Determining form and data assimilation algorithm for weakly damped and driven Korteweg-de Vries equation - Fourier modes case
Jolly, Michael S.
Titi, Edriss S.
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SubjectKdV equation; data assimilation; Korteweg–de Vries equation; differential equations; determining form; Fourier mode
We show that the global attractor of a weakly damped and driven Korteweg–de Vries equation (KdV) is embedded in the long-time dynamics of an ordinary differential equation called a determining form. In particular, there is a one-to-one identification of the trajectories in the global attractor of the damped and driven KdV and the steady state solutions of the determining form. Moreover, we analyze a data assimilation algorithm (down-scaling) for the weakly damped and driven KdV. We show that given a certain number of low Fourier modes of a reference solution of the KdV equation, the algorithm recovers the full reference solution at an exponential rate in time.