Two Perspectives on Travelling Waves and Stochasticity Christian - PowerPoint PPT Presentation

Two Perspectives on Travelling Waves and Stochasticity Christian Kuehn Vienna University of Technology Institute for Analysis and Scientific Computing

Overview Topic 1: Travelling Waves and Anomalous Diffusion ◮ Review of reaction-diffusion models ◮ Nagumo travelling waves ◮ Perturbations and Riesz-Feller operators ◮ Anomalous diffusion joint work with Franz Achleitner (TU Vienna)

Overview Topic 1: Travelling Waves and Anomalous Diffusion ◮ Review of reaction-diffusion models ◮ Nagumo travelling waves ◮ Perturbations and Riesz-Feller operators ◮ Anomalous diffusion joint work with Franz Achleitner (TU Vienna) Topic 2: Travelling Waves for the FKPP SPDE ◮ Critical transitions for SDEs ◮ Stochastic warning signs ◮ Numerics of FKPP waves

Reaction-Diffusion Models Simplest case u = u ( x , t ) satisfies ∂ t = ∂ 2 u ∂ u ∂ x 2 + f ( u ) , ( x , t ) ∈ R × [0 , ∞ ) .

Reaction-Diffusion Models Simplest case u = u ( x , t ) satisfies ∂ t = ∂ 2 u ∂ u ∂ x 2 + f ( u ) , ( x , t ) ∈ R × [0 , ∞ ) . Classical nonlinearities: (a) Nagumo/Allen-Cahn/RGL f ( u ) = u (1 − u )( u − a ), (b) Fisher-Kolmogorov-Petrovskii-Piscounov f ( u ) = u (1 − u ), (c) combustion nonlinearity f | [0 ,ρ ] ≡ 0, f | ( ρ, 1) > 0, f(1)=0. bistable-type monostable-type ignition-type 0.04 0.25 f (b) (a) f (c) f 0.1 0 −0.06 0 0 a u u ρ u 0 1 0 1 0 1

Travelling wave ansatz u ( x , t ) = U ( x − ct ), c = wave speed.

Travelling wave ansatz u ( x , t ) = U ( x − ct ), c = wave speed. ∂ t = ∂ 2 u d ξ = d 2 U ∂ u − c dU ξ = x − ct ∂ x 2 + f ( u ) − → d ξ 2 + f ( U ) .

Travelling wave ansatz u ( x , t ) = U ( x − ct ), c = wave speed. ∂ t = ∂ 2 u d ξ = d 2 U ∂ u − c dU ξ = x − ct ∂ x 2 + f ( u ) − → d ξ 2 + f ( U ) . Look for travelling front with ◮ bistable nonlinearity f ( U ) = U (1 − U )( U − a ), ◮ boundary conditions ξ →−∞ U ( ξ ) = 0 lim and ξ →∞ U ( ξ ) = 1 . lim

Travelling wave ansatz u ( x , t ) = U ( x − ct ), c = wave speed. ∂ t = ∂ 2 u d ξ = d 2 U ∂ u − c dU ξ = x − ct ∂ x 2 + f ( u ) − → d ξ 2 + f ( U ) . Look for travelling front with ◮ bistable nonlinearity f ( U ) = U (1 − U )( U − a ), ◮ boundary conditions ξ →−∞ U ( ξ ) = 0 lim and ξ →∞ U ( ξ ) = 1 . lim heteroclinic orbit ↔ front 1 U ′ c ≈ 0 . 141 U 0 0 0 U 1 ξ

Theorem (Aronson, Fife, McLeod, Nagumo, Weinberger, . . . ) For a ∈ (0 , 1) , there exists an exponentially stable travelling front u ( x , t ) = U ( x − ct ) = U ( ξ ) ∈ C 1 ( R ) to ∂ t = ∂ 2 u ∂ u ∂ x 2 + u (1 − u )( u − a ) , which is unique up to translation and satisfies | ξ |→∞ U ′ ( ξ ) = 0 , U ′ ( ξ ) > 0 . ξ →−∞ U ( ξ ) = 0 , lim ξ →∞ U ( ξ ) = 1 , lim lim

Theorem (Aronson, Fife, McLeod, Nagumo, Weinberger, . . . ) For a ∈ (0 , 1) , there exists an exponentially stable travelling front u ( x , t ) = U ( x − ct ) = U ( ξ ) ∈ C 1 ( R ) to ∂ t = ∂ 2 u ∂ u ∂ x 2 + u (1 − u )( u − a ) , which is unique up to translation and satisfies | ξ |→∞ U ′ ( ξ ) = 0 , U ′ ( ξ ) > 0 . ξ →−∞ U ( ξ ) = 0 , lim ξ →∞ U ( ξ ) = 1 , lim lim ◮ existence: ∃ c ∈ R s.t. the associated ODE has a heteroclinic. ◮ stability: ∃ κ > 0 s.t. for u ( · , 0) = u 0 ∈ L ∞ ( R ), 0 ≤ u 0 ≤ 1 � u ( · , t ) − U ( · − ct + γ ) � L ∞ ( R ) ≤ Ke − κ t , for all t ≥ 0. for some constants γ and K depending upon u 0 . ◮ uniqueness: any other pair (˜ U , ˜ c ) satisfies ˜ c = ˜ c , U ( · ) = U ( · + ξ 0 ) , for some ξ 0 ∈ R .

A Possible Generalization... Consider the abstract bistable nonlinearity f ∈ C 1 ( R ) , f (0) = f (1) = f ( a ) = 0 , f | [0 , a ) < 0 , f | ( a , 1] > 0 . and define the convolution � J ∗ S ( u ) := J ( x − y ) S ( u ( y , t )) dy . R

A Possible Generalization... Consider the abstract bistable nonlinearity f ∈ C 1 ( R ) , f (0) = f (1) = f ( a ) = 0 , f | [0 , a ) < 0 , f | ( a , 1] > 0 . and define the convolution � J ∗ S ( u ) := J ( x − y ) S ( u ( y , t )) dy . R Theorem (Chen, 1997) Let f ( u ) := G ( u , S 1 ( u ) , · · · , S n ( u )) , assume (mild) conditions for ∂ t = D ∂ 2 u ∂ u ∂ x 2 + G ( u , J 1 ∗ S 1 ( u ) , · · · , J n ∗ S n ( u )) , D ≥ 0 ⇒ existence, uniqueness, exponential stability of a front hold.

Intermezzo: Why do we bother?

Intermezzo: Why do we bother? The bistable nonlinearity f ( u ) ◮ arises from the classical double-well potential ( f ( u ) = F ′ ( u )), ◮ occurs in normal forms / amplitude equations (NLS, RGLE), ◮ appears for coarsening, oscillations, neural fields, . . . ,

Intermezzo: Why do we bother? The bistable nonlinearity f ( u ) ◮ arises from the classical double-well potential ( f ( u ) = F ′ ( u )), ◮ occurs in normal forms / amplitude equations (NLS, RGLE), ◮ appears for coarsening, oscillations, neural fields, . . . , ◮ is a “building block” e.g. in the FitzHugh-Nagumo equation � ∂ u ∂ t = ∂ 2 u ∂ x 2 + f ( u ) − v + I , I , γ ∈ R , 0 < ǫ ≪ 1 . ∂ v ∂ t = ǫ ( u − γ v ) ,

Intermezzo: Why do we bother? The bistable nonlinearity f ( u ) ◮ arises from the classical double-well potential ( f ( u ) = F ′ ( u )), ◮ occurs in normal forms / amplitude equations (NLS, RGLE), ◮ appears for coarsening, oscillations, neural fields, . . . , ◮ is a “building block” e.g. in the FitzHugh-Nagumo equation � ∂ u ∂ t = ∂ 2 u ∂ x 2 + f ( u ) − v + I , I , γ ∈ R , 0 < ǫ ≪ 1 . ∂ v ∂ t = ǫ ( u − γ v ) , V V = V ∗ fast subsyst. ǫ = 0, V = 0 0.12 0.3 { V = 0 } U ′ 0.08 { U = 0 = U ′ } 0 0.04 (b) −0.3 0 (a) 0 0.4 0.8 1.2 U −0.4 0 0.4 0.8 U

Return to 1-D Case ∂ t = ∂ 2 u ∂ u + f ( u ) , ∂ x 2 ���� � �� � reaction diffusion equation Bistable case: front is robust under reaction-term perturbation.

Return to 1-D Case ∂ t = ∂ 2 u ∂ u + f ( u ) , ∂ x 2 ���� � �� � reaction diffusion equation Bistable case: front is robust under reaction-term perturbation. Robust to perturbation of diffusion-equation part? ∂ t = ∂ 2 u ∂ u ∂ x 2 =: Lu . Replace L by ˜ L ... Question: How to do this?

Return to 1-D Case ∂ t = ∂ 2 u ∂ u + f ( u ) , ∂ x 2 ���� � �� � reaction diffusion equation Bistable case: front is robust under reaction-term perturbation. Robust to perturbation of diffusion-equation part? ∂ t = ∂ 2 u ∂ u ∂ x 2 =: Lu . Replace L by ˜ L ... Question: How to do this? Answer: Go back to probabilistic fundamentals of diffusion.

Continuous-Time Random Walks and Diffusion 4 2 x 0 −2 t 0 10

Continuous-Time Random Walks and Diffusion Choice of two distributions: 4 ◮ waiting time in ( t , t + ∆ t ) is 2 w ( t )d t x 0 ◮ jump length in ( x , x + ∆ x ) is −2 t λ ( x )d x 0 10

Continuous-Time Random Walks and Diffusion Choice of two distributions: 4 ◮ waiting time in ( t , t + ∆ t ) is 2 w ( t )d t x 0 ◮ jump length in ( x , x + ∆ x ) is −2 t λ ( x )d x 0 10 Important is the choice of moments � ∞ ◮ mean waiting time T = 0 w ( t ) t d t � ∞ ◮ jump length variance Σ 2 = 0 ( x − µ λ ) 2 λ ( x ) d x

Continuous-Time Random Walks and Diffusion Choice of two distributions: 4 ◮ waiting time in ( t , t + ∆ t ) is 2 w ( t )d t x 0 ◮ jump length in ( x , x + ∆ x ) is −2 t λ ( x )d x 0 10 Important is the choice of moments � ∞ ◮ mean waiting time T = 0 w ( t ) t d t � ∞ ◮ jump length variance Σ 2 = 0 ( x − µ λ ) 2 λ ( x ) d x Result: Assume T , Σ 2 < ∞ , then central limit theorem implies P (particle at x at time t ) = u ( x , t ) obeys ∂ 2 u ∂ u ∂ t = K 1 ∂ x 2 , K 1 = diffusion coefficient .

Some Facts on Perturbed Models... Case 1: T = ∞ , Σ 2 < ∞ , subdiffusive with long waiting time ◮ example: w ( t ) ∼ A β 1 t 1+ β with β ∈ (0 , 1), ◮ non-Markovian with “diffusion” equation ∂ 2 u ∂ u ∂ t = D 1 − β RL , t K α ∂ x 2 involving the Riemann-Liouville fractional derivative � t 1 ∂ u ( x , s ) D 1 − β RL , t u ( x , t ) := ( t − s ) 1 − β ds Γ( β ) ∂ t 0

Some Facts on Perturbed Models... Case 1: T = ∞ , Σ 2 < ∞ , subdiffusive with long waiting time ◮ example: w ( t ) ∼ A β 1 t 1+ β with β ∈ (0 , 1), ◮ non-Markovian with “diffusion” equation ∂ 2 u ∂ u ∂ t = D 1 − β RL , t K α ∂ x 2 involving the Riemann-Liouville fractional derivative � t 1 ∂ u ( x , s ) D 1 − β RL , t u ( x , t ) := ( t − s ) 1 − β ds Γ( β ) ∂ t 0 TODAY - Case 2: T < ∞ , Σ 2 = ∞ , long jumps / L´ evy flights 1 ◮ example: λ ( x ) ∼ A α | x | 1+ α with α ∈ (1 , 2), ◮ Markovian with “diffusion” equation ∂ u ∂ t = K α D α RF , x u involving the Riemann-Feller fractional operator D α RF , x .

Riesz-Feller Operators ◮ Schwartz space � � � � � x ρ ∂ γ f � < ∞ , ∀ ρ, γ ∈ N 0 f ∈ C ∞ ( R ) : sup x ∈ R S ( R ) = ∂ x γ ( x ) ◮ Fourier transform and Fourier inverse transform � � R e + i ξ x f ( x ) dx and F − 1 f ( x ) = 1 R e − i ξ x f ( ξ ) d ξ F f ( ξ ) = 2 π