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Khasminskii-Whitham averaging for randomly perturbed KdV equation. (English) Zbl 1148.35077
Summary: We consider the damped-driven KdV equation:
$\dot u-\nu u_{xx}+u_{xxx}-6uu_x=\sqrt\nu \eta(t,x),\quad x\in S^1,\quad \int u\,dx\equiv\int \eta\,dx\equiv 0,$
where $$0<\nu\leq 1$$ and the random process $$\eta$$ is smooth in $$x$$ and white in $$t$$. For any periodic function $$u(x)$$ let $$I=(I_1,I_2,\dots)$$ be the vector, formed by the KdV integrals of motion, calculated for the potential $$u(x)$$. We prove that if $$u(t,x)$$ is a solution of the equation above, then for $$0\leq t\leq \nu^{-1}$$ and $$\nu\to 0$$ the vector $$I(t)=(I_1(u(t,\cdot)),I_2(u(t,\cdot)),\dots)$$ satisfies the (Whitham) average equation.

##### MSC:
 35Q53 KdV equations (Korteweg-de Vries equations) 35R60 PDEs with randomness, stochastic partial differential equations 60H15 Stochastic partial differential equations (aspects of stochastic analysis)
##### Keywords:
KdV equation; random process; Whitham averaged equation
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##### References:
 [1] Arnold, V.; Kozlov, V.V.; Neistadt, A.I., Mathematical aspects of classical and celestial mechanics, (1989), Springer Berlin [2] Djakov, P.; Mityagin, B., Instability zones of 1D periodic Schrödinger and Dirac operators, Russ. math. surv., 61, 4, (2006) · Zbl 1128.47041 [3] Dubrovin, B.A.; Novikov, S.P., Hydrodynamics of weakly deformed soliton lattices, differential geometry and Hamiltonian theory, Russ. math. surv., 44, 35-124, (1989) · Zbl 0712.58032 [4] Da Prato, G.; Zabczyk, J., Stochastic equations in infinite dimensions, (1992), Cambridge University Press Cambridge · Zbl 0761.60052 [5] Flaschka, H.; Forest, M.G.; McLaughlin, D.W., Multiphase averaging and the inverse spectral solutions of the korteweg – de Vries equation, Comm. pure appl. math., 33, 739-784, (1980) · Zbl 0454.35080 [6] Freidlin, M.; Wentzell, A., Random perturbations of dynamical systems, (1998), Springer-Verlag New York · Zbl 0922.60006 [7] Freidlin, M.I.; Wentzell, A.D., Averaging principle for stochastic perturbations of multifrequency systems, Stochastics and dynamics, 3, 393-408, (2003) · Zbl 1050.60078 [8] Jacod, J.; Shiryaev, A.N., Limit theorems for stochastic processes, (1987), Springer-Verlag Berlin · Zbl 0830.60025 [9] Khasminskii, R.Z., On the averaging principle for ito stochastic differential equations, Kybernetika, 4, 260-279, (1968), (in Russian) · Zbl 0231.60045 [10] Korotyaev, E., Estimates for the Hill operator, II, J. diff. equations, 223, 2, 229-260, (2006) · Zbl 1098.34070 [11] Kappeler, T.; Pöschel, J., Kam & KdV, (2003), Springer · Zbl 1032.37001 [12] Krichever, I.M., The averaging method for two-dimensional “integrable” equations, Funct. anal. appl., 22, 200-213, (1988) · Zbl 0688.35088 [13] Krylov, N.V., Controlled diffusion processes, (1980), Springer · Zbl 0459.93002 [14] Kuksin, S.B., Analysis of Hamiltonian pdes, (2000), Oxford University Press Oxford · Zbl 0960.35001 [15] S.B. Kuksin, Randomly forced nonlinear PDEs and statistical hydrodynamics in 2 space dimensions, European Mathematical Society Publishing House, 2006, also see mp_arc 06-178 · Zbl 1099.35083 [16] Moser, J.; Siegel, C.L., Lectures on celestial mechanics, (1971), Springer Berlin · Zbl 0312.70017 [17] McKean, H.; Trubowitz, E., Hill’s operator and hyperelliptic function theory in the presence of infinitely many branching points, Comm. pure appl. math., 29, 143-226, (1976) · Zbl 0339.34024 [18] Veretennikov, A.Yu., On the averaging principle for systems of stochastic differential equations, Mat. USSR sb., 69, 271-284, (1991) · Zbl 0724.60069 [19] Whitham, G.B., Linear and nonlinear waves, (1974), John Wiley & Sons New York · Zbl 0373.76001 [20] Yor, M., Existence et unicité de diffusion a valeurs dans un espace de Hilbert, Ann. inst. Henri Poincaré, 10, 55-88, (1974) · Zbl 0281.60094
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