Monotone Traveling Wave Solutions Research Articles

Novel traveling waves solutions for nonlinear delayed dynamical neural networks with leakage term

In this work, the exponential stability of monotone traveling wave solutions for a class of nonlinear delayed dynamical neural networks with leakage term and distributed delays is investigated. The outcomes of this study are new and has obtained by applying of Ikeharas theorem, the weighted energy method, Taylor formula, comparison principle and the first integral mean value theorem. In addition, the leakage term is considered for the first time in this work to the dynamical model of cellular neural networks.

Traveling wave phenomena of a nonlocal reaction-diffusion equation with degenerate nonlinearity

This paper deals with traveling wave phenomena of a degenerate reaction-diffusion equation with the nonlocal effect. We study the existence of traveling wave solutions which may be non-monotonic based on the two-point boundary value problem and Schauder’s fixed point theorem. We are excited to find that the unknown positive steady state is exactly a unique positive equilibrium for the large wave speed and the monotonicity of traveling waves depends on the wave speed. On the other hand, we demonstrate the existence of monotone traveling wave solutions by using the monotone iteration and operator theory, and investigate how wave profile (e.g. being non-monotone or periodic) is affected by the enhancement of nonlocality, parameter values and variation of initial values for two different forms of convolution kernel functions.

MONOTONE TRAVELING WAVES OF NONMONOTONE INTEGRODIFFERENCE EQUATIONS

This paper deals with the existence of monotone traveling wave solutions of integrodifference equations without monotonicity. By the comparison principle in the sense of exponential ordering, we confirm the existence of monotone traveling wave solutions for integrodifference equations when the overcompensatory is weak.

Traveling wave solutions for the Richards equation with hysteresis

Abstract We investigate the one-dimensional non-equilibrium Richards equation with play-type hysteresis. It is known that regularized versions of this equation permit traveling wave solutions that show oscillations and, in particular, the physically relevant effect of a saturation overshoot. We investigate here the non-regularized hysteresis operator and combine it with a positive $\tau $-term. Our result is that the model has monotone traveling wave solutions. These traveling waves describe the behavior of fronts in a bounded domain. In a two-dimensional interpretation, the result characterizes the speed of fingers in non-homogeneous solutions.

Existence and non-monotonicity of traveling wave solutions for general diffusive predator-prey models

This paper is concerned with the existence and non-monotonicity of traveling wave solutions for general diffusive predator-prey models. By using Schauder's fixed point theorem and the existence of contracting rectangles, we obtain the existence result. Then we investigate the asymptotic behavior of positive monotone traveling wave solutions by using the modified Ikehara's Theorem. With the help of their asymptotic behavior, we provide a sufficient condition which guarantee that all positive traveling wave solutions of the system are non-monotone. Furthermore, to illustrate our main results, the existence and non-monotonicity of traveling wave solutions of Lotka-Volterra predator-prey model and modified Leslie-Gower predator-prey models with different kinds of functional responses are also discussed.

The stability of traveling wave solutions for a diffusive competition system of three species

In this article, we investigate the stability of monotone traveling wave solutions for a diffusive three species competition system. By considering the initial perturbations of the traveling waves in some weighted function spaces, the monotone three-species waves become asymptotically stable. Further stability will be determined from the asymptotic behavior of the waves. This can be achieved by using the method of super- and subsolutions.

Traveling wave solutions of a reaction-diffusion equation with state-dependent delay

This paper is concerned with the traveling wave solutions of a reaction-diffusion equation with state-dependent delay. When the birth function is monotone, the existence and nonexistence of monotone traveling wave solutions are established. When the birth function is not monotone, the minimal wave speed of nontrivial traveling wave solutions is obtained. The results are proved by the construction of upper and lower solutions and application of the fixed point theorem.

Traveling wave solutions in a Lotka-Volterra type competition recursion

This paper is concerned with the traveling wave solutions of a competitive integrodifference system with Lotka-Volterra type nonlinearity. The existence of traveling wave solutions is proved by constructing generalized upper and lower solutions. The asymptotic behavior of traveling wave solutions is established by combining the theory of asymptotic spreading with the idea of contracting rectangles. The nonexistence of monotone traveling wave solutions is also confirmed by the theory of asymptotic spreading.

Multitype Branching Brownian Motion and Traveling Waves

In this article we study the parabolic system of equations which is closely related to a multitype branching Brownian motion. Particular attention is paid to the monotone traveling wave solutions of this system. Provided with some moment conditions, we show the existence, uniqueness, and asymptotic behaviors of such waves with speed greater than or equal to a critical value c̲ and nonexistence of such waves with speed smaller than c̲.

Multitype Branching Brownian Motion and Traveling Waves

In this article we study the parabolic system of equations which is closely related to a multitype branching Brownian motion. Particular attention is paid to the monotone traveling wave solutions of this system. Provided with some moment conditions, we show the existence, uniqueness, and asymptotic behaviors of such waves with speed greater than or equal to a critical value c̲ and nonexistence of such waves with speed smaller than c̲.

Competitive Gompertz model of two species

In this paper, a competitive Gompertz model of two species is proposed. Furthermore, under a certain condition, the existence of monotone traveling wave solutions of this model is shown by the method of super- and subsolutions, which is developed in (6, 15, 16). Dedicate to my lovely sisters Wen and Chun

On traveling wave solutions to a Hamilton–Jacobi–Bellman equation with inequality constraints

The aim of this paper is to construct and analyze solutions to a class of Hamilton–Jacobi–Bellman equations with range bounds on the optimal response variable. Using the Riccati transformation we derive and analyze a fully nonlinear parabolic partial differential equation for the optimal response function. We construct monotone traveling wave solutions and identify parametric regions for which the traveling wave solution has a positive or negative wave speed.

Symmetry of Traveling Wave Solutions to the Allen–Cahn Equation in $${{\mathbb R^{2}}}$$

In this paper, we prove even symmetry of monotone traveling wave solutions to the balanced Allen-Cahn equation in the entire plane. Related results for the unbalanced Allen-Cahn equation are also discussed.

Existence and uniqueness of traveling waves in a class of unidirectional lattice differential equations

We prove the existence and uniqueness, for wave speeds sufficiently large, of monotone traveling wave solutions connecting stable to unstable spatial equilibria for a class of $N$-dimensional lattice differential equations with unidirectional coupling. This class of lattice equations includes some cellular neural networks, monotone systems, and semi-discretizations for hyperbolic conservation laws with a source term. We obtain a variational characterization of the critical wave speed above which monotone traveling wave solutions are guaranteed to exist. We also discuss non-monotone waves, and the coexistence of monotone and non-monotone waves.

Traveling wave solutions to the nonlinear evolution equation for the risk preference

A singular nonlinear partial differential equation (PDE) is introduced, which can be interpreted as the evolution of the risk preference in the optimal investment problem under the random risk process. The unknown quantity is related to the Arrow-Pratt coefficient of relative risk aversion with respect to the optimal value function. We show the existence of monotone traveling wave solutions and the nonexistence of non-monotone such solutions, which are

Propagation of second order integrodifference equations with local monotonicity

This paper is concerned with the spreading speeds and traveling wavefronts of second order integrodifference equations with local monotonicity. By introducing two auxiliary integrodifference equations, the spreading speed and traveling wave solutions are studied. In particular, we obtain the nonexistence of monotone traveling wave solutions for an example if it is local monotone. These results are applied to a model which is obtained by introducing the spatial variable to a difference equation used by the International Whaling Commission.

Traveling wave fronts in reaction-diffusion systems with spatio-temporal delay and applications

This paper is concerned with monotone traveling wave solutions of reaction-diffusion systems with spatio-temporal delay. Our approach is to use a new monotone iteration scheme based on a lower solution in the set of the profiles. The smoothness of upper and lower solutions is not required in this paper. We will apply our results to Nicholson's blowflies systems with non-monotone birth functions and show that the systems admit traveling wave solutions connecting two spatially homogeneous equilibria and the wave shape is monotone. Due to the biological realism, the positivity of the monotone traveling wave solutions can be directly obtained by the construction of suitable upper-lower solutions.

DIVERSITY OF TRAVELING WAVE SOLUTIONS IN DELAYED CELLULAR NEURAL NETWORKS

This work investigates the diversity of traveling wave solutions for a class of delayed cellular neural networks on the one-dimensional integer lattice ℤ1. The dynamics of a given cell is characterized by instantaneous self-feedback and neighborhood interaction with distributed delay due to, for example, finite switching speed and finite velocity of signal transmission. Applying the monotone iteration scheme, we can deduce the existence of monotonic traveling wave solutions provided the templates satisfy the so-called quasi-monotonicity condition. We then consider two special cases of the delayed cellular neural network in which each cell interacts only with either the nearest m left neighbors or the nearest m right neighbors. For the former case, we can directly figure out the analytic solution in an explicit form by the method of step with the help of the characteristic function and then prove that, in addition to the existence of monotonic traveling wave solutions, for certain templates there exist nonmonotonic traveling wave solutions such as camel-like waves with many critical points. For the latter case, employing the comparison arguments repeatedly, we can clarify the deformation of traveling wave solutions with respect to the wave speed. More specifically, we can describe the transition of profiles from monotonicity, damped oscillation, periodicity, unboundedness and back to monotonicity as the wave speed is varied. Some numerical results are also given to demonstrate the theoretical analysis.

Asymptotic speed of propagation and traveling wave solutions for a lattice integral equation

We investigate the asymptotic speed of propagation and monotone traveling wave solutions for a lattice integral equation which is an epidemic model while the population is distributed on one-dimensional lattice Z . It is proved that the asymptotic speed of propagation c ∗ coincides with the minimal wave speed.

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.