Large amplitude behavior of the Grinfeld instability: a variational approach

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.Comment: subm. to EPJ

Vesicle propulsion in haptotaxis : a local model

We study theoretically vesicle locomotion due to haptotaxis. Haptotaxis is referred to motion induced by an adhesion gradient on a substrate. The problem is solved within a local approximation where a Rayleigh-type dissipation is adopted. The dynamical model is akin to the Rousse model for polymers. A powerful gauge-field invariant formulation is used to solve a dynamical model which includes a kind of dissipation due to bond breaking/restoring with the substrate. For a stationary situation where the vesicle acquires a constant drift velocity, we formulate the propulsion problem in terms of a nonlinear eigenvalue (the a priori unknown drift velocity) one of Barenblat-Zeldovitch type. A counting argument shows that the velocity belongs to a discrete set. For a relatively tense vesicle, we provide an analytical expression for the drift velocity as a function of relevant parameters. We find good agreement with the full numerical solution. Despite the oversimplification of the model it allows the identification of a relevant quantity, namely the adhesion length, which turns out to be crucial also in the nonlocal model in the presence of hydrodynamics, a situation on which we have recently reported [I. Cantat, and C. Misbah, Phys. Rev. Lett. {\bf 83}, 235 (1999)] and which constitutes the subject of a forthcoming extensive study.Comment: 12 pages, 8 figures, submitted to Eur. Phys. J.

Spontaneous polarization in an interfacial growth model for actin filament networks with a rigorous mechano-chemical coupling

Many processes in eukaryotic cells, including cell motility, rely on the growth of branched actin networks from surfaces. Despite its central role the mechano-chemical coupling mechanisms which guide the growth process are poorly understood, and a general continuum description combining growth and mechanics is lacking. We develop a theory that bridges the gap between mesoscale and continuum limit and propose a general framework providing the evolution law of actin networks growing under stress. This formulation opens an area for the systematic study of actin dynamics in arbitrary geometries. Our framework predicts a morphological instability of actin growth on a rigid sphere, leading to a spontaneous polarization of the network with a mode selection corresponding to a comet, as reported experimentally. We show that the mechanics of the contact between the network and the surface plays a crucial role, in that it determines directly the existence of the instability. We extract scaling laws relating growth dynamics and network properties offering basic perspectives for new experiments on growing actin networks.Comment: 7 pages, 4 figure

Coarsening dynamics in one dimension: The phase diffusion equation and its numerical implementation

Many nonlinear partial differential equations (PDEs) display a coarsening dynamics, i.e., an emerging pattern whose typical length scale $L$ increases with time. The so-called coarsening exponent $n$ characterizes the time dependence of the scale of the pattern, $L(t)\approx t^n$, and coarsening dynamics can be described by a diffusion equation for the phase of the pattern. By means of a multiscale analysis we are able to find the analytical expression of such diffusion equations. Here, we propose a recipe to implement numerically the determination of $D(\lambda)$, the phase diffusion coefficient, as a function of the wavelength $\lambda$ of the base steady state $u_0(x)$. $D$ carries all information about coarsening dynamics and, through the relation $|D(L)| \simeq L^2 /t$, it allows us to determine the coarsening exponent. The main conceptual message is that the coarsening exponent is determined without solving a time-dependent equation, but only by inspecting the periodic steady-state solutions. This provides a much faster strategy than a forward time-dependent calculation. We discuss our method for several different PDEs, both conserved and not conserved

Modified Kuramoto-Sivashinsky equation: stability of stationary solutions and the consequent dynamics

We study the effect of a higher-order nonlinearity in the standard Kuramoto-Sivashinsky equation: \partial_x \tilde G(H_x). We find that the stability of steady states depends on dv/dq, the derivative of the interface velocity on the wavevector q of the steady state. If the standard nonlinearity vanishes, coarsening is possible, in principle, only if \tilde G is an odd function of H_x. In this case, the equation falls in the category of the generalized Cahn-Hilliard equation, whose dynamical behavior was recently studied by the same authors. Instead, if \tilde G is an even function of H_x, we show that steady-state solutions are not permissible.Comment: 4 page

Phase instability and coarsening in two dimensions

Instabilities and pattern formation is the rule in nonequilibrium systems. Selection of a persistent lengthscale, or coarsening (increase of the lengthscale with time) are the two major alternatives. When and under which conditions one dynamics prevails over the other is a longstanding problem, particularly beyond one dimension. It is shown that the challenge can be defied in two dimensions, using the concept of phase diffusion equation. We find that coarsening is related to the \lambda-dependence of a suitable phase diffusion coefficient, D_{11}(\lambda), depending on lattice symmetry and conservation laws. These results are exemplified analytically on prototypical nonlinear equations.Comment: Five pages. Introduction and summary strongly revised. One page of supplementary material has been adde