First and Second Order Semi-strong Interaction in Reaction-Diffusion - - PowerPoint PPT Presentation

Department of Mathematics

First and Second Order Semi-strong Interaction in Reaction-Diffusion Systems

IMA, Minneapolis, June 2013 Jens Rademacher

Quasi-stationary sharp interfaces

Prototype: Allen-Cahn model for phase separation Vt = ε2Vxx + V (1 − V 2), x ∈ R, 0 < ε ≪ 1.

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 0.2 0.4 0.6 0.8 1

Interface/front: on small scale y = x/ε as ε → 0.

Quasi-stationary sharp interfaces

Prototype: Allen-Cahn model for phase separation Vt = ε2Vxx + V (1 − V 2), x ∈ R, 0 < ε ≪ 1.

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 0.2 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.

Quasi-stationary sharp interfaces

Prototype: Allen-Cahn model for phase separation Vt = ε2Vxx + V (1 − V 2), x ∈ R, 0 < ε ≪ 1.

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 0.2 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.

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.

0.2 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 0.2 0.4 0.6 0.8 1

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.

0.2 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 0.2 0.4 0.6 0.8 1

Nonlocal coupling: U-component globally couples V -interfaces.

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.

0.2 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 0.2 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].

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.

0.2 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 0.2 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])

Semi-strong interaction

Interface motion: Now of order ε2.

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 −1 1 2 3 4 5 6 7 8 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 −1 1 2 3 4 5 6 7 8

ε = 0.01, T = 2000 ε = 0.005, T = 8000

Semi-strong interaction

Interface motion: Now of order ε2.

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 −1 1 2 3 4 5 6 7 8 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 −1 1 2 3 4 5 6 7 8

ε = 0.01, T = 2000 ε = 0.005, T = 8000 Semi-strong interaction laws: Leading order form

d dtrj = −ε2u0,j, ∂yv0/∂yv02 2, u0,j = aj(r1, . . . , rN)

Semi-strong interaction

Interface motion: Now of order ε2.

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 −1 1 2 3 4 5 6 7 8 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 −1 1 2 3 4 5 6 7 8

ε = 0.01, T = 2000 ε = 0.005, T = 8000 Semi-strong interaction laws: Leading order form

d dtrj = −ε2u0,j, ∂yv0/∂yv02 2, u0,j = aj(r1, . . . , rN)

Rigorously [Doelman, van Heijster, Kaper, Promislow]

Semi-strong interaction

Interface motion: Now of order ε2.

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 −1 1 2 3 4 5 6 7 8 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 −1 1 2 3 4 5 6 7 8

ε = 0.01, T = 2000 ε = 0.005, T = 8000 Semi-strong interaction laws: Leading order form

d dtrj = −ε2u0,j, ∂yv0/∂yv02 2, u0,j = aj(r1, . . . , rN)

Rigorously [Doelman, van Heijster, Kaper, Promislow]

Strong interaction: numerics as in scalar case, monotone & coarsening

The large and small scale problem

∂tU = ∂xxU − U + V ∂tV = ε2∂xxV + V (1 − V 2) + εU.

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 2

0 ).

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 2

0 ).

Small scale: y = x/ε ε2∂tu = ∂yyu − ε2(u + v) ∂tv = ∂yyv + v(1 − v2) + εu.

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 2

0 ).

Small scale: y = x/ε, assume stationary to leading order 0 = ∂yyu0 0 = ∂yyv0 + v0(1 − v2

0).

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]

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)

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.

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.

‘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’

Example: simplified Schnakenberg model

∂tU = ∂xxU + α − V ∂tV = ε2∂xxV − V + UV 2.

0.5 1 1.5 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Leading order existence, stability, interaction?

Two regimes within semi-strong regime

∂tu = ∂xxu + α − v ∂tv = ε2∂xxv − v + uv2. For Dirac-mass on x-scale set: u = ˆ u, v = ε−1ˆ v → ∂tˆ u = ∂xxˆ u + ˆ α − ε−1ˆ v ∂tˆ v = ε2∂xxˆ v − ˆ v + ε−1ˆ uˆ v2. We will see that here motion is order ε.

Two regimes within semi-strong regime

∂tu = ∂xxu + α − v ∂tv = ε2∂xxv − v + uv2. For Dirac-mass on x-scale set: u = ˆ u, v = ε−1ˆ v → ∂tˆ u = ∂xxˆ u + ˆ α − ε−1ˆ v ∂tˆ v = ε2∂xxˆ v − ˆ v + ε−1ˆ uˆ v2. We will see that here motion is order ε. Embedded motion of order ε2 analogous to front for α = √εˇ α: u = √εˇ u, v = √ε−1ˇ v → ∂tˇ u = ∂xxˇ u + ˇ α − ε−1ˇ v ∂tˇ v = ε2∂xxˇ v − ˇ v + ˇ uˇ v2.

Generally: Two regimes within semi-strong regime

∂tu = ∂xxu + α − u − uv2 ∂tv = ε2∂xxv − v + uv2.

Generally: Two regimes within semi-strong regime

∂tu = ∂xxu + α − u − uv2 ∂tv = ε2∂xxv − v + uv2. Case α = ˆ α = O(1): u = ˆ u, v = ˆ v/ε → ‘1st order standard form’ ∂tˆ u = ∂xxˆ u + ˆ α − ε−1(ˆ u + ε−1ˆ uˆ v2) ∂tˆ v = ε2∂xxˆ v − ˆ v + ε−1ˆ uˆ v2.

Generally: Two regimes within semi-strong regime

∂tu = ∂xxu + α − u − uv2 ∂tv = ε2∂xxv − v + uv2. Case α = ˆ α = O(1): u = ˆ u, v = ˆ v/ε → ‘1st order standard form’ ∂tˆ u = ∂xxˆ u + ˆ α − ε−1(ˆ u + ε−1ˆ uˆ v2) ∂tˆ v = ε2∂xxˆ v − ˆ v + ε−1ˆ uˆ v2. Case α = √εˇ α: u = √εˇ u, v = ˇ v/√ε → ‘2nd order standard form’ ∂tˇ u = ∂xxˇ u + ˇ α − ε−1(ˇ u + ˇ uˇ v2) ∂tˇ v = ε2∂xxˇ v − ˇ v + ˇ uˇ v2.

Generally: Two regimes within semi-strong regime

∂tu = ∂xxu + α − u − uv2 ∂tv = ε2∂xxv − v + uv2. Case α = ˆ α = O(1): u = ˆ u, v = ˆ v/ε → ‘1st order standard form’ ∂tˆ u = ∂xxˆ u + ˆ α − ε−1(ˆ u + ε−1ˆ uˆ v2) ∂tˆ v = ε2∂xxˆ v − ˆ v + ε−1ˆ uˆ v2. Case α = √εˇ α: u = √εˇ u, v = ˇ v/√ε → ‘2nd order standard form’ ∂tˇ u = ∂xxˇ u + ˇ α − ε−1(ˇ u + ˇ uˇ v2) ∂tˇ v = ε2∂xxˇ v − ˇ v + ˇ uˇ v2. Can distinguish interaction type in general systems via ‘standard forms’

[R. SIADS ’13]

1st order semi-strong interaction

∂tu = ∂xxu + α − uv2 ∂tv = ε2∂xxv − v + uv2.

0.5 1 1.5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.5 1 1.5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

ε = 0.01, T = 200 ε = 0.005, T = 400 α = 2.95

Asymptotics for 1st order interaction

∂tu = ∂xxu + α − uv2 ∂tv = ε2∂xxv − v + uv2. Expand u = u0 + εu1 + O(ε2), v = v0 + O(ε)

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 = ∂xx ˆ U0 + ˆ α → parabola

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 = ∂xx ˆ U0 + ˆ α → parabola Small scale: (‘core problem’) ˆ u0 = 0, ∂yyˆ u1 = ˆ u1ˆ v2 ∂yyˆ v0 = ˆ v0 − ˆ u1ˆ v2

0.

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 = ∂xx ˆ U0 + ˆ α → parabola Small scale: (‘core problem’) ˆ u0 = 0, ∂yyˆ u1 = ˆ u1ˆ v2 ∂yyˆ v0 = ˆ v0 − ˆ u1ˆ v2

0.

Matching: ˆ U0(xj) = 0 (!) ∂yˆ u1(±∞) = ∂x ˆ U0(xj ± 0) ˆ v0(±∞) = 0.

x4 x1 x2 x3

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 = ∂xx ˆ U0 + ˆ α → parabola Small scale: (‘core problem’) ˆ u0 = 0, ∂yyˆ u1 = ˆ u1ˆ v2 ∂yyˆ v0 = ˆ v0 − ˆ u1ˆ v2

0.

Matching: ˆ U0(xj) = 0 (!) ∂yˆ u1(±∞) = ∂x ˆ U0(xj ± 0) ˆ v0(±∞) = 0.

x4 x1 x2 x3

⇒ one parameter missing → allow for dxj/dt = εc + O(ε2).

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 = ∂xx ˆ U0 + ˆ α → parabola Small scale: (‘core problem’) ˆ u0 = 0, ∂yyˆ u1 = ˆ u1ˆ v2 ∂yyˆ v0 = ˆ v0 − ˆ u1ˆ v2

0+c∂yˆ

v0. Matching: ˆ U0(xj) = 0 (!) ∂yˆ u1(±∞) = ∂x ˆ U0(xj ± 0) ˆ v(±∞) = 0.

x4 x1 x2 x3

⇒ one parameter missing → allow for dxj/dt = εc + O(ε2).

Large and small scale problems

ˆ U0(xj) = ∂yˆ u1(±∞) = ∂x ˆ U0(xj ± 0)

x4 x1 x2 x3

Large and small scale problems

ˆ U0(xj) = ∂yˆ u1(±∞) = ∂x ˆ U0(xj ± 0)

x4 x1 x2 x3

Existence problem local: nearest neighbor coupling ⇒ use single small scale problem, parameters c, p± := ∂x ˆ U0(xj ± 0). ∂yyˆ u1 = ˆ u1ˆ v2 ∂yyˆ v0 = c∂yˆ v0+ˆ v0 − ˆ u1ˆ v2

0.

Motion law not projection

• nto translation mode!

large scale large scale small scale

O(ǫ) ∂xu ∼ p+ u ˆ v O(ǫ) ∂xu ∼ p−

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 .

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 .

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]

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

Hopf stable region fold Hopf

0.025 2.704 0.01 2.686

ps pa c = 0 c < 0

Crossing the boundary: Pulse-replication

pulse position

ˆ α

1 2 3 4 5 10 20 30 40 50 t=40 1 2 3 4 5 5 10 15 20 25 30 35 40 t=63 1 2 3 4 5 5 10 15 20 25 30 35 40 t=74 1 2 3 4 5 5 10 15 20 25 30 t=80

50 100 150 200 1 2 3 4 5 time pulse position

Numerics by J. Ehrt (WIAS/HU)

1st and 2nd order semi-strong interaction

with M. Wolfrum & J. Ehrt (WIAS/HU)

∂tu = ∂xxu + α − uv2 ∂tv = ε2∂xxv − v + uv2.

1 2 3 4 5 6 7 0.25 0.5 0.75 1 1.25 1.5 X Time/ε2

ε=0.05 ε=0.02 ε=0.01

1 2 3 4 5 6 7 1 2 3 4 5

T*ε2=0.25 X

1 2 3 4 5 6 7 1 2 3 4 5 6 7 X Time/ε

ε=0.05 ε=0.02 ε=0.01

x T*ε2=1.25

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

Stability boundary in 2nd order case

PDE numerics when crossing boundary: annihilation (‘overcrowding’) Numerics delicate: delayed Hopf-bifurcation...

Crossing unstable region

0.5 1 1.5 2 2.5 3 3 3.5 4 4.5 5 5.5 6 r1 r2 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 1.6 0.2 0.4 0.6 0.8 1 position of left pulse pulse amplitude

1 2 3 4 5 6 0.2 0.4 0.6 0.8 1 x v(x)

Crossing unstable region

0.5 1 1.5 2 2.5 3 3 3.5 4 4.5 5 5.5 6 r1 r2 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 1.6 0.2 0.4 0.6 0.8 1 position of left pulse pulse amplitude

1 2 3 4 5 6 0.2 0.4 0.6 0.8 1 x v(x)

Only for relatively large ε: bifurcation (appears to be) subcritical.

A class of examples

∂tu = ∂xxu + α − µu + γv − uv2 ∂tv = ε2∂xxv + β − v + uv2,

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.

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−1ˆ v 2nd order semi-strong: α = ε1/2ˇ α, u = ε1/2ˇ u, v = ε−1/2ˇ v

Literature

Second order case (α = εˇ α): 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?

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.

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.

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.

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]

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...

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!