The Annals of Applied Probability

Stability of solitons under rapidly oscillating random perturbations of the initial conditions

Ennio Fedrizzi

Full-text: Open access

Abstract

We use the inverse scattering transform and a diffusion approximation limit theorem to study the stability of soliton components of the solution of the nonlinear Schrödinger and Korteweg–de Vries equations under random perturbations of the initial conditions: for a wide class of rapidly oscillating random perturbations this problem reduces to the study of a canonical system of stochastic differential equations which depends only on the integrated covariance of the perturbation. We finally study the problem when the perturbation is weak, which allows us to analyze the stability of solitons quantitatively.

Article information

Source
Ann. Appl. Probab. Volume 24, Number 2 (2014), 616-651.

Dates
First available in Project Euclid: 10 March 2014

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1394465367

Digital Object Identifier
doi:10.1214/13-AAP931

Mathematical Reviews number (MathSciNet)
MR3178493

Zentralblatt MATH identifier
1315.35053

Subjects
Primary: 60B12: Limit theorems for vector-valued random variables (infinite- dimensional case) 35C08: Soliton solutions
Secondary: 35Q53: KdV-like equations (Korteweg-de Vries) [See also 37K10] 35Q55: NLS-like equations (nonlinear Schrödinger) [See also 37K10]

Keywords
Diffusion approximation limit theorem random perturbation of initial conditions solitons NLS equation KdV equation

Citation

Fedrizzi, Ennio. Stability of solitons under rapidly oscillating random perturbations of the initial conditions. Ann. Appl. Probab. 24 (2014), no. 2, 616--651. doi:10.1214/13-AAP931. https://projecteuclid.org/euclid.aoap/1394465367


Export citation

References

  • [1] Ablowitz, M. J. and Clarkson, P. A. (1991). Solitons, Nonlinear Evolution Equations and Inverse Scattering. London Mathematical Society Lecture Note Series 149. Cambridge Univ. Press, Cambridge.
  • [2] Ablowitz, M. J., Prinari, B. and Trubatch, A. D. (2004). Discrete and Continuous Nonlinear Schrödinger Systems. London Mathematical Society Lecture Note Series 302. Cambridge Univ. Press, Cambridge.
  • [3] Aldous, D. (1978). Stopping times and tightness. Ann. Probab. 6 335–340.
  • [4] Burzlaff, J. (1988). The soliton number of optical soliton bound states for two special families of input pulses. J. Phys. A 21 561–566.
  • [5] Derevyanko, S. A. and Prilepsky, J. E. (2008). Random input problem for the nonlinear Schrödinger equation. Phys. Rev. E (3) 78 046610, 12.
  • [6] Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes: Characterization and Convergence. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics 623. Wiley, New York.
  • [7] Fouque, J.-P., Garnier, J., Papanicolaou, G. and Sølna, K. (2007). Wave Propagation and Time Reversal in Randomly Layered Media. Stochastic Modelling and Applied Probability 56. Springer, New York.
  • [8] Karpman, V. I. (1979). Soliton evolution in the presence of perturbation. Phys. Scr. 20 462–478.
  • [9] Kivshar, Y. S. (1989). On the soliton generation in optical fibres. J. Phys. A: Math. Gen. 22 337–340.
  • [10] Kunita, H. (1984). Stochastic differential equations and stochastic flows of diffeomorphisms. In École D’été de Probabilités de Saint–Flour, XII—1982. Lecture Notes in Math. 1097 143–303. Springer, Berlin.
  • [11] Kunita, H. (1997). Stochastic Flows and Stochastic Differential Equations. Cambridge Studies in Advanced Mathematics 24. Cambridge Univ. Press, Cambridge.
  • [12] Metivier, M. (1984). Convergence faible et principe d’invariance pour des martingales à valeurs dans des espaces de Sobolev. Rapport Interne CMAP 106, Ecole Polytechnique, Paris.
  • [13] Métivier, M. (1988). Stochastic Partial Differential Equations in Infinite-Dimensional Spaces. Scuola Normale Superiore di Pisa. Quaderni. [Publications of the Scuola Normale Superiore of Pisa]. Scuola Normale Superiore, Pisa.
  • [14] Moloney, J. and Newell, A. (2004). Nonlinear Optics. Westview Press, Boulder, CO.
  • [15] Murray, A. C. (1978). Solutions of the Korteweg–de Vries equation from irregular data. Duke Math. J. 45 149–181.
  • [16] Whitham, G. B. (1974). Linear and Nonlinear Waves. Wiley, New York.