Fingering Instability in Combustion

A thin solid (e.g., paper), burning against an oxidizing wind, develops a fingering instability with two decoupled length scales. The spacing between fingers is determined by the P\'eclet number (ratio between advection and diffusion). The finger width is determined by the degree two dimensionality. Dense fingers develop by recurrent tip splitting. The effect is observed when vertical mass transport (due to gravity) is suppressed. The experimental results quantitatively verify a model based on diffusion limited transport

Nonlinear Dynamics of Moving Curves and Surfaces: Applications to Physical Systems

The subject of moving curves (and surfaces) in three dimensional space (3-D) is a fascinating topic not only because it represents typical nonlinear dynamical systems in classical mechanics, but also finds important applications in a variety of physical problems in different disciplines. Making use of the underlying geometry, one can very often relate the associated evolution equations to many interesting nonlinear evolution equations, including soliton possessing nonlinear dynamical systems. Typical examples include dynamics of filament vortices in ordinary and superfluids, spin systems, phases in classical optics, various systems encountered in physics of soft matter, etc. Such interrelations between geometric evolution and physical systems have yielded considerable insight into the underlying dynamics. We present a succinct tutorial analysis of these developments in this article, and indicate further directions. We also point out how evolution equations for moving surfaces are often intimately related to soliton equations in higher dimensions.Comment: Review article, 38 pages, 7 figs. To appear in Int. Jour. of Bif. and Chao

Theory of spiral wave dynamics in weakly excitable media: asymptotic reduction to a kinematic model and applications

In a weakly excitable medium, characterized by a large threshold stimulus, the free end of an isolated broken plane wave (wave tip) can either rotate (steadily or unsteadily) around a large excitable core, thereby producing a spiral pattern, or retract causing the wave to vanish at boundaries. An asymptotic analysis of spiral motion and retraction is carried out in this weakly excitable large core regime starting from the free-boundary limit of the reaction-diffusion models, valid when the excited region is delimited by a thin interface. The wave description is shown to naturally split between the tip region and a far region that are smoothly matched on an intermediate scale. This separation allows us to rigorously derive an equation of motion for the wave tip, with the large scale motion of the spiral wavefront slaved to the tip. This kinematic description provides both a physical picture and exact predictions for a wide range of wave behavior, including: (i) steady rotation (frequency and core radius), (ii) exact treatment of the meandering instability in the free-boundary limit with the prediction that the frequency of unstable motion is half the primary steady frequency (iii) drift under external actions (external field with application to axisymmetric scroll ring motion in three-dimensions, and spatial or/and time-dependent variation of excitability), and (iv) the dynamics of multi-armed spiral waves with the new prediction that steadily rotating waves with two or more arms are linearly unstable. Numerical simulations of FitzHug-Nagumo kinetics are used to test several aspects of our results. In addition, we discuss the semi-quantitative extension of this theory to finite cores and pinpoint mathematical subtleties related to the thin interface limit of singly diffusive reaction-diffusion models

Mean Field Theory of the Morphology Transition in Stochastic Diffusion Limited Growth

We propose a mean-field model for describing the averaged properties of a class of stochastic diffusion-limited growth systems. We then show that this model exhibits a morphology transition from a dense-branching structure with a convex envelope to a dendritic one with an overall concave morphology. We have also constructed an order parameter which describes the transition quantitatively. The transition is shown to be continuous, which can be verified by noting the non-existence of any hysteresis.Comment: 16 pages, 5 figure

Flame front propagation V: Stability Analysis of Flame Fronts: Dynamical Systems Approach in the Complex Plane

We consider flame front propagation in channel geometries. The steady state solution in this problem is space dependent, and therefore the linear stability analysis is described by a partial integro-differential equation with a space dependent coefficient. Accordingly it involves complicated eigenfunctions. We show that the analysis can be performed to required detail using a finite order dynamical system in terms of the dynamics of singularities in the complex plane, yielding detailed understanding of the physics of the eigenfunctions and eigenvalues.Comment: 17 pages 7 figure

Traveling Waves, Front Selection, and Exact Nontrivial Exponents in a Random Fragmentation Problem

We study a random bisection problem where an initial interval of length x is cut into two random fragments at the first stage, then each of these two fragments is cut further, etc. We compute the probability P_n(x) that at the n-th stage, each of the 2^n fragments is shorter than 1. We show that P_n(x) approaches a traveling wave form, and the front position x_n increases as x_n\sim n^{\beta}{\rho}^n for large n. We compute exactly the exponents \rho=1.261076... and \beta=0.453025.... as roots of transcendental equations. We also solve the m-section problem where each interval is broken into m fragments. In particular, the generalized exponents grow as \rho_m\approx m/(\ln m) and \beta_m\approx 3/(2\ln m) in the large m limit. Our approach establishes an intriguing connection between extreme value statistics and traveling wave propagation in the context of the fragmentation problem.Comment: 4 pages Revte

Perturbative Linearization of Reaction-Diffusion Equations

We develop perturbative expansions to obtain solutions for the initial-value problems of two important reaction-diffusion systems, viz., the Fisher equation and the time-dependent Ginzburg-Landau (TDGL) equation. The starting point of our expansion is the corresponding singular-perturbation solution. This approach transforms the solution of nonlinear reaction-diffusion equations into the solution of a hierarchy of linear equations. Our numerical results demonstrate that this hierarchy rapidly converges to the exact solution.Comment: 13 pages, 4 figures, latex2

Derivation of the relativistic "proper-time" quantum evolution equations from Canonical Invariance

Based on 1) the spectral resolution of the energy operator; 2) the linearity of correspondence between physical observables and quantum Hermitian operators; 3) the definition of conjugate coordinate-momentum variables in classical mechanics; and 4) the fact that the physical point in phase space remains unchanged under (canonical) transformations between one pair of conjugate variables to another, we are able to show that , the proper-time rest-energy transformation matrices, are given as a*exp[-iE_s t_s/\hbar], from which we obtain the proper-time rest -energy evolution equation i\hbar{\partial/\partial t_s} |Psi>= \hat{E_s}|Psi>. For special relativistic situations this equation can be reduced to the usual i\hbar{\partial/\partial t}|Psi>=\hat{E}|Psi> dynamical equations, where t is the "reference time" and E is the total energy. Extension of these equations to accelerating frames is then provided.Comment: J. Phys. A, accepted for publicatio

Existence of a rotating wave pattern in a disk for a wave front interaction model

[[abstract]]We study the rotating wave patterns in an excitable medium in a disk. This wave pattern is rotating along the given disk boundary with a constant angular speed. To study this pattern we use the wave front interaction model proposed by Zykov in 2007. This model is derived from the FitzHugh-Nagumo equation and it can be described by two systems of ordinary differential equations for wave front and wave back respectively. Using a delicate shooting argument with the help of the comparison principle, we derive the existence and uniqueness of rotating wave patterns for any admissible angular speed with convex front in a given disk.[[incitationindex]]SCI[[booktype]]紙本[[booktype]]電子

Flame front propagation I: The Geometry of Developing Flame Fronts: Analysis with Pole Decomposition

The roughening of expanding flame fronts by the accretion of cusp-like singularities is a fascinating example of the interplay between instability, noise and nonlinear dynamics that is reminiscent of self-fractalization in Laplacian growth patterns. The nonlinear integro-differential equation that describes the dynamics of expanding flame fronts is amenable to analytic investigations using pole decomposition. This powerful technique allows the development of a satisfactory understanding of the qualitative and some quantitative aspects of the complex geometry that develops in expanding flame fronts.Comment: 4 pages, 2 figure