3 in Fig. 1 with go = 2.4 and Is = 0.33. (b) The enlarged evolution of (a) for 2100 < z < 2500. (c) Spectrum corresponding to (b). (d) The evolution trajectory in (σFQ) space when 2200 < z < 6000. (e) The XFROG traces during the formation of the soliton molecule. (e1), (e2), (e3), (e4), (e5) and (e6) are at z = 1499.2, 1501.2, 2322.1, 2327.4, 2330.8 and 2500, respectively (see Visualization 1).
Fig. 5
Fig. 5 (a) and (b) are the evolution of temporal intensity profile and spectrum of 3000 < z < 3400 of point 8 in Fig. 1 with g0 = 2.4 and Is = 0.528, respectively. (c) Evolution trajectory in (σF, Q) space for 140 < z < 6000. (d) XFROG traces in a pulsating period. (d1), (d2), (d3) and (d4) are at z = 3005.6, 3008.4, 3011.1 and 3014.1, respectively (see Visualization 2).
Fig. 6
Fig. 6 (a) Evolution of the temporal intensity profile for the snaking pulsating soliton, point 4 in Fig. 1 with g0 = 2.3 and Is = 0.676. (b) Evolution trajectory in (σF, Q) space. The upper figure is for 2300 < z < 2700, the lower figure is for 2700 < z < 6000. (c) and (e) are the enlarged evolution of temporal intensity profiles for 3000 < z < 4000 and 3000 < z < 3200, respectively. (d) and (f) are the enlarged evolution of spectrum for 3000 < z < 4000 and 3000 < z < 3200, respectively.
Fig. 7
Fig. 7 (a) Evolution of temporal intensity profile for the snaking pulsating soliton, point 5 in Fig. 1 with g0 = 2.4 and Is = 0.506. (b) Evolution trajectory in (σF, Q) space for 1300 < z < 6000. (c) and (d) are enlarged evolution of temporal intensity profiles and spectrum for 3000 < z < 4000.
Fig. 8
Fig. 8 (a) Evolution of the temporal intensity profile for the snaking pulsating soliton, point 6 in Fig. 1 with g0 = 2.5 and Is = 0.412. (b) Evolution trajectory in (σF, Q) space. (c) Evolution of the temporal intensity profile for snaking pulsating soliton, point 7, in Fig. 1 with g0 = 2.6 and Is = 0.34. (d) Evolution trajectory in (σF, Q) space.
Fig. 9
Fig. 9 DSs multistability diagram with varying Is and initial condition. The rest of parameters are fixed: D = 1, ε = 0.58, β = 0.025, µ = −0.12, υ = −0.1, r = 2, g0 = 2.4 and Γ = 0.
Fig. 10
Fig. 10 (a) and (b) are the spatial evolution of single-period soliton, point 2 in Fig. 1 with Γ = 0.0001 and Γ = 0.001.

Equations (2)

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E z = ( g 0 1 + | E | 2 I s r ) E + ( β + i D 2 ) E t t + ( ε + i ) | E | 2 E + ( μ + i υ ) | E Γ E t ( | E | 2 | E | 2 ) d t
S ( f , τ ) = | E ( t ) g ( t τ ) exp ( i 2 π f t ) d t | 2