On long time dynamics of perturbed KdV equations

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 $ε$-quasi-invariant measure $μ$ and satisfies some additional mild assumptions. Let $u^ε(t)$ be a solution. Then on time intervals of order $ε^{-1}$, as $ε\to0$, its actions $I(u^ε(t,\cdot))$ can be approximated by solutions of a certain well-posed averaged equation, provided that the initial datum is $μ$-typical.

No reviews yet.

No comments yet.