Abstract
In this paper we consider the problem of solitary wave propagation in a weakly disordered potential. Through a series of careful numerical experiments we have observed behavior which is in agreement with the theoretical predictions of Kivshar et al., Bronski, and Gamier. In particular we observe numerically the existence of two regimes of propagation. In the first regime the mass of the solitary wave decays exponentially, while the velocity of the solitary wave approaches a constant. This exponential decay is what one would expect from known results in the theory of localization for the linear Schrödinger equation. In the second regime, where nonlinear effects dominate, we observe the anomalous behavior which was originally predicted by Kivshar et al. In this regime the mass of the solitary wave approaches a constant, while the velocity of the solitary wave displays an anomalously slow decay. For sufficiently small velocities (when the theory is no longer valid) we observe phenomena of total reflection and trapping.
Similar content being viewed by others
REFERENCES
A. C. Biswas and C. S. Warke, Nonlinear effects in two-dimensional superfluid 4He, Phys. Rev. B 22(5):2581-2584 (1980).
J. C. Bronski, Nonlinear scattering and analyticity properties of solitons, J. Nonlin. Sci. 8 (1998).
J. C. Bronski, D. W. McLaughlin, and M. J. Shelley, Stability of time-harmonic localized solutions to a random NLS equation, J. Stat. Phys. 88(5/6) (1997).
V. S. Buslaev and G. S. Perel'man, On the stability of solitary waves for nonlinear Schrödinger equations, AMS Translations 164:75-98 (1995).
C. A. Condat and R. A. Guyer, Korteweg-De Vries solitons and helium films, Phys. Rev. B 25(5):3117-3122 (1982).
P. Devillard and B. Souillard, Polynomially decaying transmission for the nonlinear Schrödinger equation in a random medium, J. Stat. Phys. 43:423-439 (1986).
B. Doucot and R. Rammal, On Anderson localization in a nonlinear random medium, Europhys. Lett. 3:969 (1987).
J. Frohlich, T. Spencer, and C. E. Wayne, Localization in disordered, nonlinear dynamical systems, J. Stat. Phys. 42:247 (1986).
J. Garnier, Solitons in milieux aléatoires, Ph.D. thesis, Ecole Polytechnique, 1996.
J. Garnier, Transmission of solitions through random media, SIAM J. Appl. Math. (1998), in press.
V. A. Hopkins, J. Keat, G. D. Meegan, T. Zhang, and J. D. Maynard, Observation of the predicted behavior of nonlinear pulse propagation in disordered media, Phys. Rev. Lett. 76(7):1102-1105 (1996).
B. A. Huberman, Superfluid solitons in helium films, Phys. Rev. Lett. 41(20):1389-1393 (1978).
Yu. S. Kivshar, S. A. Gredeskul, A. Sánchez, and L. Vázquez, Localization decay induced by strong nonlinearity in disordered systems, Phys. Rev. Lett. 64(15):1693 (1990).
R. Knapp, Transmission of solitons through random media, Physica D 85:496-508 (1995).
R. Knapp, G. Pananicolaou, and B. White, Transmission of waves by a nonlinear medium, J. Stat. Phys. 63(4):567-584 (1991).
A. Soffer and M. I. Weinstein, Resonances, radiation damping and instability in hamiltonian nonlinear wave equations. Preprint, 1997.
M. I. Weinstein, personal communication.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Bronski, J.C. Nonlinear Wave Propagation in a Disordered Medium. Journal of Statistical Physics 92, 995–1015 (1998). https://doi.org/10.1023/A:1023096627528
Issue Date:
DOI: https://doi.org/10.1023/A:1023096627528