Abstract
This paper deals with the transmission of a soliton in a random medium described by a randomly perturbed Korteweg–de Vries equation. Different kinds of perturbations are addressed, depending on their specific time or position dependences, with or without damping. We derive effective evolution equations for the soliton parameter by applying a perturbation theory of the inverse scattering transform and limit theorems of stochastic calculus. Original results are derived that are very different compared to a randomly perturbed Nonlinear Schrödinger equation. First the emission of a soliton gas is proved to be a very general feature. Second some perturbations are shown to involve a speeding-up of the soliton, instead of the decay that is usually observed in random media.
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REFERENCES
N. J. Zabuski and M. D. Kruskal, Interaction of solitons in a collisionless plasma and the recurrene of initial states, Phys. Rev. Lett. 15:240–243 (1965).
C. S. Gardner, J. M. Greene, M. D. Kruskal, and R. M. Miura, Method for solving the Korteweg–de Vries equation, Phys. Rev. Lett. 19:1095–1097 (1967).
P. D. Lax, Integrals of nonlinear equations of evolution and solitary waves, Comm. Pure Appl. Math. 21:467–490 (1968).
V. E. Zakharov and A. B. Shabat, Exact theory of two-dimensional self-focusing and onedimensional self-modulation of waves in nonlinear media, Sov. Phys. JETP 34:62–69 (1972).
F. G. Bass, Y. S. Kivshar, V. V. Konotop, and Y. A. Sinitsyn, Dynamics of solitons under random perturbations, Phys. Rep. 157:63–181 (1988).
F. Kh. Abdullaev, A. R. Bishop, and St. Pnevmatikos (eds.), Nonlinearity with Disorder(Springer, Berlin, 1991).
F. Kh. Abdullaev, Theory of Solitons in Inhomogeneous Media(Wiley, Chichester, England 1994).
Yu. S. Kivshar, S. A. Gredeskul, A. Sanchez, and L. Vasquez, Localization decay induced by strong nonlinearity in disordered systems, Phys. Rev. Lett. 64:1693–1696 (1990).
J. C. Bronski, Nonlinear scattering and analyticity properties of solitons, J. Nonlinear Sci. 8:161–182 (1998).
J. Garnier, Asymptotic transmission of solitons through random media, SIAM J. Appl. Math. 58:1969–1995 (1998).
R. Knapp, Transmission of solitons through random media, Physica D 85:496–508 (1995).
J. C. Bronski, Nonlinear wave propagation in a disordered medium, J. Statist. Phys. 92, 995–1015 (1998).
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. 76:1102–1105 (1996).
F. Kh. Abdullaev, S. A. Darmanyan, M. R. Djumaev, A. J. Majid, and M. P. Sørensen, Evolution of randomly perturbed Korteweg–de Vries solitons, Phys. Rev. E 52:3577–3583 (1995).
M. Wadati, Stochastic Korteweg–de Vries equation, J. Phys. Soc. Japan 52:2642–2648 (1983).
M. Wadati and Y. Akutsu, Stochastic Korteweg–de Vries equation with and without damping, J. Phys. Soc. Japan 53:3342–3350 (1984).
T. Iizuka, Anomalous diffusion of solitons in random systems, Phys. Lett. A 181:39–42 (1993).
M. Scalerandi, A. Romano, and C. A. Condat, Korteweg–de Vries solitons under additive stochastic perturbations, Phys. Rev. E 58:4166–4173 (1998).
V. I. Karpman, Soliton evolution in the presence of perturbations, Phys. Scripta 20:462–478 (1979).
J. Wright, Soliton production and solutions to perturbed Korteweg–de Vries equations, Phys. Rev. A 21:335–339 (1980).
P. Anderson, Absence of diffusion in certain random lattices, Phys. Rev. 109:1492–1505 (1958).
I. M. Lifshitz, S. A. Gredeskul, and L. A. Pastur, Introduction to the Theory of Disordered Systems(Wiley, New York, 1988).
S. V. Manakov, S. Novikov, J. P. Pitaevskii, and V. E. Zakharov, Theory of Solitons(Consultants Bureau, New York, 1984).
M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform(SIAM, Philadelphia, 1981).
V. E. Zakharov and L. D. Faddeev, Korteweg–de Vries equation, a completely integrable Hamiltonian system, Funct. Anal. Appl. 5:280–287 (1971).
R. L. Herman, The stochastic, damped KdV equation, J. Phys. A: Math. Gen. 23:1063–1084 (1990).
H. J. Kushner, Approximation and Weak Convergence Methods for Random Processes(MIT Press, Cambridge, 1984).
D. Middleton, Introduction to Statistical Communication Theory(McGraw–Hill, New York, 1960).
I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products(Academic Press, San Diego, 1980).
T. R. Taha and M. J. Ablowitz, Analytical and numerical aspects of certain nonlinear evolution equations. III. Numerical, Korteweg–de Vries equation, J. Comput. Phys. 55:231–253 (1984).
G. Dahlquist and A. Björk, Numerical Methods(Prenctice Hall, Englewoods Cliffs, New Jersey, 1974).
B. Fornberg and G. B. Whitham, A numerical and theoretical study of certain nonlinear wave phenomena, Philos. Trans. Roy. Soc. London Ser. A 289:373–403 (1978).
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Garnier, J. Long-Time Dynamics of Korteweg–de Vries Solitons Driven by Random Perturbations. Journal of Statistical Physics 105, 789–833 (2001). https://doi.org/10.1023/A:1013549126956
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DOI: https://doi.org/10.1023/A:1013549126956