On long time dynamics of perturbed KdV equations

Abstract

Consider perturbed KdV equations: \[u_t+u_{xxx}-6uu_x=εf(u(\cdot)),\quad x\in\mathbb{T}=\mathbb{R}/\mathbb{Z},\;\int_{\mathbb{T}}u(x,t)dx=0,\] where the nonlinearity defines analytic operators $u(\cdot )\mapsto f(u(\cdot ))$ in sufficiently smooth Sobolev spaces. Assume that the equation has an $\epsilon$-quasi-invariant measure $\mu$ and satisfies some additional mild assumptions. Let $u^{\epsilon }(t)$ be a solution. Then on time intervals of order ${\epsilon }^{-1}$, as $\epsilon \to 0$, its actions $I(u^{\epsilon }(t,\cdot ))$ can be approximated by solutions of a certain well-posed averaged equation, provided that the initial datum is $\mu$-typical.