Front interactions in a three-component systemmath.bu.edu/people/heijster/PRESENTATIONS/snowbird2009.pdf · 2011-07-07 · Front interactions in a three-component system Peter van

Frontinteractions in athree-component

system

Peter vanHeijster

Introduction

Existence

Stability

Interaction

Future work

Front interactions in a three-componentsystem

Peter van Heijster

CWI, Amsterdam

[email protected]

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)


system

Peter vanHeijster

Introduction

Existence

Stability

Interaction

Future work

Outline

1 Introduction

2 Existence

3 Stability

4 Interaction

5 Future work


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:

−1000 −800 −600 −400 −200 0 200 400 600 800 1000

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

ξ

U, V

, W

V

U

W


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.


system

Peter vanHeijster

Introduction

Existence

Stability

Interaction

Future work

Front Interaction

2-front interacting: different initial conditions

0

0.5

1

1.5

2

2.5

x 104

−500

0

500

−1

0

1

t

ξ

U

0

0.5

1

1.5

2

2.5

x 104

−200−100

0100

200

−1

0

1

t

ξ

U


system

Peter vanHeijster

Introduction

Existence

Stability

Interaction

Future work

Understood, but not today I

Stationary 2-pulse solution

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0

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1

ξ

U,V

,W

W

V

U


system

Peter vanHeijster

Introduction

Existence

Stability

Interaction

Future work

Understood, but not today II

Uniformly traveling 1-pulse solution

0

200

400

600

800

1000

1200

−500

50100

−101

time

ξ

U

τ, θ ≫ O(1)


system

Peter vanHeijster

Introduction

Existence

Stability

Interaction

Future work

Understood, but not today III

3-front interacting: different forcing parameter γ

0

1

2

3

4

x 104

−400

−200

0

200

400

−1

0

1

tξ

U

0

500

1000

1500

2000−300

−200

−100

0

100

200

300

−1

0

1

tξ

U


system

Peter vanHeijster

Introduction

Existence

Stability

Interaction

Future work

Don’t understand

Complex dynamics

0

2

4

6

8

10

12

14

2000

1000

0

−1000

−2000

−101

0

0.5

1

1.5

2

2000

1000

0

−1000

−2000

−101

ξξ

U U

t × 105t × 106

τ, θ ≫ O(1)


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)


system

Peter vanHeijster

Introduction

Existence

Stability

Interaction

Future work

Preliminaries


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)


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

«


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


system

Peter vanHeijster

Introduction

Existence

Stability

Interaction

Future work


Schematic picture

q, r

v , wu, p

q, r

v , wu, p

P−ε

P+ε

U = 1U = −1U = −1

IV

II

IIII

V

−1000 −800 −600 −400 −200 0 200 400 600 800 1000

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

ξ

U, V

, W

V

U

W

Analysis

Fenichel theory

Melnikov method: αv0 + βw0 + γ = 0

Solve SRS + matching: αe−εξ∗ + βe−ε

Dξ∗ = γ

Rigorous


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.

...


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) .


system

Peter vanHeijster

Introduction

Existence

Stability

Interaction

Future work

Construction

Eigenfunctions: again 5 regions

-200 -100 100 200

-0.8

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0.2

0.4

0.6

0.8

-200 -100 100 200

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

ξ

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)


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.


system

Peter vanHeijster

Introduction

Existence

Stability

Interaction

Future work

Saddle-node bifurcation

α > 0 > β

γ

γSN

α + β

ASN

A


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?


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∗.


system

Peter vanHeijster

Introduction

Existence

Stability

Interaction

Future work

ODE and PDE I

5000 10000 15000 20000 25000

100

200

300

400

500

600

∆Γ

∆Γ1∗

∆Γ2∗

t

0

0.5

1

1.5

2

2.5

x 104

−500

0

500

−1

0

1

t

ξ

U

0

0.5

1

1.5

2

2.5

x 104

−200−100

0100

200

−1

0

1

t

ξ

U


system

Peter vanHeijster

Introduction

Existence

Stability

Interaction

Future work

ODE and PDE II


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.


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.


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


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)

−1000 −800 −600 −400 −200 0 200 400 600 800 1000

−1

−0.8

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0

0.2

0.4

0.6

0.8

1

ξ

U, V

, W

V

U

W

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 .


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 .


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


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)

Front interactions in a three-component systemmath.bu.edu/people/heijster/PRESENTATIONS/snowbird2009.pdf · 2011-07-07 · Front interactions in a three-component system Peter van

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