Solitary pulses and periodic wave trains in a bistable FitzHugh-Nagumo model with cross diffusion and cross advection

2022 ◽  
Vol 105 (1) ◽  
Author(s):  
Evgeny P. Zemskov ◽  
Mikhail A. Tsyganov ◽  
Genrich R. Ivanitsky ◽  
Werner Horsthemke
Keyword(s):  
Periodic Wave ◽  
Cross Diffusion ◽  
Wave Trains

Stability of periodic waves in shallow water

1974 ◽  
Vol 66 (1) ◽  
pp. 81-96 ◽  
Author(s):  
P. J. Bryant
Keyword(s):  
Shallow Water ◽  
Wave Train ◽  
Periodic Wave ◽  
Fundamental Wave ◽  
Cnoidal Waves ◽  
Margin Of Stability ◽  
Fundamental Wavelength ◽  
Stokes Waves ◽  
Wave Trains ◽  
Permanent Shape

Waves of small but finite amplitude in shallow water can occur as periodic wave trains of permanent shape in two known forms, either as Stokes waves for the shorter wavelengths or as cnoidal waves for the longer wavelengths. Calculations are made here of the periodic wave trains of permanent shape which span uniformly the range of increasing wavelength from Stokes waves to cnoidal waves and beyond. The present investigation is concerned with the stability of such permanent waves to periodic disturbances of greater or equal wavelength travelling in the same direction. The waves are found to be stable to infinitesimal and to small but finite disturbances of wavelength greater than the fundamental, the margin of stability decreasing either as the fundamental wave becomes more nonlinear (i.e. contains more harmonics), or as the wavelength of the periodic disturbance becomes large compared with the fundamental wavelength. The decreasing margin of stability is associated with an increasing loss of spatial periodicity of the wave train, to the extent that small but finite disturbances can cause a form of interaction between consecutive crests of the disturbed wave train. In such a case, a small but finite disturbance of wavelength n times the fundamental wavelength converts the wave train into n interacting wave trains. The amplitude of the disturbance subharmonic is then nearly periodic, the time scale being the time taken for repetitions of the pattern of interactions. When the disturbance is of the same wavelength as the permanent wave, the wave is found to be neutrally stable both to infinitesimal and to small but finite disturbances.

RESONANCE PATTERNS IN ONE-DIMENSIONAL ARRAYS OF COUPLED NONLINEAR EXCITABLE SYSTEMS

1994 ◽  
Vol 04 (06) ◽  
pp. 1631-1638 ◽  
Author(s):  
V. PÉREZ-MUÑUZURI ◽  
M. ALONSO ◽  
L.O. CHUA ◽  
V. PÉREZ-VILLAR
Keyword(s):  
Coupling Strength ◽  
Signal Transmission ◽  
Periodic Wave ◽  
Excitable Cells ◽  
Nonmonotonic Dependence ◽  
One Dimensional ◽  
Locking Range ◽  
Excitable Systems ◽  
Wave Trains ◽  
Coupling Strengths

Periodical signal transmission of waves through a one-dimensional array of coupled nonlinear electronic excitable cells have been investigated experimentally. Periodic wave trains give rise to a full devil’s staircase. The dependence of firing numbers defined for an excitable medium, on the amplitude and frequency of forcing, excitability of the medium, and coupling strength between cells is investigated. A nonmonotonic dependence between the locking range and the excitability has been observed for various n:m resonance regions, for different coupling strengths.

Non-linear dispersion of water waves

1967 ◽  
Vol 27 (2) ◽  
pp. 399-412 ◽  
Author(s):  
G. B. Whitham
Keyword(s):  
Water Waves ◽  
Wave Train ◽  
Periodic Wave ◽  
Wave Number ◽  
Amplitude Wave ◽  
Linear Dispersion ◽  
Elliptic Case ◽  
Non Linear ◽  
Wave Trains ◽  
Linear Water Waves

The slow dispersion of non-linear water waves is studied by the general theory developed in an earlier paper (Whitham 1965b). The average Lagrangian is calculated from the Stokes expansion for periodic wave trains in water of arbitrary depth. This Lagrangian can be used for the various applications described in the above reference. In this paper, the crucial question of the ‘type’ of the differential equations for the wave-train parameters (local amplitude, wave-number, etc.) is established. The equations are hyperbolic or elliptic according to whetherkh0is less than or greater than 1.36, wherekis the wave-number per 2π andh0is the undisturbed depth. In the hyperbolic case, changes in the wave train propagate and the characteristic velocities give generalizations of the linear group velocity. In the elliptic case, modulations in the wave train grow exponentially and a periodic wave train will be unstable in this sense; thus, periodic wave trains on water will be unstable ifkh0> 1·36, The instability of deep-water waves,kh0> 1·36, was discovered in a different way by Benjamin (1966). The relation between the two approaches is explained.

Nonlinear waves in a quintic FitzHugh–Nagumo model with cross diffusion: Fronts, pulses, and wave trains

2021 ◽  
Vol 31 (3) ◽  
pp. 033141
Author(s):  
Evgeny P. Zemskov ◽  
Mikhail A. Tsyganov ◽  
Klaus Kassner ◽  
Werner Horsthemke
Keyword(s):  
Nonlinear Waves ◽  
Cross Diffusion ◽  
Wave Trains

On the modulation of water waves in the neighbourhood of kh ≈ 1.363

1977 ◽  
Vol 357 (1689) ◽  
pp. 131-141 ◽  
Keyword(s):  
Schrödinger Equation ◽  
Water Waves ◽  
Schrodinger Equation ◽  
Multiple Scales ◽  
Wave Interaction ◽  
Periodic Wave ◽  
Stokes Wave ◽  
Modulation Equation ◽  
Wave Trains ◽  
The Stability

In 1967, T. Brooke Benjamin showed that periodic wave-trains on the surface of water could be unstable. If the undisturbed depth is h , and k is the wavenumber of the fundamental, then the Stokes wave is unstable if kh ≥ σ 0 , where σ 0 ≈ 1.363. The instability is provided by the growth of waves with a wavenumber close to k . This result is associated with an almost resonant quartet wave interaction and can be obtained by examining the cubic nonlinearity in the nonlinear Schrodinger equation for the modulation of harmonic water waves: this term vanishes at kh = cr0. In this paper the multiple-scales technique is adapted in order to derive the appropriate modulation equation for the amplitude of the fundamental when kh is near to σ 0 . The resulting equation takes the form i A T - a 1 A ζζ - a 2 A | A | 2 + a 3 A | A | 4 + i( a 4 | A | 2 A ζ - a 5 A (| A | 2 ) ζ ) - a 6 Aψ T = 0 where ψ ζ = | A | 2 , and the a i are real numbers. [Coefficients a 3 - a 6 are given on kh ≈ 1.363 only.] This equation is uniformly valid in that it reduces to the classical non-linear Schrödinger equation in the appropriate limit and is correct when a 2 = 0, i.e. at kh = σ 0 . The equation is used to examine the stability of the Stokes wave and the new inequality for stability is derived: this now depends on the wave amplitude. If the wave is unstable then it is expected that soli to ns will be produced: the simplest form of soliton is therefore examined by constructing the corresponding ordinary differential equation. Some comments are made concerning the phase-plane of this equation, but more analytical details are extracted by treating the new terms as perturbations of the classical Schrodinger soliton. It is shown that the soliton is both flatter (symmetrically) and skewed forward, although the skewing eventually gives way to an oscillation above the mean level.

scholarly journals Existence of periodic wave trains for an age-structured model with diffusion

2017 ◽  
Vol 22 (11) ◽  
pp. 0-0
Author(s):  
Zhihua Liu ◽  
◽  
Yayun Wu ◽  
Xiangming Zhang ◽  
Keyword(s):  
Periodic Wave ◽  
Structured Model ◽  
Wave Trains ◽  
Age Structured ◽  
Age Structured Model
Sign in / Sign up

Export Citation Format

Share Document