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Zeitschrift für angewandte Mathematik und Physik

, Volume 66, Issue 5, pp 2343–2377 | Cite as

Factorization technique for the fourth-order nonlinear Schrödinger equation

  • Nakao HayashiEmail author
  • Pavel I. Naumkin
Article

Abstract

We consider the Cauchy problem for the cubic fourth-order nonlinear Schrödinger equation
$$\left\{\begin{array}{ll}i\partial_{t}u+\frac{1}{4}\partial_{x}^{4}u = i \lambda \partial _{x}(\left| u \right| ^{2}u),&\quad t > 0,\, x \in \mathbf{R},\\ u \left( 0,x\right) = u_{0}\left( x\right) ,&\quad x \in \mathbf{R},\end{array}\right.$$
where \({\lambda \in \mathbf{R}.}\) We introduce the factorization formula for the free evolution group to prove the global existence of solutions. Also we show that the large time asymptotics of solutions has a logarithmic correction in the phase comparing with the corresponding linear case.

Keywords

Fourth-order nonlinear Schrödinger equation Factorization formula Large time asymptotics of solutions Logarithmic phase correction Critical order of nonlinearity 

Mathematics Subject Classification

35Q55 35Q35 35Q51 

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References

  1. 1.
    Ben-Artzi M., Koch H., Saut J.C.: Dispersion estimates for fourth order Schrödinger equations. C. R. Math. Acad. Sci. 330, 87–92 (2000)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Dysthe K.B.: Note on a modification to the nonlinear Schrödinger equation for application to deep water waves. Proc. R. Soc. Lond. Ser. A 369, 105–114 (1979)CrossRefzbMATHGoogle Scholar
  3. 3.
    Fukumoto, Y.: Motion of a curved vortex filament: higher-order asymptotics. In: Proceedings of IUTAM Symposium on Geometry and Statistics of Turbulence, pp. 211–216 (2001)Google Scholar
  4. 4.
    Fedoryuk, M.V.: Asymptotics: Integrals and Series. Mathematical Reference Library, “Nauka”, Moscow (1987)Google Scholar
  5. 5.
    Guo C., Cui S.: Global existence of solutions for a fourth-order nonlinear Schrödinger equation. Appl. Math. Lett. 19, 706–711 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Hao C., Hsian L., Wang B.: Wellposedness for the fourth order nonlinear Schrödinger equations. J. Math. Anal. Appl. 320, 246–265 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Hayashi N.: Global existence of small solutions to quadratic nonlinear Schrödinger equations. Commun. Partial Differ. Equ. 18, 1109–1124 (1993)CrossRefzbMATHGoogle Scholar
  8. 8.
    Hayashi N., Naumkin P.I.: The initial value problem for the cubic nonlinear Klein–Gordon equation. Z. Angew. Math. Phys. 59(6), 1002–1028 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Hayashi N., Naumkin P.I.: Large time asymptotics for the fourth-order nonlinear Schrödinger equation. J. Differ. Equ. 258, 880–905 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Hayashi N., Ozawa T.: Scattering theory in the weighted \({\mathbf{L}^{2}({\mathbf{R}}^{n})}\) spaces for some Schr ödinger equations. Ann. I.H.P. (Phys. Théor.) 48, 17–37 (1988)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Huo Z., Jia Y.: The Cauchy problem for the fourth-order nonlinear Schrödinger equation related to the vortex filament. J. Differ. Equ. 214, 1–35 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Karpman V.I.: Stabilization of soliton instabilities by higher-order dispersion: fourth order nonlinear Schrö dinger-type equations. Phys. Rev. E 53(2), 1336–1339 (1996)CrossRefGoogle Scholar
  13. 13.
    Karpman V.I., Shagalov A.G.: Stability of soliton described by nonlinear Schrödinger-type equations with higher-order dispersion. Phys. D. 144, 194–210 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Kenig C.E., Ponce G., Vega L.: Oscillatory integrals and regularity of dispersive equations. Indiana Univ. Math. J. 40, 33–69 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Kenig C.E., Ponce G., Vega L.: Well-posedness and scattering results for the generalised Korteweg-de Vries equation via the contraction principle. Commun. Pure Appl. Math. 46, 527–620 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Pausader B.: Global well-posedness for energy critical fourth-order Schrödinger equations in the radial case. Dyn. Partial Differ. Equ. 4, 197–225 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Pausader B., Shao S.: The mass-critical fourth-order Schrödinger equation in high dimensions. J. Hyperbolic Differ. Equ. 7, 651–705 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Pausader B., Xia S.: Scattering theory for the fourth-order Schrödinger equation in low dimensions. Nonlinearity 26(8), 2175–2191 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Segata J.: Well-posedness for the fourth-order nonlinear Schrödinger type equation related to the vortex filament. Differ. Integral Equ. 16(7), 841–864 (2003)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Segata J.: Modified wave operators for the fourth-order non-linear Schrödinger-type equation with cubic non-linearity. Math. Methods Appl. Sci. 26(15), 1785–1800 (2006)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Stein E.M., Shakarchi R.: Functional Analysis. Introduction to Further Topics in Analysis. Princeton Lectures in Analysis, 4. Princeton University Press, Princeton, NJ (2011)Google Scholar
  22. 22.
    Wang Y.: Global well-posedness for the generalised fourth-order Schrödinger equation. Bull. Aust. Math. Soc. 85(3), 371–379 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Wang Y.: Nonlinear fourth-order Schrödinger equations with radial data. Nonlinear Anal. 75(4), 2534–2541 (2012)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Basel 2015

Authors and Affiliations

  1. 1.Department of Mathematics, Graduate School of ScienceOsaka UniversityOsakaJapan
  2. 2.Centro de Ciencias MatemáticasUNAM Campus MoreliaMoreliaMexico

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