Determining form and data assimilation algorithm for weakly damped and driven Korteweg–de Vries equation — Fourier modes case. (August 2017)

• Record Type: Journal Article
• Authors: Jolly, Michael S.; Sadigov, Tural; Titi, Edriss S.
• Is Part Of: Nonlinear analysis. Volume 36(2017)
• Page Start: 287
• Page End: 317
• Publication Date: 2017-08
• Subjects: KdV equation -- Determining forms -- Data assimilation; Nonlinear functional analysis -- Periodicals; Analyse fonctionnelle non linéaire -- Périodiques; Nonlinear functional analysis; Periodicals; 515.7248
• DOI: 10.1016/j.nonrwa.2017.01.010
• Languages: English
• ISSNs: 1468-1218

Abstract

Abstract: 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.