Extended Fisher Kolmogorov Research Articles

Heteroclinic solutions for the extended Fisher–Kolmogorov equation

We study the extended Fisher–Kolmogorov (EFK) equation and its variants. By variational approach, we show that, for every global minimum ξ of the potential function V, there is a pair of heteroclinic solutions, one emanating from ξ and the other terminating at ξ. We require neither superquadratic growth of V nor the presence of saddle-focus equilibria; moreover V is allowed to approach its minimal level near infinity.

A fully discrete [formula omitted] interior penalty Galerkin approximation of the extended Fisher–Kolmogorov equation

A fully discrete C0 interior penalty finite element method is proposed and analyzed for the Extended Fisher–Kolmogorov (EFK) equation ut+γΔ2u−Δu+u3−u=0 with appropriate initial and boundary conditions, where γ is a positive constant. We derive a regularity estimate for the solution u of the EFK equation that is explicit in γ and as a consequence we derive a priori error estimates that are robust in γ.

A New Linearized Crank-Nicolson Mixed Element Scheme for the Extended Fisher-Kolmogorov Equation

We present a new mixed finite element method for solving the extended Fisher-Kolmogorov (EFK) equation. We first decompose the EFK equation as the two second-order equations, then deal with a second-order equation employing finite element method, and handle the other second-order equation using a new mixed finite element method. In the new mixed finite element method, the gradient ∇u belongs to the weaker (L 2(Ω))2 space taking the place of the classical H(div; Ω) space. We prove some a priori bounds for the solution for semidiscrete scheme and derive a fully discrete mixed scheme based on a linearized Crank-Nicolson method. At the same time, we get the optimal a priori error estimates in L 2 and H 1-norm for both the scalar unknown u and the diffusion term w = −Δu and a priori error estimates in (L 2)2-norm for its gradient χ = ∇u for both semi-discrete and fully discrete schemes.

Global dynamics of stationary solutions of the extended Fisher–Kolmogorov equation

In this paper we study the fourth order differential equation $\frac{d^4u}{dt^4}+q\frac{d^2u}{dt^2}+u^3-u=0$d4udt4+qd2udt2+u3−u=0, which arises from the study of stationary solutions of the Extended Fisher–Kolmogorov equation. Denoting \documentclass[12pt]{minimal}\begin{document}$x=u,y=\frac{du}{dt},z=\frac{d^2u}{dt^2},v=\frac{d^3u}{dt^3}$\end{document}x=u,y=dudt,z=d2udt2,v=d3udt3 this equation becomes equivalent to the polynomial system \documentclass[12pt]{minimal}\begin{document}$\dot{x}=y,\dot{y} =z,\dot{z} =v,\dot{v}=x-qz-x^3$\end{document}ẋ=y,ẏ=z,ż=v,v̇=x−qz−x3 with \documentclass[12pt]{minimal}\begin{document}$(x,y,z,v)\in \mathbb {R}^4$\end{document}(x,y,z,v)∈R4 and \documentclass[12pt]{minimal}\begin{document}$q\in \mathbb {R}.$\end{document}q∈R. As usual, the dot denotes the derivative with respect to the time t. Since the system has a first integral we can reduce our analysis to a family of systems on \documentclass[12pt]{minimal}\begin{document}$\mathbb {R}^3.$\end{document}R3. We provide the global phase portrait of these systems in the Poincaré ball (i.e., in the compactification of \documentclass[12pt]{minimal}\begin{document}$\mathbb {R}^3$\end{document}R3 with the sphere \documentclass[12pt]{minimal}\begin{document}$\mathbb {S}^2$\end{document}S2 of the infinity).

Finite difference discretization of the extended Fisher–Kolmogorov equation in two dimensions

Approximate solutions are considered for the extended Fisher–Kolmogorov (EFK) equation in two space dimension with Dirichlet boundary conditions by a Crank–Nicolson type finite difference scheme. A priori bounds are proved using Lyapunov functional. Further, existence, uniqueness and convergence of difference solutions with order O ( h 2 + k 2 ) in the L ∞ -norm are proved. Numerical results are also given in order to check the properties of analytical solutions.

A second-order accurate difference scheme for an extended Fisher–Kolmogorov equation

In this paper, we present a Crank–Nicolson type finite difference scheme to approximate the nonlinear evolutionary Extended Fisher–Kolmogorov (EFK) equation. We prove the existence of the solution by using the well-known Browder fixed-point theorem. The stability of this scheme is established in L∞-norm. The uniqueness and convergence of the solution are analyzed. We discuss an iterative algorithm for solving the difference scheme and prove its convergence.

Matlab implementation of a moving grid method based on the equidistribution principle

The objective of this paper is to report on the development of a method of lines (MOL) toolbox within MATLAB, and especially, on the implementation and test of a moving grid algorithm based on the equidistribution principle. This new implementation includes various spatial approximation schemes based on finite differences and slope limiters, the choice between several monitor functions, automatic grid adaptation to the initial condition, and provides a relatively easy tuning for the non-expert user. Several issues, including the sensitivity of the numerical results to the tuning parameters, are discussed. A few test problems characterized by solutions with steep moving fronts, including the Buckley–Leverett equation and an extended Fisher–Kolmogorov equation, are investigated so as to demonstrate the algorithm and software performance.

Lower and upper solutions for a fully nonlinear beam equation

In this paper the two point fourth order boundary value problem is considered { u ( i v ) = f ( t , u , u ′ , u ″ , u ‴ ) , 0 < t < 1 , u ( 0 ) = u ′ ( 1 ) = u ″ ( 0 ) = u ‴ ( 1 ) = 0 , where f : [ 0 , 1 ] × R 4 → R is a continuous function satisfying a Nagumo-type condition. We prove the existence of a solution lying between lower and upper solutions using an a priori estimation, lower and upper solutions method and degree theory. The same arguments can be used, with adequate modifications, for any type of two-point boundary value problem, including all derivatives until order three, with the second and the third derivatives given in different end-points. An application to the extended Fisher–Kolmogorov problem will be obtained.

Periodic solutions of some fourth-order nonlinear differential equations

In this paper, using Mawhin’s continuation theorem, we obtain existence results concerning T -periodic solutions of fourth-order nonlinear differential equations of the type u ⁗ − p u ″ − g ( t , u ) = e ( t ) , where p ∈ R , g : R 2 → R is continuous, T -periodic in t and e : R → R is continuous, T -periodic with mean value zero. Applications are given for the extended Fisher–Kolmogorov and Swift–Hohenberg equations with forcing term.

Periodic solutions for extended Fisher–Kolmogorov and Swift–Hohenberg equations by truncature techniques

Combining truncature techniques with a variational approach we establish an existence result for nontrivial periodic solutions for a class of fourth-order ordinary differential equations involving extended Fisher–Kolmogorov and Swift–Hohenberg equations.

On the large time behavior of solutions of fourth order parabolic equations and ε-entropy of their attractors

We study the large time behavior of solutions of a class of fourth order parabolic equations defined on unbounded domains. Specific examples of the equations we study are the Swift–Hohenberg equation and the Extended Fisher–Kolmogorov equation. We establish the existence of a global attractor in a local topology. Since the dynamics is infinite dimensional, we use the Kolmogorov ε-entropy as a measure, deriving a sharp upper and lower bound. To cite this article: M.A. Efendiev, L.A. Peletier, C. R. Acad. Sci. Paris, Ser. I 344 (2007).

On the transition from pulled to pushed monotonic fronts of the extended Fisher–Kolmogorov equation

The extended Fisher–Kolmogorov equation u t = u xx - γ u xxxx + f ( u ) with arbitrary positive f ( u ) , satisfying f ( 0 ) = f ( 1 ) = 0 , has monotonic traveling fronts for γ < 1 12 . We find a simple lower bound on the speed of the fronts which allows to determine, for a given reaction term, when will the front of minimal speed be pushed.

Pattern forming pulled fronts: bounds and universal convergence

We analyze the dynamics of pattern forming fronts which propagate into an unstable state, and whose dynamics is of the pulled type, so that their asymptotic speed is equal to the linear spreading speed v * . We discuss a method that allows to derive bounds on the front velocity, and which, hence, can be used to prove for, among others, the Swift–Hohenberg equation, the extended Fisher–Kolmogorov equation and the cubic complex Ginzburg–Landau equation, that the dynamically relevant fronts are of the pulled type. In addition, we generalize the derivation of the universal power law convergence of the dynamics of uniformly translating pulled fronts to both coherent and incoherent pattern forming fronts. The analysis is based on a matching analysis of the dynamics in the leading edge of the front, to the behavior imposed by the nonlinear region behind it. Numerical simulations of fronts in the Swift–Hohenberg equation are in full accord with our analytical predictions.

Orthogonal cubic spline collocation method for the extended Fisher–Kolmogorov equation

A second-order splitting combined with orthogonal cubic spline collocation method is formulated and analysed for the extended Fisher–Kolmogorov equation. With the help of Lyapunov functional, a bound in maximum norm is derived for the semidiscrete solution. Optimal error estimates are established for the semidiscrete case. Finally, using the monomial basis functions we present the numerical results in which the integration in time is performed using RADAU 5 software library.

Multitransition kinks and pulses for fourth order equations with a bistable nonlinearity

We study the existence of stationary solutions of a class of diffusion equations related to the so-called extended Fisher–Kolmogorov equation and the Swift–Hohenberg equation. We prove the existence of multitransition kinks and pulses. These solutions are obtained as local minima of the associated functional. For the Swift–Hohenberg equation, our result partially proves a numerical conjecture.

Existence and numerical approximations of periodic solutions of semilinear fourth-order differential equations

A multiplicity result of existence of periodic solutions with prescribed wavelength for a class of fourth-order nonautonomous differential equations related either to the extended Fisher–Kolmogorov or to the Swift–Hohenberg equation is proved. Variational approach is used. Some numerical solutions are calculated via the finite element method.

Γ-limit for the extended Fisher–Kolmogorov equation

We consider the Lyapunov functional, of the rescaled Extended Fisher-Kolmogorov equation This is a fourth order generalization of the Fisher–Kolmogorov or Allen–Cahn equation. We prove that if ε → 0, then tends to the area functional in the sense of Γ-limits, where the transition energy is given by the one-dimensional kink of the Extended Fisher–Kolmogorov equation.

Existence and Stability of Traveling Fronts in the Extended Fisher–Kolmogorov Equation

We study traveling wave solutions to a general fourth-order differential equation that is a singular perturbation of the Fisher–Kolmogorov equation. We apply the geometric method for singularly perturbed systems to show that for every positive wavespeed there exists a traveling wave. Also, we find that there exists a critical wavespeed c* which divides these solutions into monotonic (c⩾c*) and oscillatory (c<c*) solutions. We show that the monotonic fronts are locally stable under perturbations in appropriate weighted Sobolev spaces by using various energy functionals.

Spatial chaos in a fourth-order nonlinear parabolic equation

Abstract Bounded stationary solutions for a fourth-order extended Fisher–Kolmogorov equation with cubic-like piecewise linear nonlinearity are given. Those solutions which have a finite number of zeros are characterized by means of a set of integers associated with the distance between the zeros. The value of the spatial entropy is estimated and the existence of spatial chaos is shown.

Uniqueness of the Stationary Wave for the Extended Fisher–Kolmogorov Equation

The extended Fisher–Kolmogorov equation, ut=−βuxxxx+uxx+u−u3, β>0, models a binary system near the Lifshitz critical point and is known to exhibit a stationary heteroclinic solution joining the equilibria ±1. For the classical case, β=0, the heteroclinic is u(x)=tanh(x/2) and is unique up to the obvious symmetries. We prove the conjecture that the uniqueness persists all the way to β=1/8, where the onset of spatial chaos associated with the loss of monotonicity of the stationary wave is known to occur. Our methods are non-perturbative and employ a global cross-section to the Hamiltonian flow of the stationary fourth order equation on the energy level of ±1. We also prove uniform a priori bounds on all bounded stationary solutions, valid for any β>0.