Upload
vuthien
View
219
Download
0
Embed Size (px)
Citation preview
MS: Nonlinear Wave Propagation in Singular Perturbed Systems
• P. van Heijster: Existence & stability of 2D localized structures in a 3-component model.
• Y. Nishiura: Rotational motion of traveling spots in dissipative systems.
• M. Wechselberger: A geometric twist on tactically-driven cell migration.
• P. Zegeling: Instability of travelling waves in a 2D non-equilibrium Richard’s equation.
MS: Nonlinear Wave Propagation in Singular Perturbed Systems
• Traveling spots, fronts etc.
• Different spatial scales: diffusion coefficients or reaction terms
• Gray-Scott, Schnakenberg, Gierer-Meinhardt
Action potential propagation along giant axon of the squid
Action potential propagation along giant axon of the squid
ARD model!
"uvw
#
$
t
+
!
"0
!h(u, w)g(u, w)v
#
$
x
=
!
"h(u, w)
0f(u, w)
#
$ + !
!
"uvw
#
$
xx
Experiment
Modelling PDE Theory
Geometric Theory
A
w
u
B
L
S
Sa
r
L
Rankine-Hugoniot+ Lax condition
III
III IV
MS: Nonlinear Wave Propagation in Singular Perturbed Systems
• Traveling spots, fronts etc.
• Different spatial scales: diffusion coefficients or reaction terms
• Gray-Scott, Schnakenberg, Gierer-Meinhardt
ARD model!
"uvw
#
$
t
+
!
"0
!h(u, w)g(u, w)v
#
$
x
=
!
"h(u, w)
0f(u, w)
#
$ + !
!
"uvw
#
$
xx
Experiment
Modelling PDE Theory
Geometric Theory
A
S
Sa
r
L
Rankine-Hugoniot+ Lax condition
III
III IV
MS: Nonlinear Wave Propagation in Singular Perturbed Systems
• Traveling spots, fronts etc.
• Different spatial scales: diffusion coefficients or reaction terms
• Gray-Scott, Schnakenberg, Gierer-Meinhardt
ARD model!
"uvw
#
$
t
+
!
"0
!h(u, w)g(u, w)v
#
$
x
=
!
"h(u, w)
0f(u, w)
#
$ + !
!
"uvw
#
$
xx
Experiment
Modelling PDE Theory
Geometric Theory
A
S
Sa
r
L
Rankine-Hugoniot+ Lax condition
MS: Nonlinear Wave Propagation in Singular Perturbed Systems
• Traveling spots, fronts etc.
• Different spatial scales: diffusion coefficients or reaction terms
• Gray-Scott, Schnakenberg, Gierer-Meinhardt
Action potential propagation along giant axon of the squid
Action potential propagation along giant axon of the squid
ARD model!
"uvw
#
$
t
+
!
"0
!h(u, w)g(u, w)v
#
$
x
=
!
"h(u, w)
0f(u, w)
#
$ + !
!
"uvw
#
$
xx
Experiment
Modelling PDE Theory
Geometric Theory
A
w
u
B
L
S
Sa
r
L
Rankine-Hugoniot+ Lax condition
MS: Nonlinear Wave Propagation in Singular Perturbed Systems
• Traveling spots, fronts etc.
• Different spatial scales: diffusion coefficients or reaction terms
• Gray-Scott, Schnakenberg, Gierer-Meinhardt
Research funded by:
Peter van Heijster
Collaborator: B. Sandstede
Previous collaborators (1D): A. Doelman, T.J Kaper, K. Promislow,
Existence & stability of 2D localized structures in a 3-component model
http://www.dam.brown.edu/people/heijster
DSPDES’10MS: Nonlinear Wave Propagation in Singular Perturbed Systems
Barcelona, Spain, June 2010
Outline
• Introduction
• Spot
- Existence
- Stability
• Future work
Generalized FitzHugh-Nagumo Equation:
where 0 < ε << 1; D>1; 0<τ,θ; α,β,γ are constants.
Paradigm system
• Physical background: Gas-discharge experiments by Purwins et al.
• Inspiration: Numerical collision experiments by Nishiura et al.
• Motivation: ‘Rich behavior’ and ‘transparent structure’ enables rigorous mathematical analysis.
Generalized FitzHugh-Nagumo Equation:
where 0 < ε << 1; D>1; 0<τ,θ; α,β,γ are constants.
Paradigm system
• Physical background: Gas-discharge experiments by Purwins et al.
• Inspiration: Numerical collision experiments by Nishiura et al.
• Motivation: ‘Rich behavior’ and ‘transparent structure’ enables rigorous mathematical analysis.
• Goal:
➡Understanding radially symmetric stationary spots
➡Influence of the third component
Experiments
III
III IV
Set up: Observed patterns:
0 200 400 600 800 1000
0
0.2
0.4
0.6
0.8
1
2004006008001000
0.2
0.4
0.6
0.8
1
1-
-
-
-
-
- - ---
Radially symmetric spot
1D: 1-pulse
0 200 400 600 800 1000
0
0.2
0.4
0.6
0.8
1
2004006008001000
0.2
0.4
0.6
0.8
1
1-
-
-
-
-
- - ---
Radially symmetric spot
1D: 1-pulse
Radially symmetric spot
2D: Spot
More complex structures
Ring Spot-ring
➡ Both unstable
Spot: existence
• Stationary:
• Radially symmetric:
• ODE:
• B.C.: Neumann at the core and (background)
Spot: existence
• Stationary:
• Radially symmetric:
• ODE:
• B.C.: Neumann at the core and (background)
• 1D problem with singularity at the core r=0.
Spot: existence
• Stationary:
• Radially symmetric:
• ODE:
• B.C.: Neumann at the core and (background)
• 1D problem with singularity at the core r=0.
1D result [Doelman, H., Kaper]:There exist a stationary 1-pulse solution with width xp if there exists a xp solving:
where v0, w0 are given by
Schematic
• 4 different regions➡Core : r=0; and
➡ Slow :
➡Fast : around r = R1, v,w= constant
➡ Slow : , asymptotes to
1D-spot
Idea: Use these properties to construct a spot.
Flavor: fast field
• Rescale: , s fast coordinate
• ODE:
Flavor: fast field
• Rescale: , s fast coordinate
• ODE:
Fast reduced system: ε =0
➡Centered around R1: and R1 = O(1) wrt r.
Flavor: fast field
• Rescale: , s fast coordinate
• ODE:
Fast reduced system: ε =0
➡Centered around R1: and R1 = O(1) wrt r.
➡Hamiltonian:
➡Heteroclinic connection:
Flavor: Fenichel
• ε=0: 4-dim. invariant manifolds:
• ε 0: 4-dim. locally invariant manifolds
• Also unstable and stable manifolds of persist
Flavor: Fenichel
• ε=0: 4-dim. invariant manifolds:
• ε 0: 4-dim. locally invariant manifolds
• Also unstable and stable manifolds of persist
Flavor: Fenichel
• ε=0: 4-dim. invariant manifolds:
• ε 0: 4-dim. locally invariant manifolds
• Also unstable and stable manifolds of persist
u
2 4 6 8 10
!1.0
!0.5
0.5
1.0
Flavor: slow field I
• : u=1 (to leading order)
• ODE:
• Modified Bessel functions:
Flavor: slow field I
• : u=1 (to leading order)
• ODE:
• Modified Bessel functions:
v
w
1 2 3 4
2
4
6
8
Flavor: slow field II
• Similar on : u=-1
• Bounded at infinity
• Modified Bessel functions:
Flavor: slow field II
• Similar on : u=-1
• Bounded at infinity
• Modified Bessel functions:
v
w
5 6 7 8 9 10
!0.98
!0.96
!0.94
!0.92
Flavor: core
• Scale:
• Matches smoothly with slow field
• Neumann at core:
Flavor: core
• Scale:
• Matches smoothly with slow field
• Neumann at core:
Flavor: core
• Scale:
• Matches smoothly with slow field
• Neumann at core:
v
w
1 2 3 4
2
4
6
8
Flavor: matching
• Slow components do not change over fast field:
• Match in fast field to determine four constants:
• Gives (using a Wronskian identity)
Flavor: matching
• Slow components do not change over fast field:
• Match in fast field to determine four constants:
• Gives (using a Wronskian identity)
2 4 6 8 10
!1.0
!0.5
0.5
1.0
vw• Slow components in fast field
Flavor: Radius
• Unperturbed fast system was Hamiltonian:
• In perturbed system:
• Hamiltonian on . Therefore (Melnikov integral),
• This yields:
Flavor: Radius
• Unperturbed fast system was Hamiltonian:
• In perturbed system:
• Hamiltonian on . Therefore (Melnikov integral),
• This yields:
Theorem: There exists a stationary radially symmetric spot solution with
radius R1 if there exists an R1 solving:
Spot: stability
• (U,V,W)(r,φ,t) = (us,vs,ws)(r) + (u,v,w)(r)eλt+iφl
• ODE:
Spot: stability
• (U,V,W)(r,φ,t) = (us,vs,ws)(r) + (u,v,w)(r)eλt+iφl
• ODE:
• Essential spectrum: left half plane
• Eigenvalues:
• Note: (translation invariance in y-direction)
• α,β<0: Spot is unstable wrt l=0
Spectrum
2 4 6 8
- 1.0
- 0.8
- 0.6
- 0.4
- 0.2
Eigenvectors for l=0 and l=5:
• Again 4 regions➡Core Ic : Neumann and u=0
➡ Slow :
➡Fast If : v,w constant
• Same type of analysis
Spectrum
2 4 6 8
- 1.0
- 0.8
- 0.6
- 0.4
- 0.2
Eigenvectors for l=0 and l=5:
• Again 4 regions➡Core Ic : Neumann and u=0
➡ Slow :
➡Fast If : v,w constant
• Same type of analysis
l=0
2 4 6 8 10
- 10
- 8
- 6
- 4
- 2
0
2
4
l=2
Υ
• α,β<0: • α,β>0:
2 4 6 8 10
- 8
- 6
- 4
- 2
0
2
4
6l=0
l=2
Υ
Future work: interaction
InitialCondition
StationarySolution
U
ξ
Δ Γ
Question:
Given the system-parameters and an initial condition, can we quantitatively predict how the structure evolves in time?
Future work: interaction
InitialCondition
StationarySolution
U
ξ
Δ Γ
Question:
Given the system-parameters and an initial condition, can we quantitatively predict how the structure evolves in time?
Answer:
For 1D, yes! We can derive a system of ODEs describing the motion of the separate fronts.
1-Pulse:
2D: interaction
Spot-ring
Can we derive something similar for planar structures?
2D: interaction
3 interacting spots
Renormalization group method used for the 1D problem does not work in higher dimensions
Future: traveling spot
Traveling spot (and growing)
No reduction to an ODE problem.
Thanks!
Questions?