ms: nonlinear wave propagation in singular perturbed systems...

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.


• 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


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





ARD model!

"uvw

#

$

t

+

!

"0

!h(u, w)g(u, w)v

#

$

x

=

!

"h(u, w)

0f(u, w)

#

$ + !

!

"uvw

#

$

xx

Experiment


Geometric Theory

A

S

Sa

r

L


III

III IV





ARD model!

"uvw

#

$

t

+

!

"0

!h(u, w)g(u, w)v

#

$

x

=

!

"h(u, w)

0f(u, w)

#

$ + !

!

"uvw

#

$

xx

Experiment


Geometric Theory

A

S

Sa

r

L








ARD model!

"uvw

#

$

t

+

!

"0

!h(u, w)g(u, w)v

#

$

x

=

!

"h(u, w)

0f(u, w)

#

$ + !

!

"uvw

#

$

xx

Experiment


Geometric Theory

A

w

u

B

L

S

Sa

r

L


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

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:


• ODE:


• 1D problem with singularity at the core r=0.

Spot: existence

• Stationary:


• ODE:


• 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


• ODE:

Fast reduced system: ε =0

➡Centered around R1: and R1 = O(1) wrt r.

Flavor: fast field


• 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

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:


v

w

1 2 3 4

2

4

6

8

Flavor: slow field II

• Similar on : u=-1

• Bounded at infinity


Flavor: slow field II

• Similar on : u=-1

• Bounded at infinity


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:

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?

ms: nonlinear wave propagation in singular perturbed systems...

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