Coupled complex Ginzburg–Landau systems with saturable nonlinearity and asymmetric cross-phase modulation

We formulate and study dynamics from a complex Ginzburg–Landau system with saturable nonlinearity, including asymmetric cross-phase modulation (XPM) parameters. Such equations can model phenomena described by complex Ginzburg–Landau systems under the added assumption of saturable media. When the saturation parameter is set to zero, we recover a general complex cubic Ginzburg–Landau system with XPM. We first derive conditions for the existence of bounded dynamics, approximating the absorbing set for solutions. We use this to then determine conditions for amplitude death of a single wavefunction. We also construct exact plane wave solutions, and determine conditions for their modulational instability. In a degenerate limit where dispersion and nonlinearity balance, we reduce our system to a saturable nonlinear Schrödinger system with XPM parameters, and we demonstrate the existence and behavior of spatially heterogeneous stationary solutions in this limit. Using numerical simulations we verify the aforementioned analytical results, while also demonstrating other interesting emergent features of the dynamics, such as spatiotemporal chaos in the presence of modulational instability. In other regimes, coherent patterns including uniform states or banded structures arise, corresponding to certain stable stationary states. For sufficiently large yet equal XPM parameters, we observe a segregation of wavefunctions into different regions of the spatial domain, while when XPM parameters are large and take different values, one wavefunction may decay to zero in finite time over the spatial domain (in agreement with the amplitude death predicted analytically). We also find a collection of transient features, including transient defects and what appear to be rogue waves, while in two spatial dimensions we observe highly localized pattern formation. While saturation will often regularize the dynamics, such transient dynamics can still be observed – and in some cases even prolonged – as the saturability of the media is increased, as the saturation may act to slow the timescale.