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Journal d'Analyse Mathématique

, Volume 135, Issue 2, pp 473–486 | Cite as

Orbital stability of solitary waves for derivative nonlinear Schrödinger equation

  • Soonsik Kwon
  • Yifei Wu
Article
  • 72 Downloads

Abstract

In this paper, we show the orbital stability of solitons arising in the cubic derivative nonlinear Schrödinger equations. We consider the zero mass case that is not covered by earlier works. As this case enjoys L2 scaling invariance, we expect orbital stability (up to scaling symmetry) in addition to spatial and phase translations. We also show a self-similar type blow up criterion of solutions with the critical mass 4π.

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References

  1. [1]
    M. Agueh, Sharp Gagliardo-Nirenberg inequalities and mass transport theory, J. Dyn. Differ. Equ. 18 (2006), 1069–1093.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    H. Berestycki and P. L. Lions, Nonlinear scalar field equations, I, Existence of a ground state, Arch. Ration. Mech. Anal. 82 (1983), 313–345.CrossRefzbMATHGoogle Scholar
  3. [3]
    M. Colin and M. Ohta, Stability of solitary waves for derivative nonlinear Schrödinger equation, Ann. Inst. H. Poincaré Anal. Non Linéaire 23 (2006), 753–764.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao, Global well-posedness result for Schrödinger equations with derivative, SIAM J. Math. Anal. 33 (2001), 649–669.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao, A refined global well-posedness result for Schrödinger equations with derivatives, SIAM J. Math. Anal. 34 (2001), 64–86.CrossRefzbMATHGoogle Scholar
  6. [6]
    P. Gérard, Description of the lack of compactness of a Sobolev embedding, ESAIM Control Optim. Calc. Var. 3 (1998), 213–233.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    A. Grünrock and S. Herr, Low regularity local well-posedness of the derivative nonlinear Schrödinger equation with periodic initial data, SIAM J. Math. Anal. 39 (2008), 1890–920.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    B. Guo and Y. Wu, Orbital stability of solitary waves for the nonlinear derivative Schrödinger equation, J. Differential Equations 123 (1995), 35–55.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    N. Hayashi, The initial value problem for the derivative nonlinear Schrödinger equation in the energy space, Nonlinear Anal. 20 (1983), 823–833.CrossRefzbMATHGoogle Scholar
  10. [10]
    N. Hayashi and T. Ozawa, On the derivative nonlinear Schrödinger equation, Phys. D 55 (1992), 14–36.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    N. Hayashi and T. Ozawa, Finite energy solution of nonlinear Schrödinger equations of derivative type, SIAM J. Math. Anal. 25 (1994), 1488–1503.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    S. Herr, On the Cauchy problem for the derivative nonlinear Schrödinger equation with periodic boundary condition, Int. Math. Res. Not. IMRN (2006), Art. ID 96763.Google Scholar
  13. [13]
    M. Klaus, D. E. Pelinovsky, and V. M. Rothos, Evans function for Lax operators with algebraically decaying potentials, J. Nonlinear Sci. 16 (2006), 1–44.MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    J. Liu, P. A. Perry, and C. Sulem, Global existence for the derivative nonlinear Schrödinger equation by the method of inverse scattering, Comm. Partial Differential Equations 41 (2016), 1692–1760.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    C. Miao, Y. Wu, and G. Xu, Global well-posedness for Schrödinger equation with derivative in H 12 (R), J. Differential Equations 251 (2011), 2164–2195.MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    A. R. Nahmod, T. Oh, L. Rey-Bellet, and G. Staffilani, Invariant weighted Wiener measures and almost sure global well-posedness for the periodic derivative NLS, J. Eur. Math. Soc. (JEMS) 14 (2012), 1275–1330.MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    M. Ohta, Instability of solitary waves for nonlinear Schrödinger equations of derivative type, SUT J. Math. 50 (2014), 399–415.MathSciNetzbMATHGoogle Scholar
  18. [18]
    T. Ozawa, On the nonlinear Schrödinger equations of derivative type, Indiana Univ. Math. J. 45 (1996), 137–163.MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    D. E. Pelinovsky and Y. Shimabukuro, Existence of global solutions to the derivative NLS equation with the inverse scattering transform method, Int. Math. Res. Not. IMRN (2017) rnx051.Google Scholar
  20. [20]
    H. Takaoka, Well-posedness for the one-dimensional Schrödinger equation with the derivative nonlinearity, Adv. Differential Equations 4 (1999), 561–680.MathSciNetzbMATHGoogle Scholar
  21. [21]
    H. Takaoka, Global well-posedness for Schrödinger equations with derivative in a nonlinear term and data in low-order Sobolev spaces, Electron. J. Differential Equations 42 (2001), 1–23.zbMATHGoogle Scholar
  22. [22]
    M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys. 87 (1983), 567–576.CrossRefzbMATHGoogle Scholar
  23. [23]
    Y. Wu, Global well-posedness of the derivative nonlinear Schrödinger equations in energy space, Anal. PDE 6 (2013), 1989–2002.MathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    Y. Wu, Global well-posedness on the derivative nonlinear Schrödinger equation, Anal. PDE 8 (2015), 1101–1112.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Hebrew University Magnes Press 2018

Authors and Affiliations

  1. 1.Department of Mathematical SciencesKorea Advanced Institute of Science and TechnologyDaejeonKorea
  2. 2.Center for Applied MathematicsTianjin UniversityTianjinP. R. China

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