Complex Ginzburg–Landau equation for time-varying anisotropic media

Abstract

When extending the complex Ginzburg–Landau equation (CGLE) to more than one spatial dimension, there is an underlying question of whether one is capturing all the interesting physics inherent in these higher dimensions. Although spatial anisotropy is far less studied than its isotropic counterpart, anisotropy is fundamental in applications to superconductors, plasma physics, and geology, to name just a few examples. We first formulate the CGLE on anisotropic, time-varying media, with this time variation permitting a degree of control of the anisotropy over time, focusing on how time-varying anisotropy influences diffusion and dispersion within both bounded and unbounded space domains. From here, we construct a variety of exact dissipative nonlinear wave solutions, including analogs of wavetrains, solitons, breathers, and rogue waves, before outlining the construction of more general solutions via a dissipative, nonautonomous generalization of the variational method. We finally consider the problem of modulational instability within anisotropic, time-varying media, obtaining generalizations to the Benjamin–Feir instability mechanism. We apply this framework to study the emergence and control of anisotropic spatiotemporal chaos in rectangular and curved domains. Our theoretical framework and specific solutions all point to time-varying anisotropy being a potentially valuable feature for the manipulation and control of waves in anisotropic media.