scholarly journals Periodic Traveling Waves of the Regularized Short Pulse and Ostrovsky Equations: Existence and Stability

2017 ◽  
Vol 49 (1) ◽  
pp. 674-698 ◽  
Author(s):  
Sevdzhan Hakkaev ◽  
Milena Stanislavova ◽  
Atanas Stefanov
Keyword(s):  
Traveling Waves ◽  
Short Pulse ◽  
Periodic Traveling Waves ◽  
Existence And Stability

scholarly journals Periodically kicked feedforward chains of simple excitable FitzHugh-Nagumo neurons

2021 ◽  
Author(s):  
B. Ambrosio ◽  
S.M. Mintchev
Keyword(s):  
Numerical Investigation ◽  
Traveling Waves ◽  
Small Amplitude ◽  
Mixed Mode ◽  
Filtration Property ◽  
Periodic Traveling Waves ◽  
Mixed Mode Oscillations ◽  
Mode Oscillation ◽  
Existence And Stability ◽  
Mixed Mode Oscillation

Abstract This article communicates results on regular depolarization cascades in periodically-kicked feedforward chains of excitable two-dimensional FitzHugh-Nagumo systems driven by sufficiently strong excitatory forcing at the front node. The study documents a parameter exploration by way of changes to the forcing period, upon which the dynamics undergoes a transition from simple depolarization to more complex behavior, including the emergence of mixed-mode oscillations. Both rigorous studies and careful numerical observations are presented. In particular, we provide rigorous proofs for existence and stability of periodic traveling waves of depolarization, as well as the existence and propagation of a simple mixed-mode oscillation that features depolarization and refraction in alternating fashion. Detailed numerical investigation reveals a mechanism for the emergence of complex mixed-mode oscillations featuring a potentially high number of large amplitude voltage spikes interspersed by an occasional small amplitude reset that fails to cross threshold. Further careful numerical investigation provides insights into the propagation of this complex phenomenology in the downstream, where we see an effective filtration property of the network; the latter amounts to a successive reduction in the complexity of mixed-mode oscillations down the chain.

scholarly journals Stability of periodic traveling waves for complex modified Korteweg–de Vries equation

2010 ◽  
Vol 248 (10) ◽  
pp. 2608-2627 ◽  
Author(s):  
Sevdzhan Hakkaev ◽  
Iliya D. Iliev ◽  
Kiril Kirchev
Keyword(s):  
Traveling Waves ◽  
Vries Equation ◽  
Periodic Traveling Waves ◽  
De Vries

scholarly journals On the stability of periodic traveling waves for the modified Kawahara equation

2019 ◽  
pp. 1-8
Author(s):  
Gisele Detomazi Almeida ◽  
Fabrício Cristófani ◽  
Fábio Natali
Keyword(s):  
Traveling Waves ◽  
Kawahara Equation ◽  
Periodic Traveling Waves ◽  
The Stability

scholarly journals Periodic traveling waves in SIRS endemic models

2009 ◽  
Vol 49 (1-2) ◽  
pp. 393-401 ◽  
Author(s):  
Tong Li ◽  
Yi Li ◽  
Herbert W. Hethcote
Keyword(s):  
Traveling Waves ◽  
Periodic Traveling Waves

scholarly journals Hyperbolic traveling waves driven by growth

2014 ◽  
Vol 24 (06) ◽  
pp. 1165-1195 ◽  
Author(s):  
Emeric Bouin ◽  
Vincent Calvez ◽  
Grégoire Nadin
Keyword(s):  
Nonlinear Stability ◽  
Traveling Waves ◽  
Hyperbolic Model ◽  
Kpp Equation ◽  
Transition Parameter ◽  
Minimal Speed ◽  
Traveling Fronts ◽  
Traveling Front ◽  
Maximal Speed ◽  
Existence And Stability

We perform the analysis of a hyperbolic model which is the analog of the Fisher-KPP equation. This model accounts for particles that move at maximal speed ϵ-1 (ϵ > 0), and proliferate according to a reaction term of monostable type. We study the existence and stability of traveling fronts. We exhibit a transition depending on the parameter ϵ: for small ϵ the behavior is essentially the same as for the diffusive Fisher-KPP equation. However, for large ϵ the traveling front with minimal speed is discontinuous and travels at the maximal speed ϵ-1. The traveling fronts with minimal speed are linearly stable in weighted L2 spaces. We also prove local nonlinear stability of the traveling front with minimal speed when ϵ is smaller than the transition parameter.

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