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Reaction-diffusion systems with cross-diffusion terms that depend linearly on density are studied via linear stability analysis and weakly nonlinear analysis. We obtain and analyze the conditions for the Turing instability and derive a universal form of these conditions. We discuss the features of the pattern-forming regions in parameter space for a cross activator-inhibitor system, the Brusselator model, and for a pure activator-inhibitor system, the two-variable Oregonator model. The supercritical or subcritical character of the Turing bifurcation for the Brusselator is determined by deriving an amplitude equation for patterns near the instability threshold. [ABSTRACT FROM AUTHOR]

We study a sigmoidal version of the FitzHugh-Nagumo reaction-diffusion system based on an analytic description using piecewise linear approximations of the reaction kinetics. We completely describe the dynamics of wave fronts and discuss the properties of the speed equation. The speed diagrams show front bifurcations between branches with one, three, or five fronts that differ significantly from the classical FitzHugh-Nagumo model. We examine how the number of fronts and their speed vary with the model parameters. We also investigate numerically the stability of the front solutions in a case when five fronts exist. [ABSTRACT FROM AUTHOR]

Bistable reaction-diffusion systems of activator-inhibitor type with cross advection are studied analytically and numerically using a piecewise linear approximation for the nonlinear reaction term. It is shown that a system with double symmetric cross advection produces the same characteristic equation as a system with cross diffusion in only one equation or a system exhibiting double self advection with opposite signs. For systems with cross-advection terms having opposite signs, traveling-wave solutions are derived explicitly. Fronts constructed from these solutions have characteristic wavy profiles. A bifurcation of excitation fronts induced by cross advection is discovered. Including cross diffusion, we find front solutions having both an oscillatory leading edge and an oscillating tail. [ABSTRACT FROM AUTHOR]

A stability analysis is performed analytically for the tristable reaction-diffusion equation, in which a quintic reaction term is approximated by a piecewise linear function. We obtain growth rate equations for two basic types of propagating fronts, monotonous and nonmonotonous ones. Their solutions show that the monotonous front is stable whereas the nonmonotonous one is unstable. It is found that there are two values of the growth rate for the most dangerous modes (corresponding to the longest possible wavelengths),and, for the monotonous front, so that atthe perturbation eigenfunction is positive whereas whenit changes sign. It is also noted that the eigenvaluebecomes negative in an inhomogeneous system with a particular (stabilizing) inhomogeneity. Counting arguments for the number of eigenmodes of the linear stability operator are presented. [ABSTRACT FROM AUTHOR]

In previous work, we have performed amplitude expansions of the continuum equations for the Grinfeld instability and carried them to high orders. Nevertheless, the approach turned out to be restricted to relatively small amplitudes. In this article, we use a variational approach in terms of multi-cycloid curves instead. Besides its higher precision at given order, the method has the advantages of giving a transparent physical meaning to the appearance of cusp singularities and of not being restricted to interfaces representable as single-valued functions. Using a single cycloid as ansatz function, the entire calculation can be performed analytically, which gives a good qualitative overview of the system. Taking into account several but few cycloid modes, we obtain remarkably good quantitative agreement with previous numerical calculations. With a few more modes taken into consideration, we improve on the accuracy of those calculations. Our approach extends them to situations involving gravity effects. Results on the shape of steady-state solutions are presented at both large stresses and amplitudes. In addition, their stability is investigated. [ABSTRACT FROM AUTHOR]

Presents a study on the front propagation under periodic forcing in reaction-diffusion systems. Use of a general mathematical model describing bistable media in terms of a reaction-diffusion equation; Extension of the study using a two-component model; Possibility to control the evolution of front waves by introducing external periodic forcing.

: We analyze the problem of vesicle migration in haptotaxis (a motion directed by an adhesion gradient), though most of the reasoning applies to chemotaxis as well as to a variety of driving forces. A brief account has been published on this topic [#!Cantat99a!#]. We present an extensive analysis of this problem and provide a basic discussion of most of the relevant processes of migration. The problem allows for an arbitrary shape evolution which is compatible with the full hydrodynamical flow in the Stokes limit. The problem is solved within the boundary integral formulation based on the Oseen tensor. For the sake of simplicity we confine ourselves to 2D flows in the numerical analysis. There are basically two regimes (i) the tense regime where the vesicle behaves as a “droplet” with an effective contact angle. In that case the migration velocity is given by the Stokes law. (ii) The flask regime where the vesicle has a significant (on the scale of the vesicle size) contact curvature. In that case we obtain a new migration law which substantially differs from the Stokes law. We develop general arguments in order to extract analytical laws of migration. These are in good agreement with the full numerical analysis. Finally we mention several important future issues and open questions. [ABSTRACT FROM AUTHOR]

We use the method presented in M von Kurnatowski et al (2013 Phys. Rev. E 87 042405) to solve the nonlinear problem of free dendritic growth in an Oseen flow. The growth process is assumed to be limited by thermal transport via diffusion and convection. A singular perturbation expansion is treated to lowest nontrivial order in the framework of asymptotic decomposition. The resulting complex integro-differential equation is solved using an elaborate numerical method. The approximate scaling laws and for the growth velocity and the tip radius of curvature of the dendrite, respectively, are found as a function of the forced flow velocity. The results are compared to those by Pelcé and Bouissou, constituting the only other attempt so far to treat the problem analytically. [ABSTRACT FROM AUTHOR]

A piecewise linear reaction–diffusion system including an external field is considered. Analytic solutions are obtained for the propagating front, the front velocity, the perturbed front and the growth rate of perturbations. It is found that, when the ratio of the time scales, null-cline parameter and the external field are small enough, the moving front has an oscillating behaviour, similar to that of a front in an oscillatory medium. A comparison with an oscillatory front invading an unstable state at small velocity is made and it is argued that there are crucial differences between the two cases. In particular, we find that for fronts oscillating about a stable state both concentrations must oscillate in two-variable models, whereas fronts oscillating about an unstable state can approach it with one of the two variables being monotonic. [Copyright &y& Elsevier]

: We investigate the mechanical properties of a two-dimensional amorphous solid. It is formed spontaneously by the adsorption of a protein (the β-lactoglobulin) at the surface of water. We measure its mechanical response in both elastic and plastic regimes by applying a point-like force (using a glass fiber). We compare our results with previous measurements of shear moduli using a floating torsion device. [ABSTRACT FROM AUTHOR]

Abstract: We report on step bunching and meandering dynamics on vicinal surfaces during MBE growth. We list four generic meandering scenarios during MBE growth: (i) when adatom desorption is significant, spatio-temporal chaos is found. When desorption is negligible three regimes are found; (ii) meandering with a wavelength which is frozen from the first stages of the instability, but whose amplitude diverges as a power law ; (iii) endless coarsening in the presence of elastic interactions between the steps; (iv) interrupted coarsening in presence of anisotropy. In the case of step bunching, we analyze two scenarios: (i) when desorption is important, ordered bunches appear. (ii) In the opposite limit, bunches separate, and wide terraces appear. [Copyright &y& Elsevier]