The Dynamics of

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The Dynamics of Reaction-Diffusion Patterns
Arjen Doelman (CWI U of Amsterdam) (Rob
Gardner, Tasso Kaper, Yasumasa Nishiura, Keith
Promislow, Bjorn Sandstede)
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• STRUCTURE OF THE TALK
• Motivation
• Topics that wont be discussed
• Analytical approaches
• Patterns close to equilibrium
• Localized structures
• Periodic patterns Busse balloons
• Interactions
• Discussion and more ...

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MOTIVATION
Reaction-diffusion equations are perhaps the most
simple PDEs that generate complex patterns
Reaction-diffusion equations serve as (often
over-) simplified models in many applications
Examples FitzHugh-Nagumo (FH-N) - nerve
conduction Gierer-Meinhardt (GM) -
morphogenesis
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EXAMPLE Vegetation patterns
Interaction between plants, soil (ground) water
modelled by 2- or 3-component RDEs. Some of
these are remarkably familiar ...
At the transition to desertification in Niger,
Africa.
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The Klausmeier Gray-Scott (GS) models
Meron, Rietkerk, Sherratt, ...
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TOPICS THAT WONT BE DISCUSSED

• Tools
• Maximum principles
• Gradient structure

Waves in random media Berestycki, Hamel, Xin,
...
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Fife, Brezis, Nishiura, Sternberg, ...
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Fife, Mimura, Nishiura, Bates, ...
Sandstede Scheel
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ANALYTICAL APPROACHES
Restriction/Condition We want explicit control
on the nature/structure of the solutions/patterns
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PATTERNS CLOSE TO EQUILIBRIUM
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Two typical pattern-generating bifurcations
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(spatial symm.)
(Turing)
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LOCALIZED STRUCTURES Far-from-equilibrium
patterns that are close to a trivial state,
except for a small spatial region.
A 2-pulse or 4-front in a 3-component model
A (simple) pulse in GS
D., Kaper, van Heijster
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fast
slow
slow
fast
slow
fast
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fast
fast
slow
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SPECTRAL STABILITY
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What about localized 2-D patterns?
Spots, stripes, volcanoes, ...., most (all?)
existence and stability analysis done for (or
close to) symmetric patterns
Ward,Wei,...
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PERIODIC PATTERNS BUSSE BALLOONS
A natural connection between periodic patterns
near criticality and far-from-equilibrium patterns
Region in (k,R)-space in which STABLE periodic
patterns exist
bifurcation parameter R
onset
Busse, 1978 (convection)
wave number k
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A Busse Balloon in the Gray-Scott model
From near-criticality to localized structures!
onset
k
k 0
A
Morgan, Doelman, Kaper
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A part of f-f-e tip of the GM-Busse balloon
(determined analytically)
stable
unstable
van der Ploeg, Doelman
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Rademacher, D.
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What about localized 2-D patterns?
DEFECT PATTERNS
Slow modulations of (parallel) stripe patterns
localized defects
A defect pattern in a convection experiment
Phase-diffusion equations with defects as
singularities
Cross, Newell, Ercolani, ....
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INTERACTIONS (OF LOCALIZED PATTERNS)
A hierarchy of problems

• Existence of stationary (or uniformly traveling)
  solutions
• The stability of the localized patterns
• The INTERACTIONS
• Note Its no longer possible to reduce the PDE
  to an ODE

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WEAK INTERACTIONS
General theory for exponentially small tail-tail
interactions Ei,
Promislow, Sandstede
Essential components can be treated as
particles
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SEMI-STRONG INTERACTIONS

• Pulses evolve and change in magnitude and shape.
  
• Only O(1) interactions through one component, the
  other components have negligible interactions
  

V
U
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Pulses are no particles and may push each
other through a bifurcation.
Semi-strong dynamics in two (different) modified
GM models
finite-time blow-up
a symmetry breaking bifurcation
D. Kaper 03
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Example Pulse-interactions in (regularized) GM
Doelman, Gardner, Kaper
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V
U
Intrinsically formal result Doelman, Kaper, Ward
(2 copies of the stationary pulses)
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Stability of the 2-pulse solution Q What is
linearized stability? A Freeze solution and
determine quasi-steady eigenvalues Note not
unrealistic, since 2-pulse evolves slowly
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The Evans function approach can be used to
explicitly determine the paths of the eigenvalues
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Nonlinear Asymptotic Stability Validity
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• DISCUSSION AND MORE ....
• There is a well-developed theory for simple
  patterns (localized, spatially periodic, radially
  symmetric, ...).
• More complex patterns can be studied with these
  tools.
• Challenges
• Defects in 2-dimensional stripe patterns
• Strong pulse interactions
• ....

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Strong interactions ...
V
(simulations in GS)
The pulse self-replication mechanism A generic
phenomenon, originally discovered by Pearson et
al in 93 in GS. Studied extensively, but still
not understood.
Pearson, Doelman, Kaper, Nishiura, Muratov,
Peletier, Ward, ....
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And there is more, much more ...
massive extinction
annihilation
self-replication
Ohta, in GS other systems
A structurally stable Sierpinsky gasket ...
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SPECTRAL STABILITY
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