1,720,994 research outputs found
Theory of domain patterns in systems with long-range interactions of Coulomb type
No full textWe develop a theory of the domain patterns in systems with competing short-range attractive interactions and long-range repulsive Coulomb interactions. We take an energetic approach, in which patterns are considered as critical points of a mean-field free energy functional. Close to the microphase separation transition, this functional takes on a universal form, allowing us to treat a number of diverse physical situations within a unified framework. We use asymptotic analysis to study domain patterns with sharp interfaces. We derive an interfacial representation of the pattern's free energy which remains valid in the fluctuating system, with a suitable renormalization of the Coulomb interaction's coupling constant. We also derive integro-differential equations describing stationary domain patterns of arbitrary shapes and their thermodynamic stability, coming from the first and second variations of the interfacial free energy. We show that the length scale of a stable domain pattern must obey a certain scaling law with the strength of the Coulomb interaction. We analyzed the existence and stability of localized (spots, stripes, annuli) and periodic (lamellar, hexagonal) patterns in two dimensions. We show that these patterns are metastable in certain ranges of the parameters and that they can undergo morphological instabilities leading to the formation of more complex patterns. We discuss nucleation of the domain patterns by thermal fluctuations and pattern formation scenarios for various thermal quenches. We argue that self-induced disorder is an intrinsic property of the domain patterns in the systems under consideration
A Quantitative Approximation Scheme for the Traveling Wave Solutions in the Hodgkin–Huxley Model
No full textAbstractWe introduce an approximation scheme for the Hodgkin–Huxley model of nerve conductance that allows calculation of both the speed and shape of the traveling pulses, in quantitative agreement with the solutions of the model. We demonstrate that the reduced problem for the front of the traveling pulse admits a unique solution. We obtain an explicit analytical expression for the speed of the pulses that is valid with good accuracy in a wide range of the parameters
Front propagation in infinite cylinders I. A variational approach
No full textIn their classical 1937 paper, Kolmogorov, Petrovsky and Piskunov proved that for a particular class of reaction-diffusion equations on the real line the solution of the initial value problem with the initial data in the form of a unit step propagates at long times with constant velocity equal to that of a certain special traveling wave solution. This type of a propagation result has since been established for a number of general classes of reaction-diffusion-advection problems in cylinders. Here we show that in problems without advection or in the presence of transverse advection by a potential flow these results do not rely on the specifics of the problem. Instead, they are a consequence of the fact that the equation considered is a gradient flow in an exponentially weighted L(2)-space generated by a certain functional, when the dynamics is considered in the reference frame moving with constant velocity along the cylinder axis. We show that independently of the details of the problem only three propagation scenarios are possible in the above context: no propagation, a "pulled" front, or a "pushed" front. The choice of the scenario is completely characterized via a minimization problem
Front propagation in infinite cylinders II. The sharp reaction zone limit
No full textThis paper applies the variational approach developed in part I of this work [22] to a singular limit of reaction-diffusion-advection equations which arise in combustion modeling. We first establish existence, uniqueness, monotonicity, asymptotic decay, and the associated free boundary problem for special traveling wave solutions which are minimizers of the considered variational problem in the singular limit. We then show that the speed of the minimizers of the approximating problems converges to the speed of the minimizer of the singular limit. Also, after an appropriate translation the minimizers of the approximating problems converge strongly on compacts to the minimizer of the singular limit. In addition, we obtain matching upper and lower bounds for the speed of the minimizers in the singular limit in terms of a certain area-type functional for small curvatures of the free boundary. The conclusions of the analysis are illustrated by a number of numerical examples
Existence of traveling waves of invasion for Ginzburg-Landau-type problems in infinite cylinders
No full textWe study a class of systems of reaction-diffusion equations in infinite cylinders which arise within the context of Ginzburg-Landau theories and describe the kinetics of phase transformation in second-order or weakly first-order phase transitions with non-conserved order parameters. We use a variational characterization to study the existence of a special class of traveling wave solutions which are characterized by a fast exponential decay in the direction of propagation. Our main result is a simple verifiable criterion for existence of these traveling waves under the very general assumptions of non-linearities. We also prove boundedness, regularity, and some other properties of the obtained solutions, as well as several sufficient conditions for existence or non-existence of such traveling waves, and give rigorous upper and lower bounds for their speed. In addition, we prove that the speed of the obtained solutions gives a sharp upper bound for the propagation speed of a class of disturbances which are initially sufficiently localized. We give a sample application of our results using a computer-assisted approach
Correction to: A Nonlocal Isoperimetric Problem with Dipolar Repulsion (Communications in Mathematical Physics, (2019), 372, 3, (1059-1115), 10.1007/s00220-019-03455-y)
No full textA Correction to this paper has been published: 10.1007/s00220-019-03455-y
Linear vs. nonlinear selection for the propagation speed of the solutions of scalar reaction-diffusion equations invading an unstable equilibrium
No full textWe revisit the classical problem of speed selection for the propagation of disturbances in scalar reaction-diffusion equations with one linearly stable and one linearly unstable equilibrium. For a wide class of initial data this problem reduces to finding the minimal speed of the monotone traveling wave solutions connecting these two equilibria in one space dimension. We introduce a variational characterization of these traveling wave solutions and give a necessary and sufficient condition for linear versus nonlinear selection mechanism. We obtain sufficient conditions for the linear and nonlinear selection mechanisms that are easily verifiable. Our method, also allows us to obtain efficient lower and upper bounds for the propagation speed
On equilibrium shapes of charged flat drops
Full text linkThe equilibrium shapes of two-dimensional charged, perfectly conducting liquid drops are governed by a geometric variational problem that involves a perimeter term modeling line tension and a capacitary term modeling Coulombic repulsion. Here we give a complete explicit solution to this variational problem. Namely, we show that at fixed total charge a ball of a particular radius is the unique global minimizer among all sufficiently regular sets in the plane. For sets whose area is also fixed, we show that balls are the only minimizers if the charge is less than or equal to a critical charge, while for larger charge minimizers do not exist. Analogous results hold for drops whose potential, rather than charge, is fixed. © 2017 Wiley Periodicals, Inc
An old problem resurfaces nonlocally: Gamow’s liquid drops inspire today’s research and applications
No full textTheory of 360° domain walls in thin ferromagnetic films
No full textAn analytical and computational study of 360° domain walls in thin uniaxial ferromagnetic films is presented. The existence of stable one-dimensional 360° domain wall solutions both with and without the applied field is demonstrated in a reduced thin film micromagnetic model. The wall energy is found to depend rather strongly on the orientation of the wall and the wall width significantly grows when the strength of the magnetostatic forces increases. It is also shown that a critical reverse field is required to break up a 360° domain wall into a pair of 180° walls. The stability of the 360° walls in two-dimensional films of finite extent is demonstrated numerically and the stability with respect to slow modulations in extended films is demonstrated analytically. These domain wall solutions are shown to play an important role in magnetization reversal. In particular, it is found that the presence of 360° domain walls may result in nonuniqueness of the observed magnetization patterns during repeated cycles of magnetization reversal by pulsed fields. © 2008 American Institute of Physics
- …
