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M. C. Depassier

Publications and source records attributed to M. C. Depassier.

At least 19 recordsLinked to original sources
arXiv2022

Upper and lower bounds for the speed of fronts of the reaction diffusion equation with Stefan boundary conditions

We establish two integral variational principles for the spreading speed of the one dimensional reaction diffusion equation with Stefan boundary conditions. The first principle is valid for monostable reaction terms and the second principle is valid for arbitrary reaction terms. These principles allow to obtain several upper and lower bounds for the speed. In particular, we construct a generalized Zeldovich-Frank-Kamenetskii type lower bound for the speed and upper bounds in terms of the speed of the standard reaction diffusion problem. We construct asymptotically exact lower bounds previously obtained by perturbation theory.

math-ph↗
arXiv2020

Equivalence of variational principles to determine the speed of scalar reaction diffusion fronts

The determination of the speed of travelling fronts of the scalar reaction diffusion equation has been the subject of much study. Using different approaches seemingly disconnected variational principles have been established. The purpose of this work is to show the connection between them. For monostable reaction terms, we prove that a principle established by Hadeler and Rothe in 1975 and a second one by Benguria and Depassier in 1996 are logically equivalent, that is, either can be derived from the other. Two variational principles, formulated for arbitrary reaction terms, are shown to be related by a suitable change of variables. Finally a variational principle proven for monostable reaction terms is shown to be a formulation of the two previous ones in yet another independent variable.

math.AP↗
arXiv2019

On the speed of domain walls in thin nanotubes: the transition from the linear to the magnonic regime

Numerical simulations of domain wall propagation in thin nanotubes when an external magnetic field is applied along the nanotube axis have shown an unexpected behavior described as a transition from a linear to a magnonic regime. As the applied magnetic field increases, the initial regime of linear growth of the speed with the field is followed by a sudden change in slope accompanied by the emission of spin waves. In this work an analytical formula for the speed of the domain wall that explains this behavior is derived by means of an asymptotic study of the Landau Lifshitz Gilbert equation for thin nanotubes. We show that the dynamics can be reduced to a one dimensional hyperbolic reaction diffusion equation, namely, the damped double Sine Gordon equation, which shows the transition to the magnonic regime as the domain wall speed approaches the speed of spin waves. This equation has been previously found to describe domain wall propagation in weak ferromagnets with the mobility proportional to the Dzyaloshinskii-Moriya interaction constant, for Permalloy nanotubes the mobility is proportional to the nanotube radius.

cond-mat.mes-hall↗
arXiv2018

Domain wall dynamics for an in-plane magnetized thin film with large perpendicular hard axis anisotropy including Dzyaloshinskii-Moriya interaction

We consider a thin ferromagnetic layer to which an external field or a current are applied along an in plane easy axis. The perpendicular hard axis anisotropy constant is large so that the out of plane magnetization component is smaller than the in plane components. A perturbation approach is used to obtain the profile and velocity of the moving domain wall. The dynamics of the in plane components of the magnetization is governed by a reaction diffusion equation which determines the speed of the profile. We find a simple analytic expression for the out of plane magnetization showing a symmetric distortion due to the motion in addition to the asymmetric component due to the Dzyaloshinskii--Moriya interaction. The results obtained complement previous studies in which either the Dzyalozhinskii vector or the out of plane hard axis anisotropy were assumed small. In the regime studied the Walker breakdown is not observed but the reaction diffusion dynamics predicts a slowing down of the domain wall for sufficiently large magnetic field. The transition point depends on the applied field, saturation magnetization and easy axis anisotropy.

cond-mat.mes-hall↗
arXiv2017

Variational characterization of the speed of reaction diffusion fronts for gradient dependent diffusion

We study the asymptotic speed of traveling fronts of the scalar reaction diffusion for positive reaction terms and with a diffusion coefficient depending nonlinearly on the concentration and on its gradient. We restrict our study to diffusion coefficients of the form $D(u,u_x) = m u^{m-1} u_x^{m(p-2)}$ for which existence and convergence to traveling fronts has been established. We formulate a variational principle for the asymptotic speed of the fronts. Upper and lower bounds for the speed valid for any $m\ge0, p\ge 1$ are constructed. When $m=1, p=2$ the problem reduces to the constant diffusion problem and the bounds correspond to the classic Zeldovich Frank-Kamenetskii lower bound and the Aronson-Weinberger upper bound respectively. In the special case $m(p-1) = 1$ a local lower bound can be constructed which coincides with the aforementioned upper bound. The speed in this case is completely determined in agreement with recent results.

math.AP↗
arXiv2015

Reaction diffusion dynamics and the Schryer-Walker solution for domain walls of the Landau-Lifshitz-Gilbert equation

We study the dynamics of the equation obtained by Schryer and Walker for the motion of domain walls. The reduced equation is a reaction diffusion equation for the angle between the applied field and the magnetization vector. If the hard axis anisotropy $K_d$ is much larger than the easy axis anisotropy $K_u$, there is a range of applied fields where the dynamics does not select the Schryer-Walker solution. We give analytic expressions for the speed of the domain wall in this regime and the conditions for its existence.

cond-mat.mes-hall↗
arXiv2015

Speed of field driven domain walls in nanowires with large transverse magnetic anisotropy

Recent analytical and numerical work on field driven domain wall propagation in nanowires has shown that for large transverse anisotropy and sufficiently large applied fields the Walker profile becomes unstable before the breakdown field, giving way to a slower stationary domain wall. We perform an asymptotic expansion of the Landau Lifshitz Gilbert equation for large transverse magnetic anisotropy and show that the asymptotic dynamics reproduces this behavior. At low applied field the speed increases linearly with the field and the profile is the classic Landau profile. Beyond a critical value of the applied field the domain wall slows down. The appearance of a slower domain wall profile in the asymptotic dynamics is due to a transition from a pushed to a pulled front of a reaction diffusion equation.

cond-mat.mes-hall↗
arXiv2013

Shift in the speed of reaction diffusion equation with a cut-off: pushed and bistable fronts

We study the change in the speed of pushed and bistable fronts of the reaction diffusion equation in the presence of a small cut-off. We give explicit formulas for the shift in the speed for arbitrary reaction terms f(u). The dependence of the speed shift on the cut-off parameter is a function of the front speed and profile in the absence of the cut-off. In order to determine the speed shift we solve the leading order approximation to the front profile u(z) in the neighborhood of the leading edge and use a variational principle for the speed. We apply the general formula to the Nagumo equation and recover the results which have been obtained recently by geometric analysis. The formulas given are of general validity and we also apply them to a class of reaction terms which have not been considered elsewhere.

nlin.PS↗
arXiv2007

Validity of the Brunet-Derrida formula for the speed of pulled fronts with a cutoff

We establish rigorous upper and lower bounds for the speed of pulled fronts with a cutoff. We show that the Brunet-Derrida formula corresponds to the leading order expansion in the cut-off parameter of both the upper and lower bounds. For sufficiently large cut-off parameter the Brunet-Derrida formula lies outside the allowed band determined from the bounds. If nonlinearities are neglected the upper and lower bounds coincide and are the exact linear speed for all values of the cut-off parameter.

nlin.PS↗
arXiv2007

On the speed of pulled fronts with a cutoff

We study the effect of a small cutoff $ε$ on the velocity of a pulled front in one dimension by means of a variational principle. We obtain a lower bound on the speed dependent on the cutoff, and for which the two leading order terms correspond to the Brunet Derrida expression. To do so we cast a known variational principle for the speed of propagation of fronts in new variables which makes it more suitable for applications.

nlin.PS↗
arXiv2006

Evolution equation for bidirectional surface waves in a convecting fluid

Surface waves in a heated viscous fluid exhibit a long wave oscillatory instability. The nonlinear evolution of unidirectional waves is known to be described by a modified Korteweg-deVries-Kuramoto-Sivashinsky equation. In the present work we eliminate the restriction of unidirectional waves and find that the evolution of the wave is governed by a modified Boussinesq system . A perturbed Boussinesq equation of the form $y_{tt}-y_{xx} -ε^2(y_{xxtt} + (y^2)_{xx})+ ε^3(y_{xxt}+y_{xxxxt} + (y^2)_{xxt}) =0 $ which includes instability and dissipation is derived from this system.

nlin.PS↗
arXiv2005

On the principal bifurcation branch of a third order nonlinear long-wave equation

We study the principal bifurcation curve of a third order equation which describes the nonlinear evolution of several systems with a long--wavelength instability. We show that the main bifurcation branch can be derived from a variational principle. This allows to obtain a close estimate of the complete branch. In particular, when the bifurcation is subcritical, the large amplitude stable branch can be found in a simple manner.

nlin.PS↗
arXiv2004

Variational calculation of the period of nonlinear oscillators

The problem of calculating the period of second order nonlinear autonomous oscillators is formulated as an eigenvalue problem. We show that the period can be obtained from two integral variational principles dual to each other. Upper and lower bounds on the period can be obtained to any desired degree of accuracy. The results are illustrated by an application to the Duffing equation.

nlin.CD↗
arXiv2004

Minimal speed of fronts of reaction-convection-diffusion equations

We study the minimal speed of propagating fronts of convection reaction diffusion equations of the form $u_t + μϕ(u) u_x = u_{xx} +f(u)$ for positive reaction terms with $f'(0 >0$. The function $ϕ(u)$ is continuous and vanishes at $u=0$. A variational principle for the minimal speed of the waves is constructed from which upper and lower bounds are obtained. This permits the a priori assesment of the effect of the convective term on the minimal speed of the traveling fronts. If the convective term is not strong enough, it produces no effect on the minimal speed of the fronts. We show that if $f''(u)/\sqrt{f'(0)} + μϕ'(u) < 0$, then the minimal speed is given by the linear value $2 \sqrt{f'(0)}$, and the convective term has no effect on the minimal speed. The results are illustrated by applying them to the exactly solvable case $u_t + μu u_x = u_{xx} + u (1 -u)$. Results are also given for the density dependent diffusion case $u_t + μϕ(u) u_x = (D(u)u_x)_x +f(u)$.

nlin.PS↗
arXiv2002

Linear and nonlinear marginal stability for fronts of hyperbolic reaction diffusion equations

We study travelling fronts of equations of the form $u_{tt} + ϕ(u) u_x = u_{xx} + f(u)$. A criterion for the transition from linear to nonlinear marginal stability is established for positive functions $ϕ(u)$ and for any reaction term $f(u)$ for which the usual parabolic reaction diffusion equation $u_t = u_{xx} + f(u)$ admits a front. As an application, we treat reaction diffusion systems with transport memory.

nlin.PS↗