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
Schr\"odinger 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).
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.