Diversity of traveling wave solutions in FitzHugh–Nagumo type equations

Abstract

In this work we consider the diversity of traveling wave solutions of the FitzHugh-Nagumo type equationswhere f (u, w) = u(u − a(w))(1 − u) for some smooth function a(w) and g(u, w) = u − w. When a(w) crosses zero and one, the corresponding profile equation possesses special turning points which result in very rich dynamics. In [W. Liu, E. Van Vleck, Turning points and traveling waves in FitzHugh-Nagumo type equations, J. Differential Equations 225 (2006) 381-410], Liu and Van Vleck examined traveling waves whose slow orbits lie only on two portions of the slow manifold, and obtained the existence results by using the geometric singular perturbation theory. Based on the ideas of their work, we study the co-existence of different traveling waves whose slow orbits could involve all portions of the slow manifold. There are more complicated and richer dynamics of traveling waves than those of [W. Liu, E. Van Vleck, Turning points and traveling waves in FitzHugh-Nagumo type equations, J. Differential Equations 225 (2006) 381-410]. We give a complete classification of all different fronts of traveling waves, and provide an example to support our theoretical analysis.

Dynamics of traveling waves for predator-prey systems with Allee effect and time delay

“…Among them, the geometric singular perturbation theory developed by Fenichel [12] is an effective method to prove the existence of traveling waves in evolution equations with small parameters. This theory has been applied to various equations, including Keller-Segel systems [10,25], FitzHugh-Nagumo equations [14,20,30], nonlinear Belousov-Zhabotinskii system [11], Liénard equations [18], Camassa-Holm equations [9,8], etc.…”

On the asymptotic spatial behaviour of the solutions of the nerve system

“…It is worth noting that if in this system we take = 0, we obtain the usual system of the FitzHugh-Nagumo equations. These systems have deserved much attention in recent years (see [2,10,14,17,23,26,27], among others) from a mathematical point of view. In this paper we want to analyse an aspect which has not been considered yet in the literature.…”

The exponential behavior of a stochastic Cahn-Hilliard-Navier-Stokes model with multiplicative noise

“…Traveling wave solutions correspond to heteroclinic or homoclinic orbits of related ordinary differential equations. In general, such orbits can be found by means of geometric singular perturbation theory [9], which has been used by many researchers to obtain the existence of traveling waves for different nonlinear differential equations including generalized KdV equations [16], Fisher equations [2], perturbed BBM equation [7], FitzHugh-Nagumo equation [13,25], reaction-diffusion equations [30], tissue interaction model [3], predator-prey models and epidemiology [22,11], etc. The method has also received a great deal of interests in studying semilinear elliptic equations [4], slow-fast dynamic systems [26,27], Liénard equations [21], etc.…”

“…The former is called the layer problem and the latter is called the reduced system. The problem (13) and 14are lower dimensional and can often be analysed in sufficient detail. By 'gluing' together fast and slow pieces of orbits, respectively obtained in the fast and slow limits, one can formally construct global singular structures, such as singular periodic orbits and singular homoclinic orbits.…”