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We formulate and study dynamics from a complex Ginzburg-Landau system withsaturable nonlinearity, including asymmetric cross-phase modulation (XPM)parameters. Such equations can model phenomena described by complexGinzburg-Landau systems under the added assumption of saturable media. When thesaturation parameter is set to zero, we recover a general complex cubicGinzburg-Landau system with XPM. We first derive conditions for the existenceof bounded dynamics, approximating the absorbing set for solutions. We use thisto then determine conditions for amplitude death of a single wavefunction. Wealso construct exact plane wave solutions, and determine conditions for theirmodulational instability. In a degenerate limit where dispersion andnonlinearity balance, we reduce our system to a saturable nonlinearSchr\"odinger system with XPM parameters, and we demonstrate the existence andbehavior of spatially heterogeneous stationary solutions in this limit. Usingnumerical simulations we verify the aforementioned analytical results, whilealso demonstrating other interesting emergent features of the dynamics, such asspatiotemporal chaos in the presence of modulational instability. In otherregimes, coherent patterns including uniform states or banded structures arise,corresponding to certain stable stationary states. For sufficiently large yetequal XPM parameters, we observe a segregation of wavefunctions into differentregions of the spatial domain, while when XPM parameters are large and takedifferent values, one wavefunction may decay to zero in finite time over thespatial domain (in agreement with the amplitude death predicted analytically).While saturation will often regularize the dynamics, such transient dynamicscan still be observed - and in some cases even prolonged - as the saturabilityof the media is increased, as the saturation may act to slow the timescale.

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