On Front Motion in a Burgers-Type Equation with Quadratic and Modular Nonlinearity and Nonlinear Amplification

A singularly perturbed initial–boundary value problem for a parabolic equation known in applications as a Burgers-type or reaction–diffusion–advection equation is considered. An asymptotic approximation of solutions with a moving front is constructed in the case of modular and quadratic nonlinearity and nonlinear amplification. The influence exerted by nonlinear amplification on front propagation and blowing- up is determined. The front localization and the blowing-up time are estimated.

     

Propagation and its failure in a lattice delayed differential equation with global interaction

We study the existence, uniqueness, global asymptotic stability and propagation failure of traveling wave fronts in a lattice delayed differential equation with global interaction for a single species population with two age classes and a fixed maturation period living in a spatially unbounded environment. In the bistable case, under realistic assumptions on the birth function, we prove that the equation admits a strictly monotone increasing traveling wave front. Moreover, if the wave speed does not vanish, then the wave front is unique (up to a translation) and globally asymptotic stable with phase shift. Of particular interest is the phenomenon of “propagation failure” or “pinning” (that is, wave speed c = 0), we also give some criteria for pinning in this paper.     

Front Propagation in Stirred Media

The problem of asymptotic features of front propagation in stirred media is addressed for laminar and turbulent velocity fields. In particular we consider the problem in two dimensional steady and unsteady cellular flows in the limit of very fast reaction and sharp front, i.e., in the geometrical optics limit. In the steady case we provide an analytical approximation for the front speed, v _f, as a function of the stirring intensity, U, in good agreement with the numerical results. In the unsteady (time-periodic) case, albeit the Lagrangian dynamics is chaotic, chaos in the front dynamics is relevant only for a transient. Asymptotically the front evolves periodically and chaos manifests only in the spatially wrinkled structure of the front. In addition we study front propagation of reactive fields in systems whose diffusive behavior is anomalous. The features of the front propagation depend, not only on the scaling exponent ν, which characterizes the diffusion properties, \({( \langle (x(t) - x(0))^2 \rangle \sim t^{2\nu} )}\) , but also on the detailed shape of the probability distribution of the diffusive process.     

Uniform asymptotic analysis for waves in an incompressible elastic rod II. Disturbances superimposed on a pre-stressed state

This paper studies the propagation of disturbances superimposedon a pre-stressed incompressible hyperelastic thin rod. Startingfrom the incremental equations given by Haughton and Ogden (1979,J. Mech. Phys. Solids 27, 179-212 and 489-512), we derive athree parameter-dependent one-dimensional rod equation as thegoverning equation. In particular, it is found that one parameterplays a crucial role. Depending on whether it is larger or smallerthan or equal to a critical value, the shear-wave velocity islarger or smaller than or equal to the bar-wave velocity. Inthe case that these two velocities are equal, there exist travelling-wavesolutions of arbitrary form. This implies that for this particularcase the initial disturbance would propagate along the rod withoutdistortion. To see the influence of the pre-stress in detail,we further consider an initial-value problem with an initialsingularity in the shear strain. The solutions are expressedin terms of integrals through the method of Fourier transform.We then conduct an asymptotic analysis for the solutions. Fora material point in a neighbourhood behind the shear-wave front,the phase function of these integrals has a stationary pointat infinity. Here, we use a technique of uniform asymtotic expansionto handle this case. An asymptotic expansion, correct up toorder O(t^-1), for the shear strain, which is uniformly validin a neighbourhood behind the shear-wave front, is derived.For material points in other spatial domains, the method ofstationary point is applicable, and asymptotic expansions (correctup to order O(t^-1)) are obtained. A novelty is that we are ableto deduce precise qualitative information about the waves inthe far field from our analytic results. Wave profiles for twoconcrete examples are also provided.     

Complex dynamic behavior during transition in a solid combustion model

Through examples in a free‐boundary model of solid combustion, this study concerns nonlinear transition behavior of small disturbances of front propagation and temperature as they evolve in time. This includes complex dynamics of period doubling, and quadrupling, and it eventually leads to chaotic oscillations. Within this complex dynamic domain we also observe a period six‐folding. Both asymptotic and numerical solutions are studied.We show that for special parameters our asymptotic method with some dominant modes captures the formation of coherent structures. Finally,we discuss possible methods to improve our prediction of the solutions in the chaotic case. © 2009 Wiley Periodicals, Inc. Complexity, 2009     

An asymptotic analysis of the steady process of saturation of a laminated porous material

The propagation of a steady saturation front in a double-layer porous material, situated between impenetrable walls, is investigated. The closed system of equations and boundary conditions are written on the assumption that the displacing and displaced phases, the viscosities of which differ considerably, are connected, and that the capillary pressure on the interface is constant. The features of the behaviour of the interface in the neighbourhood of the boundary of the layers are investigated in the case when the layer thicknesses differ considerably. When the permeabilities of the layers differ considerably, asymptotic expressions are obtained for the pressure and shape of the interface, and a comparison is made with the results of a numerical solution of the complete problem and with the known asymptotic relations obtained when using a simplified boundary condition at the interface.     

Shock Wave Tracking for Hyperbolic Systems Exhibiting Local Linear Degeneracy

Extending the results of our previous study [2], we now investigate the propagation of interior shocks corresponding to the signaling problem of small-amplitude, high-frequency type. We derive a formula for the shock front and show that the previously constructed asymptotic solution is valid on both sides of this front. This solution is further distinguished to a higher order in which the effects of material inhomogeneity are accounted for. Moreover, if λ = λ( u , x) represents the eigenvalue under consideration, we show that the single-wave-mode boundary disturbance of [2] can lead only to a λ-shock. We also derive an entropy condition for the shock wave. As an application of our theory, the fluid-filled hyperelastic tube problem of [7] is further examined and an example calculation made in which we show that a compressive shock wave is generated at the shock-initiation point. This demonstration is effected as a particular example of the solution to a general bifurcation problem.     

Reaction front propagation of actin polymerization in a comb-reaction system

We develop a theoretical model of anomalous transport with polymerization-reaction dynamics. We are motivated by the experimental problem of actin polymerization occurring in a microfluidic device with a comb-like geometry. Depending on the concentration of reagents, two limiting regimes for the propagation of reaction are recovered: the failure of the reaction front propagation and a finite speed of the reaction front corresponding to the Fisher-Kolmogorov-Petrovskii-Piscounov (FKPP) at the long time asymptotic regime. To predict the relevance of these regimes we obtain an explicit expression for the transient time as a function of geometry and parameters of the experimental setup. Explicit analytical expressions of the reaction front velocity are obtained as functions of the experimental setup.     

Numerical approaches for computation of fronts

Moving fronts and pulses appear in many engineering applications like flame propagation and a falling liquid film. Standard computation methods are inappropriate since the problem is defined over an infinite domain and a steady-state solution exists only for a certain front velocity. This work presents a transformation that converts the original problem into a boundary-value problem within a finite domain, in a way that preserves the behavior at the boundaries. Good low-order approximations can be obtained as demonstrated by two examples. In another approach, a central element of adjustable length is incorporated into a three-element structure where the edge-elements obey known asymptotic solutions. That yields multiplicity of travelling fronts in an infinite domain but it successfully approximates standing wave solutions in a finite domain. The approximate solutions are shown to obey the qualitative features known for the exact solutions, like asymptotic solutions or the bifurcation set–the boundary where a new solution emerges or disappears.     

Spreading–vanishing dichotomy in a diffusive logistic model with a free boundary, II

We study the diffusive logistic equation with a free boundary in higher space dimensions and heterogeneous environment. Such a model may be used to describe the spreading of a new or invasive species, with the free boundary representing the expanding front. For simplicity, we assume that the environment and the solution are radially symmetric. In the special case of one space dimension and homogeneous environment, this free boundary problem was investigated in Du and Lin (2010) [10]. We prove that the spreading-vanishing dichotomy established in Du and Lin (2010) [10] still holds in the more general and ecologically realistic setting considered here. Moreover, when spreading occurs, we obtain best possible upper and lower bounds for the spreading speed of the expanding front. When the environment is asymptotically homogeneous at infinity, these two bounds coincide. Our results indicate that the asymptotic spreading speed determined by this model does not depend on the spatial dimension.     

Propagation dynamics of a nonlocal periodic organism model with non-monotone birth rates

This work is concerned with a nonlocal reaction–diffusion system modeling the propagation dynamics of organisms owning mobile and stationary states in periodic environments. We establish the existence of the asymptotic speed of spreading for the model system with monotone birth function via asymptotic propagation theory of monotone semiflow, and then discuss the case for non-monotone birth function by using the squeezing technique. In terms of the truncated problem on a finite interval, we apply the method of super- and sub-solutions and the fixed point theorem combined with regularity estimation and limit arguments to obtain the existence of time periodic traveling waves for the model system without quasi-monotonicity. The non-existence proof is to use the results of the spreading speed. Finally, as an application, we study the spatial dynamics of the model with the birth rate function of Ricker type and numerically demonstrate analytic results.     

一类具有边界摄动的奇摄动问题

利用渐近理论,讨论了一类具有边界摄动的奇摄动问题.在适当的条件下,得出了这类问题解的存在性条件及其渐近解, 并将所得的结果应用于一类壁面波的传播问题.     

Acoustic Wave Propagation in an Underwater Sound Channel 2. Quantitative Theory

By using ray-theoretic-techniques we have developed in our previouspaper, "Acoustic Wave Propagation in an Underwater Sound Channel.1. Qualitative Theory" a certain qualitative understanding ofseveral features of SOFAR propagation in an underwater soundchannel. In the present paper a more penetrating quantitativestudy is done by means of analytical techniques on the governingequations. We study the transient problem for the Epstein profileby employing a double transform to formally derive an integralrepresentation, and we obtain several alternative representationsneeded for asymptotic results which we also present. It is believedthat several new and interesting asymptotic results have beenderived.     

Front Motion in a Problem with Weak Advection in the Case of a Continuous Source and a Modular-Type Source

Differential Equations - We obtain an asymptotic approximation to a moving inner layer (front) solution of an initial–boundary value problem for a singularly perturbed parabolic...     

Simple Diffraction Processes for Nagumo- and Fisher-type Diffusion Fronts

The diffraction of a diffusion front by concave and convex wedges is studied for Nagumo and Fisher's equations on the limit of fast reaction and small diffusion, using both the asymptotic theory and full numerical solutions. For the case of a convex corner, the full numerical solution confirms that the front evolves according to the asymptotic theories. On the other hand, for the concave corner, it is shown numerically that the diffraction produces at the corner a region of low values of the solution for both the Nagumo and Fisher's equations. Moreover, in both cases, the front eventually evolves, leaving behind a cavity. In the case of the Nagumo equation, it is shown that the long-term behavior of the diffraction front is just a traveling front, bent at the sloping wall. The bent region maintains its size as the front travels. This behavior is predicted by an exact traveling wave solution of the asymptotic equation for the front propagation. Good agreement is found between the numerical and the asymptotic solutions. On the other hand, behavior of the diffracted front for Fisher's equation is different. In this case, the front is bent at the sloping wall, but, as time passes, the bend becomes smaller and moves toward the sloping wall. This behavior is, again, predicted by the asymptotic solution. The numerics strongly suggest that the final state for the concave corner is a steady cavity-like solution with low values at the corner and high values away from it. This solution has an angular dependence that varies with the angle of the sloping wall.     

Diffusion-controlled smoulder propagation in a semi-infinite solid: The asymptotic temperature distribution

The authors consider the steady propagation of a two-dimensionaldiffusioncontrolled smouldering reaction front parallel to theplane boundary of a semi-infinite nonporous reactant. The reactionfront is assumed to be a sheet of line heat sources of variablestrength. The distribution of oxidizer concentration and temperaturein the porous burnt char and of the temperature in the reactantis determined in the form of an asymptotic expansion involvinga similarity variable. The temperature on the reaction frontin the asymptotic region is found to be constant to high order.The dependence of this temperature on the Lewis numbers associatedwith the reactant and the char is found to be in general agreementwith observations.     

A mathematical analysis of the resonance of the finite thin slots

In this article we discuss the asymptotic behaviour of the solution of a wave propagation problem in a domain including a thin slot. Several convergence rates are obtained and illustrated by numerical examples.     

Gevrey properties of the asymptotic critical wave speed in a family of scalar reaction–diffusion equations

We consider front propagation in a family of scalar reaction–diffusion equations in the asymptotic limit where the polynomial degree of the potential function tends to infinity. We investigate the Gevrey properties of the corresponding critical propagation speed, proving that the formal series expansion for that speed is Gevrey-1 with respect to the inverse of the degree. Moreover, we discuss the question of optimal truncation. Finally, we present a reliable numerical algorithm for evaluating the coefficients in the expansion with arbitrary precision and to any desired order, and we illustrate that algorithm by calculating explicitly the first ten coefficients. Our analysis builds on results obtained previously in [F. Dumortier, N. Popovi?, T.J. Kaper, The asymptotic critical wave speed in a family of scalar reaction–diffusion equations, J. Math. Anal. Appl. 326 (2) (2007) 1007–1023], and makes use of the blow-up technique in combination with geometric singular perturbation theory and complex analysis, while the numerical evaluation of the coefficients in the expansion for the critical speed is based on rigorous interval arithmetic.