Monotone Traveling Wave Solutions Research Articles

EXISTENCE OF MONOTONIC TRAVELING WAVES IN LATTICE DYNAMICAL SYSTEMS

This paper deals with the existence of monotonic traveling and standing wave solutions for a certain class of lattice differential equations. Employing the techniques of monotone iteration coupled with the concept of upper and lower solutions in the theory of monotone dynamical systems, we can classify the monotonic traveling wave solutions with various asymptotic boundary conditions. For the case of zero wave speed, a novel discrete monotone iteration scheme is established for proving the existence of monotonic standing wave solutions. Applications are made to several models including cellular neural networks, original and modified RTD-based cellular neural networks. Numerical simulations of the monotone iteration schemes are also given.

Structure of a class of traveling waves in delayed cellular neural networks

This work investigates the structure of a class of traveling wave solutions of delayed cellular neural networks distributed in the one-dimensional integer lattice $\mathbb Z^1$. The dynamics of a given cell is characterized by instantaneous self-feedback and neighborhood interaction with its two left neighbors in which one is instantaneous and the other is distributively delayed due to, for example, finite switching speed and finite velocity of signal transmission. Applying the method of step with the aid of positive roots of the corresponding characteristic function of the profile equation, we can directly figure out the solution in explicit form. We then partition the parameter space $(\alpha, \beta)$-plane into four regions such that the qualitative properties of traveling waves can be completely determined for each region. In addition to the existence of monotonic traveling wave solutions, we also find that, for certain parameters, there exist non-monotonic traveling wave solutions such as camel-like waves with many critical points.

Wave propagation in RTD-based cellular neural networks

This work investigates the existence of monotonic traveling wave and standing wave solutions of RTD-based cellular neural networks in the one-dimensional integer lattice Z 1 . For nonzero wave speed c, applying the monotone iteration method with the aid of real roots of the corresponding characteristic function of the profile equation, we can partition the parameter space ( γ, δ)-plane into four regions such that all the admissible monotonic traveling wave solutions connecting two neighboring equilibria can be classified completely. For the case of c=0, a discrete version of the monotone iteration scheme is established for proving the existence of monotonic standing wave solutions. Furthermore, if γ or δ is zero then the profile equation for the standing waves can be viewed as an one-dimensional iteration map and we then prove the multiplicity results of monotonic standing waves by using the techniques of dynamical systems for maps. Some numerical results of the monotone iteration scheme for traveling wave solutions are also presented.

Linear vs. nonlinear selection for the propagation speed of the solutions of scalar reaction‐diffusion equations invading an unstable equilibrium

AbstractWe 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. © 2004 Wiley Periodicals, Inc.

Traveling waves in time periodic lattice differential equations

The current paper concerns with traveling wave solutions to time periodic two-dimensional lattice differential equations of bistable type. Employing the so-called ‘vanishing viscosity’ approach, we obtain the existence of monotone traveling wave solutions connecting the two spatially homogeneous periodic solutions. Uniqueness of the wave speed and uniqueness of continuous wave front are shown by repeatedly utilizing comparison arguments. The existence and uniqueness results of the current paper together with the stability results proved in an author's earlier paper generalize those in the time-independent case.

Existence and Multiplicity of Traveling Waves in a Lattice Dynamical System

This work proves the existence and multiplicity results of monotonic traveling wave solutions for some lattice differential equations by using the monotone iteration method. Our results include the model of cellular neural networks (CNN). In addition to the monotonic traveling wave solutions, non-monotonic and oscillating traveling wave solutions in the delay type of CNN are also obtained.

Scaling Variables and Stability of Hyperbolic Fronts

We consider the damped hyperbolic equation $$ \epsilon u_{tt} + u_t \,=\, u_{xx} + \FF(u) , \quad x \in \mathbf{R} , \quad t \ge 0 , \leqno(1) $$ where $\epsilon$ is a positive, not necessarily small parameter. We assume that $\FF(0) = \FF(1) = 0$ and that $\FF$ is concave on the interval [0,1]. Under these hypotheses, (1) has a family of monotone traveling wave solutions (or propagating fronts) connecting the equilibria u=0 and u=1. This family is indexed by a parameter $c \ge c_*$ related to the speed of the front. In the critical case $c=c_*$, we prove that the traveling wave is asymptotically stable with respect to perturbations in a weighted Sobolev space. In addition, we show that the perturbations decay to zero like $t^{-3/2}$ as $t \to +\infty$ and approach a universal self-similar profile, which is independent of $\epsilon$, F, and the initial data. In particular, our solutions behave for large times like those of the parabolic equation obtained by setting $\epsilon = 0$ in (1). The proof of our results relies on various energy estimates for (1) rewritten in self-similar variables $x/\sqrt{t}$, $\log t$.

Traveling wave solutions for some density dependent diffusion equations

In this paper, we show the existence and stability of monotone traveling wave solutions for some nonlinear parabolic equations arising in population dynamics. Using these traveling wave solutions as comparison functions, we investigate the asymptotic behavior of solutions of the pure initial value problem and estimate the support of the solutions.

Traveling Wave and Multiple Traveling Wave Solutions of Parabolic Equations

We consider scalar equations $u_t = f(u_{xx} ,u_x ,u)$ with $\frac{\partial }{{\partial \alpha }}f(\alpha ,\beta ,\gamma ) \geq 1$. We first determine the stability of the monotonic traveling wave solutions $u(t,x) = \phi (x - ct,c)$. We then study the continued existence and bifurcations of these solutions as the wavespeed c varies. We use these continuation results to explore the connection between the initial conditions $u(0,x)$ and the wavespeed(s) of the resulting solution $u(t,x)$.