scholarly journals Travelling Waves for Adaptive Grid Discretizations of Reaction Diffusion Systems I: Well-Posedness

2021 ◽  
Author(s):  
H. J. Hupkes ◽  
E. S. Van Vleck
Keyword(s):  
Travelling Waves ◽  
Reaction Diffusion ◽  
Adaptive Grid ◽  
Discrete Analogue ◽  
Moving Mesh Method ◽  
Reaction Diffusion Systems ◽  
Effective Dynamics ◽  
Diffusion Systems ◽  
Infinite Range Interactions ◽  
Van Vleck

AbstractIn this paper we consider a spatial discretization scheme with an adaptive grid for the Nagumo PDE. In particular, we consider a commonly used time dependent moving mesh method that aims to equidistribute the arclength of the solution under consideration. We assume that the discrete analogue of this equidistribution is strictly enforced, which allows us to reduce the effective dynamics to a scalar non-local problem with infinite range interactions. We show that this reduced problem is well-posed and obtain useful estimates on the resulting nonlinearities. In the sequel papers (Hupkes and Van Vleck in Travelling waves for adaptive grid discretizations of reaction diffusion systems II: linear theory; Travelling waves for adaptive grid discretizations of reaction diffusion systems III: nonlinear theory) we use these estimates to show that travelling waves persist under these adaptive spatial discretizations.

scholarly journals Travelling Waves for Adaptive Grid Discretizations of Reaction Diffusion Systems II: Linear Theory

2021 ◽  
Author(s):  
H. J. Hupkes ◽  
E. S. Van Vleck
Keyword(s):  
Travelling Waves ◽  
Reaction Diffusion ◽  
Travelling Wave ◽  
Order Differential Operator ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Non Local ◽  
Infinite Range Interactions ◽  
Van Vleck ◽  
Adaptation Method

AbstractIn this paper we consider an adaptive spatial discretization scheme for the Nagumo PDE. The scheme is a commonly used spatial mesh adaptation method based on equidistributing the arclength of the solution under consideration. We assume that this equidistribution is strictly enforced, which leads to the non-local problem with infinite range interactions that we derived in Hupkes and Van Vleck (J Dyn Differ Equ 28:955, 2016). For small spatial grid-sizes, we establish some useful Fredholm properties for the operator that arises after linearizing our system around the travelling wave solutions to the original Nagumo PDE. In particular, we perform a singular perturbation argument to lift these properties from the natural limiting operator. This limiting operator is a spatially stretched and twisted version of the standard second order differential operator that is associated to the PDE waves.

Noise-induced suppression of periodic travelling waves in oscillatory reaction–diffusion systems

2010 ◽  
Vol 466 (2119) ◽  
pp. 1903-1917 ◽  
Author(s):  
Michael Sieber ◽  
Horst Malchow ◽  
Sergei V. Petrovskii
Keyword(s):  
Travelling Waves ◽  
Natural Populations ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Oscillatory Reaction ◽  
Stochastic Forcing ◽  
Spatial Direction ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Spatio Temporal

Ecological field data suggest that some species show periodic changes in abundance over time and in a specific spatial direction. Periodic travelling waves as solutions to reaction–diffusion equations have helped to identify possible scenarios, by which such spatio-temporal patterns may arise. In this paper, such solutions are tested for their robustness against an irregular temporal forcing, since most natural populations can be expected to be subject to erratic fluctuations imposed by the environment. It is found that small environmental noise is able to suppress periodic travelling waves in stochastic variants of oscillatory reaction–diffusion systems. Irregular spatio-temporal oscillations, however, appear to be more robust and persist under the same stochastic forcing.

Spectral stability of modulated travelling waves bifurcating near essential instabilities

2000 ◽  
Vol 130 (2) ◽  
pp. 419-448 ◽  
Author(s):  
B. Sandstede ◽  
A. Scheel
Keyword(s):  
Imaginary Axis ◽  
Travelling Waves ◽  
Reaction Diffusion ◽  
Spectral Stability ◽  
Moving Frame ◽  
Turing Patterns ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Spatially Periodic ◽  
Time Periodic

Localized travelling waves to reaction-diffusion systems on the real line are investigated. The issue addressed in this work is the transition to instability which arises when the essential spectrum crosses the imaginary-axis. In the first part of this work, it has been shown that large modulated pulses bifurcate near the onset of instability; they are a superposition of the primary pulse with spatially periodic Turing patterns of small amplitude. The bifurcating modulated pulses can be parametrized by the wavelength of the Turing patterns. Furthermore, they are time periodic in a moving frame. In the second part, spectral stability of the bifurcating modulated pulses is addressed. It is shown that the modulated pulses are spectrally stable if and only if the small Turing patterns are spectrally stable, that is, if their continuous spectrum only touches the imaginary-axis at zero. This requires an investigation of the period map associated with the time-periodic modulated pulses.

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