Here we propose mechanisms for suppressing non-steady-state motions--propagating pulses, spiral waves, spiral-wave chaos--in excitable media. Our approach is based on two points: (1) excitable media are multistable; and (2) traveling waves in excitable media can be separated into fast and slow motions, which can be considered independently. We show that weak impulses can be used to change the v...

Motivated by physical and numerical observations of time oscillatory “galloping”, “spinning”, and “cellular” instabilities of detonation waves, we study Poincaré–Hopf bifurcation of traveling-wave solutions of viscous conservation laws. The main difficulty is the absence of a spectral gap between oscillatory modes and essential spectrum, preventing standard reduction to a finite-dimensional cen...

We consider the macroscopic, second order model of Kerner–Konhäuser for traffic flow given by a system of PDE. Assuming conservation of cars, traveling waves solution of the PDE are reduced to a dynamical system in the plane. We prove that under generic conditions on the so called fundamental diagram, the surface of critical points has a fold or cusp catastrophe and each fold point gives rise t...

This paper is concerned with the existence of the traveling bands to the Keller-Segel model with cell population growth in the form of a chemical uptake kinetics. We find that when the cell growth is considered, the profile of traveling bands, the minimum wave speed and the range of the chemical consumption rate for the existence of traveling wave solutions will change. Our results reveal that ...

We have considered infinite systems of nonlinear ODEs on the one-dimensional integer lattice which describes the activity in an excitatorily coupled network of excitable cells. For an ideal nonlinearity, we calculated the speed of propagation of an activity and derived the condition for its existence. We also studied the existence and stability of the traveling wave solution and gave, in the si...

We establish the existence of traveling front solutions and small amplitude traveling wave train solutions for a reaction-diffusion system based on a predator-prey model with Holling type-II functional response. The traveling front solutions are equivalent to heteroclinic orbits in R(4) and the small amplitude traveling wave train solutions are equivalent to small amplitude periodic orbits in R...

For viscous conservation laws, solutions contain smooth but high-contrast features, which require the use of fine grids to properly resolve. On coarse grids, these high-contrast jumps resemble shocks rather than their true viscous profiles, which could lead to issues in the numerical approximation of their underlying dynamics. In many cases, the equations of motion emit traveling wave solutions...

In this paper, we consider a class of nonlocal equations where the convolution kernel is given by Bessel potential symbol order α for > 1. Based on properties operator, apply global bifurcation technique to show existence highest, even, 2 π-periodic traveling-wave solution. The regularity wave proved be exactly Lipschitz.

A modified F-expansion method to find the exact traveling wave solutions of  two-component nonlinear partial differential equations (NLPDEs) is discussed. We use this method to construct many new solutions to the nonlinear Whitham-Broer-Kaup system (1+1)-dimensional. The solutions obtained include Jacobi elliptic periodic wave solutions which exactly degenerate to the soliton solutions, triangu...

We analyze the dynamics of a spatially extended system with bistability between a homogeneous stationary state and an unstable fixed point surrounded by an oscillatory state. We show that a wave front extinguishes homogeneous oscillations, replacing them with unsteady oscillations. A traveling wave solution connects the unsteady oscillations to the stationary state. The difference in potentials...