An averaging theorem for a perturbed KdV equation

We consider a perturbed KdV equation: u̇+ u_xxx - 6uu _x = εf (x, u(·)), x ε t, ∫_t u dx = 0. For any periodic function u(x), let I (u) = I₁ (u), I₂ (u), . . .) ε r^∞₊ be the vector, formed by the KdV integrals of motion, calculated for the potential u(x). Assuming that the perturbation -f (x, u(x)) defines a smoothing mapping u(x) → f (x, u(x)) (e.g. it is a smooth function -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 ≥ t ≲ε^-1, the vector I (u(t)) may be well approximated by a solution of theaveraged equation.