128,474 research outputs found
Transverse Patterns in Nonlinear Optical Resonators
The book is devoted to the formation and dynamics of localized structures
(vortices, solitons) and extended patterns (stripes, hexagons, tilted waves) in
nonlinear optical resonators such as lasers, optical parametric oscillators,
and photorefractive oscillators. The theoretical analysis is performed by
deriving order parameter equations, and also through numerical integration of
microscopic models of the systems under investigation. Experimental
observations, and possible technological implementations of transverse optical
patterns are also discussed. A comparison with patterns found in other
nonlinear systems, i.e. chemical, biological, and hydrodynamical systems, is
given. This article contains the table of contents and the introductory chapter
of the book.Comment: 37 pages, 14 figures. Table of contents and introductory chapter of
the boo
Nonlinear Polariton Fluids in a Flatband Reveal Discrete Gap Solitons
Phase frustration in periodic lattices is responsible for the formation of
dispersionless flat bands. The absence of any kinetic energy scale makes flat
band physics critically sensitive to perturbations and interactions. We report
here on the experimental investigation of the nonlinear dynamics of cavity
polaritons in the gapped flat band of a one-dimensional Lieb lattice. We
observe the formation of gap solitons with quantized size and very abrupt
edges, signature of the frozen propagation of switching fronts. This type of
gap solitons belongs to the class of truncated Bloch waves, and had only been
observed in closed systems up to now. Here the driven-dissipative character of
the system gives rise to a complex multistability of the nonlinear domains
generated in the flat band. These results open up interesting perspective
regarding more complex 2D lattices and the generation of correlated photon
phases.Comment: 6 pages, 4 figures + supplemental material (6 pages, 6 figures
Nonlinear Analysis of the Eckhaus Instability: Modulated Amplitude Waves and Phase Chaos with Non-zero Average Phase Gradient
We analyze the Eckhaus instability of plane waves in the one-dimensional
complex Ginzburg-Landau equation (CGLE) and describe the nonlinear effects
arising in the Eckhaus unstable regime. Modulated amplitude waves (MAWs) are
quasi-periodic solutions of the CGLE that emerge near the Eckhaus instability
of plane waves and cease to exist due to saddle-node bifurcations (SN). These
MAWs can be characterized by their average phase gradient and by the
spatial period P of the periodic amplitude modulation. A numerical bifurcation
analysis reveals the existence and stability properties of MAWs with arbitrary
and P. MAWs are found to be stable for large enough and
intermediate values of P. For different parameter values they are unstable to
splitting and attractive interaction between subsequent extrema of the
amplitude. Defects form from perturbed plane waves for parameter values above
the SN of the corresponding MAWs. The break-down of phase chaos with average
phase gradient > 0 (``wound-up phase chaos'') is thus related to these
SNs. A lower bound for the break-down of wound-up phase chaos is given by the
necessary presence of SNs and an upper bound by the absence of the splitting
instability of MAWs.Comment: 24 pages, 14 figure
Stochastic reaction & diffusion on growing domains: understanding the breakdown of robust pattern formation
Many biological patterns, from population densities to animal coat markings, can be thought of as heterogeneous spatiotemporal distributions of mobile agents. Many mathematical models have been proposed to account for the emergence of this complexity, but, in general, they have consisted of deterministic systems of differential equations, which do not take into account the stochastic nature of population interactions. One particular, pertinent criticism of these deterministic systems is that the exhibited patterns can often be highly sensitive to changes in initial conditions, domain geometry, parameter values, etc. Due to this sensitivity, we seek to understand the effects of stochasticity and growth on paradigm biological patterning models. In this paper, we extend spatial Fourier analysis and growing domain mapping techniques to encompass stochastic Turing systems. Through this we find that the stochastic systems are able to realize much richer dynamics than their deterministic counterparts, in that patterns are able to exist outside the standard Turing parameter range. Further, it is seen that the inherent stochasticity in the reactions appears to be more important than the noise generated by growth, when considering which wave modes are excited. Finally, although growth is able to generate robust pattern sequences in the deterministic case, we see that stochastic effects destroy this mechanism for conferring robustness. However, through Fourier analysis we are able to suggest a reason behind this lack of robustness and identify possible mechanisms by which to reclaim it
Spatio-temporal dynamics induced by competing instabilities in two asymmetrically coupled nonlinear evolution equations
Pattern formation often occurs in spatially extended physical, biological and
chemical systems due to an instability of the homogeneous steady state. The
type of the instability usually prescribes the resulting spatio-temporal
patterns and their characteristic length scales. However, patterns resulting
from the simultaneous occurrence of instabilities cannot be expected to be
simple superposition of the patterns associated with the considered
instabilities. To address this issue we design two simple models composed by
two asymmetrically coupled equations of non-conserved (Swift-Hohenberg
equations) or conserved (Cahn-Hilliard equations) order parameters with
different characteristic wave lengths. The patterns arising in these systems
range from coexisting static patterns of different wavelengths to traveling
waves. A linear stability analysis allows to derive a two parameter phase
diagram for the studied models, in particular revealing for the Swift-Hohenberg
equations a co-dimension two bifurcation point of Turing and wave instability
and a region of coexistence of stationary and traveling patterns. The nonlinear
dynamics of the coupled evolution equations is investigated by performing
accurate numerical simulations. These reveal more complex patterns, ranging
from traveling waves with embedded Turing patterns domains to spatio-temporal
chaos, and a wide hysteretic region, where waves or Turing patterns coexist.
For the coupled Cahn-Hilliard equations the presence of an weak coupling is
sufficient to arrest the coarsening process and to lead to the emergence of
purely periodic patterns. The final states are characterized by domains with a
characteristic length, which diverges logarithmically with the coupling
amplitude.Comment: 9 pages, 10 figures, submitted to Chao
The large core limit of spiral waves in excitable media: A numerical approach
We modify the freezing method introduced by Beyn & Thuemmler, 2004, for
analyzing rigidly rotating spiral waves in excitable media. The proposed method
is designed to stably determine the rotation frequency and the core radius of
rotating spirals, as well as the approximate shape of spiral waves in unbounded
domains. In particular, we introduce spiral wave boundary conditions based on
geometric approximations of spiral wave solutions by Archimedean spirals and by
involutes of circles. We further propose a simple implementation of boundary
conditions for the case when the inhibitor is non-diffusive, a case which had
previously caused spurious oscillations.
We then utilize the method to numerically analyze the large core limit. The
proposed method allows us to investigate the case close to criticality where
spiral waves acquire infinite core radius and zero rotation frequency, before
they begin to develop into retracting fingers. We confirm the linear scaling
regime of a drift bifurcation for the rotation frequency and the core radius of
spiral wave solutions close to criticality. This regime is unattainable with
conventional numerical methods.Comment: 32 pages, 17 figures, as accepted by SIAM Journal on Applied
Dynamical Systems on 20/03/1
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