Skip to main content
SpringerLink
Log in

Stability of Schrödinger-Poisson type equations

  • Published:
Applied Mathematics and Mechanics Aims and scope Submit manuscript

Abstract

Variational methods are used to study the nonlinear Schrödinger-Poisson type equations which model the electromagnetic wave propagating in the plasma in physics. By analyzing the Hamiltonian property to construct a constrained variational problem, the existence of the ground state of the system is obtained. Furthermore, it is shown that the ground state is orbitally stable.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abdullaev, F. Kh. Dynamical chaos of solitons and nonlinear periodic waves. Phys. Reports 179(1), 1–78 (1989)

    Article  MathSciNet  Google Scholar 

  2. Gagliardo, E. Proprieta di alcune classi di funzioni in piu varibili (in French). Ricerche di Mathematica 7(1), 102–137 (1958)

    MATH  MathSciNet  Google Scholar 

  3. Gagliardo, E. Ulteriori proprieta di alcune classi di funzioni in piu varibili (in French). Ricerche di Mathematica 8(1), 24–51 (1959)

    MATH  MathSciNet  Google Scholar 

  4. Kuznetsov, E. A., Rubenchik, A. M., and Zakharov, V. E. Soliton stability in plasmas and by drodynamics. Phys. Reports 142(3), 103–165 (1986)

    Article  MathSciNet  Google Scholar 

  5. Makhankov, V. G. Dynamics of classical solutions (in non-interable systems). Phys. Reports 35(1), 1–128 (1978)

    Article  MathSciNet  Google Scholar 

  6. Zakharov, V. E. Collapse of Langmuir waves. Sovet Physics-JETP 35(5), 908–914 (1972)

    Google Scholar 

  7. Guo, Boling and Yang, Linge. The global solution and asymptotic behaviors for one class of nonlinear evolution equations. J. Partial Diff. Eqs. 10(3), 232–246 (1997)

    MATH  MathSciNet  Google Scholar 

  8. Fukuizumi, R. and Ohta, M. Stability of standing waves for nonlinear Schrödinger equations with potentials. Differential and Integral Equations 16(1), 111–128 (2003)

    MATH  MathSciNet  Google Scholar 

  9. Goncalves Rebeiro, J. M. Instability of symmetric stationary states for some nonlinear Schrödinger equations with an external magnetic field. Annales de l’Institut Henri Poincaré, Physique Théorique 54(4), 403–433 (1991)

    Google Scholar 

  10. Grillakis, M., Shatah, J., and Strauss, W. Stability theory of solitary waves in the presence of symmetry. Journal of Functional Analysis 74(1), 160–197 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  11. Shatah, J. and Strauss, W. A. Instability of nonlinear bound states. Comm. Math. Phys. 100(2), 173–190 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  12. Tsutsumi, Y. and Zhang, J. Instability of optical solitons for two-wave interaction model in cubic nonlinear media. Advances in Mathematical Sciences and Applications 8(5), 691–713 (1998)

    MATH  MathSciNet  Google Scholar 

  13. Wei, Yunyun and Chen, Guanggan. On the standing wave for a class of nonlinear Schrödinger equations. J. Math. Anal. Appl. 337(2), 1022–1030 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  14. Zhang, Jian. Stability of attractive Bose-Einstein condensates. Journal of Statistical Physics 101(3–4), 731–746 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  15. Zhang, Jian. Stability of standing waves for nonlinear Schrödinger equations with unbounded potentials. Z. Angew. Math. Phys. 51(1), 498–503 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  16. Zhang, Jian. On the standing wave in coupled nonlinear Klein-Gordon equations. Math. Meth. Appl. Sci. 26(1), 11–25 (2003)

    Article  MATH  Google Scholar 

  17. Zhang, Jian. Sharp threshold for global existence and blowup in nonlinear Schrödinger equation with harmonic potential. Communications in Partial Differential Equations 30(10–12), 1429–1443 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  18. Strauss, W. A. Existence of solitary waves in higher dimensions. Commun. Math. Phys. 55(2), 149–162 (1977)

    Article  MATH  Google Scholar 

  19. Kwong, M. K. Uniqueness of positive solutions of Δu − u + u p = 0 in R N. Archive for Rational Mechanics and Analysis 105(3), 243–266 (1989) DOI: 10.1007/BF00251502

    Article  MATH  MathSciNet  Google Scholar 

  20. Cazenave, T. and Lions, P. L. Orbital stability of standing waves for some nonlinear Schrödinger equations. Comm. Math. Phys. 85(4), 549–561 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  21. Weinstein, M. I. Nonlinear Schrödinger equations and sharp interpolations estimates. Commun. Math. Phys. 87(4), 567–576 (1983)

    Article  MATH  Google Scholar 

  22. Berestycki, H. and Cazenave, T. Instabilité des états stationnaires dans les équations de Schrödinger et de Klein-Gordon non linéarires (in French). C. R. Acad. Sci. Paris, Seire I 293(1), 489–492 (1981)

    MATH  MathSciNet  Google Scholar 

  23. Struwe, M. Varitional Methods, Springer-Verlag, Berlin/Heidelberg/New York (1996)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. College of Mathematics and Software Science, Sichuan Normal University, Chengdu, 610066, P. R. China

    Juan Huang  (黄娟), Jian Zhang  (张健) & Guang-gan Chen  (陈光淦)

Authors
  1. Juan Huang  (黄娟)
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jian Zhang  (张健)
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Guang-gan Chen  (陈光淦)
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Juan Huang  (黄娟).

Additional information

Communicated by Li-qun CHEN

Project supported by the National Natural Science Foundation of China (Nos. 10771151 and 10901115), the Scientific Research Fund of Sichuan Provincial Education Department (No. 2006A063), and the Scientific Research Fund of Science and Technology Bureau of Sichuan Province (No. 07JY029-012)

Rights and permissions

Reprints and permissions

About this article

Cite this article

Huang, J., Zhang, J. & Chen, Gg. Stability of Schrödinger-Poisson type equations. Appl. Math. Mech.-Engl. Ed. 30, 1469–1474 (2009). https://doi.org/10.1007/s10483-009-1113-y

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10483-009-1113-y

Key words

Chinese Library Classification

2000 Mathematics Subject Classification

Search

Navigation