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 x ∈ R , u ∈ R n u t = Du xx + f ( u ) , Defects and coherent structures: • time periodic in an appropriate moving frame • spatially asymptotic to periodic travelling waves c d c − c + p p wave train wave train defect time time space space

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

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: x ∈ R , u ∈ R n u t = Du xx + f ( u ) , Wave trains u ( x, t ) = u 0 ( kx − ωt ; k ) satisfy − ωu φ = k 2 Du φφ + f ( u ) u 0 ( φ ; k ) = u 0 ( φ + 2 π ; k ) Wave train Temporal frequency ω = ω nl ( k ) Wave speed c p = ω/k Group velocity c g = d ω/ d k c g c p phase velocity group velocity

Spectra of Wave Trains v t = Dv xx + f ′ ( u 0 ( kx − ωt ; k )) v v ( x, t ) = e λt + νx v per ( kx − ωt ) − → Linear dispersion relation: λ = λ ( ν ) with ν ∈ i R c g = d ω d k = − d λ � Group velocity: � d ν � ν =0 Hypothesis: Im λ λ (i γ ) Re λ Translational invariance implies that λ = 0 is contained in the spectrum: wave train u 0 translation mode v per = ∂ φ u 0

Burgers Equation for Modulated Wave Trains c g Slowly-varying modulations of the wavenumber: u ( x, t ) = u 0 ( kx − ωt + φ ( X, T ); k + ǫφ X ( X, T )) T = ǫ 2 t/ 2 where X = ǫ ( x − c g t ) , ǫ ≪ 1 and Wavenumber q = φ X satisfies viscous Burgers equation: ∂T = λ ′′ (0) ∂ 2 q ∂q ∂X 2 − ω ′′ q 2 � � nl ( k ) 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 c d c − c + g g wave train wave train defect Sink Contact defect c − g > c d > c + c − g = c d = c + g g Transmission defect Source c − g < c d < c + g c ± g < c d • Rankine–Hugoniot condition: c d = ω nl ( k + ) − ω nl ( k − ) k + − k −

Spatial Dynamics Reaction-diffusion system in ( ξ, τ ) = ( x − c d t, ω d t ) : ω d u τ = Du ξξ + c d u ξ + f ( u ) Space-time plot for τ ∈ [0 , 2 π ] : time space Modulated-wave equation for defects:      u v d  =   d ξ D − 1 [ ω d u τ − c d v − f ( u )] v where ( u, v )( ξ, · ) is time-periodic in τ with period 2 π Defects are heteroclinic orbits that connect periodic orbits Key: Group velocity determines relative dimensions of stable and unstable manifolds of periodic orbits 2 c g < c d c g = c d c g > c d

Spatial Dynamics 2 Contact defects Sinks Transmission defects Sources • 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 L 2 ( R ) : 1 1 2 Sink Contact Transmission Source Floquet spectra in exponentially weighted spaces: 1 2 Sink Transmission Source Admissible functions in weighted spaces Techniques: • Evans functions for waves with algebraic spatial decay

Rigorous Justifi cation and Proofs Modulated-wave equation:      u v d  =   d ξ D − 1 [ ω d u τ − c d v − f ( u )] v Issues: 1 2 ( S 1 ) × L 2 ( S 1 ) • Spaces: ( u, v ) ∈ H • 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 of 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