Abstract

The different aspects of few-cycle pulse dynamics governed by the regularized short pulse equation (RSPE) are reported. It is shown that the RSPE provides an accurate description of the dynamics of the few-cycle pulse whose duration is larger than a single optical period when the few-cycle pulse’s spectrum is in the medium’s anomalous dispersion regime. The approximate solutions of the RSPE are constructed from the soliton solutions of the nonlinear Schrödinger (NLS) equation. We demonstrate numerically that the stability of these few-cycle pulses strongly depends on their pulse duration. Furthermore, the interactions of the two and three few-cycle pulses are studied. When pulse parameters are suitably chosen, we show the elastic collision, inelastic collision and repulsive interaction between these multi few-cycle pulses. It is revealed that the interactions of the multi few-cycle pulses rely heavily on their pulse duration.

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

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References

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    [Crossref]
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    [Crossref]
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2017 (2)

R. Šuminas, G. Tamošauskas, V. Jukna, A. Couairon, and A. Dubietis, “Second-order cascading-assisted filamentation and controllable supercontinuum generation in birefringent crystals,” Opt. Express. 25, 6746–6756 (2017).
[Crossref] [PubMed]

P. He, Y. Liu, K. Zhao, H. Teng, X. He, P. Huang, H. Huang, S. Zhong, Y. Jiang, S. Fang, X. Hou, and Z. Wei, “High-efficiency supercontinuum generation in solid thin plates at 0.1 TW level,” Opt. Lett. 42, 474–477 (2017).
[Crossref] [PubMed]

2016 (4)

Z. Chen and A. Pukhov, “Bright high-order harmonic generation with controllable polarization from a relativistic plasma mirror,” Nat. Commun. 7, 12515 (2016).
[Crossref] [PubMed]

H. Leblond, Ph. Grelu, D. Mihalache, and H. Triki, “Few-cycle solitons in supercontinuum generation dynamics,” Eur. Phys. J. Special Topics 225, 2435–2451 (2016).
[Crossref]

D. Grossmann, M. Reininghaus, C. Kalupka, M. Kumkar, and R. Poprawe, “Transverse pump-probe microscopy of moving breakdown, filamentation and self-organized absorption in alkali aluminosilicate glass using ultrashort pulse laser,” Opt. Express. 24, 23221–23231 (2016).
[Crossref] [PubMed]

V. Shumakova, S. Ališauskas, A. Voronin, A. M. Zheltikov, D. Faccio, D. Kartashov, A. Baltuška, and A. Pugžlys, “Multi-millijoule few-cycle mid-infrared pulses through nonlinear self-compression in bulk,” Nat. Commun. 7, 12877 (2016).
[Crossref] [PubMed]

2015 (1)

2014 (4)

Y. Shen, T. P. Horikis, P. G. Kevrekidis, and D. J. Frantzeskakis, “Traveling waves of the regularized short pulse equation,” J. Phys. A 47, 315204 (2014).
[Crossref]

G. Mourou, S. Mironov, E. Khazanov, and A. Sergeev, “Single cycle thin film compressor opening the door to Zeptosecond-Exawatt physics,” Eur. Phys. J. Special Topics 223, 1181–1188 (2014).
[Crossref]

H. Leblond, Ph. Grelu, and D. Mihalache, “Models for supercontinuum generation beyond the slowly-varying-envelope approximation,” Phys. Rev. A 90, 053816 (2014).
[Crossref]

D. J. Frantzeskakis, H. Leblond, and D. Mihalache, “Nonlinear optics of intense few-cycle pulses: An overview of recent theoretical and experimental developments,” Rom. J. Phys. 59, 767–784 (2014).

2013 (1)

H. Leblond and D. Mihalache, “Models of few optical cycle solitons beyond the slowly varying envelope approximation,” Phys. Rep. 523, 61–126 (2013).
[Crossref]

2012 (2)

H. Leblond, H. Triki, F. Sanchez, and D. Mihalache, “Circularly polarized few-optical-cycle solitons in Kerr media: A complex modified Korteweg-de Vries model,” Opt. Commun. 285, 356–363 (2012).
[Crossref]

J. Jia and J. Lin, “Solitons in nonlocal nonlinear kerr media with exponential response function,” Opt. Express 20, 7469–7479 (2012).
[Crossref] [PubMed]

2011 (4)

P. Dorey, K. Mersh, T. Romanczukiewicz, and Y. Shnir, “Kink-Antikink Collisions in the ϕ6 Model,” Phys. Rev. Lett. 107, 091602 (2011).
[Crossref]

H. Leblond, H. Triki, F. Sanchez, and D. Mihalache, “Robust circularly polarized few-optical-cycle solitons in Kerr media,” Phys. Rev. A 83, 063802 (2011).
[Crossref]

S. V. Sazonov, “On the nonlinear optics of few-cycle pulses,” Bull. Rus. Acad. Sci.: Physics 75, 157–160 (2011).

A. V. Kim and S. A. Skobelev, “Few-cycle vector solitons of light,” Phys. Rev. A 83, 063832 (2011).
[Crossref]

2010 (1)

Y. Shen, F. Williams, N. Whitaker, P. G. Kevrekidis, A. Saxen, and D. J. Frantzeskakis, “On some single-hump solutions of the short-pulse equation and their periodic generalizations,” Phys. Lett. A 374, 2964–2967 (2010).
[Crossref]

2009 (4)

N. Constanzino, V. Manukian, and C. K. R. T. Jones, “Solitary waves of the regularized short pulse and Ostrovsky equations,” SIAM J. Math. Anal. 41, 2088–2106 (2009).
[Crossref]

F. Krausz and M. Ivanov, “Attosecond physics,” Rev. Mod. Phys. 81, 163–234 (2009).
[Crossref]

H. Leblond and D. Mihalache, “Few-optical-cycle solitons: Modified Korteweg–de Vries sine-Gordon equation versus other non–slowly-varying-envelope-approximation models,” Phys. Rev. A 79, 063835 (2009).
[Crossref]

A. Kumar and V. Mishra, “Single-cycle pulse propagation in a cubic medium with delayed Raman response,” Phys. Rev. A 79, 063807 (2009).
[Crossref]

2008 (4)

A. A. Voronin and A. M. Zheltikov, “Soliton-number analysis of soliton-effect pulse compression to single-cycle pulse widths,” Phys. Rev. A 78, 063834 (2008).
[Crossref]

H. Leblond, I. V. Mel’nikov, and D. Mihalache, “Interaction of few-optical-cycle solitons,” Phys. Rev. A 78, 043802 (2008).
[Crossref]

Y. Matsuno, “Periodic solutions of the short pulse model equation,” J. Math. Phys. 49, 073508 (2008).
[Crossref]

E. J. Parkes, “Some periodic and solitary travelling-wave solutions of the short-pulse equation,” Chaos Solitons Fractals 38, 154–159 (2008).
[Crossref]

2007 (3)

Y. Matsuno, “Multiloop soliton and multibreather solutions of the short pulse model equation,” J. Phys. Soc. Jpn. 76, 084003 (2007).
[Crossref]

K. K. Victor, B. B. Thomas, and T. C. Kofane, “On exact solutions of the Schä er-Wayne short pulse equation: WKI eigenvalue problem,” J. Phys. A 39, 5585 (2007).
[Crossref]

S. A. Skobelev, D. V. Kartashov, and A. V. Kim, “Few-optical-cycle solitons and pulse self-compression in a Kerr medium,” Phys. Rev. Lett. 99, 203902 (2007).
[Crossref]

2006 (3)

J. C. Brunelli, “The bi-Hamiltonian structure of the short pulse equation,” Phys. Lett. A 353, 475–478 (2006).
[Crossref]

A. Sakovich and S. Sakovich, “Solitary wave solutions of the short pulse equation,” J. Phys. A 39, 361–367 (2006).
[Crossref]

H. Leblond, S. V. Sazonov, I. V. Mel’nikov, D. Mihalache, and F. Sanchez, “Few-cycle nonlinear optics of multicomponent media,” Phys. Rev. A 74, 063815 (2006).
[Crossref]

2005 (1)

A. Sakovich and S. Sakovich, “The short pulse equation is integrable,” J. Phys. Soc. Jpn. 74, 239–241 (2005).
[Crossref]

2004 (1)

T. Schäfer and C. E. Wayne, “Propagation of ultra-short optical pulses in cubic nonlinear media,” Physica D 196, 90–105 (2004).
[Crossref]

2003 (1)

H. Leblond and F. Sanchez, “Models for optical solitons in the two-cycle regime,” Phys. Rev. A 67, 013804 (2003).
[Crossref]

2002 (1)

V. G. Bespalov, S. A. Kozlov, Y. A. Shpolyansky, and I. A. Walmsley, “Simplified field wave equations for the nonlinear propagation of extremely short light pulses,” Phys. Rev. A 66, 013811 (2002).
[Crossref]

2001 (1)

Y. Tan and J. K. Yang, “Complexity and regularity of vector-soliton collisions,” Phys. Rev. E 64, 056616 (2001).
[Crossref]

2000 (2)

J. Yang and Y. Tan, “Fractal structure in the collision of vector solitons,” Phys. Rev. Lett. 85, 3624 (2000).
[Crossref] [PubMed]

T. Brabec and F. Krausz, “Intense few-cycle laser fields: frontiers of nonlinear optics,” Rev. Mod. Phys. 72, 545–591 (2000).
[Crossref]

1999 (1)

M. A. Porras, “Propagation of single-cycle pulsed light beams in dispersive media,” Phys. Rev. A 60, 5069 (1999).
[Crossref]

1998 (1)

C. Z. Tan, “Determination of refractive index of silica glass for infrared wavelengths by IR spectroscopy,” J. Non-Cryst. Solids 223, 158–163 (1998).
[Crossref]

1997 (3)

K. E. Oughstun and H. Xiao, “Failure of the quasimonochromatic approximation for ultrashort pulse propagation in a dispersive, attenuative medium,” Phys. Rev. Lett. 78, 642 (1997).
[Crossref]

T. Brabec and F. Krausz, “Nonlinear Optical Pulse Propagation in the Single-Cycle Regime,” Phys. Rev. Lett. 78, 3282 (1997).
[Crossref]

S. A. Kozlov and S. V. Sazonov, “Nonlinear propagation of optical pulses of a few oscillations duration in dielectric media,” JETP. 84, 221–228 (1997).
[Crossref]

1979 (1)

M. J. Ablowitz, M. D. Kruskal, and J. R. Ladik, “Solitary wave collisions,” SIAM J. Appl. Math 36, 428–437 (1979).
[Crossref]

1973 (1)

R. Hirota, “Exact envelope-soliton solutions of a nonlinear wave equation,” J. Math. Phys. 14, 805 (1973).
[Crossref]

1965 (1)

Ablowitz, M. J.

M. J. Ablowitz, M. D. Kruskal, and J. R. Ladik, “Solitary wave collisions,” SIAM J. Appl. Math 36, 428–437 (1979).
[Crossref]

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic, 2001).

Ališauskas, S.

V. Shumakova, S. Ališauskas, A. Voronin, A. M. Zheltikov, D. Faccio, D. Kartashov, A. Baltuška, and A. Pugžlys, “Multi-millijoule few-cycle mid-infrared pulses through nonlinear self-compression in bulk,” Nat. Commun. 7, 12877 (2016).
[Crossref] [PubMed]

Baltuška, A.

V. Shumakova, S. Ališauskas, A. Voronin, A. M. Zheltikov, D. Faccio, D. Kartashov, A. Baltuška, and A. Pugžlys, “Multi-millijoule few-cycle mid-infrared pulses through nonlinear self-compression in bulk,” Nat. Commun. 7, 12877 (2016).
[Crossref] [PubMed]

Bespalov, V. G.

V. G. Bespalov, S. A. Kozlov, Y. A. Shpolyansky, and I. A. Walmsley, “Simplified field wave equations for the nonlinear propagation of extremely short light pulses,” Phys. Rev. A 66, 013811 (2002).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1968).

Brabec, T.

T. Brabec and F. Krausz, “Intense few-cycle laser fields: frontiers of nonlinear optics,” Rev. Mod. Phys. 72, 545–591 (2000).
[Crossref]

T. Brabec and F. Krausz, “Nonlinear Optical Pulse Propagation in the Single-Cycle Regime,” Phys. Rev. Lett. 78, 3282 (1997).
[Crossref]

Brunelli, J. C.

J. C. Brunelli, “The bi-Hamiltonian structure of the short pulse equation,” Phys. Lett. A 353, 475–478 (2006).
[Crossref]

Chen, Z.

Z. Chen and A. Pukhov, “Bright high-order harmonic generation with controllable polarization from a relativistic plasma mirror,” Nat. Commun. 7, 12515 (2016).
[Crossref] [PubMed]

Constanzino, N.

N. Constanzino, V. Manukian, and C. K. R. T. Jones, “Solitary waves of the regularized short pulse and Ostrovsky equations,” SIAM J. Math. Anal. 41, 2088–2106 (2009).
[Crossref]

Couairon, A.

R. Šuminas, G. Tamošauskas, V. Jukna, A. Couairon, and A. Dubietis, “Second-order cascading-assisted filamentation and controllable supercontinuum generation in birefringent crystals,” Opt. Express. 25, 6746–6756 (2017).
[Crossref] [PubMed]

Dorey, P.

P. Dorey, K. Mersh, T. Romanczukiewicz, and Y. Shnir, “Kink-Antikink Collisions in the ϕ6 Model,” Phys. Rev. Lett. 107, 091602 (2011).
[Crossref]

Dubietis, A.

R. Šuminas, G. Tamošauskas, V. Jukna, A. Couairon, and A. Dubietis, “Second-order cascading-assisted filamentation and controllable supercontinuum generation in birefringent crystals,” Opt. Express. 25, 6746–6756 (2017).
[Crossref] [PubMed]

Faccio, D.

V. Shumakova, S. Ališauskas, A. Voronin, A. M. Zheltikov, D. Faccio, D. Kartashov, A. Baltuška, and A. Pugžlys, “Multi-millijoule few-cycle mid-infrared pulses through nonlinear self-compression in bulk,” Nat. Commun. 7, 12877 (2016).
[Crossref] [PubMed]

Fang, S.

Frantzeskakis, D. J.

Y. Shen, T. P. Horikis, P. G. Kevrekidis, and D. J. Frantzeskakis, “Traveling waves of the regularized short pulse equation,” J. Phys. A 47, 315204 (2014).
[Crossref]

D. J. Frantzeskakis, H. Leblond, and D. Mihalache, “Nonlinear optics of intense few-cycle pulses: An overview of recent theoretical and experimental developments,” Rom. J. Phys. 59, 767–784 (2014).

Y. Shen, F. Williams, N. Whitaker, P. G. Kevrekidis, A. Saxen, and D. J. Frantzeskakis, “On some single-hump solutions of the short-pulse equation and their periodic generalizations,” Phys. Lett. A 374, 2964–2967 (2010).
[Crossref]

Grelu, Ph.

H. Leblond, Ph. Grelu, D. Mihalache, and H. Triki, “Few-cycle solitons in supercontinuum generation dynamics,” Eur. Phys. J. Special Topics 225, 2435–2451 (2016).
[Crossref]

H. Leblond, Ph. Grelu, and D. Mihalache, “Models for supercontinuum generation beyond the slowly-varying-envelope approximation,” Phys. Rev. A 90, 053816 (2014).
[Crossref]

Grossmann, D.

D. Grossmann, M. Reininghaus, C. Kalupka, M. Kumkar, and R. Poprawe, “Transverse pump-probe microscopy of moving breakdown, filamentation and self-organized absorption in alkali aluminosilicate glass using ultrashort pulse laser,” Opt. Express. 24, 23221–23231 (2016).
[Crossref] [PubMed]

Harel, E.

He, P.

He, X.

Hirota, R.

R. Hirota, “Exact envelope-soliton solutions of a nonlinear wave equation,” J. Math. Phys. 14, 805 (1973).
[Crossref]

Horikis, T. P.

Y. Shen, T. P. Horikis, P. G. Kevrekidis, and D. J. Frantzeskakis, “Traveling waves of the regularized short pulse equation,” J. Phys. A 47, 315204 (2014).
[Crossref]

Hou, X.

Huang, H.

Huang, P.

Ivanov, M.

F. Krausz and M. Ivanov, “Attosecond physics,” Rev. Mod. Phys. 81, 163–234 (2009).
[Crossref]

Jia, J.

Jiang, Y.

Jones, C. K. R. T.

N. Constanzino, V. Manukian, and C. K. R. T. Jones, “Solitary waves of the regularized short pulse and Ostrovsky equations,” SIAM J. Math. Anal. 41, 2088–2106 (2009).
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Jukna, V.

R. Šuminas, G. Tamošauskas, V. Jukna, A. Couairon, and A. Dubietis, “Second-order cascading-assisted filamentation and controllable supercontinuum generation in birefringent crystals,” Opt. Express. 25, 6746–6756 (2017).
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Kalupka, C.

D. Grossmann, M. Reininghaus, C. Kalupka, M. Kumkar, and R. Poprawe, “Transverse pump-probe microscopy of moving breakdown, filamentation and self-organized absorption in alkali aluminosilicate glass using ultrashort pulse laser,” Opt. Express. 24, 23221–23231 (2016).
[Crossref] [PubMed]

Kartashov, D.

V. Shumakova, S. Ališauskas, A. Voronin, A. M. Zheltikov, D. Faccio, D. Kartashov, A. Baltuška, and A. Pugžlys, “Multi-millijoule few-cycle mid-infrared pulses through nonlinear self-compression in bulk,” Nat. Commun. 7, 12877 (2016).
[Crossref] [PubMed]

Kartashov, D. V.

S. A. Skobelev, D. V. Kartashov, and A. V. Kim, “Few-optical-cycle solitons and pulse self-compression in a Kerr medium,” Phys. Rev. Lett. 99, 203902 (2007).
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Kevrekidis, P. G.

Y. Shen, T. P. Horikis, P. G. Kevrekidis, and D. J. Frantzeskakis, “Traveling waves of the regularized short pulse equation,” J. Phys. A 47, 315204 (2014).
[Crossref]

Y. Shen, F. Williams, N. Whitaker, P. G. Kevrekidis, A. Saxen, and D. J. Frantzeskakis, “On some single-hump solutions of the short-pulse equation and their periodic generalizations,” Phys. Lett. A 374, 2964–2967 (2010).
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Khazanov, E.

G. Mourou, S. Mironov, E. Khazanov, and A. Sergeev, “Single cycle thin film compressor opening the door to Zeptosecond-Exawatt physics,” Eur. Phys. J. Special Topics 223, 1181–1188 (2014).
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Kim, A. V.

A. V. Kim and S. A. Skobelev, “Few-cycle vector solitons of light,” Phys. Rev. A 83, 063832 (2011).
[Crossref]

S. A. Skobelev, D. V. Kartashov, and A. V. Kim, “Few-optical-cycle solitons and pulse self-compression in a Kerr medium,” Phys. Rev. Lett. 99, 203902 (2007).
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Kofane, T. C.

K. K. Victor, B. B. Thomas, and T. C. Kofane, “On exact solutions of the Schä er-Wayne short pulse equation: WKI eigenvalue problem,” J. Phys. A 39, 5585 (2007).
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Koh, C. J.

Kozlov, S. A.

V. G. Bespalov, S. A. Kozlov, Y. A. Shpolyansky, and I. A. Walmsley, “Simplified field wave equations for the nonlinear propagation of extremely short light pulses,” Phys. Rev. A 66, 013811 (2002).
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S. A. Kozlov and S. V. Sazonov, “Nonlinear propagation of optical pulses of a few oscillations duration in dielectric media,” JETP. 84, 221–228 (1997).
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M. J. Ablowitz, M. D. Kruskal, and J. R. Ladik, “Solitary wave collisions,” SIAM J. Appl. Math 36, 428–437 (1979).
[Crossref]

Kumar, A.

A. Kumar and V. Mishra, “Single-cycle pulse propagation in a cubic medium with delayed Raman response,” Phys. Rev. A 79, 063807 (2009).
[Crossref]

Kumkar, M.

D. Grossmann, M. Reininghaus, C. Kalupka, M. Kumkar, and R. Poprawe, “Transverse pump-probe microscopy of moving breakdown, filamentation and self-organized absorption in alkali aluminosilicate glass using ultrashort pulse laser,” Opt. Express. 24, 23221–23231 (2016).
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M. J. Ablowitz, M. D. Kruskal, and J. R. Ladik, “Solitary wave collisions,” SIAM J. Appl. Math 36, 428–437 (1979).
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Landau, L. D.

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, 1960).

Leblond, H.

H. Leblond, Ph. Grelu, D. Mihalache, and H. Triki, “Few-cycle solitons in supercontinuum generation dynamics,” Eur. Phys. J. Special Topics 225, 2435–2451 (2016).
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H. Leblond, Ph. Grelu, and D. Mihalache, “Models for supercontinuum generation beyond the slowly-varying-envelope approximation,” Phys. Rev. A 90, 053816 (2014).
[Crossref]

D. J. Frantzeskakis, H. Leblond, and D. Mihalache, “Nonlinear optics of intense few-cycle pulses: An overview of recent theoretical and experimental developments,” Rom. J. Phys. 59, 767–784 (2014).

H. Leblond and D. Mihalache, “Models of few optical cycle solitons beyond the slowly varying envelope approximation,” Phys. Rep. 523, 61–126 (2013).
[Crossref]

H. Leblond, H. Triki, F. Sanchez, and D. Mihalache, “Circularly polarized few-optical-cycle solitons in Kerr media: A complex modified Korteweg-de Vries model,” Opt. Commun. 285, 356–363 (2012).
[Crossref]

H. Leblond, H. Triki, F. Sanchez, and D. Mihalache, “Robust circularly polarized few-optical-cycle solitons in Kerr media,” Phys. Rev. A 83, 063802 (2011).
[Crossref]

H. Leblond and D. Mihalache, “Few-optical-cycle solitons: Modified Korteweg–de Vries sine-Gordon equation versus other non–slowly-varying-envelope-approximation models,” Phys. Rev. A 79, 063835 (2009).
[Crossref]

H. Leblond, I. V. Mel’nikov, and D. Mihalache, “Interaction of few-optical-cycle solitons,” Phys. Rev. A 78, 043802 (2008).
[Crossref]

H. Leblond, S. V. Sazonov, I. V. Mel’nikov, D. Mihalache, and F. Sanchez, “Few-cycle nonlinear optics of multicomponent media,” Phys. Rev. A 74, 063815 (2006).
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H. Leblond and F. Sanchez, “Models for optical solitons in the two-cycle regime,” Phys. Rev. A 67, 013804 (2003).
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L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, 1960).

Lin, J.

Liu, Y.

Malitson, I. H.

Manukian, V.

N. Constanzino, V. Manukian, and C. K. R. T. Jones, “Solitary waves of the regularized short pulse and Ostrovsky equations,” SIAM J. Math. Anal. 41, 2088–2106 (2009).
[Crossref]

Matsuno, Y.

Y. Matsuno, “Periodic solutions of the short pulse model equation,” J. Math. Phys. 49, 073508 (2008).
[Crossref]

Y. Matsuno, “Multiloop soliton and multibreather solutions of the short pulse model equation,” J. Phys. Soc. Jpn. 76, 084003 (2007).
[Crossref]

Mel’nikov, I. V.

H. Leblond, I. V. Mel’nikov, and D. Mihalache, “Interaction of few-optical-cycle solitons,” Phys. Rev. A 78, 043802 (2008).
[Crossref]

H. Leblond, S. V. Sazonov, I. V. Mel’nikov, D. Mihalache, and F. Sanchez, “Few-cycle nonlinear optics of multicomponent media,” Phys. Rev. A 74, 063815 (2006).
[Crossref]

Mersh, K.

P. Dorey, K. Mersh, T. Romanczukiewicz, and Y. Shnir, “Kink-Antikink Collisions in the ϕ6 Model,” Phys. Rev. Lett. 107, 091602 (2011).
[Crossref]

Mihalache, D.

H. Leblond, Ph. Grelu, D. Mihalache, and H. Triki, “Few-cycle solitons in supercontinuum generation dynamics,” Eur. Phys. J. Special Topics 225, 2435–2451 (2016).
[Crossref]

H. Leblond, Ph. Grelu, and D. Mihalache, “Models for supercontinuum generation beyond the slowly-varying-envelope approximation,” Phys. Rev. A 90, 053816 (2014).
[Crossref]

D. J. Frantzeskakis, H. Leblond, and D. Mihalache, “Nonlinear optics of intense few-cycle pulses: An overview of recent theoretical and experimental developments,” Rom. J. Phys. 59, 767–784 (2014).

H. Leblond and D. Mihalache, “Models of few optical cycle solitons beyond the slowly varying envelope approximation,” Phys. Rep. 523, 61–126 (2013).
[Crossref]

H. Leblond, H. Triki, F. Sanchez, and D. Mihalache, “Circularly polarized few-optical-cycle solitons in Kerr media: A complex modified Korteweg-de Vries model,” Opt. Commun. 285, 356–363 (2012).
[Crossref]

H. Leblond, H. Triki, F. Sanchez, and D. Mihalache, “Robust circularly polarized few-optical-cycle solitons in Kerr media,” Phys. Rev. A 83, 063802 (2011).
[Crossref]

H. Leblond and D. Mihalache, “Few-optical-cycle solitons: Modified Korteweg–de Vries sine-Gordon equation versus other non–slowly-varying-envelope-approximation models,” Phys. Rev. A 79, 063835 (2009).
[Crossref]

H. Leblond, I. V. Mel’nikov, and D. Mihalache, “Interaction of few-optical-cycle solitons,” Phys. Rev. A 78, 043802 (2008).
[Crossref]

H. Leblond, S. V. Sazonov, I. V. Mel’nikov, D. Mihalache, and F. Sanchez, “Few-cycle nonlinear optics of multicomponent media,” Phys. Rev. A 74, 063815 (2006).
[Crossref]

Mironov, S.

G. Mourou, S. Mironov, E. Khazanov, and A. Sergeev, “Single cycle thin film compressor opening the door to Zeptosecond-Exawatt physics,” Eur. Phys. J. Special Topics 223, 1181–1188 (2014).
[Crossref]

Mishra, V.

A. Kumar and V. Mishra, “Single-cycle pulse propagation in a cubic medium with delayed Raman response,” Phys. Rev. A 79, 063807 (2009).
[Crossref]

Mourou, G.

G. Mourou, S. Mironov, E. Khazanov, and A. Sergeev, “Single cycle thin film compressor opening the door to Zeptosecond-Exawatt physics,” Eur. Phys. J. Special Topics 223, 1181–1188 (2014).
[Crossref]

Oughstun, K. E.

K. E. Oughstun and H. Xiao, “Failure of the quasimonochromatic approximation for ultrashort pulse propagation in a dispersive, attenuative medium,” Phys. Rev. Lett. 78, 642 (1997).
[Crossref]

Parkes, E. J.

E. J. Parkes, “Some periodic and solitary travelling-wave solutions of the short-pulse equation,” Chaos Solitons Fractals 38, 154–159 (2008).
[Crossref]

Poprawe, R.

D. Grossmann, M. Reininghaus, C. Kalupka, M. Kumkar, and R. Poprawe, “Transverse pump-probe microscopy of moving breakdown, filamentation and self-organized absorption in alkali aluminosilicate glass using ultrashort pulse laser,” Opt. Express. 24, 23221–23231 (2016).
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Porras, M. A.

M. A. Porras, “Propagation of single-cycle pulsed light beams in dispersive media,” Phys. Rev. A 60, 5069 (1999).
[Crossref]

Pugžlys, A.

V. Shumakova, S. Ališauskas, A. Voronin, A. M. Zheltikov, D. Faccio, D. Kartashov, A. Baltuška, and A. Pugžlys, “Multi-millijoule few-cycle mid-infrared pulses through nonlinear self-compression in bulk,” Nat. Commun. 7, 12877 (2016).
[Crossref] [PubMed]

Pukhov, A.

Z. Chen and A. Pukhov, “Bright high-order harmonic generation with controllable polarization from a relativistic plasma mirror,” Nat. Commun. 7, 12515 (2016).
[Crossref] [PubMed]

Reininghaus, M.

D. Grossmann, M. Reininghaus, C. Kalupka, M. Kumkar, and R. Poprawe, “Transverse pump-probe microscopy of moving breakdown, filamentation and self-organized absorption in alkali aluminosilicate glass using ultrashort pulse laser,” Opt. Express. 24, 23221–23231 (2016).
[Crossref] [PubMed]

Romanczukiewicz, T.

P. Dorey, K. Mersh, T. Romanczukiewicz, and Y. Shnir, “Kink-Antikink Collisions in the ϕ6 Model,” Phys. Rev. Lett. 107, 091602 (2011).
[Crossref]

Sakovich, A.

A. Sakovich and S. Sakovich, “Solitary wave solutions of the short pulse equation,” J. Phys. A 39, 361–367 (2006).
[Crossref]

A. Sakovich and S. Sakovich, “The short pulse equation is integrable,” J. Phys. Soc. Jpn. 74, 239–241 (2005).
[Crossref]

Sakovich, S.

A. Sakovich and S. Sakovich, “Solitary wave solutions of the short pulse equation,” J. Phys. A 39, 361–367 (2006).
[Crossref]

A. Sakovich and S. Sakovich, “The short pulse equation is integrable,” J. Phys. Soc. Jpn. 74, 239–241 (2005).
[Crossref]

Sanchez, F.

H. Leblond, H. Triki, F. Sanchez, and D. Mihalache, “Circularly polarized few-optical-cycle solitons in Kerr media: A complex modified Korteweg-de Vries model,” Opt. Commun. 285, 356–363 (2012).
[Crossref]

H. Leblond, H. Triki, F. Sanchez, and D. Mihalache, “Robust circularly polarized few-optical-cycle solitons in Kerr media,” Phys. Rev. A 83, 063802 (2011).
[Crossref]

H. Leblond, S. V. Sazonov, I. V. Mel’nikov, D. Mihalache, and F. Sanchez, “Few-cycle nonlinear optics of multicomponent media,” Phys. Rev. A 74, 063815 (2006).
[Crossref]

H. Leblond and F. Sanchez, “Models for optical solitons in the two-cycle regime,” Phys. Rev. A 67, 013804 (2003).
[Crossref]

Saxen, A.

Y. Shen, F. Williams, N. Whitaker, P. G. Kevrekidis, A. Saxen, and D. J. Frantzeskakis, “On some single-hump solutions of the short-pulse equation and their periodic generalizations,” Phys. Lett. A 374, 2964–2967 (2010).
[Crossref]

Sazonov, S. V.

S. V. Sazonov, “On the nonlinear optics of few-cycle pulses,” Bull. Rus. Acad. Sci.: Physics 75, 157–160 (2011).

H. Leblond, S. V. Sazonov, I. V. Mel’nikov, D. Mihalache, and F. Sanchez, “Few-cycle nonlinear optics of multicomponent media,” Phys. Rev. A 74, 063815 (2006).
[Crossref]

S. A. Kozlov and S. V. Sazonov, “Nonlinear propagation of optical pulses of a few oscillations duration in dielectric media,” JETP. 84, 221–228 (1997).
[Crossref]

Schäfer, T.

T. Schäfer and C. E. Wayne, “Propagation of ultra-short optical pulses in cubic nonlinear media,” Physica D 196, 90–105 (2004).
[Crossref]

Sergeev, A.

G. Mourou, S. Mironov, E. Khazanov, and A. Sergeev, “Single cycle thin film compressor opening the door to Zeptosecond-Exawatt physics,” Eur. Phys. J. Special Topics 223, 1181–1188 (2014).
[Crossref]

Shen, Y.

Y. Shen, T. P. Horikis, P. G. Kevrekidis, and D. J. Frantzeskakis, “Traveling waves of the regularized short pulse equation,” J. Phys. A 47, 315204 (2014).
[Crossref]

Y. Shen, F. Williams, N. Whitaker, P. G. Kevrekidis, A. Saxen, and D. J. Frantzeskakis, “On some single-hump solutions of the short-pulse equation and their periodic generalizations,” Phys. Lett. A 374, 2964–2967 (2010).
[Crossref]

Shnir, Y.

P. Dorey, K. Mersh, T. Romanczukiewicz, and Y. Shnir, “Kink-Antikink Collisions in the ϕ6 Model,” Phys. Rev. Lett. 107, 091602 (2011).
[Crossref]

Shpolyansky, Y. A.

V. G. Bespalov, S. A. Kozlov, Y. A. Shpolyansky, and I. A. Walmsley, “Simplified field wave equations for the nonlinear propagation of extremely short light pulses,” Phys. Rev. A 66, 013811 (2002).
[Crossref]

Shumakova, V.

V. Shumakova, S. Ališauskas, A. Voronin, A. M. Zheltikov, D. Faccio, D. Kartashov, A. Baltuška, and A. Pugžlys, “Multi-millijoule few-cycle mid-infrared pulses through nonlinear self-compression in bulk,” Nat. Commun. 7, 12877 (2016).
[Crossref] [PubMed]

Skobelev, S. A.

A. V. Kim and S. A. Skobelev, “Few-cycle vector solitons of light,” Phys. Rev. A 83, 063832 (2011).
[Crossref]

S. A. Skobelev, D. V. Kartashov, and A. V. Kim, “Few-optical-cycle solitons and pulse self-compression in a Kerr medium,” Phys. Rev. Lett. 99, 203902 (2007).
[Crossref]

Spokoyny, B.

Šuminas, R.

R. Šuminas, G. Tamošauskas, V. Jukna, A. Couairon, and A. Dubietis, “Second-order cascading-assisted filamentation and controllable supercontinuum generation in birefringent crystals,” Opt. Express. 25, 6746–6756 (2017).
[Crossref] [PubMed]

Tamošauskas, G.

R. Šuminas, G. Tamošauskas, V. Jukna, A. Couairon, and A. Dubietis, “Second-order cascading-assisted filamentation and controllable supercontinuum generation in birefringent crystals,” Opt. Express. 25, 6746–6756 (2017).
[Crossref] [PubMed]

Tan, C. Z.

C. Z. Tan, “Determination of refractive index of silica glass for infrared wavelengths by IR spectroscopy,” J. Non-Cryst. Solids 223, 158–163 (1998).
[Crossref]

Tan, Y.

Y. Tan and J. K. Yang, “Complexity and regularity of vector-soliton collisions,” Phys. Rev. E 64, 056616 (2001).
[Crossref]

J. Yang and Y. Tan, “Fractal structure in the collision of vector solitons,” Phys. Rev. Lett. 85, 3624 (2000).
[Crossref] [PubMed]

Teng, H.

Thomas, B. B.

K. K. Victor, B. B. Thomas, and T. C. Kofane, “On exact solutions of the Schä er-Wayne short pulse equation: WKI eigenvalue problem,” J. Phys. A 39, 5585 (2007).
[Crossref]

Triki, H.

H. Leblond, Ph. Grelu, D. Mihalache, and H. Triki, “Few-cycle solitons in supercontinuum generation dynamics,” Eur. Phys. J. Special Topics 225, 2435–2451 (2016).
[Crossref]

H. Leblond, H. Triki, F. Sanchez, and D. Mihalache, “Circularly polarized few-optical-cycle solitons in Kerr media: A complex modified Korteweg-de Vries model,” Opt. Commun. 285, 356–363 (2012).
[Crossref]

H. Leblond, H. Triki, F. Sanchez, and D. Mihalache, “Robust circularly polarized few-optical-cycle solitons in Kerr media,” Phys. Rev. A 83, 063802 (2011).
[Crossref]

Victor, K. K.

K. K. Victor, B. B. Thomas, and T. C. Kofane, “On exact solutions of the Schä er-Wayne short pulse equation: WKI eigenvalue problem,” J. Phys. A 39, 5585 (2007).
[Crossref]

Voronin, A.

V. Shumakova, S. Ališauskas, A. Voronin, A. M. Zheltikov, D. Faccio, D. Kartashov, A. Baltuška, and A. Pugžlys, “Multi-millijoule few-cycle mid-infrared pulses through nonlinear self-compression in bulk,” Nat. Commun. 7, 12877 (2016).
[Crossref] [PubMed]

Voronin, A. A.

A. A. Voronin and A. M. Zheltikov, “Soliton-number analysis of soliton-effect pulse compression to single-cycle pulse widths,” Phys. Rev. A 78, 063834 (2008).
[Crossref]

Walmsley, I. A.

V. G. Bespalov, S. A. Kozlov, Y. A. Shpolyansky, and I. A. Walmsley, “Simplified field wave equations for the nonlinear propagation of extremely short light pulses,” Phys. Rev. A 66, 013811 (2002).
[Crossref]

Wayne, C. E.

T. Schäfer and C. E. Wayne, “Propagation of ultra-short optical pulses in cubic nonlinear media,” Physica D 196, 90–105 (2004).
[Crossref]

Wegener, M.

M. Wegener, Extreme Nonlinear Optics (Springer, 2006).

Wei, Z.

Whitaker, N.

Y. Shen, F. Williams, N. Whitaker, P. G. Kevrekidis, A. Saxen, and D. J. Frantzeskakis, “On some single-hump solutions of the short-pulse equation and their periodic generalizations,” Phys. Lett. A 374, 2964–2967 (2010).
[Crossref]

Williams, F.

Y. Shen, F. Williams, N. Whitaker, P. G. Kevrekidis, A. Saxen, and D. J. Frantzeskakis, “On some single-hump solutions of the short-pulse equation and their periodic generalizations,” Phys. Lett. A 374, 2964–2967 (2010).
[Crossref]

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1968).

Xiao, H.

K. E. Oughstun and H. Xiao, “Failure of the quasimonochromatic approximation for ultrashort pulse propagation in a dispersive, attenuative medium,” Phys. Rev. Lett. 78, 642 (1997).
[Crossref]

Yang, J.

J. Yang and Y. Tan, “Fractal structure in the collision of vector solitons,” Phys. Rev. Lett. 85, 3624 (2000).
[Crossref] [PubMed]

J. Yang, Nonlinear Waves in Integrable and Nonintegrable Systems (SIAM, 2010).
[Crossref]

Yang, J. K.

Y. Tan and J. K. Yang, “Complexity and regularity of vector-soliton collisions,” Phys. Rev. E 64, 056616 (2001).
[Crossref]

Zhao, K.

Zheltikov, A. M.

V. Shumakova, S. Ališauskas, A. Voronin, A. M. Zheltikov, D. Faccio, D. Kartashov, A. Baltuška, and A. Pugžlys, “Multi-millijoule few-cycle mid-infrared pulses through nonlinear self-compression in bulk,” Nat. Commun. 7, 12877 (2016).
[Crossref] [PubMed]

A. A. Voronin and A. M. Zheltikov, “Soliton-number analysis of soliton-effect pulse compression to single-cycle pulse widths,” Phys. Rev. A 78, 063834 (2008).
[Crossref]

Zhong, S.

Bull. Rus. Acad. Sci.: Physics (1)

S. V. Sazonov, “On the nonlinear optics of few-cycle pulses,” Bull. Rus. Acad. Sci.: Physics 75, 157–160 (2011).

Chaos Solitons Fractals (1)

E. J. Parkes, “Some periodic and solitary travelling-wave solutions of the short-pulse equation,” Chaos Solitons Fractals 38, 154–159 (2008).
[Crossref]

Eur. Phys. J. Special Topics (2)

G. Mourou, S. Mironov, E. Khazanov, and A. Sergeev, “Single cycle thin film compressor opening the door to Zeptosecond-Exawatt physics,” Eur. Phys. J. Special Topics 223, 1181–1188 (2014).
[Crossref]

H. Leblond, Ph. Grelu, D. Mihalache, and H. Triki, “Few-cycle solitons in supercontinuum generation dynamics,” Eur. Phys. J. Special Topics 225, 2435–2451 (2016).
[Crossref]

J. Math. Phys. (2)

Y. Matsuno, “Periodic solutions of the short pulse model equation,” J. Math. Phys. 49, 073508 (2008).
[Crossref]

R. Hirota, “Exact envelope-soliton solutions of a nonlinear wave equation,” J. Math. Phys. 14, 805 (1973).
[Crossref]

J. Non-Cryst. Solids (1)

C. Z. Tan, “Determination of refractive index of silica glass for infrared wavelengths by IR spectroscopy,” J. Non-Cryst. Solids 223, 158–163 (1998).
[Crossref]

J. Opt. Soc. Am. (1)

J. Phys. A (3)

Y. Shen, T. P. Horikis, P. G. Kevrekidis, and D. J. Frantzeskakis, “Traveling waves of the regularized short pulse equation,” J. Phys. A 47, 315204 (2014).
[Crossref]

A. Sakovich and S. Sakovich, “Solitary wave solutions of the short pulse equation,” J. Phys. A 39, 361–367 (2006).
[Crossref]

K. K. Victor, B. B. Thomas, and T. C. Kofane, “On exact solutions of the Schä er-Wayne short pulse equation: WKI eigenvalue problem,” J. Phys. A 39, 5585 (2007).
[Crossref]

J. Phys. Soc. Jpn. (2)

Y. Matsuno, “Multiloop soliton and multibreather solutions of the short pulse model equation,” J. Phys. Soc. Jpn. 76, 084003 (2007).
[Crossref]

A. Sakovich and S. Sakovich, “The short pulse equation is integrable,” J. Phys. Soc. Jpn. 74, 239–241 (2005).
[Crossref]

JETP. (1)

S. A. Kozlov and S. V. Sazonov, “Nonlinear propagation of optical pulses of a few oscillations duration in dielectric media,” JETP. 84, 221–228 (1997).
[Crossref]

Nat. Commun. (2)

Z. Chen and A. Pukhov, “Bright high-order harmonic generation with controllable polarization from a relativistic plasma mirror,” Nat. Commun. 7, 12515 (2016).
[Crossref] [PubMed]

V. Shumakova, S. Ališauskas, A. Voronin, A. M. Zheltikov, D. Faccio, D. Kartashov, A. Baltuška, and A. Pugžlys, “Multi-millijoule few-cycle mid-infrared pulses through nonlinear self-compression in bulk,” Nat. Commun. 7, 12877 (2016).
[Crossref] [PubMed]

Opt. Commun. (1)

H. Leblond, H. Triki, F. Sanchez, and D. Mihalache, “Circularly polarized few-optical-cycle solitons in Kerr media: A complex modified Korteweg-de Vries model,” Opt. Commun. 285, 356–363 (2012).
[Crossref]

Opt. Express (1)

Opt. Express. (2)

D. Grossmann, M. Reininghaus, C. Kalupka, M. Kumkar, and R. Poprawe, “Transverse pump-probe microscopy of moving breakdown, filamentation and self-organized absorption in alkali aluminosilicate glass using ultrashort pulse laser,” Opt. Express. 24, 23221–23231 (2016).
[Crossref] [PubMed]

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[Crossref]

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Figures (7)

Fig. 1
Fig. 1 (a) The black line denotes the linear permittivity curve of fused silica. The blue line denotes ϵ4 with ω0 = 100 THz, ϵ0 = 2.111336, a1 = 0.0965 and a2 = 0.000133. The red line denotes ϵ6 with ω0 = 100 THz, ϵ0 = 2.113213, a1 = 0.100369, a2 = 0.0003111 and a3 = −0.00005125. ωc = 230THz is the zero dispersion frequency of fused silica. (b) The black line denotes the linear permittivity curve of fluoride glass. The blue line denotes ϵ4 with ω0 = 100 THz, ϵ0 = 1.5507, a1 = 1.86222694 and a2 = 0.0000007620895. The red line denotes ϵ6 with ω0 = 100 THz, ϵ0 = 1.549467, a1 = 1.8470735, a2 = 0.00000172049 and a3 = −0.00000010453898. ωc = 1615 THz is the zero dispersion frequency of fused silica.
Fig. 2
Fig. 2 (a) The evolution of the FCP with n = 1.2. (b) The blue (solid) line denotes the input FCP at ξ = 0 and the red (dashed) line denotes the output FCP at ξ = 500. The location of the FCP at ξ = 500 is artificially reset to the initial location. (c) The variation of the maximum value of the pulse with ξ. (d) The evolution of the FCP with n = 1.17. (e) The blue (solid) line denotes the input pulse at ξ = 0 and the red (dashed) line denotes the output FCP at ξ = 500. The location of the FCP at ξ = 500 is artificially reset to the initial location. (f) The variation of the maximum value of the pulse with ξ. In (a), (b) and (c), α = 1, β = 1/250, n = 1.2. Dispersion length for these parameters is Ld = 2.3. In (d), (e) and (f), α = 1, β = 1/250, n = 1.17. Dispersion length for these parameters is Ld = 2.1.
Fig. 3
Fig. 3 The red line (dashed) denotes the curve of transition distance ξtr versus the number of optical cycles n and the blue (solid) line denotes an exponential growth. Here α = 1, β = 1/250, Ω = 4. For these parameters, the dispersion length is Ld = 2.6 when the FCP’s number of optical cycles is n = 1.28.
Fig. 4
Fig. 4 (a) The elastic collision of two-FCPs. (b) The blue (solid) line is the initial profile of the FCPs at ξ = −80 and the red (dashed) line denotes the final profile of the FCPs at ξ = 120. Here w11 = 5/4 and w21 = 3/4.
Fig. 5
Fig. 5 (a) The inelastic collision of two-FCPs. (b) The blue (solid) line is the initial profile of the pulses at ξ = −75 and the red (dashed) line denotes the final profile of the pulses at ξ = 80. In (a) and (b), w11 = 1 and w21 = 133/100. (c) The repulsive interaction of two FCPs. (d) The blue (solid) line is the initial profile of the pulses at ξ = −60 and the red (dashed) line denotes the final profile of the pulses at ξ = 700. In (c) and (d), w11 = 42/25 and w21 = 73/50.
Fig. 6
Fig. 6 (a) The elastic collision of three-FCPs. (b) The blue (solid) line is the initial profile of the pulses at ξ = −150 and the red (dashed) line denotes the final profile of the pulses at ξ = 300. In (a) and (b), w11 = 7/8, w21 = 3/4 and w31 = 3/4.
Fig. 7
Fig. 7 (a) The inelastic collision of three-FCPs. (b) The blue (solid) line is the initial profile of the pulses at ξ = −65 and the red (dashed) line denotes the final profile of the pulses at ξ = 250. (c) The repulsive interaction of the three pulses. (d) The blue (solid) line is the initial profile of the pulses at ξ = −90 and the red (dashed) line denotes the final profile of the pulses at ξ = 240. In (a) and (b), w11 = 17/10, w21 = 1 and w31 = 5/4. In (c) and (d), w11 = 42/25, w21 = 73/50 and w31 = 42/25.

Tables (2)

Tables Icon

Table 1 Comparison of the FCP’s spectrum and the linear permittivity’s fitting range for the fused silica

Tables Icon

Table 2 Comparison of the FCP’s spectrum and the linear permittivity’s fitting range for the fluoride glass

Equations (19)

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u ξ τ + α u + ( u 3 ) τ τ + β u τ τ τ τ = 0 ,
2 E x 2 1 c 2 2 t 2 t ϵ ( t t ) E ( t ) d t 4 π c 2 2 P n l t 2 = 0
ϵ = ϵ 0 a 1 ω 0 2 ω 2 a 2 ω 2 ω 0 2 a 3 ω 4 ω 0 4 + ,
2 u ξ τ + α u + β 4 u τ 4 + μ 6 u τ 6 + 2 u 3 τ 2 = 0
2 u ξ τ + α u + β 4 u τ 4 + 2 u τ 4 + 2 u 3 τ 2 = 0
τ 0 = τ , τ 1 = δ τ , ξ 0 = ξ , ξ 1 = δ ξ , ξ 2 = δ 2 ξ ,
u = δ u 1 + δ 2 u 2 + δ 3 u 3 + ,
( 2 τ 0 ξ 0 + β 4 τ 0 4 + α ) u j = Γ ( j ) ,
u 1 = A ( τ 1 , ξ 1 , ξ 2 , ξ 3 ) e i ( Ω τ k ξ ) + c . c .
( 2 τ 0 ξ 0 + β 4 τ 0 4 + α ) u 2 = i Ω ( A ξ 1 + 3 β Ω 4 + α Ω 2 A τ 1 ) e i ( Ω τ k ξ ) + c . c .
i ( A ξ 1 + 1 v g A τ 1 ) = 0 ,
i A ξ 2 6 β Ω 2 A τ 1   2 + 1 Ω 2 A τ 1 ξ 1 3 Ω | A | 2 A = 0 .
i U ξ + P 2 U T 2 + Q | U | 2 U = 0 ,
u = U ( T , ξ ) e i ( Ω τ k ξ ) + c . c .
u = 2 b 2 P Q sech ( b τ b ξ / v g ) cos ( Ω τ + ( P b 2 + k ) ξ )
U = m 1 e θ 1 + m 2 e θ 2 + b 11 e θ 1 + θ 2 + θ 1 * + b 12 e θ 1 + θ 2 + θ 2 * 1 + a 11 e θ 1 + θ 1 * + a 12 e θ 1 + θ 2 * + a 21 e θ 2 + θ 1 * + a 22 e θ 2 + θ 2 * + m 11 e θ 1 + t h e t a 2 + θ 1 * + θ 2 *
U = g 1 + g 3 + g 5 1 + f 2 + f 4 + f 6 ,
g 1 = m 1 e θ 1 + m 2 e θ 2 + m 3 e θ 3 , f 2 = Q 11 e θ 1 + θ 2 + θ 3 + θ 1 * + θ 2 * + θ 3 * , g 3 = F 12312 e θ 1 + θ 2 + θ 3 + θ 1 * + θ 2 * + F 12323 e θ 1 + θ 2 + θ 3 + θ 2 * + θ 3 * + F 12313 e θ 1 + θ 2 + θ 3 + θ 1 * + θ 3 * , g 5 = b 121 e θ 1 + θ 2 + θ 1 * + b 122 e θ 1 + θ 2 + θ 2 * + b 123 e θ 1 + θ 2 + θ 3 * + b 131 e θ 1 + θ 3 + θ 1 * + b 132 e θ 1 + θ 3 + θ 2 * + b 133 e θ 1 + θ 3 + θ 3 * + b 231 e θ 2 + θ 3 + θ 1 * + b 232 e θ 2 + θ 3 + θ 2 * + b 233 e θ 2 + θ 3 + θ 3 * , f 4 = a 11 e θ 1 + θ 1 * + a 12 e θ 1 + θ 2 * + a 13 e θ 1 + θ 3 * + a 21 e θ 2 + θ 1 + a 22 e θ 2 + θ 2 + a 23 e θ 2 + θ 3 * + a 31 e θ 3 + θ 1 * + a 32 e θ 3 + θ 2 * + a 33 e θ 3 + θ 3 * , f 6 = M 1212 e θ 1 + θ 2 + θ 1 * + θ 2 * + M 1213 e θ 1 + θ 2 + θ 1 * + θ 3 * + M 1223 e θ 1 + θ 2 + θ 2 * + θ 3 * + M 1312 e θ 1 + θ 3 + θ 1 * + θ 2 * + M 1313 e θ 1 + θ 3 + θ 1 * + θ 3 * + M 1323 e θ 1 + θ 3 + θ 2 * + θ 3 * + M 2312 e θ 2 + θ 3 + θ 1 * + θ 2 * + M 2313 e θ 2 + θ 3 + θ 1 * + θ 3 * + M 2323 e θ 2 + θ 3 + θ 2 * + θ 3 * .
a j l = Q m j m l 2 P ( w j + w l * ) 2 , ( j , l = 1 , 2 , 3 ) b j k l = Q m j m k m l ( w j w k ) 2 2 P ( w j + w l * ) 2 ( w k + w l * ) , ( j , k , l = 1 , 2 , 3 ) M j k l o = Q 2 m j m k m l m 0 ( w j w k ) 2 ( w l * w o * ) 2 4 P 2 ( w j + w l * ) 2 ( w k + w l * ) 2 ( w k + w o * ) 2 ( w j + w o * ) 2 , ( j , k , l , o = 1 , 2 , 3 ) F j k s l o = Q 2 m j m k m s m l m 0 ( w j w k ) 2 ( w l * w o * ) 2 ( w j w s ) 2 ( w k w s ) 2 4 P 2 ( w j + w l * ) 2 ( w k + w l * ) 2 ( w k + w o * ) 2 ( w j + w o * ) 2 ( w s + w l * ) 2 ( w s + w o * ) 2 , ( j , k , s , l , o = 1 , 2 , 3 ) Q 11 = m 1 2 m 2 2 m 3 2 Q 3 ( w 1 * w 2 * ) 2 ( w 1 w 2 ) 2 ( w 1 w 3 ) 2 ( w 1 * w 3 * ) 2 ( w 2 * w 3 * ) 2 ( w 2 w 3 ) 2 512 w 11 2 w 21 2 w 31 2 P 3 ( w 1 + w 2 * ) 2 ( w 1 + w 3 * ) 2 ( w 2 + w 2 * ) 2 ( w 1 + w 3 * ) 2 ( w 3 + w 1 * ) 2 ( w 3 + w 2 * ) 2 .

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