10-40 Sergei B. Kuksin

Damped-driven KdV and effective equations for long-time behaviour of its solutions (438K, pdf) Feb 25, 10

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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.

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