J. Lega

Universal description of laser dynamics near threshold

Abstract Complex order parameter descriptions of large aspect ratio, single longitudinal mode, two-level lasers with flat end reflectors, valid near onset of lasing and for small detunings of the laser from the peak gain, are given in terms of a complex Swift-Hohenberg equation for Class A and C lasers and by a complex Swift-Hohenberg equation coupled to a mean flow for the case of a Class B laser. The latter coupled system is a physically consistent generalized rate equation model for wide aperture stiff laser systems. These universal order parameter equations provide a connection between spatially homogeneous oscillating states of the complex Ginzburg-Landau equation description of the laser system valid for finite negative detunings, and traveling wave states, described by coupled Newell-Whitehead-Segel equations valid for finite positive detunings. One of the main conclusions of the present paper is that the usual Eckhaus instability boundary associated with a long wavelength phase instability, and which delineates the region of the stable traveling wave solutions for Class A and C lasers, no longer defines the stability boundary for the mathematically stiff Class B laser. Instead a short wavelength phase instability appears causing the stability domain to shrink as a function of increasing stiffness of the system. This prediction is consistent with the strong spatiotemporal filamentation instabilities experimentally observed in a borad area semiconductor laser, a Class B system.

A form of turbulence associated with defects

Abstract We show by means of numerical simulations of complex Ginzburg-Landau equations that phase instability leads to the spontaneous nucleation of topological defects, which disorganize the system.

Multiple Quenching Solutions of a Fourth Order Parabolic PDE with a Singular Nonlinearity Modeling a MEMS Capacitor

Finite time singularity formation in a fourth order nonlinear parabolic partial differential equation (PDE) is analyzed. The PDE is a variant of a ubiquitous model found in the field of microelectromechanical systems (MEMS) and is studied on a one-dimensional (1D) strip and the unit disc. The solution itself remains continuous at the point of singularity while its higher derivatives diverge, a phenomenon known as quenching. For certain parameter regimes it is shown numerically that the singularity will form at multiple isolated points in the 1D strip case and along a ring of points in the radially symmetric two-dimensional case. The location of these touchdown points is accurately predicted by means of asymptotic expansions. The solution itself is shown to converge to a stable self-similar profile at the singularity point. Analytical calculations are verified by use of adaptive numerical methods which take advantage of symmetries exhibited by the underlying PDE to accurately resolve solutions very close t...

Hydrodynamics of bacterial colonies

Understanding the growth and dynamics of bacterial colonies is a fascinating problem, which requires combining ideas from biology, physics and applied mathematics. We briefly review the recent experimental and theoretical literature relevant to this question and describe a hydrodynamic model (Lega and Passot 2003 Phys. Rev. E 67 031906, 2004 Chaos 14 562–70), which captures macroscopic motions within bacterial colonies, as well as the macroscopic dynamics of colony boundaries. The model generalizes classical reaction–diffusion systems and is able to qualitatively reproduce a variety of colony shapes observed in experiments. We conclude by listing open questions about the stability of interfaces as modelled by reaction–diffusion equations with nonlinear diffusion and the coupling between reaction–diffusion equations and a hydrodynamic field.

Dynamics of Bloch Walls in a Rotating Magnetic Field: a Model

We both analytically and numerically show the existence of a drift of Bloch walls when submitted to a uniform parallel-to-the-wall-plane rotating magnetic field. The drift velocity changes sign with Bloch wall handedness and is proportional to the amplitude square of the magnetic field, when the latter is small.

Traveling hole solutions to the complex Ginzburg-Landau equation as perturbations of nonlinear Schro¨dinger dark solitons

Abstract We describe complex Ginzburg-Landau (CGL) traveling hole solutions as singular perturbations of nonlinear Schrodinger (NLS) dark solitons. Modulation of the free parameters of the NLS solutions leads to a dynamical system describing the CGL dynamics in the vicinity of a traveling hole solution.

Spatio-temporal intermittency in a 1D convective pattern: theoretical model and experiments

Abstract We describe the occurence of spatio-temporal intermittency in a one-dimensional convective system that first shows time-dependent patterns. We recall experimental results and propose a model based on the normal form description of a secondary Hopf bifurcation of a stationary periodic structure. Numerical simulations of this model show spatio-temporal intermittent behaviors, which we characterize briefly and compare to those given by the experiment.

The Quenching Set of a MEMS Capacitor in Two-Dimensional Geometries

The formation of finite time singularities in a nonlinear parabolic fourth order partial differential equation (PDE) is investigated for a variety of two-dimensional geometries. The PDE is a variant of a canonical model for Micro–Electro Mechanical systems (MEMS). The singularities are observed to form at specific points in the domain and correspond to solutions whose values remain finite but whose derivatives diverge as the finite time singularity is approached. This phenomenon is known as quenching. An asymptotic analysis reveals that the quenching set can be predicted by simple geometric considerations suggesting that the phenomenon described is generic to higher order parabolic equations which exhibit finite time singularity.

Morphometric modeling of olfactory circuits in the insect antennal lobe: I. Simulations of spiking local interneurons

Inhibitory local interneurons (LNs) play a critical role in shaping the output of olfactory glomeruli in both the olfactory bulb of vertebrates and the antennal lobe of insects and other invertebrates. In order to examine how the complex geometry of LNs may affect signaling in the antennal lobe, we constructed detailed multi-compartmental models of single LNs from the sphinx moth, Manduca sexta, using morphometric data from confocal-microscopic images. Simulations clearly revealed a directionality in LNs that impeded the propagation of injected currents from the sub-micron-diameter glomerular dendrites toward the much larger-diameter integrating segment (IS) in the coarse neuropil. Furthermore, the addition of randomly-firing synapses distributed across the LN dendrites (simulating the noisy baseline activity of afferent input recorded from LNs in the odor-free state) led to a significant depolarization of the LN. Thus the background activity typically recorded from LNs in vivo could influence synaptic integration and spike transformation in LNs through voltage-dependent mechanisms. Other model manipulations showed that active currents inserted into the IS can help synchronize the activation of inhibitory synapses in glomeruli across the antennal lobe. These data, therefore, support experimental findings suggesting that spiking inhibitory LNs can operate as multifunctional units under different ambient odor conditions. At low odor intensities, (i.e. subthreshold for IS spiking), they participate in local, mostly intra-glomerular processing. When activated by elevated odor concentrations, however, the same neurons will fire overshooting action potentials, resulting in the spread of inhibition more globally across the antennal lobe. Modulation of the passive and active properties of LNs may, therefore, be a deciding factor in defining the multi-glomerular representations of odors in the brain.

Pulses, fronts and oscillations of an elastic rod

Abstract Two coupled nonlinear Klein–Gordon equations modeling the three-dimensional dynamics of a twisted elastic rod near its first bifurcation threshold are analyzed. First, it is shown that these equations are Hamiltonian and that they admit a two-parameter family of traveling wave solutions. Second, special solutions corresponding to simple deformations of the elastic rod are considered. The stability of such configurations is analyzed by means of two coupled nonlinear Schrodinger equations, which are derived from the nonlinear Klein–Gordon equations in the limit of small deformations. In particular, it is shown that periodic solutions are modulationally unstable, which is consistent with the looping process observed in the writhing instability of elastic filaments. Third, numerical simulations of the nonlinear Klein–Gordon equations suggesting that traveling pulses are stable, are presented.