Convective and absolute instabilities in the subcritical Ginzburg-Landau equation

We study the nature of the instability of the homogeneous steady states of the subcritical real Ginzburg-Landau equation in the presence of group velocity. The shift of the absolute instability threshold of the trivial steady state, induced by the destabilizing cubic nonlinearities, is confirmed by the numerical analysis of the evolution of its perturbations. It is also shown that the dynamics of these perturbations is such that finite size effects may suppress the transition from convective to absolute instability. Finally, we analyze the instability of the subcritical middle branch of steady states, and show, analytically and numerically, that this branch may be convectively unstable for sufficiently high values of the group velocity. Received 17 December 1998     

Convective and absolute Eckhaus instability leading to modulated waves in a finite box

We report an experimental study of the secondary modulational instability of a one-dimensional nonlinear traveling wave in a long bounded channel. Two qualitatively different instability regimes involving fronts of spatiotemporal defects are linked to the convective and absolute nature of the instability. Both transitions appear to be subcritical. The spatiotemporal defects control the global mode structure.     

Subcritical Kelvin-Helmholtz instability in a Hele-Shaw cell

We investigate experimentally the subcritical behavior of the Kelvin-Helmholtz instability for a gas-liquid shearing flow in a Hele-Shaw cell. The subcritical curve separating the solutions of a stable plane interface and a fully saturated nonlinear wave train is determined. Experimental results are fitted by a fifth order complex Ginzburg-Landau equation whose linear coefficients are compared to theoretical ones.     

强激光与等离子体相互作用中受激陷俘电子声波散射及相空间离子涡旋的形成

应用一维相对论电磁粒子模拟程序，研究了线性极化强激光入射到无碰撞密度均匀的次临界密度等离子体中所引起的受激陷俘电子声波散射不稳定性过程.不稳定性的早期行为与是否考虑离子动力学效应无关.当考虑离子动力学效应之后会激发一个随时间增长的离子声波，并且最终由于大振幅电磁孤立子的产生而中断.由于电磁孤立子内的静电场与电磁场所产生的离子加速与俘获效应，导致一个离子涡旋在离子相空间中形成；当电磁孤立子向后加速过程中，若干个离子涡旋结构随之形成.研究发现，离子涡旋结构同样存在于密度不均匀的次临界密度等离子体中.从拓扑的观 关键词： 粒子模拟 受激陷俘电子声波散射 电磁孤立子 离子涡旋     

Turing instability in reaction-diffusion systems with nonlinear diffusion

The Turing instability is studied in two-component reaction-diffusion systems with nonlinear diffusion terms, and the regions in parametric space where Turing patterns can form are determined. The boundaries between super- and subcritical bifurcations are found. Calculations are performed for one-dimensional brusselator and oregonator models.     

Instabilities of uniform filtration flows with phase transition

New mechanisms of instability are described for vertical flows with phase transition through horizontally extended two-dimensional regions of a porous medium. A plane surface of phase transition becomes unstable at an infinitely large wavenumber and at zero wavenumber. In the latter case, the unstable flow undergoes reversible subcritical bifurcations leading to the development of secondary flows (which may not be horizontally uniform). The evolution of subcritical modes near the instability threshold is governed by the Kolmogorov-Petrovskii-Piskunov equation. Two examples of flow through a porous medium are considered. One is the unstable flow across a water-bearing layer above a layer that carries a vapor-air mixture under isothermal conditions in the presence of capillary forces at the phase transition interface. The other is the vertical flow with phase transition in a high-temperature geothermal reservoir consisting of two high-permeability regions separated by a low-permeability stratum.     

Three-dimensional effects in directional solidification in hele-shaw cells: nonlinear evolution and pattern selection

Directional solidification of a dilute binary alloy in a Hele-Shaw cell is modeled by a long-wave nonlinear evolution equation with zero flux and contact-angle conditions at the walls. The basic steady-state solution and its linear stability criteria are found analytically, and the nonlinear system is solved numerically. Concave-down (toward the solid) interfaces under physically realistic conditions are found to be more unstable than the planar front. Weakly nonlinear analysis indicates that subcritical bifurcation is promoted, the domain of modulational instability is expanded and transition to three-dimensional patterns is delayed due to the contact-angle condition. In the strongly nonlinear regime fully three-dimensional steady-state solutions are found whose characteristic amplitude is larger than that for the two-dimensional problem. In the subcritical regime secondary bifurcation to stable solutions is promoted.     

Subcritical bifurcations and nonlinear balloons in faraday waves

Bicritical points at wave numbers k(b) larger than the critical wave numbers k(c) are found in parametric surface waves (Faraday waves) using both numerical simulations and nonlinear analysis. Because k(b)-k(c) is small, it is argued that subcritical bifurcations at k>k(b) can be easily observed in experiments. In the second part we present a generic argument predicting the existence of nonlinear states resembling a balloon outside the instability region. The prediction is confirmed in simulations and it is argued to apply to other systems with similar instability curves.     

The rotating bucket of sand: Experiment and theory

The surface shape of a bucket of sand rotating about its cylindrical axis is studied experimentally and theoretically. Focusing on fast time scales on which surface shape is determined by avalanches, we identify three regimes of behavior. At intermediate and high frequencies, the surface shape is always at its critical shape determined by the Coulomb yield condition. The low frequency behavior displays an unexpected subcritical region at the center of the bucket. To understand this central region, we adapt a continuum model of surface flow developed by Bouchaud et al. and Mehta et al. The model indicates that the subcritical region is due to a nonlinear instability mechanism. (c) 1999 American Institute of Physics.     

CONVECTIVE INSTABILITY OF A TWO-COMPONENT FLUID-FLUID SYSTEM

The stability of a plane interface of two immiscible liquids(both are finite thickness)with a perpendicular mass transfer is investigated by means of linear stability and energy methods.An analytical formula is derived for the linear stability boundary,whereas the numerical solutions are obtained for boundaries following from both linear and energy analyses. It is concluded that the difference in the chemical potential of.these two phases drives the convective flow and that a threshold Marangoni number existg for the instability to occur. It is also shown that the energy stability boundary does not coincide with that following from the linear analysis,so the subcritical instability is allowed in the region in between.     

A nonlinear stability analysis of a double-diffusive magnetized ferrofluid with magnetic field-dependent viscosity

A rigorous nonlinear stability result is derived by introducing a suitable generalized energy functional for a magnetized ferrofluid layer heated and soluted from below with magnetic field-dependent (MFD) viscosity, for stress-free boundaries. The mathematical emphasis is on how to control the nonlinear terms caused by magnetic body and inertia forces. For ferrofluids, we find that there is possibility of existence of subcritical instabilities, however, it is noted that in case of non-ferrofluid, global nonlinear stability Rayleigh number is exactly the same as that for linear instability. For lower values of magnetic parameters, this coincidence is immediately lost. The effects of magnetic parameter, M₃, solute gradient, S₁ and MFD viscosity parameter, δ, on the subcritical instability region have also been analyzed.     

Unfolding the sulcus

Sulci are localized furrows on the surface of soft materials that form by a compression-induced instability. We unfold this instability by breaking its natural scale and translation invariance, and compute a limiting bifurcation diagram for sulcfication showing that it is a scale-free, subcritical nonlinear instability. In contrast with classical nucleation, sulcification is continuous, occurs in purely elastic continua and is structurally stable in the limit of vanishing surface energy. During loading, a sulcus nucleates at a point with an upper critical strain and an essential singularity in the linearized spectrum. On unloading, it quasistatically shrinks to a point with a lower critical strain, explained by breaking of scale symmetry. At intermediate strains the system is linearly stable but nonlinearly unstable with no energy barrier. Simple experiments confirm the existence of these two critical strains.     

Instability of bound states of a nonlinear Schrödinger equation with a Dirac potential

We study analytically and numerically the stability of the standing waves for a nonlinear Schrödinger equation with a point defect and a power type nonlinearity. A major difficulty is to compute the number of negative eigenvalues of the linearized operator around the standing waves. This is overcome by a perturbation method and continuation arguments. Among others, in the case of a repulsive defect, we show that the standing-wave solution is stable in and unstable in under subcritical nonlinearity. Further we investigate the nature of instability: under critical or supercritical nonlinear interaction, we prove the instability by blowup in the repulsive case by showing a virial theorem and using a minimization method involving two constraints. In the subcritical radial case, unstable bound states cannot collapse, but rather narrow down until they reach the stable regime (a finite-width instability). In the nonradial repulsive case, all bound states are unstable, and the instability is manifested by a lateral drift away from the defect, sometimes in combination with a finite-width instability or a blowup instability.     

Nonlinear periodic waves on the charged surface of a deep low-viscosity conducting liquid

An analytical expression for the profile of a finite-amplitude wave on the free charged surface of a deep low-viscosity conducting liquid is derived in an approximation quadratic in wave amplitude-to-wavelength ratio. It is shown that viscosity causes the wave amplitude to decay with time and makes the wave profile asymmetric at surface charge densities subcritical in terms of Tonks-Frenkel instability. At supercritical values of the surface charge density, taking account of viscosity decreases the growth rate of emissive protrusions on the unstable free surface, slightly broadens them for short waves, and narrows for long ones. Analytical expressions for the wave frequencies, damping rates, and instability growth rates with regard to viscosity are found.     

Statics and dynamics of a single ferrofluid-peak

We consider a single peak of a ferrofluid resulting from the Rosensweig instability for a small fluid container. Minimizing the total energy of the system by a variational method we determine the shape of the peak in a static field as well as the characteristics of the subcritical bifurcation leading to its formation. The latter are in very good agreement with experiment. Generalizing the approach to dynamic situations we qualitatively reproduce the complicated subharmonic response of the peak to an oscillating part in the external magnetic field found in recent experiments. Received 14 December 1999 and Received in final form 31 May 2000     

Magic angles and cross-hatching instability in hydrogel fracture

The full 2D analysis of roughness profiles of fracture surfaces resulting from quasistatic crack propagation in gelatin gels reveals an original behavior characterized by (i) strong anisotropy with maximum roughness at V-independent symmetry-preserving angles and (ii) a subcritical instability leading, below a critical velocity, to a cross-hatched regime due to straight macrosteps drifting at the same magic angles and nucleated on crack-pinning network inhomogeneities. Step height values are determined by the width of the strain-hardened zone, governed by the elastic crack blunting characteristic of soft solids with breaking stresses much larger than low strain moduli.     

ADS瞬态分析中的概率论-确定论耦合方法

针对加速器驱动次临界系统(ADS)瞬态问题,采用预估校正改进准静态方法(PCQS)处理时空中子动力学方程中的时间自变量,采用蒙特卡罗方法处理相应的空间-角度-能量自变量,重点解决了低次临界度下模拟计算不稳定的问题,验证了TWGIL-Seed-Blanket动力学基准问题和小型模拟ADS问题,得到瞬态过程的功率变化结果,与基于其他方法的程序比较,经初步验证取得了较好结果,证明了该耦合方法可行。     

Nonlinear theory of mirror instability near its threshold

An asymptotic model based on a reductive perturbative expansion of the drift kinetic and Maxwell’s equations is used to demonstrate that, near the instability threshold, the nonlinear dynamics of mirror modes in a magnetized plasma with anisotropic ion temperatures involves a subcritical bifurcation, leading to the formation of small-scale structures with amplitudes comparable with the ambient magnetic field.     

Cooperative nucleation of shear dislocation loops

The mean elastic interaction between randomly distributed transient subcritical shear loops of the same sign, formed in the presence of an applied shear stress, is the "image stress." This stress is proportional to the volume density of loops and has the same sign as the applied stress. The image stress promotes the cooperative nucleation of shear loops, and leads to an instability in which the number of loops and the image stress increase rapidly, leading to the generation of stable expanding loops. The critical stress at which the instability is predicted is relatively high.     

Probing a subcritical instability with an amplitude expansion: An exploration of how far one can get

We explore methods to locate subcritical branches of spatially periodic solutions in pattern forming systems with a nonlinear finite-wavelength instability. We do so by means of a direct expansion in the amplitude of the linearly least stable mode about the appropriate reference state which one considers. This is motivated by the observation that for some equations fully nonlinear chaotic dynamics has been found to be organized around periodic solutions that do not simply bifurcate from the basic (laminar) state. We apply the method to two model equations, a subcritical generalization of the Swift–Hohenberg equation and a novel extension of the Kuramoto–Sivashinsky equation that we introduce to illustrate the abovementioned scenario in which weakly chaotic subcritical dynamics is organized around periodic states that bifurcate “from infinity” and that can nevertheless be probed perturbatively. We explore the reliability and robustness of such an expansion, with a particular focus on the use of these methods for determining the existence and approximate properties of finite-amplitude stationary solutions. Such methods obviously are to be used with caution: the expansions are often only asymptotic approximations, and if they converge their radius of convergence may be small. Nevertheless, expansions to higher order in the amplitude can be a useful tool to obtain qualitatively reliable results.