Applied Mathematics and Mechanics

, Volume 28, Issue 7, pp 981–986 | Cite as

On a class of quasilinear schrödinger equations

  • Shu Ji  (舒级)Email author
  • Zhang Jian  (张健)


A type of quasilinear Schrödinger equations in two space dimensions which describe attractive Bose-Einstein condensates in physics is discussed. By establishing the property of the equation and applying the energy method, the blowup of solutions to the equation are proved under certain conditions. At the same time, by the variational method, a sufficient condition of global existence which is related to the ground state of a classical elliptic equation is obtained.

Key words

quasilinear Schrödinger equations blowup global existence ground state Bose-Einstein condensates 

Chinese Library Classification


2000 Mathematics Subject Classification



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  1. [1]
    Laedke E W, Spatschek K H, Stenflo L. Evolution theorem for a class of perturbed envelope soliton solutions[J]. J Math Phys, 1983, 24(12):2764–2769.zbMATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    De Bouard A, Hayashi N, Saut J C. Global existence of small solutions to a relativistic nonlinear Schrödinger equation[J]. Comm Math Phys, 1997, 189(1):73–105.zbMATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    Nakamura A. Damping and modification of exciton solitary waves[J]. J Phys Soc Japan, 1977, 42(6):1824–1835.CrossRefGoogle Scholar
  4. [4]
    Porkolab M, Goddman M V. Upper-hybrid solitons and oscillating two-stream instabilities[J]. Phys Fluids, 1976, 19(6):872–881.CrossRefMathSciNetGoogle Scholar
  5. [5]
    Quispel G R W, Capel H W. Equation of motion for the Heisenberg spin chain[J]. Physica A, 1982, 110(1/2):41–80.CrossRefMathSciNetGoogle Scholar
  6. [6]
    Takeno S, Homma S. Classical planar Heisenberg ferromagnet, complex scalar fields and nonlinear excitations[J]. Progr Theoret Phys, 1981, 65(1):172–189.CrossRefMathSciNetGoogle Scholar
  7. [7]
    Hasse R W. A general method for the solution of nonlinear soliton and kink Schrödinger equations[J]. Z Physik B, 1980, 37(1):83–87.CrossRefMathSciNetGoogle Scholar
  8. [8]
    Makhankov V G, Fedyanin V K. Non-linear effects in quasi-one-dimensional models of condensed matter theory[J]. Physics Reports, 1984, 104(1):1–86.CrossRefMathSciNetGoogle Scholar
  9. [9]
    Poppenberg M. Smooth solutions for a class of fully nonlinear Schrödinger type equations[J]. Nonlinear Analysis TMA, 2001, 45(6):723–741.zbMATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    Poppenberg M. On the local well posedness of quasi-linear Schrödinger equations in arbitrary space dimension[J]. J Diff Eq, 2001, 172(1):83–115.zbMATHCrossRefMathSciNetGoogle Scholar
  11. [11]
    Poppenberg M. An inverse function theorem in Sobolev spaces and applications to quasi-linear Schrödinger equations[J]. J Math Anal Appl, 2001, 258(1):146–170.zbMATHCrossRefMathSciNetGoogle Scholar
  12. [12]
    Poppenberg M, Schmitt K, Wang Z Q. On the existence of soliton solutions to quasi-linear Schrödinger equations[J]. Calc Var Partial Differential Equations, 2002, 14(3):329–344.zbMATHCrossRefMathSciNetGoogle Scholar
  13. [13]
    Liu J Q, Wang Y Q, Wang Z Q. Soliton solutions for quasi-linear Schrödinger equations II[J]. J Diff Eq, 2003, 187(3):473–493.zbMATHCrossRefGoogle Scholar
  14. [14]
    García-Ripoll Juan J, Konotop Vladimit V, Malomed Boris, Perez-Garca V M. A quasi-local Gross-Pitaevskii equation for attractive Bose-Einstein condensates[J]. Mathematics and Computers in Simulation, 2003, 62(1/2):21–30.zbMATHCrossRefMathSciNetGoogle Scholar
  15. [15]
    Zhang Jian. The blowup properties of the initial-boundary problem for second order derivative nonlinear Schrödinger equations[J]. Acta Mathematica Scientia, 1994, 14(suppl):89–94 (in Chinese).Google Scholar
  16. [16]
    Glassey R T. On the blowup of nonlinear Schrödinger equations[J]. J Math Phys, 1977, 18(9):1794–1797.zbMATHCrossRefMathSciNetGoogle Scholar
  17. [17]
    Tsutsumi Y, Zhang Jian. Instability of optical solitions for two-wave interaction model in cubic nonlinear media[J]. Adv Math Sci Appl, 1998, 8(2):691–713.zbMATHMathSciNetGoogle Scholar
  18. [18]
    Weinstein M I. Nonlinear Schrödinger equations and sharp interpolations estimates[J]. Comm Math Phys, 1983, 87(4):567–576.zbMATHCrossRefGoogle Scholar
  19. [19]
    Kwong M K. Uniqueness of positive solutions of Δuu + u p = 0 in R N[J]. Arch Ration Mech Anal, 1989, 105(3):243–266.zbMATHCrossRefMathSciNetGoogle Scholar
  20. [20]
    Zhang Jian. Stability of attractive Bose-Einstein condensates[J]. J Stat Phys, 2000, 101(3/4):731–746.zbMATHCrossRefGoogle Scholar
  21. [21]
    Kavian O. A remark on the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger equations[J]. Trans Amer Math Soc, 1987, 299(1):193–203.zbMATHCrossRefMathSciNetGoogle Scholar
  22. [22]
    Cazenave T. An introduction to nonlinear Schrödinger equations[M]. Rio de Janeiro: Textos de Metods Matematicos, 1989.Google Scholar

Copyright information

© Editorial Committee of Appl. Math. Mech. 2007

Authors and Affiliations

  1. 1.Key Laboratory of Computer SoftwareSichuan Normal UniversityChengduP. R. China
  2. 2.College of Mathematics and Software ScienceSichuan Normal UniversityChengduP. R. China

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