December  2019, 8(4): 737-753. doi: 10.3934/eect.2019036

Discontinuous solutions for the generalized short pulse equation

1. 

Dipartimento di Meccanica, Matematica e Management, Politecnico di Bari, via E. Orabona 4, 70125 Bari, Italy

2. 

Dipartimento di Matematica, Università di Bari, via E. Orabona 4, 70125 Bari, Italy

* Corresponding author: G. M. Coclite

Received  July 2018 Revised  January 2019 Published  June 2019

Fund Project: The authors are members of the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM)

The generalized short pulse equation is a non-slowly-varying envelope approximation model that describes the physics of few-cycle-pulse optical solitons. This is a nonlinear evolution equation. In this paper, we prove the wellposedness of the Cauchy problem associated with this equation within a class of discontinuous solutions.

Citation: Giuseppe Maria Coclite, Lorenzo di Ruvo. Discontinuous solutions for the generalized short pulse equation. Evolution Equations & Control Theory, 2019, 8 (4) : 737-753. doi: 10.3934/eect.2019036
References:
[1]

S. Amiranashvili, A. G. Vladimirov and U. Bandelow, Solitary-wave solutions for few-cycle optical pulses, Phys. Rev. A, 77 (2008), 063821, URL https://link.aps.org/doi/10.1103/PhysRevA.77.063821. doi: 10.1103/PhysRevA.77.063821. Google Scholar

[2]

N. Belashenkov, A. Drozdov, S. Kozlov, Y. Shpolyanski and A. Tsypkin, Phase modulation of femtosecond light pulses whose spectra are superbroadened in dielectrics with normal group dispersion, J. Opt. Technol., 75 (2008), 611–614, URL http://jot.osa.org/abstract.cfm?URI=jot-75-10-611. doi: 10.1364/JOT.75.000611. Google Scholar

[3]

A. N. Berkovsky, S. A. Kozlov and Y. A. Shpolyanskiy, Self-focusing of few-cycle light pulses in dielectric media, Phys. Rev. A, 72 (2005), 043821, URL https://link.aps.org/doi/10.1103/PhysRevA.72.043821. doi: 10.1103/PhysRevA.72.043821. Google Scholar

[4]

V. G. Bespalov, S. A. Kozlov, Y. A. Shpolyanskiy and I. A. Walmsley, Simplified field wave equations for the nonlinear propagation of extremely short light pulses, Phys. Rev. A, 66 (2002), 013811, URL https://link.aps.org/doi/10.1103/PhysRevA.66.013811. doi: 10.1103/PhysRevA.66.013811. Google Scholar

[5]

V. BespalovS. KozlovY. Shpolyanskiy and A. N. Sutyagin, Spectral superbroadening of high-power femtosecond laser pulses and their time compression down to one period of the light field, Journal of Optical Technology, 65 (1998), 823-825. Google Scholar

[6]

T. Brabec and F. Krausz, Nonlinear optical pulse propagation in the single-cycle regime, Phys. Rev. Lett., 78 (1997), 3282–3285, URL https://link.aps.org/doi/10.1103/PhysRevLett.78.3282. doi: 10.1103/PhysRevLett.78.3282. Google Scholar

[7]

T. Brabec and F. Krausz, Intense few-cycle laser fields: Frontiers of nonlinear optics, Rev. Mod. Phys., 72 (2000), 545–591, URL https://link.aps.org/doi/10.1103/RevModPhys.72.545. doi: 10.1103/RevModPhys.72.545. Google Scholar

[8]

G. M. Coclite, L. di Ruvo and K. H. Karlsen, Some wellposedness results for the Ostrovsky-Hunter equation, in Hyperbolic Conservation Laws and Related Analysis with Applications, vol. 49 of Springer Proc. Math. Stat., Springer, Heidelberg, 2014,143–159, URL https://doi.org/10.1007/978-3-642-39007-4_7. doi: 10.1007/978-3-642-39007-4_7. Google Scholar

[9]

G. M. Coclite, J. Ridder and N. H. Risebro, A convergent finite difference scheme for the Ostrovsky-Hunter equation on a bounded domain, BIT, 57 (2017), 93–122, URL https://doi.org/10.1007/s10543-016-0625-x. doi: 10.1007/s10543-016-0625-x. Google Scholar

[10]

G. M. Coclite and L. di Ruvo, Convergence of the Ostrovsky equation to the Ostrovsky-Hunter one, J. Differential Equations, 256 (2014), 3245–3277, URL https://doi.org/10.1016/j.jde.2014.02.001. doi: 10.1016/j.jde.2014.02.001. Google Scholar

[11]

G. M. Coclite and L. di Ruvo, Dispersive and diffusive limits for Ostrovsky-Hunter type equations, NoDEA Nonlinear Differential Equations Appl., 22 (2015), 1733–1763, URL https://doi.org/10.1007/s00030-015-0342-1. doi: 10.1007/s00030-015-0342-1. Google Scholar

[12]

G. M. Coclite and L. di Ruvo, Oleinik type estimates for the Ostrovsky-Hunter equation, J. Math. Anal. Appl., 423 (2015), 162–190, URL https://doi.org/10.1016/j.jmaa.2014.09.033. doi: 10.1016/j.jmaa.2014.09.033. Google Scholar

[13]

G. M. Coclite and L. di Ruvo, Well-posedness of bounded solutions of the non-homogeneous initial-boundary value problem for the Ostrovsky-Hunter equation, J. Hyperbolic Differ. Equ., 12 (2015), 221–248, URL https://doi.org/10.1142/S021989161550006X. doi: 10.1142/S021989161550006X. Google Scholar

[14]

G. M. Coclite and L. di Ruvo, Well-posedness results for the short pulse equation, Z. Angew. Math. Phys., 66 (2015), 1529–1557, URL https://doi.org/10.1007/s00033-014-0478-6. doi: 10.1007/s00033-014-0478-6. Google Scholar

[15]

G. M. Coclite and L. di Ruvo, Wellposedness of bounded solutions of the non-homogeneous initial boundary for the short pulse equation, Boll. Unione Mat. Ital., 8 (2015), 31–44, URL https://doi.org/10.1007/s40574-015-0023-3. doi: 10.1007/s40574-015-0023-3. Google Scholar

[16]

G. M. Coclite and L. di Ruvo, Convergence of the solutions on the generalized Korteweg–de Vries equation, Math. Model. Anal., 21 (2016), 239–259, URL https://doi.org/10.3846/13926292.2016.1150358. doi: 10.3846/13926292.2016.1150358. Google Scholar

[17]

G. M. Coclite and L. di Ruvo, Well-posedness of the Ostrovsky-Hunter equation under the combined effects of dissipation and short-wave dispersion, J. Evol. Equ., 16 (2016), 365–389, URL https://doi.org/10.1007/s00028-015-0306-2. doi: 10.1007/s00028-015-0306-2. Google Scholar

[18]

G. M. Coclite and L. di Ruvo, Convergence of the regularized short pulse equation to the short pulse one, Math. Nachr., 291 (2018), 774–792, URL https://doi.org/10.1002/mana.201600301. doi: 10.1002/mana.201600301. Google Scholar

[19]

G. M. Coclite and L. di Ruvo, Well-posedness and dispersive/diffusive limit of a generalized Ostrovsky-Hunter equation, Milan J. Math., 86 (2018), 31–51, URL https://doi.org/10.1007/s00032-018-0278-0. doi: 10.1007/s00032-018-0278-0. Google Scholar

[20]

G. M. Coclite, L. di Ruvo and K. H. Karlsen, The initial-boundary-value problem for an Ostrovsky-Hunter type equation, in Non-linear Partial Differential Equations, Mathematical Physics, and Stochastic Analysis, EMS Ser. Congr. Rep., Eur. Math. Soc., Zürich, 2018, 97–109. Google Scholar

[21]

G. M. Coclite, H. Holden and K. H. Karlsen, Wellposedness for a parabolic-elliptic system, Discrete Contin. Dyn. Syst., 13 (2005), 659–682, URL https://doi.org/10.3934/dcds.2005.13.659. doi: 10.3934/dcds.2005.13.659. Google Scholar

[22]

J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Sharp global well-posedness for KdV and modified KdV on $\Bbb R$ and $\Bbb T$, J. Amer. Math. Soc., 16 (2003), 705–749, URL https://doi.org/10.1090/S0894-0347-03-00421-1. doi: 10.1090/S0894-0347-03-00421-1. Google Scholar

[23]

N. Costanzino, V. Manukian and C. K. R. T. Jones, Solitary waves of the regularized short pulse and Ostrovsky equations, SIAM J. Math. Anal., 41 (2009), 2088–2106, URL https://doi.org/10.1137/080734327. doi: 10.1137/080734327. Google Scholar

[24]

M. Davidson, Continuity properties of the solution map for the generalized reduced Ostrovsky equation, J. Differential Equations, 252 (2012), 3797–3815, URL https://doi.org/10.1016/j.jde.2011.11.013. doi: 10.1016/j.jde.2011.11.013. Google Scholar

[25]

L. di Ruvo, Discontinuous solutions for the Ostrovsky–Hunter equation and two phase flows, Phd Thesis, University of BariGoogle Scholar

[26]

C. E. Kenig, G. Ponce and L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Comm. Pure Appl. Math., 46 (1993), 527–620, URL https://doi.org/10.1002/cpa.3160460405. doi: 10.1002/cpa.3160460405. Google Scholar

[27]

S. A. Kozlov and S. V. Sazonov, Nonlinear propagation of optical pulses of a few oscillations duration in dielectric media, Journal of Experimental and Theoretical Physics, 84 (1997), 221–228, URL https://doi.org/10.1134/1.558109. doi: 10.1134/1.558109. Google Scholar

[28]

S. N. Kružkov, First order quasilinear equations with several independent variables, Mat. Sb. (N.S.), 81(123) (1970), 228-255. Google Scholar

[29]

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 (2009), 063835, URL https://link.aps.org/doi/10.1103/PhysRevA.79.063835. doi: 10.1103/PhysRevA.79.063835. Google Scholar

[30]

H. Leblond and D. Mihalache, Models of few optical cycle solitons beyond the slowly varying envelope approximation, Physics Reports, 523 (2013), 61–126, URL http://www.sciencedirect.com/science/article/pii/S0370157312003511, Models of few optical cycle solitons beyond the slowly varying envelope approximation. doi: 10.1016/j.physrep.2012.10.006. Google Scholar

[31]

H. Leblond and F. Sanchez, Models for optical solitons in the two-cycle regime, Phys. Rev. A, 67 (2003), 013804, URL https://link.aps.org/doi/10.1103/PhysRevA.67.013804. doi: 10.1103/PhysRevA.67.013804. Google Scholar

[32]

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 (2006), 063815, URL https://link.aps.org/doi/10.1103/PhysRevA.74.063815. doi: 10.1103/PhysRevA.74.063815. Google Scholar

[33]

P. G. LeFloch and R. Natalini, Conservation laws with vanishing nonlinear diffusion and dispersion, Nonlinear Anal., 36 (1999), 213–230, URL https://doi.org/10.1016/S0362-546X(98)00012-1. doi: 10.1016/S0362-546X(98)00012-1. Google Scholar

[34]

Y. Liu, D. Pelinovsky and A. Sakovich, Wave breaking in the short-pulse equation, Dyn. Partial Differ. Equ., 6 (2009), 291–310, URL https://doi.org/10.4310/DPDE.2009.v6.n4.a1. doi: 10.4310/DPDE.2009.v6.n4.a1. Google Scholar

[35]

F. Murat, L'injection du cône positif de H-1 dans W-1, q est compacte pour tout q < 2, J. Math. Pures Appl. (9), 60 (1981), 309–322. Google Scholar

[36]

A. Nazarkin, Nonlinear optics of intense attosecond light pulses, Phys. Rev. Lett., 97 (2006), 163904, URL https://link.aps.org/doi/10.1103/PhysRevLett.97.163904. doi: 10.1103/PhysRevLett.97.163904. Google Scholar

[37]

D. Pelinovsky and A. Sakovich, Global well-posedness of the short-pulse and sine-gordon equations in energy space, Communications in Partial Differential Equations, 35 (2010), 613–629, URL https://doi.org/10.1080/03605300903509104. doi: 10.1080/03605300903509104. Google Scholar

[38]

D. Pelinovsky and G. Schneider, Rigorous justification of the short-pulse equation, Nonlinear Differential Equations and Applications NoDEA, 20 (2013), 1277–1294, URL https://doi.org/10.1007/s00030-012-0208-8. doi: 10.1007/s00030-012-0208-8. Google Scholar

[39]

N. N. Rosanov, V. E. Semenov and N. V. Vysotina, Few-cycle dissipative solitons in active nonlinear optical fibres, Quantum Electronics, 38 (2008), 137, URL http://stacks.iop.org/1063-7818/38/i=2/a=A08. doi: 10.1070/QE2008v038n02ABEH013568. Google Scholar

[40]

N. N. Rosanov, V. E. Semenov and N. V. Vyssotina, Collisions of few-cycle dissipative solitons in active nonlinear fibers, Laser Physics, 17 (2007), 1311, URL https://doi.org/10.1134/S1054660X07110072. doi: 10.1134/S1054660X07110072. Google Scholar

[41]

T. Schäfer and C. Wayne, Propagation of ultra-short optical pulses in cubic nonlinear media, Physica D: Nonlinear Phenomena, 196 (2004), 90–105, URL http://www.sciencedirect.com/science/article/pii/S0167278904002064. doi: 10.1016/j.physd.2004.04.007. Google Scholar

[42]

M. E. Schonbek, Convergence of solutions to nonlinear dispersive equations, Comm. Partial Differential Equations, 7 (1982), 959–1000, URL https://doi.org/10.1080/03605308208820242. doi: 10.1080/03605308208820242. Google Scholar

[43]

A. Scrinzi, M. Y. Ivanov, R. Kienberger and D. M. Villeneuve, Attosecond physics, Journal of Physics B: Atomic, Molecular and Optical Physics, 39 (2006), R1, URL http://stacks.iop.org/0953-4075/39/i=1/a=R01.Google Scholar

[44]

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 (2007), 203902, URL https://link.aps.org/doi/10.1103/PhysRevLett.99.203902. doi: 10.1103/PhysRevLett.99.203902. Google Scholar

[45]

A. Stefanov, Y. Shen and P. G. Kevrekidis, Well-posedness and small data scattering for the generalized Ostrovsky equation, J. Differential Equations, 249 (2010), 2600–2617, URL https://doi.org/10.1016/j.jde.2010.05.015. doi: 10.1016/j.jde.2010.05.015. Google Scholar

[46]

X. Tan, X. Fan, Y. Yang and D. Tong, Time evolution of few-cycle pulse in a dense v-type three-level medium, Journal of Modern Optics, 55 (2008), 2439–2448, URL https://doi.org/10.1080/09500340802130670. doi: 10.1080/09500340802130670. Google Scholar

[47]

T. Tao, Nonlinear Dispersive Equations, vol. 106 of CBMS Regional Conference Series in Mathematics, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2006, URL https://doi.org/10.1090/cbms/106, Local and global analysis. doi: 10.1090/cbms/106. Google Scholar

[48]

L. Tartar, Compensated compactness and applications to partial differential equations, in Nonlinear Analysis and Mechanics: Heriot-Watt Symposium, Vol. IV, vol. 39 of Res. Notes in Math., Pitman, Boston, Mass.-London, 1979,136–212. Google Scholar

[49]

M. V. Tognetti and H. M. Crespo, Sub-two-cycle soliton-effect pulse compression at 800 nm in photonic crystal fibers, J. Opt. Soc. Am. B, 24 (2007), 1410–1415, URL http://josab.osa.org/abstract.cfm?URI=josab-24-6-1410. doi: 10.1364/JOSAB.24.001410. Google Scholar

[50]

N. Tsitsas, T. Horikis, Y. Shen, P. Kevrekidis, N. Whitaker and D. Frantzeskakis, Short pulse equations and localized structures in frequency band gaps of nonlinear metamaterials, Physics Letters A, 374 (2010), 1384–1388, URL http://www.sciencedirect.com/science/article/pii/S0375960110000150. doi: 10.1016/j.physleta.2010.01.004. Google Scholar

[51]

A. A. Voronin and A. M. Zheltikov, Soliton-number analysis of soliton-effect pulse compression to single-cycle pulse widths, Phys. Rev. A, 78 (2008), 063834, URL https://link.aps.org/doi/10.1103/PhysRevA.78.063834. doi: 10.1103/PhysRevA.78.063834. Google Scholar

[52]

M. Wegener, Extreme Nonlinear Optics, Advanced Texts in Physics, Springer-Verlag, Berlin, 2005, An introduction. Google Scholar

show all references

References:
[1]

S. Amiranashvili, A. G. Vladimirov and U. Bandelow, Solitary-wave solutions for few-cycle optical pulses, Phys. Rev. A, 77 (2008), 063821, URL https://link.aps.org/doi/10.1103/PhysRevA.77.063821. doi: 10.1103/PhysRevA.77.063821. Google Scholar

[2]

N. Belashenkov, A. Drozdov, S. Kozlov, Y. Shpolyanski and A. Tsypkin, Phase modulation of femtosecond light pulses whose spectra are superbroadened in dielectrics with normal group dispersion, J. Opt. Technol., 75 (2008), 611–614, URL http://jot.osa.org/abstract.cfm?URI=jot-75-10-611. doi: 10.1364/JOT.75.000611. Google Scholar

[3]

A. N. Berkovsky, S. A. Kozlov and Y. A. Shpolyanskiy, Self-focusing of few-cycle light pulses in dielectric media, Phys. Rev. A, 72 (2005), 043821, URL https://link.aps.org/doi/10.1103/PhysRevA.72.043821. doi: 10.1103/PhysRevA.72.043821. Google Scholar

[4]

V. G. Bespalov, S. A. Kozlov, Y. A. Shpolyanskiy and I. A. Walmsley, Simplified field wave equations for the nonlinear propagation of extremely short light pulses, Phys. Rev. A, 66 (2002), 013811, URL https://link.aps.org/doi/10.1103/PhysRevA.66.013811. doi: 10.1103/PhysRevA.66.013811. Google Scholar

[5]

V. BespalovS. KozlovY. Shpolyanskiy and A. N. Sutyagin, Spectral superbroadening of high-power femtosecond laser pulses and their time compression down to one period of the light field, Journal of Optical Technology, 65 (1998), 823-825. Google Scholar

[6]

T. Brabec and F. Krausz, Nonlinear optical pulse propagation in the single-cycle regime, Phys. Rev. Lett., 78 (1997), 3282–3285, URL https://link.aps.org/doi/10.1103/PhysRevLett.78.3282. doi: 10.1103/PhysRevLett.78.3282. Google Scholar

[7]

T. Brabec and F. Krausz, Intense few-cycle laser fields: Frontiers of nonlinear optics, Rev. Mod. Phys., 72 (2000), 545–591, URL https://link.aps.org/doi/10.1103/RevModPhys.72.545. doi: 10.1103/RevModPhys.72.545. Google Scholar

[8]

G. M. Coclite, L. di Ruvo and K. H. Karlsen, Some wellposedness results for the Ostrovsky-Hunter equation, in Hyperbolic Conservation Laws and Related Analysis with Applications, vol. 49 of Springer Proc. Math. Stat., Springer, Heidelberg, 2014,143–159, URL https://doi.org/10.1007/978-3-642-39007-4_7. doi: 10.1007/978-3-642-39007-4_7. Google Scholar

[9]

G. M. Coclite, J. Ridder and N. H. Risebro, A convergent finite difference scheme for the Ostrovsky-Hunter equation on a bounded domain, BIT, 57 (2017), 93–122, URL https://doi.org/10.1007/s10543-016-0625-x. doi: 10.1007/s10543-016-0625-x. Google Scholar

[10]

G. M. Coclite and L. di Ruvo, Convergence of the Ostrovsky equation to the Ostrovsky-Hunter one, J. Differential Equations, 256 (2014), 3245–3277, URL https://doi.org/10.1016/j.jde.2014.02.001. doi: 10.1016/j.jde.2014.02.001. Google Scholar

[11]

G. M. Coclite and L. di Ruvo, Dispersive and diffusive limits for Ostrovsky-Hunter type equations, NoDEA Nonlinear Differential Equations Appl., 22 (2015), 1733–1763, URL https://doi.org/10.1007/s00030-015-0342-1. doi: 10.1007/s00030-015-0342-1. Google Scholar

[12]

G. M. Coclite and L. di Ruvo, Oleinik type estimates for the Ostrovsky-Hunter equation, J. Math. Anal. Appl., 423 (2015), 162–190, URL https://doi.org/10.1016/j.jmaa.2014.09.033. doi: 10.1016/j.jmaa.2014.09.033. Google Scholar

[13]

G. M. Coclite and L. di Ruvo, Well-posedness of bounded solutions of the non-homogeneous initial-boundary value problem for the Ostrovsky-Hunter equation, J. Hyperbolic Differ. Equ., 12 (2015), 221–248, URL https://doi.org/10.1142/S021989161550006X. doi: 10.1142/S021989161550006X. Google Scholar

[14]

G. M. Coclite and L. di Ruvo, Well-posedness results for the short pulse equation, Z. Angew. Math. Phys., 66 (2015), 1529–1557, URL https://doi.org/10.1007/s00033-014-0478-6. doi: 10.1007/s00033-014-0478-6. Google Scholar

[15]

G. M. Coclite and L. di Ruvo, Wellposedness of bounded solutions of the non-homogeneous initial boundary for the short pulse equation, Boll. Unione Mat. Ital., 8 (2015), 31–44, URL https://doi.org/10.1007/s40574-015-0023-3. doi: 10.1007/s40574-015-0023-3. Google Scholar

[16]

G. M. Coclite and L. di Ruvo, Convergence of the solutions on the generalized Korteweg–de Vries equation, Math. Model. Anal., 21 (2016), 239–259, URL https://doi.org/10.3846/13926292.2016.1150358. doi: 10.3846/13926292.2016.1150358. Google Scholar

[17]

G. M. Coclite and L. di Ruvo, Well-posedness of the Ostrovsky-Hunter equation under the combined effects of dissipation and short-wave dispersion, J. Evol. Equ., 16 (2016), 365–389, URL https://doi.org/10.1007/s00028-015-0306-2. doi: 10.1007/s00028-015-0306-2. Google Scholar

[18]

G. M. Coclite and L. di Ruvo, Convergence of the regularized short pulse equation to the short pulse one, Math. Nachr., 291 (2018), 774–792, URL https://doi.org/10.1002/mana.201600301. doi: 10.1002/mana.201600301. Google Scholar

[19]

G. M. Coclite and L. di Ruvo, Well-posedness and dispersive/diffusive limit of a generalized Ostrovsky-Hunter equation, Milan J. Math., 86 (2018), 31–51, URL https://doi.org/10.1007/s00032-018-0278-0. doi: 10.1007/s00032-018-0278-0. Google Scholar

[20]

G. M. Coclite, L. di Ruvo and K. H. Karlsen, The initial-boundary-value problem for an Ostrovsky-Hunter type equation, in Non-linear Partial Differential Equations, Mathematical Physics, and Stochastic Analysis, EMS Ser. Congr. Rep., Eur. Math. Soc., Zürich, 2018, 97–109. Google Scholar

[21]

G. M. Coclite, H. Holden and K. H. Karlsen, Wellposedness for a parabolic-elliptic system, Discrete Contin. Dyn. Syst., 13 (2005), 659–682, URL https://doi.org/10.3934/dcds.2005.13.659. doi: 10.3934/dcds.2005.13.659. Google Scholar

[22]

J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Sharp global well-posedness for KdV and modified KdV on $\Bbb R$ and $\Bbb T$, J. Amer. Math. Soc., 16 (2003), 705–749, URL https://doi.org/10.1090/S0894-0347-03-00421-1. doi: 10.1090/S0894-0347-03-00421-1. Google Scholar

[23]

N. Costanzino, V. Manukian and C. K. R. T. Jones, Solitary waves of the regularized short pulse and Ostrovsky equations, SIAM J. Math. Anal., 41 (2009), 2088–2106, URL https://doi.org/10.1137/080734327. doi: 10.1137/080734327. Google Scholar

[24]

M. Davidson, Continuity properties of the solution map for the generalized reduced Ostrovsky equation, J. Differential Equations, 252 (2012), 3797–3815, URL https://doi.org/10.1016/j.jde.2011.11.013. doi: 10.1016/j.jde.2011.11.013. Google Scholar

[25]

L. di Ruvo, Discontinuous solutions for the Ostrovsky–Hunter equation and two phase flows, Phd Thesis, University of BariGoogle Scholar

[26]

C. E. Kenig, G. Ponce and L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Comm. Pure Appl. Math., 46 (1993), 527–620, URL https://doi.org/10.1002/cpa.3160460405. doi: 10.1002/cpa.3160460405. Google Scholar

[27]

S. A. Kozlov and S. V. Sazonov, Nonlinear propagation of optical pulses of a few oscillations duration in dielectric media, Journal of Experimental and Theoretical Physics, 84 (1997), 221–228, URL https://doi.org/10.1134/1.558109. doi: 10.1134/1.558109. Google Scholar

[28]

S. N. Kružkov, First order quasilinear equations with several independent variables, Mat. Sb. (N.S.), 81(123) (1970), 228-255. Google Scholar

[29]

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 (2009), 063835, URL https://link.aps.org/doi/10.1103/PhysRevA.79.063835. doi: 10.1103/PhysRevA.79.063835. Google Scholar

[30]

H. Leblond and D. Mihalache, Models of few optical cycle solitons beyond the slowly varying envelope approximation, Physics Reports, 523 (2013), 61–126, URL http://www.sciencedirect.com/science/article/pii/S0370157312003511, Models of few optical cycle solitons beyond the slowly varying envelope approximation. doi: 10.1016/j.physrep.2012.10.006. Google Scholar

[31]

H. Leblond and F. Sanchez, Models for optical solitons in the two-cycle regime, Phys. Rev. A, 67 (2003), 013804, URL https://link.aps.org/doi/10.1103/PhysRevA.67.013804. doi: 10.1103/PhysRevA.67.013804. Google Scholar

[32]

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 (2006), 063815, URL https://link.aps.org/doi/10.1103/PhysRevA.74.063815. doi: 10.1103/PhysRevA.74.063815. Google Scholar

[33]

P. G. LeFloch and R. Natalini, Conservation laws with vanishing nonlinear diffusion and dispersion, Nonlinear Anal., 36 (1999), 213–230, URL https://doi.org/10.1016/S0362-546X(98)00012-1. doi: 10.1016/S0362-546X(98)00012-1. Google Scholar

[34]

Y. Liu, D. Pelinovsky and A. Sakovich, Wave breaking in the short-pulse equation, Dyn. Partial Differ. Equ., 6 (2009), 291–310, URL https://doi.org/10.4310/DPDE.2009.v6.n4.a1. doi: 10.4310/DPDE.2009.v6.n4.a1. Google Scholar

[35]

F. Murat, L'injection du cône positif de H-1 dans W-1, q est compacte pour tout q < 2, J. Math. Pures Appl. (9), 60 (1981), 309–322. Google Scholar

[36]

A. Nazarkin, Nonlinear optics of intense attosecond light pulses, Phys. Rev. Lett., 97 (2006), 163904, URL https://link.aps.org/doi/10.1103/PhysRevLett.97.163904. doi: 10.1103/PhysRevLett.97.163904. Google Scholar

[37]

D. Pelinovsky and A. Sakovich, Global well-posedness of the short-pulse and sine-gordon equations in energy space, Communications in Partial Differential Equations, 35 (2010), 613–629, URL https://doi.org/10.1080/03605300903509104. doi: 10.1080/03605300903509104. Google Scholar

[38]

D. Pelinovsky and G. Schneider, Rigorous justification of the short-pulse equation, Nonlinear Differential Equations and Applications NoDEA, 20 (2013), 1277–1294, URL https://doi.org/10.1007/s00030-012-0208-8. doi: 10.1007/s00030-012-0208-8. Google Scholar

[39]

N. N. Rosanov, V. E. Semenov and N. V. Vysotina, Few-cycle dissipative solitons in active nonlinear optical fibres, Quantum Electronics, 38 (2008), 137, URL http://stacks.iop.org/1063-7818/38/i=2/a=A08. doi: 10.1070/QE2008v038n02ABEH013568. Google Scholar

[40]

N. N. Rosanov, V. E. Semenov and N. V. Vyssotina, Collisions of few-cycle dissipative solitons in active nonlinear fibers, Laser Physics, 17 (2007), 1311, URL https://doi.org/10.1134/S1054660X07110072. doi: 10.1134/S1054660X07110072. Google Scholar

[41]

T. Schäfer and C. Wayne, Propagation of ultra-short optical pulses in cubic nonlinear media, Physica D: Nonlinear Phenomena, 196 (2004), 90–105, URL http://www.sciencedirect.com/science/article/pii/S0167278904002064. doi: 10.1016/j.physd.2004.04.007. Google Scholar

[42]

M. E. Schonbek, Convergence of solutions to nonlinear dispersive equations, Comm. Partial Differential Equations, 7 (1982), 959–1000, URL https://doi.org/10.1080/03605308208820242. doi: 10.1080/03605308208820242. Google Scholar

[43]

A. Scrinzi, M. Y. Ivanov, R. Kienberger and D. M. Villeneuve, Attosecond physics, Journal of Physics B: Atomic, Molecular and Optical Physics, 39 (2006), R1, URL http://stacks.iop.org/0953-4075/39/i=1/a=R01.Google Scholar

[44]

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 (2007), 203902, URL https://link.aps.org/doi/10.1103/PhysRevLett.99.203902. doi: 10.1103/PhysRevLett.99.203902. Google Scholar

[45]

A. Stefanov, Y. Shen and P. G. Kevrekidis, Well-posedness and small data scattering for the generalized Ostrovsky equation, J. Differential Equations, 249 (2010), 2600–2617, URL https://doi.org/10.1016/j.jde.2010.05.015. doi: 10.1016/j.jde.2010.05.015. Google Scholar

[46]

X. Tan, X. Fan, Y. Yang and D. Tong, Time evolution of few-cycle pulse in a dense v-type three-level medium, Journal of Modern Optics, 55 (2008), 2439–2448, URL https://doi.org/10.1080/09500340802130670. doi: 10.1080/09500340802130670. Google Scholar

[47]

T. Tao, Nonlinear Dispersive Equations, vol. 106 of CBMS Regional Conference Series in Mathematics, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2006, URL https://doi.org/10.1090/cbms/106, Local and global analysis. doi: 10.1090/cbms/106. Google Scholar

[48]

L. Tartar, Compensated compactness and applications to partial differential equations, in Nonlinear Analysis and Mechanics: Heriot-Watt Symposium, Vol. IV, vol. 39 of Res. Notes in Math., Pitman, Boston, Mass.-London, 1979,136–212. Google Scholar

[49]

M. V. Tognetti and H. M. Crespo, Sub-two-cycle soliton-effect pulse compression at 800 nm in photonic crystal fibers, J. Opt. Soc. Am. B, 24 (2007), 1410–1415, URL http://josab.osa.org/abstract.cfm?URI=josab-24-6-1410. doi: 10.1364/JOSAB.24.001410. Google Scholar

[50]

N. Tsitsas, T. Horikis, Y. Shen, P. Kevrekidis, N. Whitaker and D. Frantzeskakis, Short pulse equations and localized structures in frequency band gaps of nonlinear metamaterials, Physics Letters A, 374 (2010), 1384–1388, URL http://www.sciencedirect.com/science/article/pii/S0375960110000150. doi: 10.1016/j.physleta.2010.01.004. Google Scholar

[51]

A. A. Voronin and A. M. Zheltikov, Soliton-number analysis of soliton-effect pulse compression to single-cycle pulse widths, Phys. Rev. A, 78 (2008), 063834, URL https://link.aps.org/doi/10.1103/PhysRevA.78.063834. doi: 10.1103/PhysRevA.78.063834. Google Scholar

[52]

M. Wegener, Extreme Nonlinear Optics, Advanced Texts in Physics, Springer-Verlag, Berlin, 2005, An introduction. Google Scholar

[1]

Rudong Zheng, Zhaoyang Yin. The Cauchy problem for a generalized Novikov equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 3503-3519. doi: 10.3934/dcds.2017149

[2]

Evgeny Yu. Panov. On a condition of strong precompactness and the decay of periodic entropy solutions to scalar conservation laws. Networks & Heterogeneous Media, 2016, 11 (2) : 349-367. doi: 10.3934/nhm.2016.11.349

[3]

Young-Sam Kwon. On the well-posedness of entropy solutions for conservation laws with source terms. Discrete & Continuous Dynamical Systems - A, 2009, 25 (3) : 933-949. doi: 10.3934/dcds.2009.25.933

[4]

. Adimurthi, Siddhartha Mishra, G.D. Veerappa Gowda. Existence and stability of entropy solutions for a conservation law with discontinuous non-convex fluxes. Networks & Heterogeneous Media, 2007, 2 (1) : 127-157. doi: 10.3934/nhm.2007.2.127

[5]

Stefano Bianchini, Elio Marconi. On the concentration of entropy for scalar conservation laws. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 73-88. doi: 10.3934/dcdss.2016.9.73

[6]

Zhi-Qiang Shao. Lifespan of classical discontinuous solutions to the generalized nonlinear initial-boundary Riemann problem for hyperbolic conservation laws with small BV data: shocks and contact discontinuities. Communications on Pure & Applied Analysis, 2015, 14 (3) : 759-792. doi: 10.3934/cpaa.2015.14.759

[7]

Hongbin Chen, Yi Li. Existence, uniqueness, and stability of periodic solutions of an equation of duffing type. Discrete & Continuous Dynamical Systems - A, 2007, 18 (4) : 793-807. doi: 10.3934/dcds.2007.18.793

[8]

Shaoyong Lai, Yong Hong Wu. The asymptotic solution of the Cauchy problem for a generalized Boussinesq equation. Discrete & Continuous Dynamical Systems - B, 2003, 3 (3) : 401-408. doi: 10.3934/dcdsb.2003.3.401

[9]

Defu Chen, Yongsheng Li, Wei Yan. On the Cauchy problem for a generalized Camassa-Holm equation. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 871-889. doi: 10.3934/dcds.2015.35.871

[10]

Hernan R. Henriquez. Generalized solutions for the abstract singular Cauchy problem. Communications on Pure & Applied Analysis, 2009, 8 (3) : 955-976. doi: 10.3934/cpaa.2009.8.955

[11]

Alberto Bressan, Marta Lewicka. A uniqueness condition for hyperbolic systems of conservation laws. Discrete & Continuous Dynamical Systems - A, 2000, 6 (3) : 673-682. doi: 10.3934/dcds.2000.6.673

[12]

Yanbo Hu, Wancheng Sheng. The Riemann problem of conservation laws in magnetogasdynamics. Communications on Pure & Applied Analysis, 2013, 12 (2) : 755-769. doi: 10.3934/cpaa.2013.12.755

[13]

Zijuan Wen, Meng Fan, Asim M. Asiri, Ebraheem O. Alzahrani, Mohamed M. El-Dessoky, Yang Kuang. Global existence and uniqueness of classical solutions for a generalized quasilinear parabolic equation with application to a glioblastoma growth model. Mathematical Biosciences & Engineering, 2017, 14 (2) : 407-420. doi: 10.3934/mbe.2017025

[14]

Tracy L. Stepien, Hal L. Smith. Existence and uniqueness of similarity solutions of a generalized heat equation arising in a model of cell migration. Discrete & Continuous Dynamical Systems - A, 2015, 35 (7) : 3203-3216. doi: 10.3934/dcds.2015.35.3203

[15]

Lorena Bociu, Petronela Radu. Existence of weak solutions to the Cauchy problem of a semilinear wave equation with supercritical interior source and damping. Conference Publications, 2009, 2009 (Special) : 60-71. doi: 10.3934/proc.2009.2009.60

[16]

Xin Lai, Xinfu Chen, Mingxin Wang, Cong Qin, Yajing Zhang. Existence, uniqueness, and stability of bubble solutions of a chemotaxis model. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 805-832. doi: 10.3934/dcds.2016.36.805

[17]

Zhaohui Huo, Boling Guo. The well-posedness of Cauchy problem for the generalized nonlinear dispersive equation. Discrete & Continuous Dynamical Systems - A, 2005, 12 (3) : 387-402. doi: 10.3934/dcds.2005.12.387

[18]

Hiroshi Watanabe. Existence and uniqueness of entropy solutions to strongly degenerate parabolic equations with discontinuous coefficients. Conference Publications, 2013, 2013 (special) : 781-790. doi: 10.3934/proc.2013.2013.781

[19]

Kenneth Hvistendahl Karlsen, Nils Henrik Risebro. On the uniqueness and stability of entropy solutions of nonlinear degenerate parabolic equations with rough coefficients. Discrete & Continuous Dynamical Systems - A, 2003, 9 (5) : 1081-1104. doi: 10.3934/dcds.2003.9.1081

[20]

Georges Bastin, B. Haut, Jean-Michel Coron, Brigitte d'Andréa-Novel. Lyapunov stability analysis of networks of scalar conservation laws. Networks & Heterogeneous Media, 2007, 2 (4) : 751-759. doi: 10.3934/nhm.2007.2.751

2018 Impact Factor: 1.048

Metrics

  • PDF downloads (17)
  • HTML views (79)
  • Cited by (0)

Other articles
by authors

[Back to Top]