system May 19, 2009 MS69: New Developments in Pulse Interactions SIAM Conference on Applications of Dynamical Systems Snowbird, Utah, USA

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1 CWI, Amsterdam May 9, 29 MS69: New Developments in Pulse s SIAM Conference on Applications of Dynamical Systems Snowbird, Utah, USA Joint work: A. Doelman (CWI/UvA), T.J. Kaper (BU), K. Promislow (MSU)

2 Outline

3 s of localized structures Localized structures are solutions to a PDE that are close to a trivial background state, except in one or more localized spatial regions Weak interaction regime: well-developed mathematical theory Strong interaction regime: no mathematical theory Semi-strong interaction regime:.8 U.6.4 U, V, W V.8 W ξ

4 Paradigm Three component 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 mathematical analysis. Goal: understanding the semi-strong interaction regime.

5 2-front interacting: different initial conditions U 2 ξ x 4.5 t U 5 ξ x 4.5 t

6 Understood, but not today I Stationary 2-pulse solution U,V,W W U V ξ

7 Understood, but not today II Uniformly traveling -pulse solution U 5 ξ time τ,θ O()

8 Understood, but not today III 3-front interacting: different forcing parameter γ U U 4 2 ξ t 3 4 x ξ t

9 Don t understand Complex dynamics U U ξ t ξ 2 2 τ, θ O() t 5

10 Geometric singular perturbation theory Idea: 5 regions I II III IV V Region I: U = + O(ε) Region II: V = V + O(ε), W = W + O(ε) Region III: U = + O(ε) Region IV: V = V + O(ε), W = W + O(ε) Region V: U = + O(ε)

11 Preliminaries Stationary -pulse solution No movement in time: t = 6-dimensional ODE: nonlinear but autonomous u ξ = p p ξ = u + u 3 + ε(αv + βw + γ) v ξ = εq q ξ = ε(v u) w ξ = ε D r r ξ = ε D (w u) Fixed points P ± ε = (±,, ±,, ±,) + O(ε), P ε = (,,,,,) + O(ε)

12 Regions II and IV: fast reduced ε { uξ = p and (v,q,w,r) = (v,q,w,r ) p ξ = u + u 3 Hamiltonian H(u,p) = 2 (u2 + p 2 ) 4 (u4 + ) Phase portrait p (u het, p het ) 2 u «u het(ξ) ± = tanh 2ξ, 2 p het(ξ) ± = «2sech 2 2ξ (u + het, p+ het )

13 Regions I, III, and V: slow reduced (SRS) U = ± (to leading order) v ξξ = ε 2 (v ) = v(ξ) = A i e εξ + B i e εξ ± w ξξ = ε 2 D (w ) = w(ξ) = Ãie ε D ξ + B i e ε D ξ ± Phase portrait U = q, r q, r U = P ε v, w P + ε v, w

14 Stationary -pulse solution ξ Schematic picture U = U = P ε I V II IV III P + ε U, V, W U q, r v, w u, p u, p q, r v, w V W Analysis Fenichel theory Melnikov method: αv + βw + γ = Solve SRS + matching: αe εξ + βe ε D ξ = γ Rigorous

15 result Theorem There exists a stationary -pulse solution if is a ξ (, ) which solves αe εξ + βe ε D ξ = γ. () Moreover, ξ corresponds to the width of the pulse. Corollary Equation () has at most 2 solutions. If sgn(α) = sgn(β), then () has at most solution. If γ > α + β, then () has no solutions. If sgn(α) sgn(β) and αd > β, then there is a saddle-node bifurcation of stationary -pulse solutions..

16 : Evans function Essential spectrum Stable and O() gap to imaginary axis. Discrete spectrum U(ξ,t) = u h (ξ;ε) + u(ξ)e λt, V(ξ,t) = v h (ξ;ε) + v(ξ)e λt, W(ξ,t) = w h (ξ;ε) + w(ξ)e λt, problem: nonautonomous but linear u ξξ + u( 3uh 2 λ) = ε(αv + βw) v ξξ = ε 2 (( + τλ)v u) w ξξ = ε2 D (( + θλ)w u). 2

17 Construction Eigenfunctions: again 5 regions ṽ w u ξ Ψ + (ξ) ṽ w Ψ (ξ) u ξ Region I: u = + O(ε) Region II: ṽ = ṽ + O(ε), w = w + O(ε) Region III: u = + O(ε) Region IV: ṽ = ṽ + O(ε), w = w + O(ε) Region V: u = + O(ε)

18 theorem Analysis Solving the corresponding FRS and SRS + matching Rigorous: Evans function Theorem The stationary -pulse solution with width ξ is stable if and only if αe εξ + β D e ε D ξ >. Corollary The -pulse solution is stable if sgn(α) = sgn(β) = and unstable if sgn(α) = sgn(β) =. One of the branch of the saddle-node bifurcation is stable, the other one is unstable.

19 Saddle-node bifurcation α > > β γ γ SN α + β A SN A

20 interaction Stationary Solution Initial Condition Γ Question Given the -parameters and an initial condition, can we predict how the structure evolves in time?

21 2- Dynamics Theorem The distance Γ between two fronts of a -pulse solution evolves according to Γ t = 3 2ε ( αe ε Γ + βe ε D Γ γ ). (2) Fixed points of (2) are precisely the solutions to the existence condition (). The fronts Γ(t) asymptote to a stationary -pulse solution with width Γ iff the -pulse solution is stable and there is no unstable stationary -pulse solution determined by Γ 2 with Γ() < Γ 2 < Γ or Γ() > Γ 2 > Γ.

22 ODE and PDE I Γ Γ 2 Γ t 25 U 2 ξ x 4.5 t U 5 ξ x 4.5 t

23 ODE and PDE II

24 General N-front dynamics Lemma Stationary N-front solutions do not exist for N odd. Uniformly travelling N-front solutions do not exist for N even.

25 Sketch of proof I Formally: Easy, introduce 2 co-moving frames and distinguish again 5 different regions = ODE analysis. Rigorous: Hard, real PDE analysis. Proof is based on a Renormalization group method.

26 s for τ,θ large - Problems with the essential spectrum Spatial inhomogeneities - Work in progress with K.-I. Ueda & Y. Nishiura Two space dimensions - Rubicon research at Brown University

27 Sketch of proof II ξ Skeleton solution Φ Γ (ξ) Φ Γ (ξ) := U (ξ;γ) G V U (ξ;γ) G W U (ξ;γ) U, V, W U V W U (ξ,γ) = + tanh ( 2 2(ξ Γ ) ) tanh ( 2 2(ξ Γ2 ) ), where Γ i is the location of the i-th front ( Γ = Γ 2 Γ ). G V = 2 εe ε ξ ; the Green s function associated to = ε 2 V ξξ + U V.

28 Sketch of proof III Decomposition: Φ 2 = Φ Γ + Z (Assumption in initial condition!) Going to show: Φ 2 evolves according to (2) Z χ = O(ε) Z t + Φ Γ Γ Γ t = R + L Γ Z + N(Z), where R =residual, L =linear part, N =nonlinear part. Freeze time t = t : Z t + Φ Γ Γ Γ t = R + L Γ Z + (L Γ L Γ )Z + N(Z) Define π Γ as the projection on the space spanned by the the eigenfunctions of L Γ associated to the small eigenvalues; π Γ = I π Γ.

29 Sketch of proof IV Projection by π Γ yields/confirms the ODE for Γ t (since Z can assumed to be perpendicular to π Γ : π Γ Z = ). Projection by π Γ shows that Z χ stays O(ε) small for a finite time t (this is due to the secular growth ). t = O(log ε) (Hard work) Renormalize: freeze a new time: t = t + t such that Z(t ) χ = O(ε). Repeat above procedure

30 Φ Z(t) Φ 2 Z(t2) π Γ π Γ 2 Z(t) Φ Γ Γ π Γ Γ(t + t) Γ Γ(t + t)γ 2 Γ