Research article

Discontinuous solutions for the short-pulse master mode-locking equation

  • Received: 09 January 2019 Accepted: 18 April 2019 Published: 08 May 2019
  • MSC : 35G15, 35L65, 35L05, 35A05

  • The short-pulse master mode-locking equation is a model for ultrafast pulse propagation in a mode-locked laser cavity in the few-femtosecond pulse regime, that 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 short-pulse master mode-locking equation[J]. AIMS Mathematics, 2019, 4(3): 437-462. doi: 10.3934/math.2019.3.437

    Related Papers:

  • The short-pulse master mode-locking equation is a model for ultrafast pulse propagation in a mode-locked laser cavity in the few-femtosecond pulse regime, that 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.
    加载中


    [1] S. Amiranashvili, A. G. Vladimirov and U. Bandelow, A model equation for ultrashort optical pulses around the zero dispersion frequency, Eur. Phys. J. D, 58 (2010), 219-226. doi: 10.1140/epjd/e2010-00010-3
    [2] S. Amiranashvili, A. G. Vladimirov and U. Bandelow, Solitary-wave solutions for few-cycle optical pulses, Phys. Rev. A, 77 (2008), 063821.
    [3] R. Beals, M. Rabelo and K. Tenenblat, Bäcklund transformations and inverse scattering solutions for some pseudospherical surface equations, Stud. Appl. Math., 81 (1989), 125-151. doi: 10.1002/sapm1989812125
    [4] N. R. Belashenkov, A. A. Drozdov, S. A. Kozlov, et al. Phase modulation of femtosecond light pulses whose spectra are superbroadened in dielectrics with normal group dispersion, J. Opt. Technol., 75 (2008), 611-614. doi: 10.1364/JOT.75.000611
    [5] Y. Chung, C. K. R. T. Jones, T. Schäfer, et al. Ultra-short pulses in linear and nonlinear media, Nonlinearity, 18 (2005), 1351-1374. doi: 10.1088/0951-7715/18/3/021
    [6] G. M. Coclite and L. di Ruvo, Discontinuous solutions for the generalized short pulse equation, Evol. Equ. Control Theory, 2019, in press.
    [7] G. M. Coclite and L. di Ruvo, A note on the non-homogeneous initial boundary problem for an Ostrovsky-Hunter type equation, 2019, in press.
    [8] G. M. Coclite and L. di Ruvo, Convergence of the Ostrovsky equation to the Ostrovsky-Hunter one, J. Differ Equations, 256 (2014), 3245-3277. doi: 10.1016/j.jde.2014.02.001
    [9] G. M. Coclite and L. di Ruvo, Dispersive and diffusive limits for Ostrovsky-Hunter type equations, NoDEA Nonlinear Differ., 22 (2015), 1733-1763. doi: 10.1007/s00030-015-0342-1
    [10] G. M. Coclite and L. di Ruvo, Oleinik type estimates for the Ostrovsky-Hunter equation, J. Math. Anal. Appl., 423 (2015), 162-190. doi: 10.1016/j.jmaa.2014.09.033
    [11] G. M. Coclite and L. di Ruvo, Well-posedness results for the short pulse equation, Z. Angew. Math. Phys., 66 (2015), 1529-1557. doi: 10.1007/s00033-014-0478-6
    [12] 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. doi: 10.1007/s40574-015-0023-3
    [13] G. M. Coclite and L. di Ruvo, On the well-posedness of the exp-Rabelo equation, Ann. Mat. Pura Appl., 195 (2016), 923-933. doi: 10.1007/s10231-015-0497-8
    [14] 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. doi: 10.1007/s00028-015-0306-2
    [15] 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. doi: 10.1002/mana.201600301
    [16] 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. doi: 10.1007/s00032-018-0278-0
    [17] G. M. Coclite, L. di Ruvo and K. H. Karlsen, Some wellposedness results for the Ostrovsky-Hunter equation, In: Chen GQ., Holden H., Karlsen K. Editors, Hyperbolic Conservation Laws and Related Analysis with Applications, Heidelberg: Springer, 49 (2014), 143-159. doi: 10.1007/978-3-642-39007-4_7
    [18] 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.
    [19] G. M. Coclite, H. Holden and K. H. Karlsen, Global weak solutions to a generalized hyperelastic-rod wave equation, SIAM J. Math. Anal., 37 (2005), 1044-1069. doi: 10.1137/040616711
    [20] G. M. Coclite, H. Holden and K. H. Karlsen, Wellposedness for a parabolic-elliptic system, Discrete Contin. Dyn. Syst., 13 (2005), 659-682. doi: 10.3934/dcds.2005.13.659
    [21] G. M. Coclite, J. Ridder and N. H. Risebro, A convergent finite difference scheme for the Ostrovsky-Hunter equation on a bounded domain, BIT Numer. Math., 57 (2017), 93-122. doi: 10.1007/s10543-016-0625-x
    [22] 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. doi: 10.1137/080734327
    [23] S. T. Cundiff, Femtosecond comb technology, J. Korean Phys. Soc., 48 (2006), 1181-1187.
    [24] S. T. Cundiff, Better by half, Nat. Phys., 3 (2007), 16 pages.
    [25] S. T. Cundiff, Rulers of light, Nat. Photonics, 13 (2019), 137-137. doi: 10.1038/s41566-019-0388-4
    [26] M. Davidson, Continuity properties of the solution map for the generalized reduced Ostrovsky equation, J. Differ. Equations, 252 (2012), 3797-3815. doi: 10.1016/j.jde.2011.11.013
    [27] L. di Ruvo, Discontinuous solutions for the Ostrovsky-Hunter equation and two phase flows, PhD Thesis, University of Bari, 2013.
    [28] J. M. Dudley, G. Genty and S. Coen, Supercontinuum generation in photonic crystal fiber, Rev. Mod. Phys., 78 (2006), 1135-1184. doi: 10.1103/RevModPhys.78.1135
    [29] R. Ell, G. Angelow, W. Seitz, et al. Quasi-synchronous pumping of modelocked few-cycle Titanium Sapphire lasers, Opt. Express, 13 (2005), 9292-9298. doi: 10.1364/OPEX.13.009292
    [30] E. D. Farnum and J. N. Kutz, Master mode-locking theory for few-femtosecond pulses, Opt. Lett., 35 (2010), 3033-3035. doi: 10.1364/OL.35.003033
    [31] E. D. Farnum and J. N. Kutz, Short-pulse perturbation theory, J. Opt. Soc. Am. B, 30 (2013), 2191-2198. doi: 10.1364/JOSAB.30.002191
    [32] E. D. Farnum and J. N. Kutz, Dynamics of a low-dimensional model for short pulse mode locking, Photonics, 2 (2015), 865-882. doi: 10.3390/photonics2030865
    [33] H. A. Haus, Mode-locking of lasers, IEEE J. Sel. Top. Quantum Electron., 6 (2000), 1173-1185. doi: 10.1109/2944.902165
    [34] H. A. Haus, J. G. Fujimoto and E. P. Ippen, Structures for additive pulse mode locking, J. Opt. Soc. Am. B, 8 (1991), 2068-2076. doi: 10.1364/JOSAB.8.002068
    [35] M. Hentschel, R. Kienberger, C. Spielmann, et al. Attosecond metrology, Nature, 414 (2001), 509-513. doi: 10.1038/35107000
    [36] F. X. Kärtner, U. Morgner, R. Ell, et al. Ultrabroadband double-chirped mirror pairs for generation of octave spectra, J. Opt. Soc. Am. B, 18 (2001), 882-885. doi: 10.1364/JOSAB.18.000882
    [37] U. Keller, Ultrafast solid-state lasers, In: Progress in Optics, Elsevier, 46 (2004), 1-115. doi: 10.1016/S0079-6638(03)46001-0
    [38] S. A. Kozlov and S. V. Sazonov, Nonlinear propagation of optical pulses of a few oscillations duration in dielectric media, J. Exp. Theor. Phys., 84 (1997), 221-228. doi: 10.1134/1.558109
    [39] F. Krausz and M. Ivanov, Attosecond physics, Rev. Mod. Phys., 81 (2009), 163-234. doi: 10.1103/RevModPhys.81.163
    [40] S. N. Kružkov, First order quasilinear equations with several independent variables, Mat. Sb. (N.S.), 81 (1970), 228-255.
    [41] C. Lattanzio and P. Marcati, Global well-posedness and relaxation limits of a model for radiating gas, J. Differ. Equations, 190 (2003), 439-465. doi: 10.1016/S0022-0396(02)00158-4
    [42] 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.
    [43] H. Leblond and D. Mihalache, Models of few optical cycle solitons beyond the slowly varying envelope approximation, Phys. Rep., 523 (2013), 61-126. doi: 10.1016/j.physrep.2012.10.006
    [44] H. Leblond and F. Sanchez, Models for optical solitons in the two-cycle regime, Phys. Rev. A, 67 (2003), 013804-0138048. doi: 10.1103/PhysRevA.67.013804
    [45] P. G. LeFloch and R. Natalini, Conservation laws with vanishing nonlinear diffusion and dispersion, Nonlinear Anal., 36 (1999), 213-230. doi: 10.1016/S0362-546X(98)00012-1
    [46] Y. Liu, D. Pelinovsky and A. Sakovich, Wave breaking in the short-pulse equation, Dynam. Part. Differ. Eq., 6 (2009), 291-310. doi: 10.4310/DPDE.2009.v6.n4.a1
    [47] F. Murat, L'injection du cȏne positif de $H^{-1}$ dans $W^{-1,\,q}$ est compacte pour tout $q<2$, J. Math. Pures Appl., 60 (1981), 309-322.
    [48] S. P. Nikitenkova, Y. A. Stepanyants and L. M. Chikhladze, Solutions of a modified Ostrovskiĭ equation with a cubic nonlinearity, J. Appl. Math. Mech., 64 (2000), 267-274. doi: 10.1016/S0021-8928(00)00048-4
    [49] D. Pelinovsky and A. Sakovich, Global well-posedness of the short-pulse and sine-Gordon equations in energy space, Commun. Partial Differ. Equations, 35 (2010), 613-629. doi: 10.1080/03605300903509104
    [50] D. Pelinovsky and G. Schneider, Rigorous justification of the short-pulse equation, NoDEA Nonlinear Differ., 20 (2013), 1277-1294. doi: 10.1007/s00030-012-0208-8
    [51] M. Pietrzyk, I. Kanattšikov and U. Bandelow, On the propagation of vector ultra-short pulses, J. Nonlinear Math. Phys., 15 (2008), 162-170. doi: 10.2991/jnmp.2008.15.2.4
    [52] M. L. Rabelo, On equations which describe pseudospherical surfaces, Stud. Appl. Math., 81 (1989), 221-248. doi: 10.1002/sapm1989813221
    [53] A. Sakovich and S. Sakovich, On transformations of the Rabelo equations, SIGMA, 3 (2007), Article ID: 086, 1-8.
    [54] K. J. Schafer, M. B. Gaarde, A. Heinrich, et al. Strong field quantum path control using attosecond pulse trains, Phys. Rev. Lett., 92 (2004), 023003.
    [55] T. Schäfer and C. E. Wayne, Propagation of ultra-short optical pulses in cubic nonlinear media, Phys. D, 196 (2004), 90-105. doi: 10.1016/j.physd.2004.04.007
    [56] M. E. Schonbek, Convergence of solutions to nonlinear dispersive equations, Commun. Partial Differ. Equations, 7 (1982), 959-1000. doi: 10.1080/03605308208820242
    [57] D. Serre, $L^1$-stability of constants in a model for radiating gases, Commun. Math. Sci., 1 (2003), 197-205. doi: 10.4310/CMS.2003.v1.n1.a12
    [58] Y. Silberberg, Physics at the attosecond frontier, Nature, 414 (2001), 494-495. doi: 10.1038/35107171
    [59] 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.
    [60] A. Stefanov, Y. Shen and P. G. Kevrekidis, Well-posedness and small data scattering for the generalized Ostrovsky equation, J. Differ. Equations, 249 (2010), 2600-2617. doi: 10.1016/j.jde.2010.05.015
    [61] L. Tartar, Compensated compactness and applications to partial differential equations, In: Nonlinear Analysis and Mechanics: Heriot-Watt Symposium, Vol. IV, Boston: Pitman, Mass.-London, 39 (1979), 136-212.
    [62] N. Tsitsas, T. Horikis, Y. Shen, et al. Short pulse equations and localized structures in frequency band gaps of nonlinear metamaterials, Phys. Lett. A, 374 (2010), 1384-1388. doi: 10.1016/j.physleta.2010.01.004
    [63] K. K. Victor, B. B. Thomas and T. C. Kofane, On the conversion of high-frequency soliton solutions to a (1+1)-dimensional nonlinear partial differential evolution equation, Chinese Phys. Lett., 25 (2008), 1972-1975. doi: 10.1088/0256-307X/25/6/014

    © 2019 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
  • Reader Comments
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(580) PDF downloads(578) Cited by(3)

Article outline

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog