Applied Koopman Theory for Partial Differential Equations and Data-Driven Modeling of Spatio-Temporal Systems

Figure 6

Spatio-temporal breather solution of the CQGLE equation for the parameter regime , , , , , and . Although the dynamics illustrated (a) look relatively simple, the singular value decay (b) shows that a large number of modes are required to reconstruct the fine spatio-temporal features of the nonlinear evolution. Moreover, the temporal dynamics of the first four POD modes, which are extracted from the columns of the matrix of the SVD (c), characterize a complicated temporal behavior for the individual modes. Unlike the NLS example, a low-rank approximation does not work well for reconstructing the dynamics. The simulation was performed over with 301 equally spaced snapshots taken. The domain was discretized with points on a domain .

(a)

(b)

(c)