Research Page
Spike Dynamics and Stability for Reaction-Diffusion Systems in
One-Dimension
description to be written; pictures and movies
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The Stability of Spike Solutions to the One-Dimensional Gierer-Meinhardt
Model
with D. Iron, J. Wei (Physica D, Vol. 150, (2001) pp. 25-62.)
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Asymmetric Spike Patterns for the One-Dimensional Gierer-Meinhardt
Model: Equilibria and Stability
with J. Wei (European J. Appl. Math., Vol. 13, No. 3, (2002),
pp. 283-320.)
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Hopf Bifurcation and Oscillatory Instabilities of Spike Solutions for
the One-Dimensional Gierer-Meinhardt Model
with J. Wei (J. Nonlinear Science, Vol. 13, No. 2, (2003),
pp. 209-264.)
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The Existence and Stability of Spike Equilibria in the One-Dimensional
Gray-Scott Model: The Low Feed Rate Regime
with T. Kolokolnikov and J. Wei (Studies in Appl. Math., Vol. 115,
No. 1, (2005), pp. 21-71.)
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The Existence and Stability of Spike Equilibria in the One-Dimensional
Gray-Scott Model: The Pulse-Splitting Regime
with T. Kolokolnikov and J. Wei (Physica D, Vol. 202, No. 3-4,
(2005), pp. 258-293.)
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Slow Translational Instabilities of Spike Patterns in the One-Dimensional
Gray-Scott Model
with T. Kolokolnikov and J. Wei (Interfaces and Free
Boundaries, Vol. 8, No. 2, (2006), pp. 185-222.)
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The Dynamics of Multi-Spike Solutions for the One-Dimensional
Gierer-Meinhardt Model
with D. Iron (SIAM J. Appl. Math., Vol. 62, No. 6, (2002),
pp. 1924-1951.)
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Numerical Challenges for Resolving Spike Dynamics for Two
Reaction-Diffusion Systems
with W. Sun, T. Tang, and J. Wei (Studies in Appl. Math., Vol. 111,
No. 1, (2003), pp. 41-84.)
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The Slow Dynamics of Two-Spike Solutions for the Gray-Scott and
Gierer-Meinhardt Systems: Competition and Oscillatory Instabilities
with W. Sun and R. Russell (SIAM J. Appl. Dyn. Systems., Vol. 4, No. 4,
(2005), pp. 904-953.)
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Oscillatory Instabilities of Multi-Spike Patterns for the One-Dimensional
Gray-Scott Model
with W. Chen (European J. Appl. Math., Vol. 20, No. 2, (2009),
pp. 187-214).
- The Dynamics and Stability of
Spike-Type Solutions to the Gierer-Meinhardt Model with Subdiffusion
with Y. Nec (Physica D, Vol. 241, No. 10, (2012), pp. 947-963).
- Slowly Varying Control
Parameters, Delayed Bifurcations, and the Stability of Spikes in
Reaction-Diffusion Systems
with J. Tzou and T. Kolokolnikov, (Physica D, Vol. 290, (2015),
pp. 24-43).
- The Stability of Localized
Spikes for the 1-D Brusselator Reaction-Diffusion Model
with J. C. Tzou and Y. Nec, (Europ. J. Appl. Math., Vol. 24,
No. 4, (2013), pp. 515-564).
- An Explicitly Solvable Nonlocal
Eigenvalue Problem and the Stability of a Spike for a Class of
Reaction-Diffusion System with Sub-Diffusion
with Y. Nec, (Mathematical Modeling of Natural Phenomena (special
issue on anomalous diffusion), Vol. 8, No. 2, (2013), pp. 55-87).
- The Stability and Slow Dynamics of
Two-Spike Patterns for a Class of Reaction Diffusion System
with Y. Nec, (Mathematical Modeling of Natural Phenomena (special issue
on Bifurcation), Vol. 8, No. 5, (2013), pp. 206-232).
- Delayed Reaction-Kinetics and the
Stability of Spikes for the Gierer-Meinhardt Model
with Nabil Fadai and Juncheng Wei (SIAM J. Appl. Math. Vol. 77, No. 2,
(2017), pp. 664-696).
- Transition to Blowup in a
Reaction-Diffusion Model with Localized Spike Solutions
with Vivi Rottschafer and Justin Tzou
(Europ. J. Appl. Math., Vol. 28, No. 6, (2017), pp. 1015-1055).
- A Time-Delay in the Activator
Kinetics Enhances the Stability of a Spike Solution to the Gierer-Meinhardt
Model
with Nabil Fadai and Juncheng Wei (DCDS-B, Vol. 23, No. 4, (2018),
pp. 1431-1458.)
- Mathematical Modeling of
Plant Root Hair Initiation: Dynamics of Localized Patches
Victor Brena-Medina, A. Champneys, C. Grierson, M.J. Ward, (
SIAM J. Appl. Dyn. Sys., Vol. 13, No. 1, (2014), pp. 210-248).