An Averaging Theorem for Perturbed KdV Equation

We consider a perturbed KdV equation: [\dot{u}+u_{xxx} - 6uu_x = \epsilon f(x,u(\cdot)), \quad x\in \mathbb{T}, \quad\int_\mathbb{T} u dx=0.] For any periodic function $u(x)$, let $I(u)=(I_1(u),I_2(u),...)\in\mathbb{R}_+^{\infty}$ be the vector, formed by the KdV integrals of motion, calculated for the potential $u(x)$. Assuming that the perturbation $\epsilon f(x,u(\cdot))$ is a smoothing mapping (e.g. it is a smooth function $\epsilon f(x)$, independent from $u$), and that solutions of the perturbed equation satisfy some mild a-priori assumptions, we prove that for solutions $u(t,x)$ with typical initial data and for $0\leqslant t\lesssim \epsilon^{-1}$, the vector $I(u(t))$ may be well approximated by a solution of the averaged equation