First and Second Order Semi-strong Interaction in Reaction ... · Department of Mathematics First...
Transcript of First and Second Order Semi-strong Interaction in Reaction ... · Department of Mathematics First...
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Department of Mathematics
First and Second Order Semi-strong Interactionin Reaction-Diffusion Systems
IMA, Minneapolis, June 2013
Jens Rademacher
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Quasi-stationary sharp interfaces
Prototype: Allen-Cahn model for phase separation
Vt = ε2Vxx + V (1− V 2),
x ∈ R, 0 < ε ≪ 1.
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Interface/front: on small scale y = x/ε as ε → 0.
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Quasi-stationary sharp interfaces
Prototype: Allen-Cahn model for phase separation
Vt = ε2Vxx + V (1− V 2),
x ∈ R, 0 < ε ≪ 1.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
−1
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−0.6
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Interface/front: on small scale y = x/ε as ε → 0.
Weak interface interaction: through exponentially small tails – motion is
exponentially slow in ε−1. Carr & Pego, Fusca & Hale.
More general: Ei; Sandstede; Promislow; Zelik & Mielke.
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Quasi-stationary sharp interfaces
Prototype: Allen-Cahn model for phase separation
Vt = ε2Vxx + V (1− V 2),
x ∈ R, 0 < ε ≪ 1.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
−1
−0.8
−0.6
−0.4
−0.2
0
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0.4
0.6
0.8
1
Interface/front: on small scale y = x/ε as ε → 0.
Weak interface interaction: through exponentially small tails – motion is
exponentially slow in ε−1. Carr & Pego, Fusca & Hale.
More general: Ei; Sandstede; Promislow; Zelik & Mielke.
Global dynamics: motion gradient-like and coarsening.
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Quasi-stationary ‘semi-sharp’ interfaces
Weak coupling to linear equation (FitzHugh-Nagumo type system):
∂tU = ∂xxU − U + V
∂tV = ε2∂xxV + V (1− V 2) + εU.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
−1
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Quasi-stationary ‘semi-sharp’ interfaces
Weak coupling to linear equation (FitzHugh-Nagumo type system):
∂tU = ∂xxU − U + V
∂tV = ε2∂xxV + V (1− V 2) + εU.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
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1
Nonlocal coupling: U -component globally couples V -interfaces.
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Quasi-stationary ‘semi-sharp’ interfaces
Weak coupling to linear equation (FitzHugh-Nagumo type system):
∂tU = ∂xxU − U + V
∂tV = ε2∂xxV + V (1− V 2) + εU.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Nonlocal coupling: U -component globally couples V -interfaces.
Interface: problem on both slow/large x-scale and fast/small y-scale.
Multiple steady patterns: replacing εU by εg(U) arbitrary
singularities can be imbedded in existence problem [manuscript].
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Quasi-stationary ‘semi-sharp’ interfaces
Weak coupling to linear equation (FitzHugh-Nagumo type system):
∂tU = ∂xxU − U + V
∂tV = ε2∂xxV + V (1− V 2) + εU.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Nonlocal coupling: U -component globally couples V -interfaces.
Interface: problem on both slow/large x-scale and fast/small y-scale.
Multiple steady patterns: replacing εU by εg(U) arbitrary
singularities can be imbedded in existence problem [manuscript].
Stability: Evans function in singular limit (‘NLEP’) [Doelman, Gardner,
Kaper]; for this system: van Heijster’s results. (For other singular perturbation
regime: SLEP method [Nishiura, Ikeda & Fuji, 80-90’s])
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Semi-strong interaction
Interface motion: Now of order ε2.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−1
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ε = 0.01, T = 2000 ε = 0.005, T = 8000
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Semi-strong interaction
Interface motion: Now of order ε2.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−1
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0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−1
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ε = 0.01, T = 2000 ε = 0.005, T = 8000
Semi-strong interaction laws: Leading order form
d
dtrj = −ε2〈u0,j, ∂yv0〉/‖∂yv0‖22, u0,j = aj(r1, . . . , rN )
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Semi-strong interaction
Interface motion: Now of order ε2.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−1
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0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−1
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ε = 0.01, T = 2000 ε = 0.005, T = 8000
Semi-strong interaction laws: Leading order form
d
dtrj = −ε2〈u0,j, ∂yv0〉/‖∂yv0‖22, u0,j = aj(r1, . . . , rN )
Rigorously [Doelman, van Heijster, Kaper, Promislow]
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Semi-strong interaction
Interface motion: Now of order ε2.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−1
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0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−1
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ε = 0.01, T = 2000 ε = 0.005, T = 8000
Semi-strong interaction laws: Leading order form
d
dtrj = −ε2〈u0,j, ∂yv0〉/‖∂yv0‖22, u0,j = aj(r1, . . . , rN )
Rigorously [Doelman, van Heijster, Kaper, Promislow]
Strong interaction: numerics as in scalar case, monotone & coarsening
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The large and small scale problem
∂tU = ∂xxU − U + V
∂tV = ε2∂xxV + V (1− V 2) + εU.
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The large and small scale problem
∂tU = ∂xxU − U + V
∂tV = ε2∂xxV + V (1− V 2) + εU.
Large scale: Assume stationary to leading order in ε
0 = ∂xxU0 − U0 + V0
0 = V0(1− V 20 ).
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The large and small scale problem
∂tU = ∂xxU − U + V
∂tV = ε2∂xxV + V (1− V 2) + εU.
Large scale: Assume stationary to leading order in ε
0 = ∂xxU0 − U0 + V0
0 = V0(1− V 20 ).
Small scale: y = x/ε
ε2∂tu = ∂yyu− ε2(u+ v)
∂tv = ∂yyv + v(1− v2) + εu.
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The large and small scale problem
∂tU = ∂xxU − U + V
∂tV = ε2∂xxV + V (1− V 2) + εU.
Large scale: Assume stationary to leading order in ε
0 = ∂xxU0 − U0 + V0
0 = V0(1− V 20 ).
Small scale: y = x/ε, assume stationary to leading order
0 = ∂yyu0
0 = ∂yyv0 + v0(1− v20).
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A 3-component FHN-type system
τ∂tU = ∂xxU − U + V
θ∂tW = ∂xxW −W + V
∂tV = ε2∂xxV + V (1− V 2) + ε(γ + αU + βW ).
Front patterns studied in semi-strong regime by van Heijster
(with Doelman, Kaper, Promislow; also in 2D with Sandstede).
Already single front behaves different from Allen-Cahn:
‘butterfly catastrophe’ and Hopf bifurcation.
[Chirilius-Bruckner, Doelman, van Heijster, R.; manuscript]
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Is this typical for localised solutions?
Consider (u, v) ∈ RN+M and systems of the form
∂tu = Du∂xxu+ F (u, v; ε)
∂tv = ε2Dv∂xxv +G(u, v; ε)
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Is this typical for localised solutions?
Consider (u, v) ∈ RN+M and systems of the form
∂tu = Du∂xxu+ F (u, v; ε)
∂tv = ε2Dv∂xxv +G(u, v; ε)
Fronts: localisation to jump
in v as ε → 0.
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Is this typical for localised solutions?
Consider (u, v) ∈ RN+M and systems of the form
∂tu = Du∂xxu+ F (u, v; ε)
∂tv = ε2Dv∂xxv +G(u, v; ε)
Fronts: localisation to jump
in v as ε → 0.Pulse: localisation to Dirac mass
in v as ε → 0.
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‘Semi-sharp’ pulses / spikes
A major motivation for semi-strong regime: Pulse motion and
pulse-splitting in Gray-Scott model.
Numerics and asymptotic matching by Reynolds, Pearson &
Ponce-Dawson in early 90’s. Continued by Osipov, Doelman, Kaper,
Ward, Wei, ...
Weak interaction: ‘edge splitting’
Semi-strong interaction: ‘2n-splitting’
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Example: simplified Schnakenberg model
∂tU = ∂xxU + α− V
∂tV = ε2∂xxV − V + UV 2.
0 0.5 1 1.50
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0.1
0.15
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0.25
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0.35
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0.5
Leading order existence, stability, interaction?
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Two regimes within semi-strong regime
∂tu = ∂xxu+ α− v
∂tv = ε2∂xxv − v + uv2.
For Dirac-mass on x-scale set: u = u, v = ε−1v →
∂tu = ∂xxu+ α− ε−1v
∂tv = ε2∂xxv − v + ε−1uv2.
We will see that here motion is order ε.
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Two regimes within semi-strong regime
∂tu = ∂xxu+ α− v
∂tv = ε2∂xxv − v + uv2.
For Dirac-mass on x-scale set: u = u, v = ε−1v →
∂tu = ∂xxu+ α− ε−1v
∂tv = ε2∂xxv − v + ε−1uv2.
We will see that here motion is order ε.
Embedded motion of order ε2 analogous to front for α =√εα:
u =√εu, v =
√ε−1
v →
∂tu = ∂xxu+ α− ε−1v
∂tv = ε2∂xxv − v + uv2.
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Generally: Two regimes within semi-strong regime
∂tu = ∂xxu+ α− u− uv2
∂tv = ε2∂xxv − v + uv2.
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Generally: Two regimes within semi-strong regime
∂tu = ∂xxu+ α− u− uv2
∂tv = ε2∂xxv − v + uv2.
Case α = α = O(1): u = u, v = v/ε → ‘1st order standard form’
∂tu = ∂xxu+ α− ε−1(u+ ε−1uv2)
∂tv = ε2∂xxv − v + ε−1uv2.
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Generally: Two regimes within semi-strong regime
∂tu = ∂xxu+ α− u− uv2
∂tv = ε2∂xxv − v + uv2.
Case α = α = O(1): u = u, v = v/ε → ‘1st order standard form’
∂tu = ∂xxu+ α− ε−1(u+ ε−1uv2)
∂tv = ε2∂xxv − v + ε−1uv2.
Case α =√εα: u =
√εu, v = v/
√ε → ‘2nd order standard form’
∂tu = ∂xxu+ α− ε−1(u+ uv2)
∂tv = ε2∂xxv − v + uv2.
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Generally: Two regimes within semi-strong regime
∂tu = ∂xxu+ α− u− uv2
∂tv = ε2∂xxv − v + uv2.
Case α = α = O(1): u = u, v = v/ε → ‘1st order standard form’
∂tu = ∂xxu+ α− ε−1(u+ ε−1uv2)
∂tv = ε2∂xxv − v + ε−1uv2.
Case α =√εα: u =
√εu, v = v/
√ε → ‘2nd order standard form’
∂tu = ∂xxu+ α− ε−1(u+ uv2)
∂tv = ε2∂xxv − v + uv2.
Can distinguish interaction type in general systems via ‘standard forms’
[R. SIADS ’13]
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1st order semi-strong interaction
∂tu = ∂xxu+ α− uv2
∂tv = ε2∂xxv − v + uv2.
0 0.5 1 1.50
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1
ε = 0.01, T = 200 ε = 0.005, T = 400
α = 2.95
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Asymptotics for 1st order interaction
∂tu = ∂xxu+ α− uv2
∂tv = ε2∂xxv − v + uv2.
Expand u = u0 + εu1 +O(ε2), v = v0 +O(ε)
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Asymptotics for 1st order interaction
∂tu = ∂xxu+ α− uv2
∂tv = ε2∂xxv − v + uv2.
Expand u = u0 + εu1 +O(ε2), v = v0 +O(ε)
Large scale: V0 = 0 , 0 = ∂xxU0 + α → parabola
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Asymptotics for 1st order interaction
∂tu = ∂xxu+ α− uv2
∂tv = ε2∂xxv − v + uv2.
Expand u = u0 + εu1 +O(ε2), v = v0 +O(ε)
Large scale: V0 = 0 , 0 = ∂xxU0 + α → parabola
Small scale:
(‘core problem’)
u0 = 0, ∂yyu1 = u1v20
∂yy v0 = v0 − u1v20 .
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Asymptotics for 1st order interaction
∂tu = ∂xxu+ α− uv2
∂tv = ε2∂xxv − v + uv2.
Expand u = u0 + εu1 +O(ε2), v = v0 +O(ε)
Large scale: V0 = 0 , 0 = ∂xxU0 + α → parabola
Small scale:
(‘core problem’)
u0 = 0, ∂yyu1 = u1v20
∂yy v0 = v0 − u1v20 .
Matching:
U0(xj) = 0 (!)
∂yu1(±∞) = ∂xU0(xj ± 0)
v0(±∞) = 0. x4x1 x2 x3
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Asymptotics for 1st order interaction
∂tu = ∂xxu+ α− uv2
∂tv = ε2∂xxv − v + uv2.
Expand u = u0 + εu1 +O(ε2), v = v0 +O(ε)
Large scale: V0 = 0 , 0 = ∂xxU0 + α → parabola
Small scale:
(‘core problem’)
u0 = 0, ∂yyu1 = u1v20
∂yy v0 = v0 − u1v20 .
Matching:
U0(xj) = 0 (!)
∂yu1(±∞) = ∂xU0(xj ± 0)
v0(±∞) = 0. x4x1 x2 x3
⇒ one parameter missing → allow for dxj/dt = εc+O(ε2).
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Asymptotics for 1st order interaction
∂tu = ∂xxu+ α− uv2
∂tv = ε2∂xxv − v + uv2.
Expand u = u0 + εu1 +O(ε2), v = v0 +O(ε)
Large scale: V0 = 0 , 0 = ∂xxU0 + α → parabola
Small scale:
(‘core problem’)
u0 = 0, ∂yyu1 = u1v20
∂yy v0 = v0 − u1v20+c∂y v0.
Matching:
U0(xj) = 0 (!)
∂yu1(±∞) = ∂xU0(xj ± 0)
v(±∞) = 0. x4x1 x2 x3
⇒ one parameter missing → allow for dxj/dt = εc+O(ε2).
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Large and small scale problems
U0(xj) = 0
∂yu1(±∞) = ∂xU0(xj ± 0)x4x1 x2 x3
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Large and small scale problems
U0(xj) = 0
∂yu1(±∞) = ∂xU0(xj ± 0)x4x1 x2 x3
Existence problem local:
nearest neighbor coupling
⇒ use single small scale problem,
parameters c, p± := ∂xU0(xj ± 0).
∂yyu1 = u1v20
∂yy v0 = c∂yv0+v0 − u1v20 .
Motion law not projection
onto translation mode!large scale large scalesmall scale
O(ǫ)
∂xu ∼ p+
u
v
O(ǫ)
∂xu ∼ p−
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Small-scale pulse manifold
Numerically compute by continuation in ps = p+ − p−, pa = p+ + p−:
2.600 2.625 2.650 2.675 2.700
75.
76.
77.
78.
79.
80.
norm
c > 0
c < 0
ps
unimodal
bimodal
p∗s
ps
0.000 0.005 0.010 0.015 0.020 0.025
2.660
2.665
2.670
2.675
2.680
2.685
2.690
2.695
2.700
folds of bimodal
folds of unimodal
c < 0
ps
pa
c = 0
pa = 0
.
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Small-scale pulse manifold
Numerically compute by continuation in ps = p+ − p−, pa = p+ + p−:
2.600 2.625 2.650 2.675 2.700
75.
76.
77.
78.
79.
80.
norm
c > 0
c < 0
ps
unimodal
bimodal
p∗s
ps
0.000 0.005 0.010 0.015 0.020 0.025
2.660
2.665
2.670
2.675
2.680
2.685
2.690
2.695
2.700
folds of bimodal
folds of unimodal
c < 0
ps
pa
c = 0
pa = 0
Reduced 1-pulse motion towards symmetric configuration for ps < p∗s .
Pulse-splitting near pa = 0 for ps > p∗s .
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Small-scale pulse manifold
Numerically compute by continuation in ps = p+ − p−, pa = p+ + p−:
2.600 2.625 2.650 2.675 2.700
75.
76.
77.
78.
79.
80.
norm
c > 0
c < 0
ps
unimodal
bimodal
p∗s
ps
0.000 0.005 0.010 0.015 0.020 0.025
2.660
2.665
2.670
2.675
2.680
2.685
2.690
2.695
2.700
folds of bimodal
folds of unimodal
c < 0
ps
pa
c = 0
pa = 0
Reduced 1-pulse motion towards symmetric configuration for ps < p∗s .
Pulse-splitting near pa = 0 for ps > p∗s .
Monotonicity of c in pa ⇒ Abstract theorem applies: e.g. largest pulse
distance is Lyapunov functional (until splitting). [R. SIADS ‘13]
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Existence & stability map for pulse patterns
Solve boundary value problem formulation for eigenfunctions again by
numerical continuation:
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
Hopfstable region
foldHopf
2.686
2.688
2.69
2.692
2.694
2.696
2.698
2.7
2.702
2.704
0.01 0.012 0.014 0.016 0.018 0.02 0.022 0.024
0.025
2.704
0.012.686
ps
pa
c = 0
c < 0
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Crossing the boundary: Pulse-replication
pulse position
α
0 1 2 3 4 50
10
20
30
40
50t=40
0 1 2 3 4 50
5
10
15
20
25
30
35
40t=63
0 1 2 3 4 50
5
10
15
20
25
30
35
40t=74
0 1 2 3 4 50
5
10
15
20
25
30t=80
0 50 100 150 2000
1
2
3
4
5
time
pu
lse
po
sitio
n
Numerics by J. Ehrt (WIAS/HU)
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1st and 2nd order semi-strong interaction
with M. Wolfrum & J. Ehrt
(WIAS/HU)
∂tu = ∂xxu+ α− uv2
∂tv = ε2∂xxv − v + uv2.
0 1 2 3 4 5 6 70
0.25
0.5
0.75
1
1.25
1.5
X
Tim
e/ε
2
ε=0.05
ε=0.02
ε=0.01
0 1 2 3 4 5 6 70
1
2
3
4
5T*ε2=0.25
X
0 1 2 3 4 5 6 70
1
2
3
4
5
6
7
X
Tim
e/ε
ε=0.05
ε=0.02
ε=0.01
0 1 2 3 4 5 6 70
1
2
3
4
5
x
T*ε2=1.25
0 1 2 3 4 5 6 70
0.5
1
1.5
2T*ε=1
x
0 1 2 3 4 5 6 70
0.2
0.4
0.6
0.8
1
x
T*ε=6
1st order semi-strong:
velocity c = O(ε), α = 0.9.
2nd order semi-strong:
velocity c = O(ε2), α = 1.3√ε.
Small ‘production’: slow motion and coarsening,Large ‘production’: fast motion and splitting
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Stability boundary in 2nd order case
PDE numerics when crossing boundary: annihilation (‘overcrowding’)
Numerics delicate: delayed Hopf-bifurcation...
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Crossing unstable region
0 0.5 1 1.5 2 2.5 33
3.5
4
4.5
5
5.5
6
r1
r 2
alpha=1.2
0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.60
0.2
0.4
0.6
0.8
1
position of left pulse
pu
lse
am
plitu
de
0 1 2 3 4 5 60
0.2
0.4
0.6
0.8
1
x
v(x
)
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Crossing unstable region
0 0.5 1 1.5 2 2.5 33
3.5
4
4.5
5
5.5
6
r1
r 2
alpha=1.2
0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.60
0.2
0.4
0.6
0.8
1
position of left pulse
pu
lse
am
plitu
de
0 1 2 3 4 5 60
0.2
0.4
0.6
0.8
1
x
v(x
)
Only for relatively large ε: bifurcation (appears to be) subcritical.
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A class of examples
∂tu = ∂xxu+ α− µu+ γv − uv2
∂tv = ε2∂xxv + β − v + uv2,
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A class of examples
∂tu = ∂xxu+ α− µu+ γv − uv2
∂tv = ε2∂xxv + β − v + uv2,
Schnakenberg model: µ = γ = 0,
Gray-Scott model: α = µ, γ = β = 0,
Brusselator model: α = µ = 0.
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A class of examples
∂tu = ∂xxu+ α− µu+ γv − uv2
∂tv = ε2∂xxv + β − v + uv2,
Schnakenberg model: µ = γ = 0,
Gray-Scott model: α = µ, γ = β = 0,
Brusselator model: α = µ = 0.
Scalings:
1st order semi-strong: v = ε−1v
2nd order semi-strong: α = ε1/2α, u = ε1/2u, v = ε−1/2v
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Literature
Second order case (α = εα):
Existence and stability from Doelman-Kaper ‘normal form’ approach.
For fronts in FHN-type system: van Heijster.
Interaction laws rigorously for FHN-type and Gierer-Meinhardt model
variants [Doelman, Kaper, Promislow; van Heijster; Bellsky].
First order case (α = O(1)):
Numerically: ‘core problem’ existence up to critical value (fold) – proof?
Rich solution set [Doelman, Kaper, Peletier ’06].
Interaction laws: asymptotics for Schnakenberg model [Ward et al].
Proofs?
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Model independent view
Semi-strong interaction can occur for 0 < ε ≪ 1 in systems of the form
∂tu = Du∂xxu+ F (u, v; ε)
∂tv = ε2Dv∂xxv +G(u, v; ε)v + εE(u, v; ε)
Pulse: localisation to Dirac
mass in v as ε → 0.
Fronts: localisation to jump in v as
ε → 0.
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Model independent view
Semi-strong interaction can occur for 0 < ε ≪ 1 in systems of the form
∂tu = Du∂xxu+ F (u, v; ε)
∂tv = ε2Dv∂xxv +G(u, v; ε)v + εE(u, v; ε)
Pulse: localisation to Dirac
mass in v as ε → 0.
Fronts: localisation to jump in v as
ε → 0.
Expand and apply natural constraints to obtain boundedness as ε → 0:
∂tu = Du∂xxu+H(u, v; ε) + ε−1(F s(u, v) + ε−1F f(u, v)u)v,
∂tv = ε2Dv∂xxv + εE(u, v; ε) +Gs(u, v)v + ε−1Gf(u, v)uv.
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Model independent view
Semi-strong interaction can occur for 0 < ε ≪ 1 in systems of the form
∂tu = Du∂xxu+ F (u, v; ε)
∂tv = ε2Dv∂xxv +G(u, v; ε)v + εE(u, v; ε)
Pulse: localisation to Dirac
mass in v as ε → 0.
Fronts: localisation to jump in v as
ε → 0.
Expand and apply natural constraints to obtain boundedness as ε → 0:
∂tu = Du∂xxu+H(u, v; ε) + ε−1(F s(u, v) + ε−1F f(u, v)u)v,
∂tv = ε2Dv∂xxv + εE(u, v; ε) +Gs(u, v)v + ε−1Gf(u, v)uv.
Second order semi-strong interaction: F f ≡ Gf ≡ 0.
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Summary
• Semi-strong interaction comes in different types.
• Unified framework for fronts, pulses and 1st, 2nd order interaction.
• Can read off the laws of motion (formally).
• In 1st order case: conditions for gradient-like pulse interaction.
[R. SIADS ’13]
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Summary
• Semi-strong interaction comes in different types.
• Unified framework for fronts, pulses and 1st, 2nd order interaction.
• Can read off the laws of motion (formally).
• In 1st order case: conditions for gradient-like pulse interaction.
[R. SIADS ’13]
Sidenote: In semi-strong regime also rich single interface bifurcations
& pencil and paper analysis possible also for nonlocalized solutions...
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Summary
• Semi-strong interaction comes in different types.
• Unified framework for fronts, pulses and 1st, 2nd order interaction.
• Can read off the laws of motion (formally).
• In 1st order case: conditions for gradient-like pulse interaction.
[R. SIADS ’13]
Sidenote: In semi-strong regime also rich single interface bifurcations
& pencil and paper analysis possible also for nonlocalized solutions...
Thank you!