Existence of the solitary wave solutions supported by the modified FitzHugh–Nagumo system

Abstract

We study a system of nonlinear differential equations simulating transport phenomena in active media. The model we are interested in is a generalization of the celebrated FitzHugh–Nagumo system describing the nerve impulse propagation in axon. The modeling system is shown to possesses soliton-like solutions under certain restrictions on the parameters. The results of theoretical studies are backed by the direct numerical simulation.

Soliton-like solutions supported by refined hydrodynamic-type model of an elastic medium with soft inclusions

“…obtained by substituting the modified state equation (6) into system (1), mainly concentrating on stating the existence of the soliton-like TW solutions and studying how does the addition of terms with higher derivatives affects the stability of soliton profiles and their dynamical features. Note that in what follows, we will also keep conditions ( 10), (11), and (12), which guarantee the presence of solitary waves in the limiting case κ = 0. Note also that for σ < 0, conditions ( 10) and ( 11) act as a necessary conditions for the existence of soliton-like solutions [24].…”

“…for the reason that conditions (10), (11) in the case when σ < 0 and, most likely, condition (12) in the case when σ > 0 serve as necessary conditions for the existence of the trajectories biasymptotic to the stationary point P [24].…”

“…As in the vast majority of works devoted to the study of asymptotic models with soliton solutions, the question of how the discarded terms affect the existence of solitons and their stability remains open. But there is some evidence that, with certain modifications of the equation of state [17], as well as the incorporation of nonlocal effects [12], soliton structures are preserved, however, these changes radically affect the stability of localized solutions and their dynamic properties.…”