Abstract
This is the first study of one of the transmission problems associate to the non-linear Schrödinger equation with a random potential. We show that for almost every realization of the medium the rate of transmission vanishes when increasing the size of the medium; however, whereas it decays exponentially in the linear regime, it decays polynomially in the nonlinear one.
Similar content being viewed by others
References
R. J. Rubin,J. Math. Phys. 9:2252 (1968).
J. Hori,Bussei Kenkyu 10:93 (1968).
H. Matsuda and K. Ishii,Prog. Theor. Phys. Supp. 45:56 (1970).
K. Ishii,Prog. Theor. Phys. Supp. 53:77 (1973).
R. Rosales and G. C. Papanicolaou,Stud. Appl. Math. 68:89 (1983).
P. Devillard,Thèse de Troisième Cycle (Université de Paris, France, 1985).
D. J. Thouless,Proceedings, Ill Condensed Matter (Les Houches, North Holland, 1979).
B. Souillard,Phys. Rep. 103:41 (1984).
B. Simon and B. Souillard,J. Stat. Phys. 36:273 (1984).
F. Delyon, Y. Levy, and B. Souillard,Phys. Rev. Lett. 55:618 (1985).
H. Furstenberg,Trans. Amer. Math. Soc. 108:377 (1963).
Ph. Bougerol and J. Lacroix,Products of Random Matrices and Applications, Birkhaüser, (1985).
A. Baylis, G. C. Papanicolaou, and B. White, Scattering by a nonlinear random medium, to appear.
P. L. Sulem and U. Frisch,J. Plasma Phys. 8:217 (1972).
P. R. Halmos,Lectures on Ergodic Theory (Chelsea, 1956).
S. Kakutani, inRandom ergodic theorems and Markoff processes with a stable distribution, Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability (1950) (University of California Press, California, 1951), p. 247–261.
A. Carverhill,Flows of Stochastic Dynamical Systems: Ergodic Theory, preprint (Mathematics Institute, Warwick, 1985).
Y. Kifer,A Multiplicative Ergodic Theorem for Random Transformations, preprint (Institute of Mathematics, Jerusalem, 1985).
Author information
Authors and Affiliations
Additional information
This work is part of a Thèse de Troisième Cycle by P. Devillard.(6)
Rights and permissions
About this article
Cite this article
Devillard, P., Souillard, B. Polynomially decaying transmission for the nonlinear Schrödinger equation in a random medium. J Stat Phys 43, 423–439 (1986). https://doi.org/10.1007/BF01020646
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01020646