Soliton perturbations and the random Kepler problem. (English) Zbl 0936.35171

Summary: We consider the influence of randomly varying parameters on the propagation of solitons for the one-dimensional nonlinear Schrödinger equation. This models, for example, optical soliton propagation in a fiber whose properties vary with distance along the fiber. By using an averaged Lagrangian approach we obtain a system of stochastic modulation equations for the evolution of the soliton parameters, which takes the form of a randomly perturbed Kepler problem. We use the action-angle formulation of the Kepler problem to calculate the statistics of the escape time. The mean escape time for the Kepler problem corresponds, in the optical context, to the expected distance until the soliton disintegrates.


35Q55 NLS equations (nonlinear Schrödinger equations)
35R60 PDEs with randomness, stochastic partial differential equations
37L55 Infinite-dimensional random dynamical systems; stochastic equations
Full Text: DOI


[1] F.Kh. Abdullaev, S.A. Darmanyan, P.K. Khabibullaev, Optical Solitons, Springer, Heidelberg, 1993.
[2] Bronski, J.C, J. nonlinear sci., 8, 161, (1998)
[3] Bronski, J.C, J. stat. phys., 92, 5-6, (1998)
[4] Garnier, J; SIAM, J, Appl. math., 58, 1969, (1998)
[5] J. Garnier, Ph.D. Thesis, Ecole Polytechnique.
[6] R. Knapp, G. Papanicolaou, B. White, in: A.R. Bishop, D. Campbell, St. Pnevmatikos (Eds.), Nonlinearity and Disorder, Springer, Heidelberg, 1989.
[7] F.Kh. Abdullaev, in: A.R. Bishop, L. Vasquez (Eds.), Fluctuations: Nonlinearity and Disorder, World Scientific, Singapore, 1995.
[8] Abdullaev, F.Kh; Abdumalikov, A.A; Baizakov, B.B, Opt. commun., 138, 49, (1997)
[9] G. Fibich, G. Papanicolaou, SIAM, in press (1999).
[10] Wai, P.K; Menyuk, C.R; Chen, H.H, Opt. lett., 16, 1231, (1991)
[11] Elgin, J.N, Opt. lett., 18, 10, (1993)
[12] Yu. Kodama, A. Hasegawa, Solitons in Optical Communications, Clarendon Press, Oxford, 1995. · Zbl 0840.35092
[13] Kath, W; Ueda, T, Physica D, 55, 166, (1992)
[14] Kath, W; Ueda, T, Josa b, 11, 818, (1994)
[15] Gredeskul, S.A; Kivshar, Yu.S, Phys. rep., 216, 1, (1992)
[16] Kutz, J.N; Wai, P.K, Opt. lett., 23, 13, (1998)
[17] G. Whitham, Linear and Nonlinear Waves, Wiley, New York, 1974. · Zbl 0373.76001
[18] Anderson, D; Lisak, M; Reichel, T, Josa b, 5, 207, (1988)
[19] Karpman, V.I, Phys. scripta, 20, 462, (1978)
[20] Keener, J.P; McLaughlin, D.W, J. math. phys., 16, 777, (1977)
[21] Kath, W.L; Smyth, N.F, Phys. rev. E, 51, 1484, (1995)
[22] L.D. Landau, E.M. Lifshitz, Mechanics, Pergamon Press, New York, 1975. · Zbl 0081.22207
[23] H. Goldstein, Classical Mechanics, Wiley, New York, 1965.
[24] W. Feller, Probability Theory and Its Applications, Wiley, New York, 1950. · Zbl 0039.13201
[25] C.W. Gardner, Handbook of Stochastic Methods, Springer, Berlin, 1983.
[26] Malomed, B.A, J. opt. soc. am. B, 13, 677, (1996)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.