Defects in oscillatory media towards a classification Bjrn - - PDF document

Defects in oscillatory media — towards a classification —

Björn Sandstede (Ohio State University) Joint work with Arnd Scheel (U Minnesota)

Modulated Waves

ut = Duxx + f(u), x ∈ R, u ∈ Rn Defects and coherent structures:

• time periodic in an appropriate moving frame
• spatially asymptotic to periodic travelling waves

cd c−

p

c+

p

defect wave train wave train space time space time

Coherent Structures in Experiments

CIMA-reaction [Perraud, De Wit, Dulos, De Kepper, Dewel, Borckmans] Heated-wire experiments [Pastur, Westra, van de Water] Period-doubled spiral wave Belousov–Zhabotinsky reaction [Yoneyama, Fujii, Maeda] Line defect [Goryachev, Kapral]

Overview

Goals:

• Multiplicity and robustness
• Bifurcations from and to defects
• Stability and interaction
• Model-independent approach to coherent structures

Issues:

• Defects are genuine PDE solutions
• Essential spectrum touches the imaginary axis:

Outline:

• Wave trains
• Classification
• Spatial dynamics
• Multiplicity and robustness

Wave Trains and Group Velocity

Reaction-diffusion system: ut = Duxx + f(u), x ∈ R, u ∈ Rn Wave trains u(x, t) = u0(kx − ωt; k) satisfy −ωuφ = k2Duφφ + f(u) Wave train u0(φ; k) = u0(φ + 2π; k) Temporal frequency ω = ωnl(k) Wave speed cp = ω/k Group velocity cg = dω/dk

cp cg phase velocity group velocity

Spectra of Wave Trains

vt = Dvxx + f ′(u0(kx − ωt; k))v − → v(x, t) = eλt+νx vper(kx − ωt) Linear dispersion relation: λ = λ(ν) with ν ∈ iR Group velocity: cg = dω dk = −dλ dν

• ν=0

Hypothesis: λ(iγ) Re λ Im λ Translational invariance implies that λ = 0 is contained in the spectrum: wave train u0 translation mode vper = ∂φu0

Burgers Equation for Modulated Wave Trains

cg

Slowly-varying modulations of the wavenumber: u(x, t) = u0(kx − ωt + φ(X, T); k + ǫφX(X, T)) where X = ǫ(x − cgt), T = ǫ2t/2 and ǫ ≪ 1 Wavenumber q = φX satisfies viscous Burgers equation: ∂q ∂T = λ′′(0) ∂2q ∂X2 − ω′′

nl(k)

• q2

X

Validity of Burgers equations over natural time scale [0, ǫ−2]: [Doelman, Sandstede, Scheel, Schneider]

Classifi cation of Coherent Structures

• time periodic in an appropriate moving frame
• spatially asymptotic to wave trains

cd c−

g

c+

g

defect wave train wave train Sink c−

g > cd > c+ g

Contact defect c−

g = cd = c+ g

Transmission defect c±

g < cd

Source c−

g < cd < c+ g

• Rankine–Hugoniot condition:

cd = ωnl(k+) − ωnl(k−) k+ − k−

Spatial Dynamics

Reaction-diffusion system in (ξ, τ) = (x − cdt, ωdt): ωduτ = Duξξ + cduξ + f(u) Space-time plot for τ ∈ [0, 2π]:

space time

Modulated-wave equation for defects: d dξ   u v   =   v D−1[ωduτ − cdv − f(u)]   where (u, v)(ξ, ·) is time-periodic in τ with period 2π Defects are heteroclinic orbits that connect periodic orbits Key: Group velocity determines relative dimensions of stable and unstable manifolds of periodic orbits cg < cd cg = cd cg > cd 2

Spatial Dynamics

Sinks Contact defects Transmission defects Sources

2

• Sinks:
• Wavenumbers (k−, k+) are free
• Defect speed determined by Rankine–Hugoniot condition
• Contact defects:
• Wavenumber k− = k+ is free
• Defect speed equal to group velocity
• Transmission defects:
• Wavenumber k− = k+ is free
• Speed selected by defect

(Rankine–Hugoniot condition is violated)

• Sources:
• Defect speed and wavenumbers selected

Floquet Spectra of Defects

Floquet spectra in L2(R):

Sink Contact Transmission Source

1 2 1

Floquet spectra in exponentially weighted spaces:

Sink Transmission Source Admissible functions in weighted spaces

1 2

Techniques:

• Evans functions for waves with algebraic spatial decay

Rigorous Justifi cation and Proofs

Modulated-wave equation: d dξ   u v   =   v D−1[ωduτ − cdv − f(u)]   Issues:

• Spaces: (u, v) ∈ H

1 2 (S1) × L2(S1)

• Initial-value problem is ill-posed
• Both stable and unstable manifolds are infinite-dimensional
• Contact defects: no asymptotic phase

− →

• Exponential dichotomies to construct stable and unstable

manifolds for ill-posed elliptic equations

• Relate Evans function in spatial dynamics to spectral stability
• f defects

[Peterhof, Sandstede, Scheel], [Sandstede, Scheel]

Comments:

• Classifi cation extends counting arguments for complex

Ginzburg–Landau equations by [van Saarloos, Hohenberg]

• Nonlinear stability:
• Sinks (phase matching)
• Transmission defects [Gallay, Schneider, Uecker]
• Bifurcations to defects
• Essential instabilities of standing and travelling pulses

− → sources and transmission defects

• Small-amplitude shocks in Burgers equation −

→ sinks

• Period doubling of homogeneous oscillations

− → sources and contact defects

• Spatial inhomogeneities −

→ sources and contact defects

• Bifurcations from defects:
• Contact defects versus sinks
• In general: homoclinic and heteroclinic bifurcations

Future Directions:

• Nonlinear stability of sources and contact defects
• Interactions of source-sinks pairs, transmission and contact defects:
• Relevance of roots of Evans functions
• Description by coupled ODEs and Burgers equations
• First insights: Interaction on circles