Skip to main content
SpringerLink
Log in

Global existence for a system of weakly coupled nonlinear Schrödinger equations

  • Published:
Mathematical Notes Aims and scope Submit manuscript

Abstract

This paper discusses the weakly coupled nonlinear Schrödinger equations in the supercritical case. With the best constant of the Gagliardo-Nirenberg inequality, we derive a sufficient condition for the global existence of solutions; this condition is expressed in terms of stationary solutions (nonlinear ground state).

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. L. Bergé, “Wave collapse in physics: principles and applications to light and plasma waves,” Phys. Rep. 303(5–6), 259–370 (1998).

    Article  MathSciNet  Google Scholar 

  2. G. Fibich and G. Papanicolaou, “Self-focusing in the perturbed and unperturbed nonlinear Schrödinger equation in critical dimension,” SIAM J. Appl.Math. 60(1), 183–240 (2000).

    Article  MathSciNet  Google Scholar 

  3. M. Lakshmanan, T. Kanna, and R. Radhakrishnan, “Shape-changing collisions of coupled bright solitons in birefringent optical fibers,” Rep. Math. Phys. 46(1–2), 143–156 (2000).

    Article  MATH  MathSciNet  Google Scholar 

  4. C. Menyuk, “Nonlinear pulse propagation in birefringent optical fibers,” IEEE J. Quantum Electron. 23(2), 174–176 (1987).

    Article  MathSciNet  Google Scholar 

  5. J. Ginibre and G. Velo, “On a class of nonlinear Schrödinger equations. I. The Cauchy problem, general case,” J. Funct. Anal. 32(1), 1–32 (1979); “On a class of nonlinear Schrödinger equations. II. Scattering theory, general case,” J. Funct. Anal. 32 (1), 33–71 (1979).

    Article  MATH  MathSciNet  Google Scholar 

  6. T. Kato, “On nonlinear Schrödinger equations,” Ann. Inst.H. Poincaré Phys.Théor. 46(1), 113–129 (1987).

    MATH  Google Scholar 

  7. R. T. Glassey, “On the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger equations,” J.Math. Phys. 18(9), 1794–1797 (1977).

    Article  MATH  MathSciNet  Google Scholar 

  8. T. Ogawa and Y. Tsutsumi, “Blow-up of H 1 solution for the nonlinear Schrödinger equation,” J. Differential Equations 92(2), 317–330 (1991).

    Article  MATH  MathSciNet  Google Scholar 

  9. T. Ogawa and Y. Tsutsumi, “Blow-up of H 1 solutions for the one-dimensional nonlinear Schrödinger equation with critical power nonlinearity,” Proc. Amer. Math. Soc. 111(2), 487–496 (1991).

    Article  MATH  MathSciNet  Google Scholar 

  10. H. Berestycki and T. Cazenave, “Instabilitédes états stationnaires dans les équations de Schrödinger et de Klein-Gordon non linéaires,” C. R. Acad. Sci. Paris Sér. I Math. 293(9), 489–492 (1981).

    MATH  MathSciNet  Google Scholar 

  11. M. I. Weinstein, “Nonlinear Schrödinger equations and sharp interpolation estimates,” Comm. Math. Phys. 87(4), 567–576 (1983).

    Article  MATH  Google Scholar 

  12. J. Zhang, “Sharp conditions of global existence for nonlinear Schrödinger and Klein-Gordon equations,” Nonlinear Anal. Ser. A. Theory Methods Appl. 48(2), 191–207 (2002).

    Article  MATH  Google Scholar 

  13. J. Zhang, “Cross-constrained variational problem and nonlinear Schrödinger equation,” in Foundations of Computational Mathematics, Proceedings of Smalefest 2000, Hong Kong, July 13–17, 2000 (World Sci. Publ., River Edge, NJ, 2002), pp. 457–469.

    Chapter  Google Scholar 

  14. T. Cazenave, An Introduction to Nonlinear Schrödinger Equations, in Textos deMetodosMatematicos (Univ. Federal Publ., Rio de Janeiro, 1989), Vol. 22.

    Google Scholar 

  15. J. Shu and J. Zhang, “Sharp conditions of global existence for a two-wave interaction model in cubic nonlinear media,” Chinese Ann. Math. Ser. A 28(6), 843–852 (2007).

    MATH  MathSciNet  Google Scholar 

  16. Y. Tsutsumi and J. Zhang, “Instability of optical solitons for two-wave interaction model in cubic nonlinear media,” Adv. Math. Sci. Appl. 8(2), 691–713 (1998).

    MATH  MathSciNet  Google Scholar 

  17. A. Ambrosetti and E. Colorado, “Standing waves of some coupled nonlinear Schrödinger equations,” J. London Math. Soc. (2) 75(1), 67–82 (2007).

    Article  MATH  MathSciNet  Google Scholar 

  18. L. A. Maia, E. Montefusco, and B. Pellacci, “Positive solutions for a weakly coupled nonlinear Schrödinger system,” J. Differential Equations 229(2), 743–767 (2006).

    Article  MATH  MathSciNet  Google Scholar 

  19. P. Bégout, “Necessary conditions and sufficient conditions for global existence in the nonlinear Schrödinger equation,” Adv. Math. Sci. Appl. 12(2), 817–827 (2002).

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Visual Computing and Virtual Reality Key Laboratory of Sichuan Province, Sichuan Normal University, Chengdu, 610066, P.R. China

    J. Shu

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

    J. Shu & J. Zhang

Authors
  1. J. Shu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J. Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to J. Shu.

Additional information

Published in Russian in Matematicheskie Zametki, 2009, Vol. 86, No. 5, pp. 686–691.

The text was submitted by the authors in English.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Shu, J., Zhang, J. Global existence for a system of weakly coupled nonlinear Schrödinger equations. Math Notes 86, 650–654 (2009). https://doi.org/10.1134/S0001434609110078

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0001434609110078

Key words

Search

Navigation