Khasminskii–Whitham averaging for randomly perturbed KdV equation
- Kuksin, Sergei B.
- Piatnitski, Andrey L.
DOIAbstract
AbstractWe consider the damped-driven KdV equation:u˙−νuxx+uxxx−6uux=νη(t,x),x∈S1,∫udx≡∫ηdx≡0, where 0<ν⩽1 and the random process η is smooth in x and white in t. For any periodic function u(x) let I=(I1,I2,…) 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⩽t≲ν−1 and ν→0 the vector I(t)=(I1(u(t,⋅)),I2(u(t,⋅)),…) satisfies the (Whitham) averaged equation
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