Determining Nodes for the Damped Forced Periodic Korteweg-de Vries Equation

Abstract

We show that solutions of the periodic KdV equations

$$\begin{aligned} u_t+\gamma u +u_{xxx}+uu_x=f, \end{aligned}$$

are asymptotically determined by their values at three points. That is if there exists $x_1,x_2,x_3$ such that $0< x_3-x_2<<x_3-x_1<<1$ and

$$\begin{aligned} \lim _{t\rightarrow +\infty } |u_1(t,x_j)-u_2(t,x_j)|=0, \; \mathrm{for} \; j=1,2,3, \end{aligned}$$

for two solutions $u_1,u_2$ of the KdV equation above, then

$$\begin{aligned} \lim _{t\rightarrow +\infty }||u_1(t)-u_2(t)||_{H^1}=0. \end{aligned}$$