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:
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show all references

References:
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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

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[5]

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[8]

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[10]

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[12]

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[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]

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[15]

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[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

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[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]

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

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[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

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

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