Compacton formation under Allen–Cahn dynamics (original) (raw)
Related papers
Stability of the stationary profiles of the Allen-Cahn equation with not constant stiffness
arXiv: Pattern Formation and Solitons, 2020
We study the solutions of a generalized Allen-Cahn equation deduced from a Landau energy functional, endowed with a non-constant higher order stiffness. We assume the stiffness to be a positive function of the field and we discuss the stability of the stationary solutions proving both linear and local non-linear stability.
On an anisotropic Allen-Cahn system
Cubo (Temuco), 2015
Our aim in this paper is to prove the existence and uniqueness of solutions for an Allen-Cahn type system based on a modification of the Ginzburg-Landau free energy proposed in [11]. In particular, the free energy contains an additional term called Willmore regularization and takes into account anisotropy effects. RESUMEN Nuestro propósito en este trabajo es probar la existencia y unicidad de soluciones para un Sistema de tipo Allen-Cahn basados en una modificación de la energía libre Ginzburg-Landau propuesta en [11]. En particular, la energía libre contiene un término adicional llamado regularización de Willmore y considera efectos de anisotropía.
Physica D: Nonlinear Phenomena, 2006
We study the phase dynamics of a chain of autonomous, self-sustained, dispersively coupled oscillators. In the quasicontinuum limit the basic discrete model reduces to a Korteveg-de Vries-like equation, but with a nonlinear dispersion. The system supports compactons -solitary waves with a compact support -and kovatons -compact formations of glued together kink-antikink pairs that propagate with a unique speed, but may assume an arbitrary width. We demonstrate that lattice solitary waves, though not exactly compact, have tails which decay at a superexponential rate. They are robust and collide nearly elastically and together with wave sources are the building blocks of the dynamics that emerges from typical initial conditions. In finite lattices, after a long time, the dynamics becomes chaotic. Numerical studies of the complex Ginzburg-Landau lattice show that the non-dispersive coupling causes a damping and deceleration, or growth and acceleration, of compactons. A simple perturbation method is applied to study these effects.
The non-isothermal Allen-Cahn equation with dynamic boundary conditions
We consider a model of nonisothermal phase transitions taking place in a bounded spatial region. The order parameter ψ is governed by an Allen-Cahn type equation which is coupled with the equation for the temperature θ. The former is subject to a dynamic boundary condition recently proposed by some physicists to account for interactions with the walls. The latter is endowed with a boundary condition which can be a standard one (Dirichlet, Neumann or Robin) or a dynamic one of Wentzell type. We thus formulate a class of initial and boundary value problems whose local existence and uniqueness is proven by means of a fixed point argument. The local solution becomes global owing to suitable a priori estimates. Then we analyze the asymptotic behavior of the solutions within the theory of infinite-dimensional dynamical systems. In particular, we demonstrate the existence of the global attractor as well as of an exponential attractor.
Stability and dynamical properties of Rosenau-Hyman compactons using Padé approximants
2010
Since their discovery by Rosenau and Hyman (RH) in 1993 [1], compactons have found diverse applications in physics in the analysis of patterns on liquid surfaces [2], in approximations for thin viscous films [3], ocean dynamics [4], magma dynamics [5, 6], and medicine [7]. Compactons are also the object of study in brane cosmology [8] as well as mathematical physics [9, 10] and the dynamics of nonlinear lattices [11–14] to model the dispersive coupling of a chain of oscillators [14–17].
The Allen-Cahn equation with dynamic boundary conditions and mass constraints
Mathematical Methods in the Applied Sciences, 2014
The Allen-Cahn equation, coupled with dynamic boundary conditions, has recently received a good deal of attention. The new issue of this paper is the setting of a rather general mass constraint which may involve either the solution inside the domain or its trace on the boundary. The system of nonlinear partial differential equations can be formulated as variational inequality. The presence of the constraint in the evolution process leads to additional terms in the equation and the boundary condition containing a suitable Lagrange multiplier. A well-posedness result is proved for the related initial value problem.
Study of stability and dynamical properties of Rosenau-Hyman compactons
Bulletin of the American Physical Society, 2010
We present a systematic approach for calculating higher-order derivatives of smooth functions on a uniform grid using Padé approximants. We illustrate our findings by deriving higher-order approximations using traditional second-order finite-differences formulas as our starting point. We employ these schemes to study the stability and dynamical properties of K(2, 2) Rosenau-Hyman (RH) compactons including the collision of two compactons and resultant shock formation. Our approach uses a differencing scheme involving only nearest and next-to-nearest neighbors on a uniform spatial grid. The partial differential equation for the compactons involves first, second and third partial derivatives in the spatial coordinate and we concentrate on four different fourthorder methods which differ in the possibility of increasing the degree of accuracy (or not) of one of the spatial derivatives to sixth order. A method designed to reduce roundoff errors was found to be the most accurate approximation in stability studies of single solitary waves, even though all derivates are accurate only to fourth order. Simulating compacton scattering requires the addition of fourth derivatives related to artificial viscosity. For those problems the different choices lead to different amounts of "spurious" radiation and we compare the virtues of the different choices.
Phase Compactons in Chains of Dispersively Coupled Oscillators
Physical Review Letters, 2005
We study the phase dynamics of a chain of autonomous oscillators with a dispersive coupling. In the quasicontinuum limit the basic discrete model reduces to a Korteveg-de Vries-like equation, but with a nonlinear dispersion. The system supports compactons: solitary waves with a compact support and kovatons which are compact formations of glued together kink-antikink pairs that may assume an arbitrary width. These robust objects seem to collide elastically and, together with wave trains, are the building blocks of the dynamics for typical initial conditions. Numerical studies of the complex Ginzburg-Landau and Van der Pol lattices show that the presence of a nondispersive coupling does not affect kovatons, but causes a damping and deceleration or growth and acceleration of compactons.
A class of quasi-linear Allen–Cahn type equations with dynamic boundary conditions
Nonlinear Analysis, 2017
In this paper, we consider a class of coupled systems of PDEs, denoted by (ACE) ε for ε ≥ 0. For each ε ≥ 0, the system (ACE) ε consists of an Allen-Cahn type equation in a bounded spacial domain Ω, and another Allen-Cahn type equation on the smooth boundary Γ := ∂Ω, and besides, these coupled equations are transmitted via the dynamic boundary conditions. In particular, the equation in Ω is derived from the nonsmooth energy proposed by Visintin in his monography "Models of phase transitions": hence, the diffusion in Ω is provided by a quasilinear form with singularity. The objective of this paper is to build a mathematical method to obtain meaningful L 2-based solutions to our systems, and to see some robustness of (ACE) ε with respect to ε ≥ 0. On this basis, we will prove two Main Theorems 1 and 2, which will be concerned with the well-posedness of (ACE) ε for each ε ≥ 0, and the continuous dependence of solutions to (ACE) ε for the variations of ε ≥ 0, respectively.