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.
Similar content being viewed by others
REFERENCES
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).
R. T. Glassey, On the blowing up solution to the Cauchy problem for nonlinear Schrödinger equations, J. Math. Phys. 18:1794–1797 (1979).
J. Ginibre and G. Velo, On a class of nonlinear Schrödinger equations. I, II, J. Funct. Anal. 32:1–71 (1979).
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.
J. L. Lebowitz, H. A. Rose, and E. R. Speer, Statistical Mechanics of the nonlinear Schrödinger equation, J. Stat. Phys. 50:657–687 (1988).
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).
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).
H. Nawa, “Mass concentration” phenomenon for the nonlinear Schrödinger equation with the critical power nonlinearity, Funk. Ekva. 35:1–18 (1992).
H. Nawa, “Mass concentration” phenomenon for the nonlinear Schrödinger equation with the critical power nonlinearity. II, Kodai Math. J. 13:333–348 (1990).
H. Nawa, Formation of singularities in solutions of the nonlinaer Schrödinger equation, Proc. Japan Acad. 67(A):29–34 (1991).
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).
H. Nawa, Asymptotic profiles of blow-up solutions of the nonlinear Schrödinger equation with critical power nonlinearity II, preprint (1997).
H. Nawa, Asymptotic profiles of blow-up solutions of the nonlinear Schrödinger equation with critical power nonlinearity III, preprint (1997).
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).
T. Ogawa and Y. Tsutsumi, Blow-up of H 1-solution for the nonlinear Schrödinger equation, J. Differential Equations 92:317–330 (1991).
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).
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.
M. Tsutsumi, Nonexistence and instability of solutions of nonlinear Schrödinger equations, unpublished.
M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Commun. Math. Phys. 87:511–517 (1983).
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).
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).
V. E. Zakharov, Collaps of Langmuir waves, Sov. Phys. JETP 35:908–914 (1972).
Rights 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
Issue Date:
DOI: https://doi.org/10.1023/A:1023012709647