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Large time asymptotics of solutions to the short-pulse equation

  • Mamoru OkamotoEmail author
Article

Abstract

We consider the long-time behavior of solutions to the short-pulse equation. Using the method of testing by wave packets, we prove small data global existence and modified scattering.

Keywords

Reduced Ostrovsky equation Short-pulse equation Modified scattering 

Mathematics Subject Classification

Primary 35Q53 Secondary 35B40 

References

  1. 1.
    Boyd, J.P.: Ostrovsky and Hunter’s generic wave equation for weakly dispersive waves: matched asymptotic and pseudospectral study of the paraboloidal travelling waves (corner and near-corner waves). Eur. J. Appl. Math. 16(1), 65–81 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Harrop-Griffiths, B.: Long time behavior of solutions to the mKdV. Commun. Partial Differ. Equ. 41(2), 282–317 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Harrop-Griffiths, B., Ifrim, M., Tataru, D.: The lifespan of small data solutions to the KP-I. Int. Math. Res. Not. (2016). doi: 10.1093/imrn/rnw017 Google Scholar
  4. 4.
    Hayashi, N., Naumkin, P.I.: On the generalized reduced Ostrovsky equation. SUT J. Math. 50(2), 67–101 (2014)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Hayashi, N., Naumkin, P.I.: Large time asymptotics for the reduced Ostrovsky equation. Commun. Math. Phys. 335(2), 713–738 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Hayashi, N., Naumkin, P.I., Niizato, T.: Asymptotics of solutions to the generalized Ostrovsky equation. J. Differ. Equ. 255(8), 2505–2520 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Hayashi, N., Naumkin, P.I., Niizato, T.: Nonexistence of the usual scattering states for the generalized Ostrovsky–Hunter equation. J. Math. Phys. 55(5), 11 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Hunter, J.K.: Numerical solutions of some nonlinear dispersive wave equations. In: Allgower, E. (ed.) Computational Solution of Nonlinear Systems of Equations. Amer. Math. Soc. vol. 26, pp. 301–316 (1900)Google Scholar
  9. 9.
    Ifrim, M., Tataru, D.: Global bounds for the cubic nonlinear Schrödinger equation (NLS) in one space dimension. Nonlinearity 28(8), 2661–2675 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Ifrim, M., Tataru, D.: Two dimensional water waves in holomorphic coordinates II: global solutions. Bull. Soc. Math. Fr. 144(2), 369–394 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Klainerman, S.: Global existence of small amplitude solutions to nonlinear Klein–Gordon equations in four space-time dimensions. Commun. Pure Appl. Math. 38(5), 631–641 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Liu, Y., Pelinovsky, D., Sakovich, A.: Wave breaking in the short-pulse equation. Dyn. Partial Differ. Equ. 6(4), 291–310 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Liu, Y., Pelinovsky, D., Sakovich, A.: Wave breaking in the Ostrovsky–Hunter equation. SIAM J. Math. Anal. 42(5), 1967–1985 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Niizato, T.: Asymptotic behavior of solutions to the short pulse equation with critical nonlinearity. Nonlinear Anal. 111, 15–32 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Pelinovsky, D., Sakovich, A.: Global well-posedness of the short-pulse and sine-Gordon equations in energy space. Commun. Partial Differ. Equ. 35(4), 613–629 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Schäfer, T., Wayne, C.E.: Propagation of ultra-short optical pulses in cubic nonlinear media. Phys. D 196(1–2), 90–105 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Stefanov, A., Shen, Y., Kevrekidis, P.G.: Well-posedness and small data scattering for the generalized Ostrovsky equation. J. Differ. Equ. 249(10), 2600–2617 (2010)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Division of Mathematics and Physics Faculty of EngineeringShinshu UniversityNagano CityJapan

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