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We show that the global attractor of a weakly damped and driven Korteweg-deVries equation (KdV) is embedded in the long-time dynamics of an ordinarydifferential equation called a determining form. In particular, there is aone-to-one identification of the trajectories in the global attractor of thedamped and driven KdV and the steady state solutions of the determining form.Moreover, we analyze a data assimilation algorithm (down-scaling) for theweakly damped and driven KdV. We show that given a certain number of lowFourier modes of a reference solution of the KdV equation, the algorithmrecovers the full reference solution at an exponential rate in time.

Determining form and data assimilation algorithm for weakly damped and driven Korteweg-de Vries equaton- Fourier modes case