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 1 0.8 0.6 V t = ε 2 V xx + V (1 − V 2 ) , 0.4 0.2 0 −0.2 x ∈ R , 0 < ε ≪ 1 . −0.4 −0.6 −0.8 −1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Interface/front: on small scale y = x/ε as ε → 0 .

Quasi-stationary sharp interfaces Prototype: Allen-Cahn model for phase separation 1 0.8 0.6 V t = ε 2 V xx + V (1 − V 2 ) , 0.4 0.2 0 −0.2 x ∈ R , 0 < ε ≪ 1 . −0.4 −0.6 −0.8 −1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 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 1 0.8 0.6 V t = ε 2 V xx + V (1 − V 2 ) , 0.4 0.2 0 −0.2 x ∈ R , 0 < ε ≪ 1 . −0.4 −0.6 −0.8 −1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 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): 1 0.8 ∂ t U = ∂ xx U − U + V 0.6 0.4 0.2 0 ∂ t V = ε 2 ∂ xx V + V (1 − V 2 ) + εU. −0.2 −0.4 −0.6 −0.8 −1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Quasi-stationary ‘semi-sharp’ interfaces Weak coupling to linear equation (FitzHugh-Nagumo type system): 1 0.8 ∂ t U = ∂ xx U − U + V 0.6 0.4 0.2 0 ∂ t V = ε 2 ∂ xx V + V (1 − V 2 ) + εU. −0.2 −0.4 −0.6 −0.8 −1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Nonlocal coupling: U -component globally couples V -interfaces.

Quasi-stationary ‘semi-sharp’ interfaces Weak coupling to linear equation (FitzHugh-Nagumo type system): 1 0.8 ∂ t U = ∂ xx U − U + V 0.6 0.4 0.2 0 ∂ t V = ε 2 ∂ xx V + V (1 − V 2 ) + εU. −0.2 −0.4 −0.6 −0.8 −1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 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): 1 0.8 ∂ t U = ∂ xx U − U + V 0.6 0.4 0.2 0 ∂ t V = ε 2 ∂ xx V + V (1 − V 2 ) + εU. −0.2 −0.4 −0.6 −0.8 −1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 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 ε 2 . 8 8 7 7 6 6 5 5 4 4 3 3 2 2 1 1 0 0 −1 −1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 ε = 0 . 01 , T = 2000 ε = 0 . 005 , T = 8000

Semi-strong interaction Interface motion: Now of order ε 2 . 8 8 7 7 6 6 5 5 4 4 3 3 2 2 1 1 0 0 −1 −1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 ε = 0 . 01 , T = 2000 ε = 0 . 005 , T = 8000 Semi-strong interaction laws: Leading order form d t r j = − ε 2 � u 0 ,j , ∂ y v 0 � / � ∂ y v 0 � 2 d 2 , u 0 ,j = a j ( r 1 , . . . , r N )

Semi-strong interaction Interface motion: Now of order ε 2 . 8 8 7 7 6 6 5 5 4 4 3 3 2 2 1 1 0 0 −1 −1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 ε = 0 . 01 , T = 2000 ε = 0 . 005 , T = 8000 Semi-strong interaction laws: Leading order form d t r j = − ε 2 � u 0 ,j , ∂ y v 0 � / � ∂ y v 0 � 2 d 2 , u 0 ,j = a j ( r 1 , . . . , r N ) Rigorously [Doelman, van Heijster, Kaper, Promislow]

Semi-strong interaction Interface motion: Now of order ε 2 . 8 8 7 7 6 6 5 5 4 4 3 3 2 2 1 1 0 0 −1 −1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 ε = 0 . 01 , T = 2000 ε = 0 . 005 , T = 8000 Semi-strong interaction laws: Leading order form d t r j = − ε 2 � u 0 ,j , ∂ y v 0 � / � ∂ y v 0 � 2 d 2 , u 0 ,j = a j ( r 1 , . . . , r N ) Rigorously [Doelman, van Heijster, Kaper, Promislow] Strong interaction: numerics as in scalar case, monotone & coarsening

The large and small scale problem ∂ t U = ∂ xx U − U + V ∂ t V = ε 2 ∂ xx V + V (1 − V 2 ) + εU.

The large and small scale problem ∂ t U = ∂ xx U − U + V ∂ t V = ε 2 ∂ xx V + V (1 − V 2 ) + εU. Large scale: Assume stationary to leading order in ε 0 = ∂ xx U 0 − U 0 + V 0 0 = V 0 (1 − V 2 0 ) .

The large and small scale problem ∂ t U = ∂ xx U − U + V ∂ t V = ε 2 ∂ xx V + V (1 − V 2 ) + εU. Large scale: Assume stationary to leading order in ε 0 = ∂ xx U 0 − U 0 + V 0 0 = V 0 (1 − V 2 0 ) . Small scale: y = x/ε ε 2 ∂ t u = ∂ yy u − ε 2 ( u + v ) ∂ t v = ∂ yy v + v (1 − v 2 ) + εu.

The large and small scale problem ∂ t U = ∂ xx U − U + V ∂ t V = ε 2 ∂ xx V + V (1 − V 2 ) + εU. Large scale: Assume stationary to leading order in ε 0 = ∂ xx U 0 − U 0 + V 0 0 = V 0 (1 − V 2 0 ) . Small scale: y = x/ε , assume stationary to leading order 0 = ∂ yy u 0 0 = ∂ yy v 0 + v 0 (1 − v 2 0 ) .

A 3-component FHN-type system τ∂ t U = ∂ xx U − U + V θ∂ t W = ∂ xx W − W + V ∂ t V = ε 2 ∂ xx V + 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 ) ∈ R N + M and systems of the form ∂ t u = D u ∂ xx u + F ( u, v ; ε ) ε 2 D v ∂ xx v + G ( u, v ; ε ) ∂ t v =

Is this typical for localised solutions? Consider ( u, v ) ∈ R N + M and systems of the form ∂ t u = D u ∂ xx u + F ( u, v ; ε ) ε 2 D v ∂ xx v + G ( u, v ; ε ) ∂ t v = Fronts: localisation to jump in v as ε → 0 .

Is this typical for localised solutions? Consider ( u, v ) ∈ R N + M and systems of the form ∂ t u = D u ∂ xx u + F ( u, v ; ε ) ε 2 D v ∂ xx v + G ( u, v ; ε ) ∂ t v = Fronts: localisation to jump Pulse: localisation to Dirac mass in v as ε → 0 . 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: ‘ 2 n -splitting’

Example: simplified Schnakenberg model 0.5 0.45 0.4 ∂ t U = ∂ xx U + α − V 0.35 0.3 0.25 ∂ t V = ε 2 ∂ xx V − V + UV 2 . 0.2 0.15 0.1 0.05 0 0 0.5 1 1.5 Leading order existence, stability, interaction?

Two regimes within semi-strong regime ∂ t u = ∂ xx u + α − v ε 2 ∂ xx v − v + uv 2 . ∂ t v = u , v = ε − 1 ˆ For Dirac-mass on x -scale set: u = ˆ v → α − ε − 1 ˆ ∂ t ˆ u = ∂ xx ˆ u + ˆ v ε 2 ∂ xx ˆ v + ε − 1 ˆ v 2 . ∂ t ˆ v = v − ˆ u ˆ We will see that here motion is order ε .

Two regimes within semi-strong regime ∂ t u = ∂ xx u + α − v ε 2 ∂ xx v − v + uv 2 . ∂ t v = u , v = ε − 1 ˆ For Dirac-mass on x -scale set: u = ˆ v → α − ε − 1 ˆ ∂ t ˆ u = ∂ xx ˆ u + ˆ v ε 2 ∂ xx ˆ v + ε − 1 ˆ v 2 . ∂ t ˆ v = v − ˆ u ˆ We will see that here motion is order ε . Embedded motion of order ε 2 analogous to front for α = √ ε ˇ α : u = √ ε ˇ u , v = √ ε − 1 ˇ v → α − ε − 1 ˇ ∂ t ˇ u = ∂ xx ˇ u + ˇ v ε 2 ∂ xx ˇ v 2 . ∂ t ˇ v = v − ˇ v + ˇ u ˇ

Generally: Two regimes within semi-strong regime ∂ xx u + α − u − uv 2 ∂ t u = ε 2 ∂ xx v − v + uv 2 . ∂ t v =

Generally: Two regimes within semi-strong regime ∂ xx u + α − u − uv 2 ∂ t u = ε 2 ∂ xx v − v + uv 2 . ∂ t v = Case α = ˆ α = O (1) : u = ˆ u , v = ˆ v/ε → ‘1st order standard form’ α − ε − 1 (ˆ u + ε − 1 ˆ v 2 ) ∂ t ˆ u = ∂ xx ˆ u + ˆ u ˆ v + ε − 1 ˆ ε 2 ∂ xx ˆ v 2 . ∂ t ˆ v = v − ˆ u ˆ