Low-Frequency Multidimensional Instabilities for Reacting Shock Waves

This paper presents a weakly nonlinear analysis for one scenario for the development of transversal instabilities in detonation waves in two space dimensions. The theory proposed and developed here is most appropriate for understanding the behavior of regular and chaotically irregular pulsation instabilities that occur in detonation fronts in condensed phases and occasionally in gases. The theory involves low-frequency instabilities and through suitable asymptotics yields a complex Ginzburg-Landau equation that describes simultaneously the evolution of the detonation front and the nonlinear interactions behind this front. The asymptotic theory mimics the familiar theory of nonlinear hydrodynamic instability in outline; however, there are several novel technical aspects in the derivation because the phenomena studied here involve a complex free boundary problem for a system of nonlinear hyperbolic equations with source terms.     

Stability analysis of an interface between immiscible liquids in Hele-Shaw flow in the presence of a magnetic field

Hydrodynamic instabilities may occur when a viscous fluid is driven by a less viscous one through a porous medium. These penetrations are common in enhanced oil recovery, dendrite formation and aquifer flow. Recent studies have shown that the use of magnetic suspensions allow the external control of the instability. The problem is nonlinear and some further improvements of both theory and experimental observations are still needed and continue being a current source of investigation. In this paper we present a generalized Darcy law formulation in order to examine the growth of finger instabilities as a magnetic field is applied to the interface between the fluids in a Hele-Shaw cell. A new linear stability analysis is performed in the presence of magnetic effects and provides a stability criterion in terms of the non-dimensional physical parameters of the examined flow and the wavenumber of the finger disturbances. The interfacial tension inhibits small wavelength instabilities. The magnetic field contributes to the interface stability for moderate wavelength as it is applied parallel to the liquid-interface. In particular, we find an explicit expression, as a function of the susceptibility, for a critical angle between the interface and the magnetic field direction, in which its effect on the interface is neutral. We have developed a new asymptotic solution for the flow problem in a weak nonlinear regime. The first correction captures the second order nonlinear effects of the magnetic field, which tends to align the fingers with the field orientation and have a destabilizing effect. The analysis predicts that the non-linear effects at second order can counterbalance the first order stabilizing effect of a parallel magnetic field which results in a loss of effectiveness for controlling the investigated finger instabilities. The relevant physical parameters for controlling these finger instabilities are clearly identified by our non-dimensional analysis.     

Stable chaos in fluctuation driven neural circuits

We study the dynamical stability of pulse coupled networks of leaky integrate-and-fire neurons against infinitesimal and finite perturbations. In particular, we compare mean versus fluctuations driven networks, the former (latter) is realized by considering purely excitatory (inhibitory) sparse neural circuits. In the excitatory case the instabilities of the system can be completely captured by an usual linear stability (Lyapunov) analysis, whereas the inhibitory networks can display the coexistence of linear and nonlinear instabilities. The nonlinear effects are associated to finite amplitude instabilities, which have been characterized in terms of suitable indicators. For inhibitory coupling one observes a transition from chaotic to non chaotic dynamics by decreasing the pulse-width. For sufficiently fast synapses the system, despite showing an erratic evolution, is linearly stable, thus representing a prototypical example of stable chaos.     

An Alternative Approach to Nonlinear Instabilities in Hydrodynamics

The conventional theoretical methods used in the study of nonlinear hydrodynamic instabilities are known to suffer from severe amplitude limitations. In this paper a modified asymptotic theory is developed which is more general and less restrictive. Such an approach may be relevant to many nonlinear instability phenomena.     

Reactive Infiltration Instabilities

When a fluid flow is imposed on a porous medium, the infiltrationflow may interact with the reaction-induced porosity variationswithin the medium and may lead to fingering instabilities. Anonlinear model of such interaction is developed and morphologicalinstability of a planar dissolution front is demonstrated usinga linear stability analysis of a moving-free-boundary problem.The fully nonlinear model is also examined numerically usingfinite-difference methods. The numerical simulations confirmthe predictions of linear stability theory and, more importantly,reveal the growth of dissolution fingers that emerge as a resultof these instabilities     

Some results on the stability and dynamics of finite difference approximations to nonlinear partial differential equations

A miscellany of results on the nonlinear instability and dynamics of finite difference discretizations of the Burgers and Kortweg de Vries equations is obtained using a variety of phase-plane, functional analytic, and regularity methods. For the semidiscrete (space-discrete, time-continuous) schemes, large-wave-numer instabilities occurring in special exact solutions are investigated, and parameter values for which the semidiscrete scheme is monotone are considered. For fully discrete schemes (space and time discrete), large-wave-number instabilities introduced by various time-stepping schemes such as forward Euler, leapfrog, and Runge–Kutta schemes are analyzed. Also, a time step restriction for the monotonicity of the forward-Euler time-stepping scheme, and regularity of a 4-stage monotone/conservative Runge–Kutta time stepping are investigated. The techniques used here may be employed, in conjunction with bifurcation-theoretic and weakly nonlinear analyses, to analyze the stability of numerical schemes for other nonlinear partial differential equations of both dissipative and dispersive varieties. © 1993 John Wiley & Sons, Inc.     

Convection in a variable gravity field

The problem of convection in a variable gravity field is studied by using methods of linear instability theory and nonlinear energy theory. It is shown that the decreasing or increasing of gravity in a specific direction can be stabilizing or destabilizing and it is further shown how to quantify this effect. Specific results are presented for the situation where gravity decreases linearly throughout a plane layer. The nonlinear results are found to be very close to the linear ones and define a small band where possible subcritical instabilities may arise.     

Vlasov equilibria: Varying the temperature or the density distributions

Stationary selfconsistent solutions of the Vlasov–Maxwell system in a magnetized inhomogeneous plasma (so called Vlasov equilibria) provide the natural starting point for the investigation of plasma stability and of the nonlinear development of plasma instabilities in collisionless or weakly collisional regimes. In view of the different mechanisms that drive these instabilities, we discuss Vlasov equilibria with both density and temperature gradients.     

Dynamics of Semi-Discretizations of the Defocusing Nonlinear Schrodinger Equation

It has been demonstrated that the nonlinear Schrödinger(NLS) equation is sensitive to discretizations. In the focusingcase this is due to the homoclinic structure associated withthe NLS equation. In this paper we show that various numericalschemes for the defocusing case are also prone to instabilities,although not as severe as those of the focusing equation. Anintegrable discretization due to Ablowitz and Ladik does notsuffer from the same instabilities. However, it is shown thatit develops a focusing singularity if a threshold conditionis exceeded. Numerical examples illustrating the phenomena pertainingto the defocusing equation are given.     

A Mean Flow First Harmonic Theory for Hydrodynamic Instabilities

A general theory is presented for nonlinear instabilities arising in steady hydrodynamic motions. For quasiparallel flows at high values of the Reynolds number it is found that for relatively small disturbance levels the usual ideas concerning the generation of higher harmonics and the subsequent modification of the fundamental may be overwhelmed by three dimensional interactions between the evolving mean flow and the first harmonic wave. The differences from and similarities to existing asymptotic and numerical studies are discussed. The theory developed applies to a variety of flow configurations. Numerical results are given for Poiseuille flow and the Blasius boundary layer. In addition the theory developed here is applied to simulate the instabilities produced in a boundary layer due to the presence of free stream disturbances.