Symmetry-breaking Instability Research Articles

Numerical computations of the flow in a finite diverging channel

The flow in a finite diverging channel opening into a large space and resembling the experimental prototype of Putkaradze and Vorobieff (2006) was numerically investigated. The effects of the Reynolds number, initial condition, intersection angle, length of the wedge edges, and the outer boundary condition were examined. The numerical results showed that the flow in the wedge undergoes a change from symmetrical flow to unsymmetrical flow with a weak backflow, then a vortical (circulation) flow and finally an unsteady jet flow as the Reynolds number is increased for an intersection angle of 32° and a wedge edge of length 30 times the width of the inlet slit. For the unsteady flow, the jet attached to one side of the wedge constantly loses stability and rolls up into a mushroom-shaped vortex-pair near the outlet of the wedge. As the intersection angle is increased to 50°, a stable jet flow is observed as a new regime between the vortex and unsteady regimes. Both the intersection angle and the wedge length have negative effects on the stability of the flow, although the effect of the wedge length on the critical Reynolds number for the symmetry-breaking instability is not pronounced. The outer boundary condition was found not to affect the flow patterns inside the wedge significantly. At a certain Re regime above the onset of symmetry-breaking instability, the flows evolve into steady state very slowly except for the initial stage in the case of decreasing flow flux. Two different solutions can be observed within the normal observation time for the experiment, providing a possible explanation for the hysteresis phenomenon in the experiment.

Symmetry breaking, snap-through and pull-in instabilities under dynamic loading ofmicroelectromechanical shallow arches

Arch-shaped microelectromechanical systems (MEMS) have been used as mechanicalmemories, micro-relays, micro-valves, optical switches and digital micro-mirrors. A bi-stablestructure, such as an arch, is characterized by a multivalued load deflection curve. Here westudy the symmetry breaking, the snap-through instability and the pull-in instability of abi-stable arch-shaped MEMS under static and dynamic electric loads. Unlike a mechanicalload, the electric load is a nonlinear function of the a priori unknown deformedshape of the arch. The nonlinear partial differential equation governing transientdeformations of the arch is solved numerically using the Galerkin method and atime integration scheme that adaptively adjusts the time step to compute thesolution within the prescribed tolerance. For the static problem, the displacementcontrol and the pseudo-arc-length continuation methods are used to obtain thebifurcation curve of the arch’s displacement versus a load parameter. The displacementcontrol method fails to compute the arch’s asymmetric deformations that arefound by the pseudo-arc-length continuation method. For the dynamic problem,two distinct mechanisms of the snap-through instability are found. It is shownthat critical loads and geometric parameters for instabilities of an arch under anelectric load with and without consideration of mechanical inertia effects arequite different. A phase diagram between a critical load parameter and the archheight is constructed to delineate different regions of instabilities. We compareresults from the present model with those from a continuum mechanics basedapproach, and with results of other models and experiments available in the literature.

Skew-symmetric vortices and solitons in crossed-lattice potentials

We introduce a wave-propagation model for a pair of linearly coupled two-dimensional (2D) cores with intrinsic cubic nonlinearity—self-focusing (attractive) or self-defocusing (repulsive). Each core carries a periodic potential corresponding to a 1D lattice, the two lattices being mutually perpendicular. Localized states in the model are classified according to their skew-symmetry or skew-antisymmetry, that combine the interchange of the cores and the rotation of coordinates by 90°. Such a setting would be difficult to implement in ordinary optics, but it has a straightforward realization in terms of atom-wave optics, using a Bose–Einstein condensate confined to a pair of tunnel-coupled parallel 'pancake-shaped' traps, which are equipped with crossed 1D optical lattices (an example of a greater versatility of matter-wave optics, as compared to the optics of photons). For the attractive nonlinearity in both cores, symmetry-breaking bifurcations are found, with the help of the variational approximation and, chiefly, by means of numerical methods, in families of skew-symmetric fundamental solitons, dipoles, and vortices. A new species of topologically organized states, which is specific to the dual-core system with crossed 1D lattices, is reported in the form of crosses. They are built as crossed dipole pairs in the two cores, with a phase shift of π/2. Actually, the crosses represent another species of skew-symmetric vortices, different from the ordinary ones. All topologically structured modes in the system exist in pairs: straight and diagonal dipoles, rhombus- and square-shaped vortices (alias on- and off-site ones), and straight and oblique crosses. A noteworthy dynamical feature is the fact that the instability which breaks the skew-symmetry of square- and rhombus-shaped vortices does not lead to their transformation into stable asymmetric counterparts, even though such states are found as stationary solutions. Actually, unstable squares and rhombuses always transform themselves, respectively, into stable crosses of the oblique and straight types. Fundamental solitons and vortices, which do not feature the symmetry-breaking instability, are also found in the first bandgap of the repulsive system, and in the semi-infinite gap of the mixed attractive–repulsive model.

Study and Simulation of Reaction–Diffusion Systems Affected by Interacting Signaling Pathways

Possible effects of interaction (cross-talk) between signaling pathways is studied in a system of Reaction-Diffusion (RD) equations. Furthermore, the relevance of spontaneous neurite symmetry breaking and Turing instability has been examined through numerical simulations. The interaction between Retinoic Acid (RA) and Notch signaling pathways is considered as a perturbation to RD system of axon-forming potential for N2a neuroblastoma cells. The present work suggests that large increases to the level of RA-Notch interaction can possibly have substantial impacts on neurite outgrowth and on the process of axon formation. This can be observed by the numerical study of the homogeneous system showing that in the absence of RA-Notch interaction the unperturbed homogeneous system may exhibit different saddle-node bifurcations that are robust under small perturbations by low levels of RA-Notch interactions, while large increases in the level of RA-Notch interaction result in a number of transitions of saddle-node bifurcations into Hopf bifurcations. It is speculated that near a Hopf bifurcation, the regulations between the positive and negative feedbacks change in such a way that spontaneous symmetry breaking takes place only when transport of activated Notch protein takes place at a faster rate.

Simulation study of the symmetry-breaking instability and the dipole field reversal in a rotating spherical shell dynamo

The reversal mechanism of a dipole magnetic field generated by dynamo action in a rotating spherical shell is investigated by a three-dimensional nonlinear magnetohydrodynamic simulation as well as a linear stability analysis. The emphasis of the study is on understanding the relationship between dipole reversal and the symmetry properties of the dynamo solution. As a result, first, it is found that there is a threshold of the magnetic Prandtl number, below which the dipole field is never reversed, and above which the reversal occurs at irregular intervals like the paleomagnetic evolution of the geodynamo. Second, it is shown that the dynamo process responsible for the generation of a dipole field (called “a-dynamo” in this paper) consists only of the antimirror symmetric magnetic field and the mirror symmetric velocity field with respect to the equatorial plane. Third, it is found that the components of the opposite symmetry to the a-dynamo grow only during the polarity reversal events and quickly decay afterwards. This indicates that the dipole field reversal and the loss of equatorial symmetry are tightly connected. In fact, it is clearly demonstrated by numerical analyses that the a-dynamo process is linearly unstable for the perturbation of opposite symmetry when the magnetic Prandtl number exceeds the threshold for dipole reversal. Mode coupling between the longitudinal Fourier components plays a crucial role in creating the instability. Based on the above results, it is proposed that symmetry-breaking instability could be the mechanism for dipole field reversal in the geodynamo process. The energy conversion between components of different symmetry is also analyzed in the quasistable polarity phase and in the polarity reversal phase, respectively.

Symmetry-breaking instabilities of generalized elliptical solitons

Elliptical solitons in 2D nonlinear Schödinger equations are studied numerically with a more-generalized formulation. New families of solitons, vortices, and soliton rings with elliptical symmetry are found and investigated. With a suitable symmetry-breaking parameter, we show that perturbed elliptical solitons tend to move transversely owing to the existences of dipole excitation modes, which are totally suppressed in circularly symmetric solitons. Furthermore, by numerical evolutions we demonstrate that elliptical vortices and soliton rings collapse into a pair of stripes and clusters, respectively, revealing the experimental observations in the literature.

Symmetry Breaking of Vortex Patterns in a Rotating Harmonic Potential

We numerically study the symmetry breaking instabilities of vortex patterns in a rotating harmonic potential using a type of Ginzburg-Landau equation. The configurations of vortex lattices change markedly by the symmetry-breaking instabilities, and then, some vortices move away from the confinement potential, which leads to the annihilation of vortices. The symmetry-breaking instabilities and the instabilities of vortex nucleation determine the parameter region of stable vortex patterns. We verify that the symmetry-breaking instabilities also occur in a type of complex Ginzburg-Landau equation.

Suppression of multi-period instabilities by third-order dispersion in mode-locked Yb-doped fiber lasers

Multi-period pulsing instabilities in a Yb-doped ring fiber laser with a mode-locking mechanism provided by nonlinear polarization rotation are studied by means of numerical simulations. The impact of the third-order dispersion (TOD) of cavity elements on dynamic instabilities of the stretched-pulse mode-locked regime is elucidated. Different instability types with symmetry-preserving and symmetry-breaking features are found in a wide range of cavity group velocity dispersion (GVD). At a large anomalous cavity GVD, where a stretched-pulse mode-locked regime borders the soliton regime, multi-period pulsing instabilities result from the development of new resonant subsidebands associated with unstable dispersive waves. When a cavity GVD approaches zero and resonant sidebands are suppressed, multi-period pulsing is caused by parametric instability of the frequencies near the spectrum center. It is demonstrated that TOD suppresses both symmetry-preserving and symmetry-breaking multi-period instabilities in the stretched-pulse regime. From the analysis of the spectral sidebands, we conclude that instability suppression is related with a modification of the resonant coupling between the mode-locked pulse and the phase-velocity as well as group-velocity-matched dispersive waves.

The Shape of Charged Drops over a Solid Surface and Symmetry-Breaking Instabilities

We study the static shape of charged drops of a conducting fluid placed over a solid substrate, surrounded by a gas, and in absence of gravitational forces. The question can be formulated as a variational problem where a certain energy involving the areas of the solid-liquid interface and of the liquid-gas interface, as well as the electric capacity of the drop, has to be minimized. As a function of two parameters, Young's angle and the potential at the drop's surface , we find the axisymmetric minimizers of the energy and describe their shape. We also discuss the existence of symmetry-breaking bifurcations such that, for given values of and , configurations for which the axial symmetry is lost are energetically more favorable than axially symmetric configurations. We prove the existence of such bifurcations in the limits of very flat and almost spherical equilibrium shapes. All other cases are studied numerically with a boundary integral method. One conclusion of this study is that axisymmetric drops cannot spread indefinitely by introducing sufficient amount of electric charges, but can reach only a limiting (saturation) size, after which the axial symmetry would be lost and finger-like shapes energetically preferred.

Gas-liquid Like Phase Transition in Granular Gases Under Zero Gravity

颗粒体系是远离平衡态的复杂散耗体系，在外力驱动下，能表现出类似于固体、液体或气体的特性。在外力驱动下，能表现出类似于固体、液体、或气体的特性，在稀疏的颗粒气体中，由于耗散，体系中常常会形成颗粒在局部的凝聚。这种凝聚行为具有自发对称性破缺的特征，使得颗粒气体中自动形成高密度区域和低密度区域共存的非均匀状态，类似于气体中液滴的形成。本文利用零重力条件下的三维颗粒气体理论模型进行计算和分析，揭示了颗粒气体这一相变行为的不稳定性根源及类气液相变本质，给出了颗粒气体中这种相分离发生的临界条件，并通过了分子动力学模拟进行检验，结果从定性上和定量上都能够很好地吻合，为进一步的空间实验提供了理论依据和相关实验参数。

Experimental observation of a nonpitchfork acceleration instability in a nematic liquid crystal

The velocity of domain walls caused by a symmetry-breaking instability is experimentally investigated for the case of electroconvection in a nematic liquid crystal. It turns out that the velocity increases linearly with the distance from the bifurcation creating the domains. This scaling behavior does not agree with the drift pitchfork bifurcation. It has, however, been predicted by a minimal model describing our scenario.

Creeping solitons in dissipative systems and their bifurcations

We present a detailed numerical study of creeping solitons in dissipative systems. A bifurcation diagram has been constructed for the region of transition between solitons and fronts. It shows a rich variety of transitions between various types of localized solutions. For the first time, we have found a sequence of period-doubling bifurcations of creeping solitons, and also a symmetry-breaking instability of creeping solitons. Creeping solitons may involve many frequencies in their dynamics, and this can result, in particular, in a multiplicity of zig-zag motions.

Symmetry-breaking instabilities in perturbed optical lattices: Nonlinear nonreciprocity and macroscopic self-trapping

We develop an asymptotic analysis of nonlinear energy propagation in lattices subject to slowly varying perturbations, investigating symmetry breaking and its effects. We derive a general set of evolution equations and study them by using catastrophe theory, revealing a wealth of system dynamics. Below a power threshold, symmetry breaking drives nonreciprocal oscillations; beyond that, symmetry breaking yields an effect of ``macroscopic'' self-trapping, which supports a self-maintained energy imbalance between Bloch bands. We numerically verify the theoretical results and discuss their possible implementation in waveguide arrays.

Synthetic Turing protocells: vesicle self-reproduction through symmetry-breaking instabilities

The reproduction of a living cell requires a repeatable set of chemical events to be properly coordinated. Such events define a replication cycle, coupling the growth and shape change of the cell membrane with internal metabolic reactions. Although the logic of such process is determined by potentially simple physico-chemical laws, modelling of a full, self-maintained cell cycle is not trivial. Here we present a novel approach to the problem that makes use of so-called symmetry breaking instabilities as the engine of cell growth and division. It is shown that the process occurs as a consequence of the breaking of spatial symmetry and provides a reliable mechanism of vesicle growth and reproduction. Our model opens the possibility of a synthetic protocell lacking information but displaying self-reproduction under a very simple set of chemical reactions.

RENORMALIZATION GROUP APPROACH TO THE PAIRING INSTABILITIES

By means of the continuous unitary transformation similar to a general scheme of the Renormalization Group (RG) procedure we study the issue of symmetry breaking and pairing instability in the system of interacting fermions. Constructing a generalized version of the Bogoliubov transformation we show that formation of the fermion pairs and their superconductivity/superfluidity can appear at different temperatures. It is shown that strong quantum fluctuations can destroy the long-range order without breaking the fermion pairs which may still exist as incoherent and/or damped entities. Such unusual phase is characterized by a partial suppression of the density of states near the Fermi energy and by residual collective features like the sound-wave mode in the fermion pair spectrum.

Symmetry breaking of discrete solitons and its suppression by partial incoherence

We study the symmetry breaking instability of discrete solitons with even parity in a 1-D waveguide array, and find that such instability can be suppressed by adding spatial incoherence. This is true for both staggered and unstaggered modes.

Transitions of axisymmetric flow between two corotating disks in an enclosure

Transitions of flow between two corotating disks in an enclosure are investigated numerically. The outer cylindrical boundary of the flow field is fixed, whereas the inner cylinder rotates together with the two disks. The flow is not only symmetric with respect to the inter-disk midplane but also axisymmetric around the axis of rotation at small Reynolds numbers although it becomes unstable to disturbances at large Reynolds numbers. Two kinds of instability modes are known, one of which breaks the axisymmetry of the flow to yield a polygonal flow pattern in the radial–tangential plane. Such instability occurs for small length ratios, the ratio of the length of cylinders (gap between two disks) to the width of the annulus. The other instability is a symmetry-breaking instability with respect to the inter-disk midplane retaining the axisymmetry, which occurs for intermediate length ratios. We investigate the latter symmetry-breaking instability assuming the axisymmetry of the flow field, in which we obtain steady axisymmetric flows numerically and analyze their linear stability. It is found that oscillatory flow as well as steady asymmetric flow appears resulting from the first instability even in the intermediate length ratios where the flow remains axisymmetric. Numerical simulations are also performed to obtain time-periodic flows, and bifurcation diagrams of the flow are depicted for various values of length ratio. The critical Reynolds numbers for the Hopf and symmetry-breaking pitchfork instabilities are evaluated and a transition diagram is obtained.

Switching and instabilities of optical vortices in nonlinear dual-core photonic crystal fibre couplers

We study switching of an optical vortex launched into one core of a dual-core waveguide coupler in a photonic crystal fibre with self-focusing nonlinearity. We analyse how the beam power and the angular momentum associated with the vortex mode transfer to the second core of the coupler in both linear and nonlinear regimes. We describe three major scenarios of the vortex dynamics and reveal novel symmetry-breaking instabilities associated with the vortex nonzero angular momentum.

Modeling bond-order wave instabilities in doped frustrated antiferromagnets: Valence bond solids at fractional filling

We explore both analytically and numerically the properties of doped t-J models on a class of highly frustrated lattices, such as the kagome and the pyrochlore lattice. Focussing on a particular sign of the hopping integral and antiferromagnetic exchange, we find a generic symmetry breaking instability towards a twofold degenerate ground state at a fractional filling below half filling. These states show modulated bond strengths and only break lattice symmetries. They can be seen as a generalization of the well-known valence bond solid states to fractional filling.

Dynamics of two-dimensional coherent structures in nonlocal nonlinear media

We study stability and dynamics of the single cylindrically symmetric solitary structures and dipolar solitonic molecules in spatially nonlocal media. The main properties of the solitons, vortex solitons, and dipolar solitons are investigated analytically and numerically. The vortices and higher-order solitons show the transverse symmetry-breaking azimuthal instability below some critical power. We find the threshold of the vortex soliton stabilization using the linear stability analysis and direct numerical simulations. The higher-order solitons, which have a central peak and one or more surrounding rings, are also demonstrated to be stabilized in nonlocal nonlinear media. Using direct numerical simulations, we find a class of radially asymmetric, dipolelike solitons and show that, at sufficiently high power, these structures are stable.