Pulse motion in the semi-strong limit of the Schnakenberg model Newton Institute 2005 Jens Rademacher, Weierstraß Institut Berlin joint work with Michael Ward (UBC) Angelfish 2, 6, 12 months old [Kondo, S. and Asai, R., Nature 1995]
Numerical simulations [Crampin 00] Turing-unstable Schnakenberg model (two species) x t Stable periodic wave train undergoes self-similar sequence of spatial period doublings Logistic growth:
Uniformly exponentially growing line Model for growth rate ρ : additional dilution term [Crampin 00] U t U t + ρ(xu) x Paradigm: growing reaction diffusion system U t = DU xx ρ(xu) x + F (U). Points of view: Weakly nonlinear: Turing instability - not pursued here Strongly nonlinear, here semi-strong limit [Doelman et al 97] Quasi-stationary approximation for slow growth ρ: time as continuation parameter
Slow growth, continuation, onset of splitting Dilution term cancels in rescaling to fixed domain: U t = g(t) 2 DU xx + F (U) ρu, g(t) = e ρt Quasi-stationary approximation: Slow parameter variation of stable solution is continuation, splitting must stem from instability. time independent bifurcation problem in g to find instability. Similarity to previously studied pulse splitting [Reynolds et al 94, Nishiura et al 99] Approximate further to U t = gd xx + F (U)
Interaction and localization u t = u xx + a uv 2 Schnakenberg model v t = δv xx + b + uv 2 v Turing instability: no localization, small sinusoidal pattern, strong interaction but weakly nonlinear. (a = 0.9, b = 0.1, δ 0.11) Semi-strong interaction: v more localized than u. Occurs for δ 0, δ = 0.01, natural limit of Turing case: Weak interaction: same strong localization of u and v. Wave trains near stable pulse - monotone path, no instability no splitting under slow growth.
Splitting in Schnakenberg model For δ = 0.01 numerically: self-similar splitting sequence stable wave trains for large range of periods - not pure modes Not directly related to Turing [Crampin 00] ρ = 0.001, t [1000, 1500], x [0, 1.3]e ρt. instability
Bifurcation and basin of attraction 10 U 9 stable 2 pulse 3.5 3.0 2.5 2.0 1.5 1.0 8 7 0.5 0.0 0.00 0.20 0.40 0.60 0.80 1.00 0.10 0.30 0.50 0.70 0.90 3.5 3.0 2.5 6 5 stable 1 pulse part of basin of attraction 2.0 1.5 1.0 0.5 4 0.0 0.00 0.20 0.40 0.60 0.80 1.00 0.10 0.30 0.50 0.70 0.90 3 0.6 0.8 1 1.2 1.4 1.6 1.8 units of Turing pattern period Quasi-stationary approximation breaks down at the fold point - splitting stems from basin of attraction of 2-pulse... Similar to other semi-strong splitting [Nishura, Doelman...]. But n 2n splitting, n 2: more solutions available
Growth of Turing pattern Near Turing instability solutions stay close to those of real Ginzburg-Landau equation: A t = A xx + µa A A 2 Modulated Turing patterns: A k (x)e ix = r(k, µ)e i(1+k)x, k 2 µ. Domain [0, 2π/g], Neumann b.c. become sin((1 + k)2π/g) = 0 k = gn/2 1, n Z Stable against e ilx for 3k 2 l 2 /2 < µ Interested in solution with k = g 1 as g. Eckhaus instability: g = 1 µ/3 t(k, l). growth µ unstable stable k [Barkley,Tuckerman 90]: instability is a subcritical pitchfork. Quasi-stationary: splitting sequence seems not very robust
Motion of pulses Building block of splitting: motion of pulses when placed away from symmetric, stationary location. Idea: Seek expression for instantaneous speeds Objectives: determine speed depending on location and domain size seek stable equilibria for quasi-stationary approximation relation to splitting onset Ansatz: formal asymptotics in semi-strong limit for slow motion
Formal asymptotics for spike speed Rescale to x ( 1, 1) via ɛ = δ/l: u t = 1/L 2 u xx + a uv 2 v t = ɛ 2 v xx + b + uv 2 v Spike position ansatz: x 0 = x 0 (ɛt), v x (x 0 ) = 0, speed c = ẋ 0. [Doelman et al 00]: Gray-Scott model on R, asymptotic ODE for c Here: obtain asymptotic boundary value problem for c cheap numerics, computations possible beyond single pulse.
Inner problem Rescale: τ = ɛt ɛy = x x 0 (τ) = d/dy = d/dτ v = (Lɛ) 1 V u = Lɛ U ɛ V ẋ 0 V = V V + UV 2 + blɛ ɛ 3 L 2 U ɛl 2 ẋ 0 U = U UV 2 + alɛ For ɛ = 0: asymmetric core problem (similar to Gray-Scott) cv 0 = V 0 V 0 + U 0 V0 2 0 = U 0 U 0 V0 2 Boundary conditions of V 0 : V 0(0) = 0, V 0 0 as y. Boundary conditions of U 0 via outer problem.
Outer problem Distributional approximation of nonlinear term: ( ) x x0 uv 2 = (Lɛ) 1 (U 0 V 0 + O(ɛ)) ɛ 1 f ɛ R f(y)dyδ x0 (x) uv 2 = A/Lδ x0 (x) + O(ɛ), A = R U 0V 2 0 dy. Substitute in u equation: L 2 u xx + a uv 2 = 0 L 2 u xx + a A/Lδ x0 (x) = O(ɛ) ɛ = 0 u xx + al 2 ALδ x0 (x) = 0
Outer problem Approximate u equation: u xx + al 2 ALδ x0 (x) = 0. Integrate: 1 1 u xx dx + 2aL 2 = AL u x 1 1 + 2aL 2 = AL a = A/2L Approximate u equation: u xx + AL/2 ALδ x0 (x) = 0. Solve by Green s function G xx = 1/2 δ x0 (x): u = u c ALG(x; x 0 )
Matching Matching one-sided Taylor series (recall u = ɛlu): u c ALG(x 0 ; x 0 ) (x x 0 )ALG x (x ± 0 ; x 0) ɛlu(y) x x 0 = ɛy u c = ALG(x 0 ; x 0 ) U 0 (y) AG x (x ± 0 ; x 0)y as y ± Here G(x; x 0 ) = x 2 /4 x x 0 /2 + G 0 G x (x ± 0 ; x 0) = (x 0 1)/2 Hence b.c. for U 0 : U 0(± ) = ±A/2(1 x 0 ) = ±al(1 x 0 )
Boundary conditions for U 0 B.c. for U 0 : U 0(± ) = ±al(1 x 0 ) outer region inner region asymptotic b.c. O( ε)
Boundary value problem for speed V + cv + UV 2 = V V U = UV 2 (0) = 0 V (± ) = 0 U (± ) = ±al(1 x 0 ) Spike dynamics: solve BVP and ẋ 0 = c with L = L(ρt). Expect validity for 1 x 0 > O(ɛ), ρ = O(ɛ). Quasi-stationary: spike motion determined by c(x 0, al), in particular the equilibrium curves c(x 0, al) = 0. Symmetry: c( x 0, al) = c(x 0, al) c(0, al) = 0. Self-similarity: same BVP arises for motion of single spike in chain of n spikes, e.g. n = 2 on ( 1, 0): x 0 = (x 0 1)/2, L = 2L.
Computation Continuation with AUTO [Doedel et al]: Alternate between solving BVP and updating x 0, L Map out c(x 0, al) and its domain of definition. Note: V (0) = 0 is interior condition. Resolved by splitting domain into ( 1, 0) for (U, V ) and (0, 1) for (U +, V + ) with b.c.s U + (0) = U (0), V + (0) = V (0) and V ±(0) = 0.
Spike motion before splitting Quasi-stationary: onset of splitting is fold point of spike (x 0 = 0) at al 1.347 [Muratov, Osipov 00,...], stability [Ward et al 03]. c(x 0, al) undefined for al > 1.347. spike should move to x 0 = 0 for small x 0. 0.9 Indeed al = 1: Motion for ρ = 0 and ρ = 0.01: 0.8 0.7 0.6 x 0 0.5 0.4 0.3 0.2 0.1 0 0 1 2 3 4 5 6 7 8 t
Singularities of c(x 0, al) for 1-spikes Onset of splitting: fold point at al 1.347. Continuation of bifurcating branch of fold points in (al, x 0, c): 0 x 0-0.2 fold of 1 spikes Outside of range of validity: -0.4 x = 1 0 U/10-0.6-0.8-1 shifted 1 spikes Invalid in O( ε) strip 1 1.2 1.4 1.6 1.8 2 al motion V
c 0.10 0.00-0.10-0.20 Extending c - other bifurcations x = 0 0 fold of 1 spikes 0.10 x 0 0.00-0.10 2 spikes c=0 c>0 fold of 1 spikes -0.30-0.40 (U/50) -0.20 c=0-0.50 * -0.30-0.60-0.40 1.20 1.30 1.40 1.50 1.60 1.70 1.80 1.90 0.50 2.00 0.75 1.00 1.25 1.50 1.75 2.00 al al Relevance for splitting? If stable: growth is slow motion along curve c = 0... Expect unstable: PDE dynamics after onset is motion to x 0 = ±1.
Todo check against PDE simulations stability of components make (partially) rigorous... (geometric theory [Doelman et al]?, monotonicity) relation to previous work on velocity for Gray-Scott model asymptotics on half-line: onset of splitting?
Conclusion Quasi-stationary approximation of domain length links to previous splitting studies Semi-strong limit yields BVP for spike velocity Motion as expected even from x 0 = ±1 Approach invalid at onset of splitting: speed large and location at boundary Bifurcations at onset of splitting fold curve, velocities towards x 0 = 0 other solutions moving away from x 0 = 0 Bifurcation of c = 0 curve off 2-spike before onset - should be unstable solutions