system CWI, Amsterdam May 21, 2008 Dynamic Analysis Seminar Vrije Universiteit

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1 CWI, Amsterdam May 21, 2008 Dynamic Analysis Seminar Vrije Universiteit Joint work: A. Doelman (CWI/UvA), T.J. Kaper (BU), K. Promislow (MSU)

2 Outline

3 Outline

4 Paradigm U t = U ξξ + U U 3 ε(αv + βw + γ) τv t = 1 ε V 2 ξξ + U V θw t = D2 ε W 2 ξξ + U W - 0 < ε 1; τ, θ > 0; D > 1; α, β, γ R; all parameters are O(1) - ξ R, and (U, V, W ) are bounded - 3 coupled reaction-diffusion equations; coupling is weak - Singular perturbed, slow-fast nature - Bi-stable

5 Paradigm U t = U ξξ + U U 3 ε(αv + βw + γ) τv t = 1 ε V 2 ξξ + U V θw t = D2 ε W 2 ξξ + U W - 0 < ε 1; τ, θ > 0; D > 1; α, β, γ R; all parameters are O(1) - ξ R, and (U, V, W ) are bounded - 3 coupled reaction-diffusion equations; coupling is weak - Singular perturbed, slow-fast nature - Bi-stable C.P. Schenk, M. Or-Guil, M. Bode, H.-G. Purwins Interacting Pulses in Three-Component Reaction-Diffusion Systems on Two-Dimensional Domains. PRL, 78 (1997). Y. Nishiura, T. Teramoto, K. Ueda Scattering and separators in dissipative s. PRE, 67 (2003).

6 Stationary 1-pulse

7 Stationary 2-pulse

8 Traveling 1-pulse

9 Rich dynamics

10 2-front dynamics

11 3-front dynamics

12 3-front dynamics

13 4-front dynamics

14 Outline

15 Solutions Theorem For ε small enough, the PDE possesses a stationary 1-pulse solution if there exists an A (0, 1) which solves The 1-pulse is stable if αa 2 + βa 2 D = γ. αa 2 + β D A 2 D > 0. Moreover, the width of the pulse is Γ = 2 log A. ε

16 Solutions Theorem For ε small enough, the PDE possesses a stationary 2-pulse solution if there exists 0 < A 1 < A 2 < 1 which solve ( 0 = α(a 1 A 2) 2 + β(a 1/D 1 A 1/D 2 ) 2, 2γ = α(a 2 2 A 2 1) 2αA 1A β(a 2/D 2 A 2/D 1 ) 2βA 1/D 1 A 1/D 2. The 2-pulse is stable if λ 2,3,4 (α, β, D, γ, A 1, A 2 ) < 0. Moreover, both pulses have the same width Γ = 1 ε log A 2 A 1.

17 Eigenvalues Eigenvalues λ 2,3,4

18 Outline

19 1-Front dynamics Theorem A 1-front solution is stable and its front evolves according to PDE vs ODE Γ t = 3 2 2εγ

20 2-Front dynamics 2-Front dynamics The two fronts Γ 1 < Γ 2 of a 2-front solution evolve according to (Γ 1 ) t = 3 2 2ε ( γ αe ε(γ 2 Γ 1) βe ε D (Γ2 Γ1)), (Γ 2 ) t = 3 2 2ε ( γ αe ε(γ 2 Γ 1) βe ε D (Γ2 Γ1)). : Introduce 2 co-moving frames. Determine jump conditions by solving in the fast fields. Determine boundary conditions by solving in the slow fields. KEY Speed no leading order influence in the slow fields.

21 Rigorous Theorem There exist a moving 2-front solution which can be decomposed into Φ 2f (ξ, t) = Φ Γ(t) (ξ) + Z(ξ, t). The two fronts Γ i (t) evolve according to the formally derived ODE, while the remainder Z(ξ, t) stays O(ε)-small.

22 Rigorous Theorem There exist a moving 2-front solution which can be decomposed into Φ 2f (ξ, t) = Φ Γ(t) (ξ) + Z(ξ, t). The two fronts Γ i (t) evolve according to the formally derived ODE, while the remainder Z(ξ, t) stays O(ε)-small. Φ Γ (ξ) = U 0(ξ; Γ) G 2 U 0 (ξ; Γ) G 3 U 0 (ξ; Γ), where ( 2 ) ( 2 ) U 0 (ξ, Γ) = 1 + tanh 2 (ξ Γ 1) tanh 2 (ξ Γ 2), G 2 (ξ) = 1 2 ε exp ( ε ξ ), G 3(ξ) = 1 ε 2 D exp ( ε D ξ ).

23 Ansatz Φ Γ (ξ)

24 χ-norm χ-norm U 3 3 χ = u i χ = ( χu i L 1 + ξ u i L 1), i=1 i=1 with χ(ξ) = 1 2 exp ( ξ ) (positive, mass 1, exponentially decaying) Properties u L u χ, G u χ C u χ, u 2 χ 2 u 2 χ.

25 Proof Initializing Freeze basepoint: Γ = Γ 0 Decompose: Φ 2f (ξ, t) = Φ Γ(t) (ξ) + Z(ξ, t), such that Z(ξ, t) X Γ 0 for all t < t Z(ξ, t) χ ε for all t < t PDE: Z t + Φ Γ Γ Γ t = R(Φ Γ ) + L Γ 0Z + (L Γ L Γ 0)Z + N(Z)

26 Proof Initializing Freeze basepoint: Γ = Γ 0 Decompose: Φ 2f (ξ, t) = Φ Γ(t) (ξ) + Z(ξ, t), such that Z(ξ, t) X Γ 0 for all t < t Z(ξ, t) χ ε for all t < t PDE: Z t + Φ Γ Γ Γ t = R(Φ Γ ) + L Γ 0Z + (L Γ L Γ 0)Z + N(Z) R(Φ Γ ) = L Γ = ( ) (U0) ξξ + U 0 (U 0) 3 ε (α(g 2 U 0) + β(g 3 U 0) + γ) 0 0 ξ U2 0 (Γ) ( εα ) εβ τ τ ε 2 2 ξ 1 0 ( ) 1 1 D θ 0 2 θ ε 2 2 ξ 1 N(Z) = ( 3Φ 1 Z 2 1 Z ) t

27 Projecting I Projecting on X c Γ 0 π Γ 0Z t = 0 and π Γ 0L Γ 0Z = 0 ( PDE: π ΦΓ Γ 0 Γ Γ t) = πγ 0 (R + LZ + N(Z)) Dynamics of Γ: (Γ 1 ) t = 1 ψ 1 (R [ LZ] 1 + [N(Z)] 1, ψ 1 ) L 2(1 + O(ε)) L 2 (Γ 2 ) t = 1 ψ 2 (R [ LZ] 1 + [N(Z)] 1, ψ 2 ) L 2(1 + O(ε)) L 2 where ψ 1,2 (ξ) = 2 1 2sech 2 ( 1 2 2(ξ Γ1,2 ) )

28 Projecting II Projecting on X Γ 0 π Γ 0Z t = Z t and π Γ 0L Γ 0Z = L Γ 0Z ( PDE: Z t = L Γ 0Z + π Γ 0 R Φ Γ Γ Γ t + LZ + N(Z) ) Variation of Constants: Z t Z(ξ, t) = S(t)Z 0 + S(t s) π Γ 0 R Φ «Γ Γt + LZ + N(Z) ds Γ 0 where S is the semigroup generated by L Γ 0.

29 Projecting III Bounds S(t)F χ Ce νt F χ if F X Γ 0 ( π Γ 0 R Φ Γ Γ Γ t) χ = O(ε) LZ χ C Γ Γ 0 Z χ N(Z) χ C { Z 1 2 χ + Z 1 3 χ} C Z 2 χ Remainder Z(ξ, t) Z(ξ, t) χ C { e νt Z(ξ, 0) χ + ε } = εc for t [ ) 1 0, log ε 4ν

30 Step Sketch

31 Iteration At t = t : Choose new basepoint Γ 1 Decompose: Φ 2f (ξ, t) = Φ Γ(t) (ξ) + Z(ξ, t), such that Z(ξ, t) X Γ 1 for all t t < 2t Γ 1 Γ(t ) C 1 Γ(t ) Γ 0 Z(ξ, t ) χ New basepoint doesn t increase the asymptotic magnitude of the remainder: Z 1 χ = O(ε) Repeat previous analysis to prove Z(ξ, t) χ = O(ε) for 0 t < 2t Final Result Z(ξ, t) χ εc for all t ( 3 (Γ 1 ) t = 2 2ε γ αe ε(γ 2 Γ 1) βe ε (Γ2 Γ1)) D ( (Γ 2 ) t = 3 2 2ε γ αe ε(γ 2 Γ 1) βe ε (Γ2 Γ1)) D

32 N-front ODE N-front ODE The same theorem can be proven for the dynamics of the N fronts of a N-front solution. For example, the 4-front ODE reads 8 Γ 1(t) = 3 2 2ε γ + α e ε(γ 1 Γ 2 ) + e ε(γ 1 Γ 3 ) e ε(γ 1 Γ 4 ) + β e D ε (Γ 1 Γ 2 ) + e D ε (Γ 1 Γ 3 ) e D ε (Γ 1 Γ 4 ), Γ 2(t) = 3 2 2ε γ + α e ε(γ 1 Γ 2 ) + e ε(γ 2 Γ 3 ) e ε(γ 2 Γ 4 ) >< + β e D ε (Γ 1 Γ 2 ) + e D ε (Γ 2 Γ 3 ) e D ε (Γ 2 Γ 4 ), Γ 3(t) = 3 2 2ε γ + α e ε(γ 1 Γ 3 ) + e ε(γ 2 Γ 3 ) e ε(γ 3 Γ 4 ) + β e D ε (Γ 1 Γ 3 ) + e D ε (Γ 2 Γ 3 ) e D ε (Γ 3 Γ 4 ), 2ε >: Γ 4(t) = β γ + α e D ε (Γ 1 Γ 4 ) + e D ε (Γ 2 Γ 4 ) e D ε (Γ 3 Γ 4 ) e ε(γ 1 Γ 4 ) + e ε(γ 2 Γ 4 ) e ε(γ 3 Γ 4 ).

33 Theorem There exists no stationary N-front solution for N odd. While a traveling N-front solution (all fronts with the same speed) doesn t exists for N even.

34 PDE vs ODE 2-front dynamics

35 PDE vs ODE 3-front dynamics

36 PDE vs ODE 4-front dynamics

37 4-front dynamics Theorem The manifold M 0 := {Γ 4 (t) = Γ 1 (t), Γ 3 (t) = Γ 2 (t)}, is an invariant manifold of the 4-front ODE. The manifold M 0 is linear stable as long as ( ) αa 1 (A 2 A 1 2 ) + βa 1 D 1 A 1 D 2 A 1 D 2 < 0, with A i := e εγ i. Moreover, the fixed points on M 0 coincide with the 2-pulse solutions.

38 Outline

39 Summary Existence and Stability results of several pulse-type solutions Front dynamics of traveling N-front solutions

40 Summary Existence and Stability results of several pulse-type solutions Front dynamics of traveling N-front solutions Future work ODE dynamics RG for τ, θ large Higher dimensional spatial variable Heterogenous background

41 Thank you for your attention!! For a preprint of the existence and stability analysis: doelman/publist.html