Reconstructing various regimes of the complex Ginzburg--Landau model from data

The complex Ginzburg--Landau equation (CGLE) is widely used in fluid mechanics, superconductivity, liquid crystal and string theory. CGLE also models the amplitude variation of traveling wave solution occurring at low frequencies, which often results in modulational instability. Relative significance of linear and nonlinear dispersion is described by the Benjamin--Feir criterion with which various regimes of CGLE are characterized. Those regimes exhibit stark qualitative differences, rendering the data-based eduction of CGLE less straightforward. This study applies a sparse regression method (PDE-FIND) to discover CGLE out of high-fidelity simulation data and show the robustness of the algorithm. The one-dimensional cubic CGLE is solved using a non-dissipative high-order compact finite difference scheme. Several distinct regimes of CGLE characterized by regular plane wave, spatio-temporal chaos, intermittency and coherent structures, respectively, are simulated by altering the linear and nonlinear dispersion parameters. The numerical data are exploited to discover CGLE. Excellent reconstruction accuracy is obtained regardless of the regimes of CGLE.