Existence Of Wave Front Solutions Research Articles

Front transition in higher order diffusion equations with a general reaction nonlinearity

In this paper, we investigate the wave front solutions of a class of higher order reaction diffusion equations with a general reaction nonlinearity. Linear stability analysis with a modulated travelling wave perturbation is used to prove the existence of wave front solutions. We proved that the studied equation supports both monotonic translating front and patterned front solutions. Also, a minimal front speed and the condition for a transition between these front types (monotonic and patterned) are determined. Two numerical examples are discussed (the extended Fisher-Kolmogorov equation with two different reaction nonlinearities) to support the obtained results.

Wavefront solutions of quasilinear reaction–diffusion systems with mixed quasi-monotonicity

ABSTRACTWe study the existence of wavefront solutions in a general class of mixed quasi-monotone reaction–diffusion systems with nonlinear diffusions in the form for . Each function is quasi-monotone increasing for some components of and decreasing for other components of . Such systems model reaction–diffusion processes with a density driven diffusion mechanism. Under certain general conditions we prove the existence of a traveling wave solution that is between a pair of coupled upper and lower solutions. As examples, we discuss a predator–prey model and a multi-species competition model, both with nonlinear diffusions. In both models, the presences of wavefront solutions flowing towards the coexistence states are established as illustrations for applications of our main result, with numerical simulations.

On existence of wavefront solutions in mixed monotone reaction-diffusion systems

In this article, we give an existence-comparison theorem for wavefront solutions in a general class of reaction-diffusion systems. With mixed quasi-monotonicity and Lipschitz condition on the set bounded by coupled upper-lower solutions, the existence of wavefront solution is proven by applying the Schauder Fixed Point Theorem on a compact invariant set. Our main result is then applied to well-known examples: a ratio-dependent predator-prey model, a three-species food chain model of Lotka-Volterra type and a three-species competition model of Lotka-Volterra type. For each model, we establish conditions on the ecological parameters for the presence of wavefront solutions flowing towards the coexistent states through suitably constructed upper and lower solutions. Numerical simulations on those models are also demonstrated to illustrate our theoretical results.

Existence of Wave Front Solutions of an Integral Differential Equation in Nonlinear Nonlocal Neuronal Network

An integral-differential model equation arising from neuronal networks with very general kernel functions is considered in this paper. The kernel functions we study here include pure excitations, lateral inhibition, lateral excitations, and more general synaptic couplings (e.g., oscillating kernel functions). The main goal of this paper is to prove the existence and uniqueness of the traveling wave front solutions. The main idea we apply here is to reduce the nonlinear integral-differential equation into a solvable differential equation and test whether the solution we get is really a wave front solution of the model equation.

Existence of wave front solutions and estimates of wave speed for a competition-diffusion system

Existence of wave front solutions and estimates of wave speed for a competition-diffusion system