2,480 research outputs found
Swift-Hohenberg equation with broken reflection symmetry
The bistable Swift-Hohenberg equation possesses a variety of time-independent spatially localized solutions organized in the so-called snakes-and-ladders structure. This structure is a consequence of a phenomenon known as homoclinic snaking, and is in turn a consequence of spatial reversibility of the equation. We examine here the consequences of breaking spatial reversibility on the snakes-and-ladders structure. We find that the localized states now drift, and show that the snakes-and-ladders
structure breaks up into a stack of isolas. We explore the evolution of this new structure with increasing reversibility breaking and study the dynamics of the system outside of the snaking region using a combination of numerical and analytical techniques
Solidification in soft-core fluids: disordered solids from fast solidification fronts
Using dynamical density functional theory we calculate the speed of
solidification fronts advancing into a quenched two-dimensional model fluid of
soft-core particles. We find that solidification fronts can advance via two
different mechanisms, depending on the depth of the quench. For shallow
quenches, the front propagation is via a nonlinear mechanism. For deep
quenches, front propagation is governed by a linear mechanism and in this
regime we are able to determine the front speed via a marginal stability
analysis. We find that the density modulations generated behind the advancing
front have a characteristic scale that differs from the wavelength of the
density modulation in thermodynamic equilibrium, i.e., the spacing between the
crystal planes in an equilibrium crystal. This leads to the subsequent
development of disorder in the solids that are formed. For the one-component
fluid, the particles are able to rearrange to form a well-ordered crystal, with
few defects. However, solidification fronts in a binary mixture exhibiting
crystalline phases with square and hexagonal ordering generate solids that are
unable to rearrange after the passage of the solidification front and a
significant amount of disorder remains in the system.Comment: 18 pages, 14 fig
Bifurcation of standing waves into a pair of oppositely traveling waves with oscillating amplitudes caused by a three-mode interaction
A novel flow state consisting of two oppositely travelling waves (TWs) with
oscillating amplitudes has been found in the counterrotating Taylor-Couette
system by full numerical simulations. This structure bifurcates out of axially
standing waves that are nonlinear superpositions of left and right handed
spiral vortex waves with equal time-independent amplitudes. Beyond a critical
driving the two spiral TW modes start to oscillate in counterphase due to a
Hopf bifurcation. The trigger for this bifurcation is provided by a nonlinearly
excited mode of different symmetry than the spiral TWs. A three-mode coupled
amplitude equation model is presented that captures this bifurcation scenario.
The mode-coupling between two symmetry degenerate critical modes and a
nonlinearly excited one that is contained in the model can be expected to occur
in other structure forming systems as well.Comment: 4 pages, 5 figure
Amplitude equations for a system with thermohaline convection
The multiple scale expansion method is used to derive amplitude equations for
a system with thermohaline convection in the neighborhood of Hopf and Taylor
bifurcation points and at the double zero point of the dispersion relation. A
complex Ginzburg-Landau equation, a Newell-Whitehead-type equation, and an
equation of the type, respectively, were obtained. Analytic
expressions for the coefficients of these equations and their various
asymptotic forms are presented. In the case of Hopf bifurcation for low and
high frequencies, the amplitude equation reduces to a perturbed nonlinear
Schr\"odinger equation. In the high-frequency limit, structures of the type of
"dark" solitons are characteristic of the examined physical system.Comment: 21 pages, 8 figure
Competition between Traveling Fluid Waves of Left and Right Spiral Vortices and Their Different Amplitude Combinations
Stability, bifurcation properties, and the spatiotemporal behavior of
different nonlinear combination structures of spiral vortices in the counter
rotating Taylor-Couette system are investigated by full numerical simulations
and by coupled amplitude equation approximations. Stable cross-spiral
structures with continuously varying content of left and right spiral modes are
found. They provide a stability transferring connection between the initially
stable, axially counter propagating wave states of pure spirals and the axially
standing waves of so-called ribbons that become stable slightly further away
from onset of vortex flow.Comment: 4 pages, 5 figure
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