Group Method. December 16, Oberwolfach workshop Dynamics of Patterns

Transcription

1 CWI, Amsterdam December 6, 28 Oberwolfach workshop Dynamics of Patterns Joint work: A. Doelman (CWI/UvA), T.J. Kaper (BU), K. Promislow (MSU)

2 Outline 2 3 4

3 Interactions of localized structures Localized structures are solutions to PDE that are close to a trivial background state, except in one or more localized spatial regions Weak interaction regime: well-developed mamatical ory Strong interaction regime: no mamatical ory Semi-strong interaction regime: U,V,W ξ

4 Paradigm system Three component system U t = U ξξ + U U 3 ε(αv + βw + γ) τv t = ε V 2 ξξ + U V θw t = D2 ε 2 W ξξ + U W where < ε ;D > ; < τ,θ and O(); α,β,γ R and O() (with respect to ε); and (ξ,t) R R +. Physical background: Gas-discharge experiments by Purwins et al. Inspiration: Numerical collision experiments by Nishiura et al. Motivation: Rich behavior and transparent structure enables rigorous mamatical analysis. Goal: Understanding semi-strong interaction regime.

5 Interacting s U 4 2 ξ t 3 4 x 4 U ξ t

6 Complex Dynamics 2.5 U U ξ t ξ 2 2 τ, θ O() t 5

7 Building blocks: Existence U, V, W : Algebraic conditions that determine existence Stationary pulse: A (,) solving αa + βa /D = γ Proofs are based on geometric singular perturbation ory ξ U 5 ξ time τ,θ O()

8 Building blocks: Stability : Algebraic conditions that determine stability Stationary pulse: αa + β D A/D > Proofs are based on Evans function techniques

9 ODE analysis PDE analysis U x/ε ξ No dynamics Stable 2-front U.2.2?? Dynamics ξx/ε Γ Γ2

10 U 3 3 χ = u i χ = ( χu i L + ξ u i L ), i= i= with χ(ξ) = 2 exp ( ξ ) (positive, mass, exponentially decaying) O(eps) gap

11 Properties u L u χ G u χ 2 G L u χ (convolution: (G u)(ξ) = R G(x)u(ξ x)dx ) uv χ 2 u χ v χ Proof of I: u(x) u(y) = u(x) u(x) u(x) Z Z Z Z x y u ξ dξ = u ξ dξ + u(y) = χ(y)dy u ξ dξ + Z Z Z u ξ dξ u L u ξ L + χu L = u χ u(y)χ(y) dy = Z χ(y)dy + u(y)χ(y) dy =

12 Skeleton Solution Skeleton solution Φ Γ (ξ) Φ Γ (ξ) := U (ξ;γ) G V U (ξ;γ) G W U (ξ;γ) U, V, W ξ U (ξ,γ) = + tanh ( 2 2(ξ Γ ) ) tanh ( 2 2(ξ Γ2 ) ) with Γ < Γ 2 and Γ 2 Γ = O(ε ) ε D e ε D ξ G V (ξ) = 2 εe ε ξ, G W (ξ) = 2 Example: G V U is exact solution of = ε 2 V ξξ + U V

13 Manifold and time bound M 2, is 2-dimensional manifold spanned by Φ Γ (ξ); it has a non-trivial boundary M 2, (Γ Γ 2 ). Γ2 M2, M 2, Γ t m := min{t : Γ 2 (t) Γ (t) < ε /2 }, where Γ,2 evolve according to Γ = 3 2 2ε ( γ αe ε(γ 2 Γ ) βe ε D (Γ2 Γ)), Γ 2 = 3 2 2ε ( γ αe ε(γ 2 Γ ) βe ε D (Γ2 Γ)). Note: often t m =

14 Let Φ 2(ξ, t) be a solution of PDE that is O(ε) close to 2-front manifold M 2, at t =, i.e., re is a Γ() such that Φ 2(ξ, ) Φ Γ() (ξ) χ < Cε, for some C >. Then, Φ 2(ξ, t) remains O(ε) close to M 2, for t < t m and its evolution is governed by leading order dynamics of fronts of Φ Γ(t) (ξ), 3 Γ = 2 2ε γ αe ε(γ 2 Γ ) βe D ε (Γ 2 Γ ), Γ 2 = 3 2 2ε γ αe ε(γ 2 Γ ) βe D ε (Γ 2 Γ ). Thus, Φ 2(ξ, t) can be decomposed into Φ 2(ξ, t) = Φ Γ(t) (ξ) + Z(ξ,t) with Z(ξ,t) χ Cε for all t < t m.

15 Initializing Freeze basepoint: Γ = (Γ,Γ 2 ) = (Γ,Γ 2 ) = Γ Decompose: Φ 2 (ξ,t) = Φ Γ(t) (ξ) + Z (ξ,t), such that Z (ξ,t) X Γ for all t < t X Γ is function space associated to dynamics NOT on M 2, Z (ξ,t) χ Cε for all t < t

16 PDE PDE: Zt + Φ Γ Γ Γ t = R(Φ Γ ) + L Γ Z + LZ + N(Z ), where - Residual: R(Φ Γ ) ε (α(g V U ) + β(g W U ) + γ) + exp.small - Linear operator: ξ 2 + 3U 2 ` εα εβ B L Γ 2 τ τ ε 2 ξ θ D 2 2 θ ε 2 ξ A C A - Secular term: L = L Γ L Γ - Nonlinear term: N(Z ) = ` 3U (Z ) 2 (Z ) 3 t

17 Determining t : I Projecting on X c Γ π Γ Zt = and π Γ L Γ Z = ( ODE: π ΦΓ Γ Γ Γ ( t) = πγ R + LZ + N(Z ) ) Dynamics of Γ: Γ = ψ 2 L 2 (R + [ LZ ] + [N(Z )],ψ ) L 2( + O(ε)) Γ 2 = ψ 2 2 L 2 (R + [ LZ ] + [N(Z )],ψ 2 ) L 2( + O(ε)) where ψ,2 (ξ) = 2 2sech 2 ( 2 2(ξ Γ,2 ) ), derivative of skeleton solution.

18 Determining t : II Projecting on X Γ π Γ Zt = Zt and π Γ L Γ Z = L Γ Z ( PDE: Zt = L Γ Z + π Γ R Φ Γ Γ Γ t + LZ + N(Z ) ) Variation of Constants: Z t Z (ξ, t) = S(t)Z + S(t s) π Γ R Φ «Γ Γ Γt + LZ + N(Z ) ds where S is semigroup generated by L Γ restricted to X Γ, and Z is initial condition of PDE.

19 Determining t : III τ,θ = O() O( ε) σ ess ν I(λ) small eigenvalues R(λ) τ,θ = O(ε 2 ) σ ess I(λ) small eigenvalues R(λ) C

20 Determining t : IV Bounds: S(t)F χ Ce νt F χ if F X Γ ( π Γ R Φ Γ Γ Γ t) χ = O(ε) LZ χ C Γ Γ Z χ N(Z ) χ C Z 2 χ Remainder Z (ξ,t) Z (ξ,t) χ C { e νt Z χ + ε } = εc for t [ ], log ε, 4ν thus t = 4ν log ε.

21 Renormalize Φ X Γ Z M 2, Z X Γ Z (t ) Γ Γ(t ) Γ Γ

22 Renormalize Iteration At t = t : Choose new basepoint Γ = (Γ,Γ 2 ) Decompose: Φ 2 (ξ,t) = Φ Γ(t) (ξ) + Z (ξ,t), such that Z (ξ,t) X Γ for all t t t + t Γ Γ(t ) C Γ(t ) Γ Z (ξ,t ) χ New basepoint doesn t increase asymptotic magnitude of remainder: Z χ = O(ε) Repeat previous analysis to prove Z (ξ,t) χ = O(ε) for t t 2t (thus t = t )

23 Final Result Z(ξ,t) χ εc for all t t m ( 3 Γ = 2 2ε γ αe ε(γ 2 Γ ) βe ε (Γ2 Γ)) D ( Γ 2 = 3 2 2ε γ αe ε(γ 2 Γ ) βe ε (Γ2 Γ)) D What goes wrong when t t m : (ψ,ψ 2 ) L 2 exp.small

24 Interactions Forcing : 3 2 εγ Back: 3 2 εγ Back Note N = : Γ = 3 2 εγ (uniformly travelling stable -front) Interactions, (j > i) Γ i Back Γ j : 3 2 ε(αe ε(γ i Γ j ) + βe ε D (Γ i Γ j ) ) Γ i Γ j : 3 2 ε(αe ε(γ i Γ j ) + βe ε D (Γ i Γ j ) )

25 4-front ODE 8 >< >: Γ (t) = 3 2 2ε γ + α e ε(γ Γ 2 ) + e ε(γ Γ 3 ) e ε(γ Γ 4 ) Γ 2(t) + β e D ε (Γ Γ 2 ) + e D ε (Γ Γ 3 ) e D ε (Γ Γ 4 ), = 3 2 2ε γ + α e ε(γ Γ 2 ) + e ε(γ 2 Γ 3 ) e ε(γ 2 Γ 4 ) Γ 3(t) + β e D ε (Γ Γ 2 ) + e D ε (Γ 2 Γ 3 ) e D ε (Γ 2 Γ 4 ), = 3 2 2ε γ + α e ε(γ Γ 3 ) + e ε(γ 2 Γ 3 ) e ε(γ 3 Γ 4 ) Γ 4(t) + β e D ε (Γ Γ 3 ) + e D ε (Γ 2 Γ 3 ) e D ε (Γ 3 Γ 4 ), = 3 2 2ε γ + α e ε(γ Γ 4 ) + e ε(γ 2 Γ 4 ) e ε(γ 3 Γ 4 ) + β e D ε (Γ Γ 4 ) + e D ε (Γ 2 Γ 4 ) e D ε (Γ 3 Γ 4 ).

26 N- Dynamics I Lemma For N odd, re exists no stationary N-front solution. For N even, re exists no uniformly travelling N-front solution with O(ε)-speed. Proof: N Γ i = i= { for N even, 3 2 εγ for N odd.

27 N- Dynamics II Conjecture For N even, re exists a stationary N-front solution. For N odd, re exists a uniformly traveling N-front solution with O(ε)-speed. Proved up to N = 5, for larger N algebra becomes too involved t U 5 ξ t x 4

28 Odd-front dynamics Lemma Assume that N is odd. Then, for γ > at least one front of an N-front solution travels to +, while for γ < at least one front has to travel to.

29 2-front dynamics Lemma The fronts of a 2-front solution asymptote to a standing -pulse solution with width Γ if and only if a -pulse solution with width Γ is stable and re is no unstable standing -pulse solution in between. unstable stable U 2 ξ x 4.5 t U 5 ξ x 4.5 t

30 4-front dynamics Lemma When does front dynamics become pulse dynamics? The 2-pulse sub-manifold M := {Γ 4 (t) = Γ (t), Γ 3 (t) = Γ 2 (t)}, is an invariant manifold of 4-front ODE. The manifold M is normally attracting as long as ( ) αa (A 2 A 2 ) + βa D A D 2 A D 2 >, with A i := e εγ i (,). Γ Γ2 ξ = Γ3 Γ4 Moreover, fixed points on M coincide with stationary 2-pulse solutions.

31 PDE vs ODE t time ξ 2 2 ξ

32 Interactions for τ,θ = O(ε 2 ) I(λ) σ ess small eigenvalues R(λ) Spatial inhomogeneities - Work in progress with K.-I. Ueda & Y. Nishiura Two space dimensions

33 Thank you for your attention!! Preprints: doelman/publist.html