Skip to main content
Log in

Two Points Blow-up in Solutions of the Nonlinear Schrödinger Equation with Quartic Potential on \(\mathbb{R} \)

  • Published:
Journal of Statistical Physics Aims and scope Submit manuscript

Abstract

We consider the blow-up problem for the nonlinear Schrödinger equation with quartic self-interacting potential on \(\mathbb{R}\). We exhibit a class of initial data leading to the blow-up solutions which have at least two L 2-concentration points.

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

Access this article

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

Instant access to the full article PDF.

Similar content being viewed by others

REFERENCES

  1. G. Fibich and G. Papanicolaou, Self-focusing in the perturbed and unperturbed nonlinear Schrödinger equation in critical dimension, preprint (submitted to SIAM J. Appl. Math.), (1997).

  2. R. T. Glassey, On the blowing up solution to the Cauchy problem for nonlinear Schrödinger equations, J. Math. Phys. 18:1794–1797 (1979).

    Google Scholar 

  3. J. Ginibre and G. Velo, On a class of nonlinear Schrödinger equations. I, II, J. Funct. Anal. 32:1–71 (1979).

    Google Scholar 

  4. T. Kato, Nonlinear Schrödinger Equations, Springer Lecture Notes in Physics, H. Holden and A. Jensen, eds., Schrödinger Operators, Vol. 345 (Springer-Verlag, Berlin/Heiderberg/New York, 1989), pp. 236–251.

    Google Scholar 

  5. J. L. Lebowitz, H. A. Rose, and E. R. Speer, Statistical Mechanics of the nonlinear Schrödinger equation, J. Stat. Phys. 50:657–687 (1988).

    Google Scholar 

  6. D. W. McLaughlin, C. Papanicolaou, C. Sulem, and P. L. Sulem, Focusing singularity of the cubic Schrödinger equation, Phys. Rev. A 34:1200–1210 (1986).

    Google Scholar 

  7. F. Merle, Construction of solutions with exactly k blow-up points for the Schrödinger equation with the critical power nonlinearity, Commun. Math. Phys. 129:223–240 (1990).

    Google Scholar 

  8. H. Nawa, “Mass concentration” phenomenon for the nonlinear Schrödinger equation with the critical power nonlinearity, Funk. Ekva. 35:1–18 (1992).

    Google Scholar 

  9. H. Nawa, “Mass concentration” phenomenon for the nonlinear Schrödinger equation with the critical power nonlinearity. II, Kodai Math. J. 13:333–348 (1990).

    Google Scholar 

  10. H. Nawa, Formation of singularities in solutions of the nonlinaer Schrödinger equation, Proc. Japan Acad. 67(A):29–34 (1991).

    Google Scholar 

  11. H. Nawa, Asymptotic profiles of blow-up solutions of the nonlinear Schrödinger equation with critical power nonlinearity, J. Math. Soc. Japan 46:557–586 (1994).

    Google Scholar 

  12. H. Nawa, Asymptotic profiles of blow-up solutions of the nonlinear Schrödinger equation with critical power nonlinearity II, preprint (1997).

  13. H. Nawa, Asymptotic profiles of blow-up solutions of the nonlinear Schrödinger equation with critical power nonlinearity III, preprint (1997).

  14. H. Nawa, Limiting profiles of blow-up solutions of the nonlinear Schrödinger equation with critical power nonlinearity, Proc. Japan Acad. 73(A):171–175 (1997).

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

  17. T. Ogawa and Y. Tsutsumi, Blow-up of solutions for the nonlinear Schrödinger equation with quartic potential and periodic boundary condition, in Springer Lecture Notes in Mathematics, Vol. 1450 (Springer-Verlag, Berlin/Heidelberg/New York, 1990), pp. 236–251.

    Google Scholar 

  18. M. Tsutsumi, Nonexistence and instability of solutions of nonlinear Schrödinger equations, unpublished.

  19. M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Commun. Math. Phys. 87:511–517 (1983).

    Google Scholar 

  20. M. I. Weinstein, On the structure and formation singularities in solutions to nonlinear dispersive evolution equations, Commun. in Partial Diffrential Equations 11:545–565 (1986).

    Google Scholar 

  21. M. I. Weinstein, The nonlinear Schrödinger equation—Singularity formation, Stability and dispersion, The Connection between Infinite and Finite Dimensional Dynamical Systems, Contemporary Math. 99:213–232 (1989).

    Google Scholar 

  22. V. E. Zakharov, Collaps of Langmuir waves, Sov. Phys. JETP 35:908–914 (1972).

    Google Scholar 

Download references

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Nawa, H. Two Points Blow-up in Solutions of the Nonlinear Schrödinger Equation with Quartic Potential on \(\mathbb{R} \) . Journal of Statistical Physics 91, 439–458 (1998). https://doi.org/10.1023/A:1023012709647

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1023012709647

Navigation