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Damped-driven KdV and effective equation for long-time behaviour of its solutions

For the damped-driven KdV equation $$ \dot u-ν{u_{xx}}+u_{xxx}-6uu_x=\sqrtνη(t,x), x\in S^1, \int u dx\equiv \intηdx\equiv0, $$ with $0<ν\le1$ and smooth in $x$ white in $t$ random force $η$, we study the limiting long-time behaviour of the KdV integrals of motions $(I_1,I_2,...)$, evaluated along a solution $u^ν(t,x)$, as $ν\to0$. We prove that %if $u=u^ν(t,x)$ is a solution of the equation above, for $0\leτ:= νt \lesssim1$ the vector $ I^ν(τ)=(I_1(u^ν(τ,\cdot)),I_2(u^ν(τ,\cdot)),...), $ converges in distribution to a limiting process $I^0(τ)=(I^0_1,I^0_2,...)$. The $j$-th component $I_j^0$ equals $\12(v_j(τ)^2+v_{-j}(τ)^2)$, where $v(τ)=(v_1(τ), v_{-1}(τ),v_2(τ),...)$ is the vector of Fourier coefficients of a solution of an {\it effective equation} for the dam-ped-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.