Determining form and data assimilation algorithm for weakly damped and driven Korteweg-de Vries equaton- Fourier modes case
Dynamical Systems
2015-10-12 v1 Analysis of PDEs
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.
Cite
@article{arxiv.1510.02730,
title = {Determining form and data assimilation algorithm for weakly damped and driven Korteweg-de Vries equaton- Fourier modes case},
author = {Michael S. Jolly and Tural Sadigov and Edriss S. Titi},
journal= {arXiv preprint arXiv:1510.02730},
year = {2015}
}