Determining nodes for the damped forced periodic Korteweg-de Vries equation (Q2419944)

The main result of this paper is that for the damped, space periodic Korteweg-de Vries equation with forcing the number of determining modes is three. More precisely, if there exist $x_1$, $x_2$, $x_3$ such that $0 < x_3 - x_2 \ll x_3 - x_1 \ll 1$ and for two solutions of the equation $u_t+\gamma u+u_{xxx}+uu_x=f$ the asymptotic relation $\lim_{t\to\infty} |u_1(t,x_j)-u_2(t,x_j)|=0$ holds for each $j=1,\, 2,\, 3$, then $\lim_{t\to\infty} \|u_1(t)-u_2(t)\|_{H^1}=0$. The proof involves subtle properties of norms of solutions in Gevrey spaces.