First and Second Order Semi-strong Interaction in Reaction ... · Department of Mathematics First and Second Order Semi-strong Interaction in Reaction-Diffusion Systems IMA, Minneapolis,

Department of Mathematics

First and Second Order Semi-strong Interactionin 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 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

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



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

0.2

0.4

0.6

0.8

1


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.



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

0.2

0.4

0.6

0.8

1


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 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



∂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.



∂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


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].



∂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


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 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−1

0

1

2

3

4

5

6

7

8

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−1

0

1

2

3

4

5

6

7

8

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



0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−1

0

1

2

3

4

5

6

7

8

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−1

0

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 = −ε2〈u0,j, ∂yv0〉/‖∂yv0‖22, u0,j = aj(r1, . . . , rN )



0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−1

0

1

2

3

4

5

6

7

8

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−1

0

1

2

3

4

5

6

7

8

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


d


Rigorously [Doelman, van Heijster, Kaper, Promislow]



0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−1

0

1

2

3

4

5

6

7

8

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−1

0

1

2

3

4

5

6

7

8

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


d


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.


∂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 ).


∂tU = ∂xxU − U + V

∂tV = ε2∂xxV + V (1− V 2) + εU.


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.


∂tU = ∂xxU − U + V

∂tV = ε2∂xxV + V (1− V 2) + εU.


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).

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)





Fronts: localisation to jump

in v as ε → 0.





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 0.5 1 1.50

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 = u, v = ε−1v →

∂tu = ∂xxu+ α− ε−1v

∂tv = ε2∂xxv − v + ε−1uv2.

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 = u, v = 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 →


∂tv = ε2∂xxv − v + uv2.

Generally: Two regimes within semi-strong regime

∂tu = ∂xxu+ α− u− uv2

∂tv = ε2∂xxv − v + 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.



∂tv = ε2∂xxv − v + uv2.


∂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.



∂tv = ε2∂xxv − v + uv2.


∂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]

1st order semi-strong interaction

∂tu = ∂xxu+ α− uv2

∂tv = ε2∂xxv − v + uv2.

0 0.5 1 1.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.5 1 1.50

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


∂tv = ε2∂xxv − v + uv2.

Expand u = u0 + εu1 +O(ε2), v = v0 +O(ε)



∂tv = ε2∂xxv − v + uv2.


Large scale: V0 = 0 , 0 = ∂xxU0 + α → parabola



∂tv = ε2∂xxv − v + uv2.



Small scale:

(‘core problem’)

u0 = 0, ∂yyu1 = u1v20

∂yy v0 = v0 − u1v20 .



∂tv = ε2∂xxv − v + uv2.



Small scale:


u0 = 0, ∂yyu1 = u1v20

∂yy v0 = v0 − u1v20 .

Matching:

U0(xj) = 0 (!)

∂yu1(±∞) = ∂xU0(xj ± 0)

v0(±∞) = 0. x4x1 x2 x3



∂tv = ε2∂xxv − v + uv2.



Small scale:


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).



∂tv = ε2∂xxv − v + uv2.



Small scale:


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).

Large and small scale problems

U0(xj) = 0

∂yu1(±∞) = ∂xU0(xj ± 0)x4x1 x2 x3

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−

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

.



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 .



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

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

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)

1st and 2nd order semi-strong interaction

with M. Wolfrum & J. Ehrt

(WIAS/HU)


∂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

Stability boundary in 2nd order case

PDE numerics when crossing boundary: annihilation (‘overcrowding’)

Numerics delicate: delayed Hopf-bifurcation...

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

)

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.

A class of examples

∂tu = ∂xxu+ α− µu+ γv − uv2

∂tv = ε2∂xxv + β − v + uv2,

A class of examples


∂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


∂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

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?

Model independent view

Semi-strong interaction can occur for 0 < ε ≪ 1 in systems of the form


∂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.








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.








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





[R. SIADS ’13]

Sidenote: In semi-strong regime also rich single interface bifurcations

& pencil and paper analysis possible also for nonlocalized solutions...

Summary





[R. SIADS ’13]

Sidenote: In semi-strong regime also rich single interface bifurcations

& pencil and paper analysis possible also for nonlocalized solutions...

Thank you!

Documents

First and Second Order Semi-strong Interaction in Reaction ... · Department of Mathematics First and Second Order Semi-strong Interaction in Reaction-Diffusion Systems IMA, Minneapolis,