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The Dynamics of Reaction-Diffusion Patterns

The Dynamics of Reaction-Diffusion Patterns. Arjen Doelman (CWI & U of Amsterdam) (Rob Gardner, Tasso Kaper, Yasumasa Nishiura, Keith Promislow, Bjorn Sandstede). STRUCTURE OF THE TALK Motivation Topics that won’t be discussed Analytical approaches Patterns close to equilibrium

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The Dynamics of Reaction-Diffusion Patterns

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  1. The Dynamics of Reaction-Diffusion Patterns Arjen Doelman (CWI & U of Amsterdam) (Rob Gardner, Tasso Kaper, Yasumasa Nishiura, Keith Promislow, Bjorn Sandstede)

  2. STRUCTURE OF THE TALK • Motivation • Topics that won’t be discussed • Analytical approaches • Patterns close to equilibrium • Localized structures • Periodic patterns & Busse balloons • Interactions • Discussion and more ...

  3. 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’ ………

  4. 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.

  5. The Klausmeier & Gray-Scott (GS) models [Meron, Rietkerk, Sherratt, ...]

  6. TOPICS THAT WON’T BE DISCUSSED: • `Tools’: • Maximum principles • Gradient structure `Waves in random media’ [Berestycki, Hamel, Xin, ...]

  7. [Fife, Brezis, Nishiura, Sternberg, ...]

  8. [Fife, Mimura, Nishiura, Bates, ...] [Sandstede & Scheel]

  9. ANALYTICAL APPROACHES Restriction/Condition: `We’ want explicit control on the nature/structure of the solutions/patterns

  10. PATTERNS CLOSE TO EQUILIBRIUM

  11. Two typical pattern-generating bifurcations

  12. (spatial symm.) (Turing)

  13. 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]

  14. fast slow slow fast slow fast

  15. fast fast slow

  16. SPECTRAL STABILITY

  17. What about localized 2-D patterns? Spots, stripes, `volcanoes’, ...., most (all?) existence and stability analysis done for (or `close to’) `symmetric’ patterns [Ward,Wei,...]

  18. 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

  19. A Busse Balloon in the Gray-Scott model: From near-criticality to localized structures! onset k k = 0 A [Morgan, Doelman, Kaper]

  20. A part of f-f-e tip of the GM-Busse balloon (determined analytically) stable unstable [van der Ploeg, Doelman]

  21. [Rademacher, D.]

  22. 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, ....]

  23. INTERACTIONS (OF LOCALIZED PATTERNS) A hierarchy of problems • Existence of stationary (or uniformly traveling) solutions • The stability of the localized patterns • The INTERACTIONS Note: It’s no longer possible to reduce the PDE to an ODE

  24. WEAK INTERACTIONS General theory for exponentially small tail-tail interactions [Ei, Promislow, Sandstede] Essential: components can be treated as `particles’

  25. 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

  26. 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]

  27. Example: Pulse-interactions in (regularized) GM [Doelman, Gardner, Kaper]

  28. V U Intrinsically formal result[Doelman, Kaper, Ward] (2 `copies’ of the stationary pulses)

  29. 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

  30. The Evans function approach can be used to explicitly determine the paths of the eigenvalues

  31. Nonlinear Asymptotic Stability & Validity

  32. 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 • ....

  33. 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, ....]

  34. And there is more, much more ... `massive extinction’ annihilation self-replication [Ohta, in GS & other systems] A structurally stable Sierpinsky gasket ...

  35. SPECTRAL STABILITY

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