Sergei B. Kuksin
Damped-driven KdV and effective equations for long-time behaviour of its
solutions
(438K, pdf)
ABSTRACT. For the damped-driven KdV equation
$$
\dot u-\nu{u_{xx}}+u_{xxx}-6uu_x=\sqrt\nu\,\eta(t,x),\; x\in S^1,\
\int u\,dx\equiv \int\eta\,dx\equiv0\,,
$$
with $0<\nu\le1$ and smooth in $x$
white in $t$ random force $\eta$, we study the limiting
long-time behaviour of the KdV integrals of motions
$(I_1,I_2,\dots)$, evaluated along a solution $u^\nu(t,x)$, as $\nu\to0$. We prove that for $0\le\tau:= \nu t \lesssim1$ the vector
$
I^\nu( \tau)=(I_1(u^\nu( \tau,\cdot)),I_2(u^\nu( \tau,\cdot)),\dots),
$
converges in distribution to a limiting
process $I^0(\tau)=(I^0_1,I^0_2,\dots)$. The $j$-th component $I_j^0$
equals $\12(v_j(\tau)^2+v_{-j}(\tau)^2)$, where
$v(\tau)=(v_1(\tau), v_{-1}(\tau),v_2(\tau),\dots)$ is the vector of Fourier coefficients of a solution of an
{\it effective equation} for the damped-driven KdV. This new equation
is a quasilinear stochastic heat equation with a non-local
nonlinearity, written in the Fourier coefficients. It is well posed.