Front interactions in a three-component...
Transcript of Front interactions in a three-component...
Frontinteractions in athree-component
system
Peter vanHeijster
Introduction
Existence
Stability
Interaction
Future work
Front interactions in a three-componentsystem
Peter van Heijster
CWI, Amsterdam
May 19, 2009MS69: New Developments in Pulse Interactions
SIAM Conference on Applications of Dynamical Systems
Snowbird, Utah, USA
Joint work: A. Doelman (CWI/UvA), T.J. Kaper (BU), K. Promislow (MSU)
Frontinteractions in athree-component
system
Peter vanHeijster
Introduction
Existence
Stability
Interaction
Future work
Outline
1 Introduction
2 Existence
3 Stability
4 Interaction
5 Future work
Frontinteractions in athree-component
system
Peter vanHeijster
Introduction
Existence
Stability
Interaction
Future work
Interactions of localized structures
Localized structures are solutions to a PDE that are close to a trivialbackground 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:
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U, V
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V
U
W
Frontinteractions in athree-component
system
Peter vanHeijster
Introduction
Existence
Stability
Interaction
Future work
Paradigm system
Three component system
Ut = Uξξ + U − U3 − ε(αV + βW + γ)
τVt = 1ε2 Vξξ + U − V
θWt = D2
ε2 Wξξ + U − W
where 0 < ε ≪ 1;D > 1; 0 < τ, θ and O(1); α, β, γ ∈ R and O(1)(with respect to ε); and (ξ, t) ∈ R × R
+.
Physical background: gas-discharge experiments by Purwins etal.
Inspiration: numerical collision experiments by Nishiura et al.
Motivation: ‘rich behavior’ and ‘transparent structure’ enablesrigorous mathematical analysis.
Goal: understanding the semi-strong interaction regime.
Frontinteractions in athree-component
system
Peter vanHeijster
Introduction
Existence
Stability
Interaction
Future work
Front Interaction
2-front interacting: different initial conditions
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Frontinteractions in athree-component
system
Peter vanHeijster
Introduction
Existence
Stability
Interaction
Future work
Understood, but not today I
Stationary 2-pulse solution
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Frontinteractions in athree-component
system
Peter vanHeijster
Introduction
Existence
Stability
Interaction
Future work
Understood, but not today II
Uniformly traveling 1-pulse solution
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U
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Frontinteractions in athree-component
system
Peter vanHeijster
Introduction
Existence
Stability
Interaction
Future work
Understood, but not today III
3-front interacting: different forcing parameter γ
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Frontinteractions in athree-component
system
Peter vanHeijster
Introduction
Existence
Stability
Interaction
Future work
Don’t understand
Complex dynamics
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t × 105t × 106
τ, θ ≫ O(1)
Frontinteractions in athree-component
system
Peter vanHeijster
Introduction
Existence
Stability
Interaction
Future work
Geometric singular perturbation theory
Idea: 5 regions
II IVIII VI
Region I: U = −1 + O(ε)
Region II: V = V0 + O(ε), W = W0 + O(ε)
Region III: U = 1 + O(ε)
Region IV: V = V0 + O(ε), W = W0 + O(ε)
Region V: U = −1 + O(ε)
Frontinteractions in athree-component
system
Peter vanHeijster
Introduction
Existence
Stability
Interaction
Future work
Preliminaries
Stationary 1-pulse solution
No movement in time: ∂∂t· = 0
6-dimensional ODE: nonlinear but autonomous
uξ = p
pξ = −u + u3 + ε(αv + βw + γ)vξ = εq
qξ = ε(v − u)wξ = ε
Dr
rξ = εD
(w − u)
Fixed points
P±ε = (±1, 0,±1, 0,±1, 0) + O(ε) , P0
ε = (0, 0, 0, 0, 0, 0) + O(ε)
Frontinteractions in athree-component
system
Peter vanHeijster
Introduction
Existence
Stability
Interaction
Future work
Regions II and IV: fast reduced system
ε ↓ 0{
uξ = p
pξ = −u + u3
and (v , q,w , r) = (v0, q0,w0, r0)
Hamiltonian
H(u, p) = 12 (u2 + p2) − 1
4 (u4 + 1)
Phase portrait
-2 -1 1 2
-2
-1
1
2
p
u
(u−
het, p−
het)
(u+het
, p+het
)
u±
het(ξ) = ∓ tanh
„
1
2
√2ξ
«
,
p±
het(ξ) = ∓1
2
√2sech2
„
1
2
√2ξ
«
Frontinteractions in athree-component
system
Peter vanHeijster
Introduction
Existence
Stability
Interaction
Future work
Regions I, III, and V: slow reduced system (SRS)
U = ±1 (to leading order)
vξξ = ε2(v ∓ 1) =⇒ v(ξ) = Aieεξ + Bie
εξ ± 1
wξξ = εD
2(w ∓ 1) =⇒ w(ξ) = Aieε
Dξ + Bie
ε
Dξ ± 1
Phase portrait
v , wP−
ε P+ε
U = 1U = −1
v , w
q, r q, r
Frontinteractions in athree-component
system
Peter vanHeijster
Introduction
Existence
Stability
Interaction
Future work
Stationary 1-pulse solution
Schematic picture
q, r
v , wu, p
q, r
v , wu, p
P−ε
P+ε
U = 1U = −1U = −1
IV
II
IIII
V
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Analysis
Fenichel theory
Melnikov method: αv0 + βw0 + γ = 0
Solve SRS + matching: αe−εξ∗ + βe−ε
Dξ∗ = γ
Rigorous
Frontinteractions in athree-component
system
Peter vanHeijster
Introduction
Existence
Stability
Interaction
Future work
Existence result
Theorem
There exists a stationary 1-pulse solution if is a ξ∗ ∈ (0,∞) whichsolves
αe−εξ∗ + βe−ε
Dξ∗ = γ. (1)
Moreover, ξ∗ corresponds to the width of the pulse.
Corollary
Equation (1) has at most 2 solutions.
If sgn(α) = sgn(β), then (1) has at most 1 solution.
If |γ| > |α| + |β|, then (1) has no solutions.
If sgn(α) 6= sgn(β) and |αD| > |β|, then there is a saddle-nodebifurcation of stationary 1-pulse solutions.
...
Frontinteractions in athree-component
system
Peter vanHeijster
Introduction
Existence
Stability
Interaction
Future work
Stability: Evans function
Essential spectrum
Stable and O(1) gap to imaginary axis.
Discrete spectrum
U(ξ, t) = uh(ξ; ε) + u(ξ)eλt ,
V (ξ, t) = vh(ξ; ε) + v(ξ)eλt ,
W (ξ, t) = wh(ξ; ε) + w(ξ)eλt ,
Stability problem: nonautonomous but linear
uξξ + u(1 − 3u2h− λ) = ε(αv + βw)
vξξ = ε2((1 + τλ)v − u)
wξξ = ε2
D2 ((1 + θλ)w − u) .
Frontinteractions in athree-component
system
Peter vanHeijster
Introduction
Existence
Stability
Interaction
Future work
Construction
Eigenfunctions: again 5 regions
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ξ
v
w
u
v
w
u
Ψ+(ξ) Ψ−(ξ)
Region I: u = 0 + O(ε)
Region II: v = v0 + O(ε), w = w0 + O(ε)
Region III: u = 0 + O(ε)
Region IV: v = v0 + O(ε), w = w0 + O(ε)
Region V: u = 0 + O(ε)
Frontinteractions in athree-component
system
Peter vanHeijster
Introduction
Existence
Stability
Interaction
Future work
Stability theorem
Analysis
Solving the corresponding FRS and SRS + matching
Rigorous: Evans function
Theorem
The stationary 1-pulse solution with width ξ∗ is stable if and only if
αe−εξ∗ +β
De−
ε
Dξ∗ > 0 .
Corollary
The 1-pulse solution is stable if sgn(α) = sgn(β) = 1 andunstable if sgn(α) = sgn(β) = −1.
One of the branch of the saddle-node bifurcation is stable, theother one is unstable.
Frontinteractions in athree-component
system
Peter vanHeijster
Introduction
Existence
Stability
Interaction
Future work
Saddle-node bifurcation
α > 0 > β
γ
γSN
α + β
ASN
A
Frontinteractions in athree-component
system
Peter vanHeijster
Introduction
Existence
Stability
Interaction
Future work
Front interaction
InitialCondition
StationarySolution
∆Γ
Question
Given the system-parameters and an initial condition, can we predicthow the structure evolves in time?
Frontinteractions in athree-component
system
Peter vanHeijster
Introduction
Existence
Stability
Interaction
Future work
2-Front Dynamics
Theorem
The distance ∆Γ between two fronts of a 1-pulse solution evolvesaccording to
∆Γt = 3√
2ε(
αe−ε∆Γ + βe−ε
D∆Γ − γ
)
. (2)
Fixed points of (2) are precisely the solutions to the existencecondition (1).
The fronts ∆Γ(t) asymptote to a stationary 1-pulse solutionwith width ∆Γ1
∗ iff the 1-pulse solution is stable and there is nounstable stationary 1-pulse solution determined by ∆Γ2
∗ with∆Γ(0) < ∆Γ2
∗ < ∆Γ1∗ or ∆Γ(0) > ∆Γ2
∗ > ∆Γ1∗.
Frontinteractions in athree-component
system
Peter vanHeijster
Introduction
Existence
Stability
Interaction
Future work
ODE and PDE I
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∆Γ1∗
∆Γ2∗
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Frontinteractions in athree-component
system
Peter vanHeijster
Introduction
Existence
Stability
Interaction
Future work
ODE and PDE II
Frontinteractions in athree-component
system
Peter vanHeijster
Introduction
Existence
Stability
Interaction
Future work
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.
Frontinteractions in athree-component
system
Peter vanHeijster
Introduction
Existence
Stability
Interaction
Future work
Sketch of proof I
Formally: Easy, introduce 2 co-moving frames and distinguishagain 5 different regions =⇒ ODE analysis.
Rigorous: Hard, real PDE analysis. Proof is based on aRenormalization group method.
Frontinteractions in athree-component
system
Peter vanHeijster
Introduction
Existence
Stability
Interaction
Future work
Future work
Interactions 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
Frontinteractions in athree-component
system
Peter vanHeijster
Introduction
Existence
Stability
Interaction
Future work
Sketch of proof II
Skeleton solution ΦΓ(ξ)
ΦΓ(ξ) :=
U0(ξ; Γ)GV ∗ U0(ξ; Γ)GW ∗ U0(ξ; Γ)
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U0(ξ, Γ) = −1 + tanh(
12
√2(ξ − Γ1)
)
− tanh(
12
√2(ξ − Γ2)
)
,where Γi is the location of the i-th front (∆Γ = Γ2 − Γ1).
GV = − 12εe−ε|ξ|; the Green’s function associated to
0 =1
ε2Vξξ + U0 − V .
Frontinteractions in athree-component
system
Peter vanHeijster
Introduction
Existence
Stability
Interaction
Future work
Sketch of proof III
Decomposition: Φ2 = ΦΓ + Z (Assumption in initial condition!)
Going to show:
Φ2 evolves according to (2)||Z ||χ = O(ε)
Zt +∂ΦΓ
∂ΓΓt = R + LΓZ + N(Z ) ,
where R =residual, L =linear part, N =nonlinear part.
‘Freeze’ time t = t0:
Zt +∂ΦΓ
∂ΓΓt = R + LΓ0Z + (LΓ − LΓ0)Z + N(Z )
Define πΓ0 as the projection on the space spanned by the theeigenfunctions of LΓ0 associated to the small eigenvalues;πΓ0 = I − πΓ0 .
Frontinteractions in athree-component
system
Peter vanHeijster
Introduction
Existence
Stability
Interaction
Future work
Sketch of proof IV
Projection by πΓ0 yields/confirms the ODE for Γt (since Z canassumed to be ‘perpendicular’ to πΓ0 : πΓ0Z = 0).
Projection by πΓ0 shows that ||Z ||χ stays O(ε) small for a finitetime ∆t (this is due to the ‘secular growth’).
∆t = O(log ε) (Hard work)
Renormalize: ‘freeze’ a new time: t1 = t0 + ∆t such that||Z (t1)||χ = O(ε).
Repeat above procedure
Frontinteractions in athree-component
system
Peter vanHeijster
Introduction
Existence
Stability
Interaction
Future work
Z (t0)
πΓ0πΓ2
πΓ1
Γ
Φ
ΦΓ
Φ2
Γ(t0 + ∆t) Γ(t1 + ∆t) Γ2Γ1Γ0
Z (t1)
Z (t2)