1. No category

Nonlinear Stability of Sources
Björn Sandstede
Arnd Scheel
Margaret Beck
Toan Nguyen
Kevin Zumbrun
Spiral waves
[Li, Ouyang, Petrov, Swinney]
[Nettesheim, von Oertzen,
Rotermund, Ertl]
– Dynamics of core / spiral tip
– Modulations of wave trains in far field
[Li, Ouyang, Petrov, Swinney]
One-dimensional defects
space
time
Chloride-iodide-malonic acid reaction (CIMA)
[Perraud, De Wit, Dulos, De Kepper, Dewel, Borckmans]
Standing time-periodic
structures
space
One-dimensional defects
Surface waves [Pastur et al.]
space
Light-sensitive BZ-reaction
[Yoneyama, Fujii, Maeda]
One-dimensional defects
wave train
defect
wave train
time
– asymptotically periodic in space
– time-periodic in co-moving frame
Overview:
– wave trains & group velocities
– sources
– existence & bifurcations
– spectral & nonlinear stability
space
Dynamics of wave trains
c
k wavenumber
ω=ω(k) temporal frequency
wave train
local wavenumber
slowly varying modulations of wavenumber
Spectrum of
wave trains
λ(iγ)
Im λ
Re λ
λ(iγ) = -icgγ - dγ2 + Ο(|γ|3)
Dynamics of wave trains
c
k wavenumber
ω=ω(k) temporal frequency
wave train
cg
q(x,t)
qt = cg vx
local wavenumber
group velocity:
direction of transport
slowly varying modulations of wavenumber
Spectrum of
wave trains
Im λ
λ(iγ)
Re λ
λ(iγ) = -icgγ - dγ2 + Ο(|γ|3)
Dynamics of wave trains
cg
q(X,T)
local wavenumber
slowly varying modulations of wavenumber near k0
on scale X=ε(x-cgt) and T=ε2t/2 for 0<ε<<1
Viscous Burgers equation:
•
•
•
@q
=
@T
00
(0)
@2 q
@X2
!00 (k0 ) q2
X
Formal derivation: [Howard & Kopell], [Kuramoto]
Validity over natural time scale 1/ε2: [Doelman, S., Scheel, Schneider]
Stability of wave trains: [S., Scheel, Schneider, Uecker], [Johnson, Zumbrun]
Anticipated dynamics:
• Zero-mean perturbations converge to zero
• Lax shocks and rarefaction waves
Sources
cg
Sources: outgoing transport
group velocities point away
from core
•
•
•
cg
transport
Existence: how do sources arise?
Spectral and linear stability: linearized equation is time-periodic
Nonlinear stability: previous methods do not apply
Essential Hopf instabilities of pulses
Im λ
standing pulse
+
Re λ
wave trains
Hopf instability
of rest state
Theorem [S., Scheel]
k
flip-flop
target
μ
source
target
flip-flop
time
Spatial dynamics
space
wave train =
periodic orbit
ux
= v
vx
= D 1 [ut
defect =
heteroclinic
orbit
cd v
✓ ◆
1
u
2 H 2 (S1 ) ⇥ L2 (S1 )
v
f(u)]
wave train =
periodic orbit
Reaction-diffusion system:
Standing sources are time-periodic:
Floquet spectrum determines spectral stability
Spectrum of
wave trains
2
time
Spectra of sources
space
Evans-function analysis
(eigenfunctions ux and ut)
L2 space
2
exponential
weight
t>>1
cg>0
cg<0
t>>1
exponentially weighted
L2 space
x
defect core
Expected dynamics
exponential adjustment
of position and phase
Burgers equation with
advection in far field
2
2
t
x=-cgt
x=cgt
position/phase
adjustment
Gaussians
error terms
x
defect core
Nonlinear stability
Theorem [Beck, Nguyen, S., Zumbrun]: Assume u*(x,t)
is a spectrally stable source and let u(x,0)=u*(x,0)+v0(x)
where ||v0(x)exp(x2/M)||<ε is sufficiently small. Then there
are constants |p∞|, |φ∞|<ε such that
|u(x,t)-u*(x-p∞,t-φ∞)| < εC exp(-ηt) for (x,t) in Ω1 and
|u(x,t)-u*(x,t)|
< εC exp(-ηt) for (x,t) in Ω2.
x=-cgt
x=cgt
t
Ω1
Ω2
Ω2
x
defect core
Nonlinear stability proofs
•
Define appropriate offset from source:
•
Derive equation for offset:
•
Variation-of-constants formula:
•
Fixed-point argument in appropriate function space:
2
2
No decay in
L2 spaces
Decay in weighted L2 spaces,
but nonlinearity not well defined
Caveats
Long-time dynamics for small localized initial data
Heat equation
Reaction-diffusion equation
Burgers equation
Heat equation
Reaction diffusion
Burgers equation
Reason: Gaussian * Gaussian2 ≠ Gaussian
Differentiated Gaussian * Gaussian2 ≈ Gaussian
Nonlinear stability proofs
•
Define appropriate offset from source:
•
Derive equation for offset:
•
Variation-of-constants formula:
•
Fixed-point argument in appropriate function space:
2
2
No decay in
L2 spaces
Decay in weighted L2 spaces,
but nonlinearity not well defined
Define offsets
• Let u(x,t) be a solution of ut = Duxx + cux + f(u) near a given defect u*(x,t)
• Define p(x,t) and φ(x,t) so that
• p(x,t) and φ(x,t): space-time shift
v(x,t):
profile changes
• Substitution gives the following system for the functions p, φ, and v:
Solve linearized system: Green’s function
• Linearization about defect:
• Solve via Green’s function:
• Expansion of Green’s function:
x=-cgt
projection
x=cgt
error function plateau
t
y
• aj(x,t;y,s):
• GR(x,t;y,s):
Gaussians
error terms
x
scalar functions composed of error function plateaus in (x,t-s)
times a localized projection function in y plus Gaussian errors
sum of moving “differentiated” Gaussians
Variation-of-constants formula
• Equation for offsets:
• Variation-of-constants formula:
• Expansion of Green’s function:
• Collect terms:
Nonlinear iterations
• Equation for offsets: “differentiated” Gaussian * Gaussian2 ≈ Gaussian
• Expansion of Green’s function:
x=-cgt
x=cgt
wj(y)
err(x,t-s)
t
• aj(x,t;y,s) = err(x,t-s) wj(y) + Gaussians
• GR(x,t;y,s) ≈ “differentiated” Gaussians
Gaussians
x
y
Nonlinear iterations
• Equation for offsets: “differentiated” Gaussian * Gaussian2 ≈ Gaussian
• Expansion of Green’s function:
• aj(x,t;y,s) = err(x,t-s) wj(y) + Gaussians
• GR(x,t;y,s) ≈ “differentiated” Gaussians
• Nonlinear iteration using templates:
• Completes nonlinear stability result …
v and (p,φ)x ≈ sum of moving Gaussians
Expansion of Green’s function
• Green’s function G(x,t;y,s) satisfies
• Laplace transform:
• Resolvent kernel (x,t;y,s,λ) is 2π-periodic in (t,s) and satisfies
• Makes connection with spatial-dynamics formulation of defects:
ux
= v
vx
= D 1 [ut
cd v
f(u)]
✓ ◆
1
u
2 H 2 (S1 ) ⇥ L2 (S1 )
v
• Linearize about defect and expand in λ gives bounds needed for (x,t;y,s,λ)
Summary + Outlook
Summary:
•
Proved that spectrally stable sources are nonlinearly stable in an
appropriate sense
Outlook:
•
Obtain expansions of dynamics in
wedge-shape interface regions:
•
Stability analysis of contact defects (cg=0):
contact
defect
•
Interaction of sink-source pairs: