Mathematical Models of Reaction Diﬀusion Systems, their ...dotten/files/pde/PDE.pdf ·...

Mathematical Modelsof Reaction Diffusion Systems,their Numerical Solutions and

the Freezing Methodwith Comsol Multiphysics

Denny Otten1

Department of MathematicsBielefeld University33501 BielefeldGermany

Date: December 19, 2010

1e-mail: [email protected], homepage: http://www.math.uni-bielefeld.de/~dotten/index_eng.shtml

http://www.math.uni-bielefeld.de/~dotten/index_eng.shtml

Contents

1 Mathematical Models 3

1.1 Fisher’s equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2 Nagumo equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.3 Quintic Nagumo equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.4 FitzHugh-Nagumo model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271.5 Purwins model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331.6 Barkley model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341.7 Schrödinger equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391.8 Gross-Pitaevskii equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411.9 Complex Ginzburg-Landau equation (CGL) . . . . . . . . . . . . . . . . . . . . . . . . 441.10 Quintic Complex Ginzburg-Landau equation (QCGL) . . . . . . . . . . . . . . . . . . 461.11 λ-ω system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 671.12 Autocatalysis model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

Bibliography 73

Index 79

1 Mathematical Models

In this chapter I want to give a large assortment of examples for reaction-diffusion equations, theirphenomena and some background stuff. The focus of this investigations will be the freezing method([12]) and their numerical computations with Comsol Multiphysics 3.5a. For this purpose we use thefollowing notions:

d spatial dimension, d ∈ {1, 2, 3}ℝd d-dimensional euclidean spacex spatial variable, x ∈ ℝ

d

t temporal variable, t ∈ [0,∞[R ratio, R ∈ ℝ with R > 0BR(0) BR(0) = {x ∈ ℝ

d | ‖x‖2 ⩽ R}, ball in ℝd with ratio R and center 0

∂BR(0) ∂BR(0) = {x ∈ ℝd | ‖x‖2 = R}, boundary of BR(0)

△x spatial discretization parameter, spatial stepsize△t temporal discretization parameter, temporal stepsizeRe(u) real part of uIm(u) imaginary part of ui imaginary unit∂∂n

normal derivative, derivative in outer direction

4 1 Mathematical Models

1.1 Fisher’s equation

Name: Fisher’s equation (sometimes called Kolmogorov-Petrovsky-Piskounov equation

(KPP) or Fisher-Kolmogorov equation)

Equations: ut = △u+ au (1− u)

where u = u(x, t) ∈ ℝ, x ∈ ℝd, d ∈ {1, 2, 3}, t ∈ [0,∞[ and a ∈ ℝ.

Notations:

u : frequency of the mutant genea : system parameter

Short description: The Fisher’s equation ([31]), named after Ronald Aylmer Fisher (1890-1962)and Andrey Nicolaevich Kolmogorov (1903-1987), describes the spreading of biological populations,chemical reaction processes, heat and mass transfer ([28]) and is also used in genetics. This verysimple model exhibits traveling front solutions.

Phenomena:

• Traveling 1-front (traveling front)

Set of parameter values:

d a R △x △t Boundary Phenomena

1 1 75 0.1 0.1 ∂u∂n

= 0 on ∂BR(0) 1-front

Numerical results:• Traveling 1-front (traveling front):I. Nonfrozen solution: Consider the nonfrozen system

ut = uxx + au(1− u) , x ∈ BR(0), t ∈ [0,∞[∂u∂n

= 0 , x ∈ ∂BR(0), t ∈ [0,∞[u(0) = u0 , x ∈ BR(0), t = 0

where d = 1 (i. e. BR(0) = [−R,R]). For the numerical computations we use parameters a = 1,R = 75 and initial data u0(x) = 1

2 tanh(x) + 12 . Moreover, for the spatial discretization we use FEM

(continuous piecewise linear finite elements) with △x = 0.1. For the temporal discretization we useBDF(2) with △t = 0.1, rtol = 10−2, atol = 10−3 and intermediate timesteps.Nonfrozen solution:

1.1 Fisher’s equation 5

Spatial-temporal pattern:

II. Frozen system (with freezing method ([12])): Consider the associated frozen system

vt = vxx + av(1− v) + λ1vx , x ∈ BR(0), t ∈ [0,∞[∂v∂n

= 0 , x ∈ ∂BR(0), t ∈ [0,∞[v(0) = u0 , x ∈ BR(0), t = 0γt = λ1 , t ∈ [0,∞[γ(0) = 0 , t = 00 = 〈v̂x, v − v̂〉L2(BR(0),ℝ) , t ∈ [0,∞[

where d = 1 (i. e. BR(0) = [−R,R]). For the numerical computations we use again parameters a = 1,R = 75, initial data u0(x) = 1

2 tanh(x) + 12 and reference function v̂(x) = u0(x). Moreover, for the

spatial discretization we use FEM (continuous piecewise linear finite elements) with △x = 0.1. Forthe temporal discretization we use BDF(2) with △t = 0.01, rtol = 10−3, atol = 10−4 and intermediatetimesteps.Frozen solution, velocity and spatial-temporal pattern:

0 20 40 60 80 100−2

−1.5

−1

−0.5

Vel

ocity

t

Velocity

lm1

Convergence error of freezing method and eigenvalues of the linearization about the frozen solution:

0 20 40 60 80 10010

−150

10−100

10−50

100

Convergence error

t

||vt|| L2

−60 −40 −20 0−15

−10

−5

0

5

10

15

Eigenvalues

Re

Im


The increasing convergence error indicates that Fisher’s-Front cannot be frozen by the freezing method.This can be recognized by the essential spectra. Because to ensure the stability of the front the essentialspectra must exhibit a spectral gab, which doesn’t exist.

Explicit solutions:• 1d:

u(x, t) =1

(

1 + C exp(

−56λt± x6

√6λ))2 , if λ > 0

u(x, t) =1 + 2C exp

(

−56λt± x6

√−6λ

)

(

1 + C exp(

−56λt± x6

√−6λ

))2 , if λ < 0

are two possible solutions of the nonfrozen system with parameter λ ∈ ℝ and an arbitrary constantC ∈ ℝ ([1],[51],[48]).

Literature: [28], [1], [51], [22], [31], [48]

1.2 Nagumo equation 7

1.2 Nagumo equation

Name: Nagumo equation

Equations: ut = △u+ u (1− u) (u− α)

u = u(x, t) ∈ ℝ, x ∈ ℝd, d ∈ {1, 2, 3}, t ∈ [0,∞[, α ∈ ]0, 1[.

Notations: not available

Short description: The Nagumo equation ([62]), named after Jin-Ichi Nagumo (1926-1999), de-scribes propagation of nerve pulses in a nerve axon ([62]), spread of genetic traits, shape and speedof pulses in the nerve and is widely used in biology, circuit theory, heat and mass transfer and otherfields. This model exhibits traveling front and traveling multifront solutions as well as sources andsinks ([4]).

Phenomena:


• Traveling 2-front (traveling multifront)

• Front interaction (3-front collision)


d α R △x △t Boundary Phenomena

1 14 75 0.1 0.1 ∂u

∂n= 0 on ∂BR(0) 1-front

1 14 100 0.1 0.1 ∂u


1 14 150 0.3 0.1 ∂u

∂n= 0 on ∂BR(0) 3-front collision


ut = uxx + u(1− u)(u− α) , x ∈ BR(0), t ∈ [0,∞[∂u∂n

= 0 , x ∈ ∂BR(0), t ∈ [0,∞[u(0) = u0 , x ∈ BR(0), t = 0

where d = 1 (i. e. BR(0) = [−R,R]). For the numerical computations we use parameters α = 14 ,

R = 50 and initial data u0(x) = 12 tanh(x) + 1

2 (also u0 =

{

0 x < 0xRx ⩾ 0

possible). Moreover, for the

spatial discretization we use FEM (continuous piecewise linear finite elements) with △x = 0.1. Forthe temporal discretization we use BDF(2) with △t = 0.1, rtol = 10−2, atol = 10−3 and intermediatetimesteps ([13],[12]).Nonfrozen solution:




vt = vxx + v(1− v)(v − α) + λ1vx , x ∈ BR(0), t ∈ [0,∞[∂v∂n


where d = 1 (i. e. BR(0) = [−R,R]). For the numerical computations we use again parameters α = 14 ,

R = 50, initial data u0(x) = 12 tanh(x) + 1

2 and reference function v̂(x) = u0(x). Moreover, for thespatial discretization we use FEM (continuous piecewise linear finite elements) with △x = 0.1. Forthe temporal discretization we use BDF(2) with △t = 0.1, rtol = 10−3, atol = 10−4 and intermediatetimesteps.Frozen solution, velocity and spatial-temporal pattern:

0 50 100 150−0.4

−0.35

−0.3

−0.25

−0.2

−0.15

−0.1

−0.05

Vel

ocity

t

Velocity

lm1



0 50 100 15010

−100

10−80

10−60

10−40

10−20

100

Convergence error

t

||vt|| L2

−150 −100 −50 0−1

−0.5

0

0.5

1Eigenvalues

Re

Im• Traveling 2-front (traveling multifront):I. Nonfrozen solution: Consider the nonfrozen system


= 0 , x ∈ ∂BR(0), t ∈ [0,∞[u(0) = u0 , x ∈ BR(0), t = 0


R = 100 and initial data u0(x) = −12 tanh (x− 50) + 1

2 +

{

12 tanh (x+ 50)− 1

2 x ⩽ 0

0 x > 0(also u0 =

exp(

−x2

R

)

possible). Moreover, for the spatial discretization we use FEM (continuous piecewise linear

finite elements) with △x = 0.1. For the temporal discretization we use BDF(2) with △t = 0.1,rtol = 10−2, atol = 10−3 and intermediate timesteps ([11])).Nonfrozen solution:


II. Frozen system (with freezing method ([11],[73])): Consider the associated frozen system (j = 1, 2)


vj,t = vj,xx + ϕ(·)∑

2

k=1ϕ(·−gk+gj)

· f(

∑2k=1 vk(· − gk + gj , ·)

)

+µjvj,x , x ∈ BR(0), t ∈ [0,∞[∂vj∂n

= 0 , x ∈ ∂BR(0), t ∈ [0,∞[vj(0) = uj0 , x ∈ BR(0), t = 0gj,t = µj , t ∈ [0,∞[gj(0) = gj0 , t = 00 = 〈(v̂j)x, vj − v̂j〉L2(BR(0),ℝ) , t ∈ [0,∞[

where f(v) = v(1− v)(v − α) and d = 1 (i. e. BR(0) = [−R,R]). For the numerical computations weuse again parameters α = 1

4 , R = 50, initial data u10(x) = 12 tanh(x) + 1

2 , u20(x) = −12 tanh(x) − 1

2 ,initial positions γ10 = −50, γ20 = 50, reference functions v̂1(x) = u10(x), v̂2(x) = u20(x) and bumpfunction ϕ(x) = 2

exp( 1

2x)+exp(− 1

2x)

. Moreover, for the spatial discretization we use FEM (continuous

piecewise linear finite elements) with △x = 0.1. For the temporal discretization we use BDF(2) with△t = 0.1, rtol = 10−3, atol = 10−4 and intermediate timesteps.Frozen solutions:

Velocities and positions:

0 50 100 150−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

Vel

ocity

t

Velocities

mu1mu2

0 50 100 150−150

−100

−50

0

50

100

150

Pos

ition

t

Positions

g1g2

Spatial-temporal patterns:



0 50 100 15010

−100

10−80

10−60

10−40

10−20

100

Convergence error

t

||vt|| L2

−40 −30 −20 −10 0

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

Eigenvalues

ReIm

• Front interaction (3-front collision):I. Nonfrozen solution: Consider the nonfrozen system


= 0 , x ∈ ∂BR(0), t ∈ [0,∞[u(0) = u0 , x ∈ BR(0), t = 0


R = 150 and initial data u0 =

12 tanh(x+ 75) + 1

2 x ⩽ −50

−12 tanh(x) + 1

2 x ∈]− 50, 50[12 tanh(x+ 75) + 1

2 x ⩾ 50

possible). Moreover, for the

spatial discretization we use FEM (continuous piecewise linear finite elements) with △x = 0.3. Forthe temporal discretization we use BDF(2) with △t = 0.1, rtol = 10−2, atol = 10−3 and intermediatetimesteps ([13],[12]).Nonfrozen solution:



II. Frozen system (with freezing method ([11])): Consider the associated frozen system (j = 1, 2, 3)


3

k=1ϕ(·−gk+gj)

· f(

∑3k=1 vk(· − gk + gj , ·)

)

+µjvj,x , x ∈ BR(0), t ∈ [0,∞[∂vj∂n


where f(v) = v(1 − v)(v − α) and d = 1 (i. e. BR(0) = [−R,R]). For the numerical computations

we use again parameters α = 14 , R = 50, initial data u10(x) =

1 x > 25150x+ 1

2 x ∈ [−25, 25]

0 x < 25

, u20(x) =

−1 x > 25

− 150x− 1

2 x ∈ [−25, 25]

0 x < 25

, u30(x) = u10(x), initial positions γ10 = −50, γ20 = 0, γ30 = 50, reference

functions v̂1(x) = u10(x), v̂2(x) = u20(x), v̂3(x) = u30(x) and bump function ϕ(x) = 2exp( 1

2x)+exp(− 1

2x)

.

Moreover, for the spatial discretization we use FEM (continuous piecewise linear finite elements) with△x = 0.1. For the temporal discretization we use BDF(2) with △t = 0.1, rtol = 10−2, atol = 10−2

and intermediate timesteps.Frozen solutions:

Velocities, positions and Spatial-temporal patterns:

0 20 40 60 80 100−3

−2

−1

0

1

2

3

Vel

ocity

t

Velocities

mu1mu2mu3

0 20 40 60 80 100−100

−50

0

50

Pos

ition

t

Positions

g1g2g3




0 20 40 60 80 10010

−50

10−40

10−30

10−20

10−10

100

Convergence error

t

||vt|| L2

−40 −30 −20 −10 0−1

−0.5

0

0.5

1Eigenvalues

Re

Im

Explicit solutions:• 1d: u(x) = u(ξ) with ξ = x− µt where

u(ξ) =

(

1 + exp

(

− ξ√2

))−1

with speed µ = −√

2

(

1

2− α

)

u(ξ) =

(

1 + exp

(

ξ√2

))−1

with speed µ =√

2

(

1

2− α

)

are two possible profiles u with their velocity µ and parameter α ∈]0, 12 [ ([23],[4]).

• 1d:

u(x, t) =A exp

(

±√

22 x+

(

12 − α

)

t)

+ αB exp(

±√

22 αx+ α

(

α2 − 1

)

t)

A exp(

±√

22 x+

(

12 − α

)

t)

+B exp(

±√

22 αx+ α

(

α2 − 1

)

t)

+ C

where A,B,C are arbitrary constants ([49]).

Additional informations: The Nagumo equation (sometimes called Allen-Cahn model or Zel-

dovich equation arising in combustion theory ([86])) is a simplified form of the FitzHugh-Nagumo

equations (to this later). Especially for α = −1 we receive a special case of the Chafee-Infante

equation (sometimes called Newell-Whitehead-Segel equation describing Rayleigh-Benardconvections ([63],[72]))

ut = △u+ λ(u− u3), λ ∈ ℝ

or more general

ut = △u+ au− bu3, a, b ∈ ℝ

which in general possesses an attractor as well as a pitchfork bifurcation ([65],[43]).

Literature: [62], [4], [13], [11], [23], [49], [12]


1.3 Quintic Nagumo equation

Name: Quintic Nagumo equation

Equations: ut = △u+ u (1− u) (u− α1) (u− α2) (u− α3)

u = u(x, t) ∈ ℝ, x ∈ ℝd, d ∈ {1, 2, 3}, t ∈ [0,∞[, α1, α2, α3 ∈ ]0, 1[ with α1 < α2 < α3.


Short description: The Quintic Nagumo equation, named after Jin-Ichi Nagumo (1926-1999) de-scribes an extension of the Nagumo equation and was developed only for mathematical investigations([11],[73]). This model exhibits traveling front and traveling multifront solutions.

Phenomena:



• Front interaction (2-front collision)



• Front interaction (4-front double collision)


d α1 α2 α3 R △x △t Boundary Phenomena

1 25

12

1720 100 0.3 0.3 ∂u


1 116

12

710 100 0.3 0.3 ∂u


1 116

12

710 250 0.4 1.0 ∂u


1 116

12

710 150 0.3 0.5 ∂u



ut = uxx + u(1− u)(u− α1)(u− α2)(u− α3) , x ∈ BR(0), t ∈ [0,∞[∂u∂n

= 0 , x ∈ ∂BR(0), t ∈ [0,∞[u(0) = u0 , x ∈ BR(0), t = 0

where d = 1 (i. e. BR(0) = [−R,R]). For the numerical computations we use parameters α1 = 25 ,

α2 = 12 , α3 = 17

20 , R = 100 and initial data u0(x) = 12 tanh(x) + 1

2 . Moreover, for the spatialdiscretization we use FEM (continuous piecewise linear finite elements) with △x = 0.3. For thetemporal discretization we use BDF(2) with △t = 0.3, rtol = 10−2, atol = 10−3 and intermediatetimesteps.Nonfrozen solution:

1.3 Quintic Nagumo equation 15



vt = vxx + v(1− v)(v − α1)(v − α2)(v − α3) + λ1vx , x ∈ BR(0), t ∈ [0,∞[∂v∂n


where d = 1 (i. e. BR(0) = [−R,R]). For the numerical computations we use again parametersα1 = 2

5 , α2 = 12 , α3 = 17

20 , R = 100, initial data u0(x) = 12 tanh(x) + 1

2 and reference functionv̂(x) = u0(x). Moreover, for the spatial discretization we use FEM (continuous piecewise linear finiteelements) with △x = 0.3. For the temporal discretization we use BDF(2) with △t = 0.2, rtol = 10−3,atol = 10−4 and intermediate timesteps.Frozen solution, velocity and spatial-temporal pattern:

0 100 200 300 400 500 6000

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

Vel

ocity

t

Velocity

lm1



0 100 200 300 400 500 60010

−150

10−100

10−50

100

Convergence error

t

||vt|| L2

−50 −40 −30 −20 −10−1

−0.5

0

0.5

1Eigenvalues

Re

Im• Traveling 2-front (traveling multifront):I. Nonfrozen solution: Consider the nonfrozen system


= 0 , x ∈ ∂BR(0), t ∈ [0,∞[u(0) = u0 , x ∈ BR(0), t = 0


α2 = 12 , α3 = 7

10 , R = 100 and initial data u0(x) = 14 tanh

(

x+255

)

+ 14 tanh

(

x−255

)

+ 12 (also u0 =

12 tanh

(

x5

)

+ 12 possible). Moreover, for the spatial discretization we use FEM (continuous piecewise

linear finite elements) with △x = 0.3. For the temporal discretization we use BDF(2) with △t = 0.3,rtol = 10−2, atol = 10−3 and intermediate timesteps.Nonfrozen solution:


II. Frozen system (with freezing method ([11])): Consider the associated frozen system (j = 1, 2)



2

k=1ϕ(·−gk+gj)

· f(

∑2k=1 vk(· − gk + gj , ·)

)

+µjvj,x , x ∈ BR(0), t ∈ [0,∞[∂vj∂n


where f(v) = v(1 − v)(v − α1)(v − α2)(v − α3) and d = 1 (i. e. BR(0) = [−R,R]). For thenumerical computations we use again parameters α1 = 1

16 , α2 = 12 , α3 = 7

10 , R = 50, initial datau10(x) = 1

4 tanh(

x5

)

+ 14 , u20(x) = 1

4 tanh(

x5

)

+ 14 , initial positions γ10 = −50, γ20 = 50, reference

functions v̂1(x) = u10(x), v̂2(x) = u20(x) and bump function ϕ(x) = 2exp( 1

2x)+exp(− 1

2x)

. Moreover, for

the spatial discretization we use FEM (continuous piecewise linear finite elements) with △x = 0.3. Forthe temporal discretization we use BDF(2) with △t = 0.3, rtol = 10−3, atol = 10−4 and intermediatetimesteps.Frozen solutions:


0 500 1000 1500−0.16

−0.14

−0.12

−0.1

−0.08

−0.06

−0.04

Vel

ocity

t

Velocities

mu1mu2

0 500 1000 1500−300

−250

−200

−150

−100

−50

0

50

Pos

ition

t

Positions

g1g2




0 500 1000 150010

−20

10−15

10−10

10−5

Convergence error

t

||vt|| L2

−50 −40 −30 −20 −10 0

−8

−6

−4

−2

0

2

4

6

8x 10

−3 Eigenvalues

Re

Im

• Front interaction (2-front collision):I. Nonfrozen solution: Consider the nonfrozen system


= 0 , x ∈ ∂BR(0), t ∈ [0,∞[u(0) = u0 , x ∈ BR(0), t = 0


α2 = 12 , α3 = 3


(

x+255

)

+ 14 tanh

(

x−255

)

+ 12 (also u0 =

12 tanh

(

x5

)

+ 12 possible). Moreover, for the spatial discretization we use FEM (continuous piecewise






2

k=1ϕ(·−gk+gj)

· f(

∑2k=1 vk(· − gk + gj , ·)

)

+µjvj,x , x ∈ BR(0), t ∈ [0,∞[∂vj∂n



4 , α2 = 12 , α3 = 3


4 tanh(

x5

)

+ 14 , u20(x) = 1

4 tanh(

x5

)

+ 14 , initial positions γ10 = −50, γ20 = 50, reference

functions v̂1(x) = u10(x), v̂2(x) = u20(x) and bump function ϕ(x) = 2exp( 1

2x)+exp(− 1

2x)

. Moreover, for

the spatial discretization we use FEM (continuous piecewise linear finite elements) with △x = 0.3.For the temporal discretization we use BDF(2) with △t = 1.5, rtol = 10−3, atol = 5 · 10−4 andintermediate timesteps.Frozen solutions:


0 500 1000 1500 2000 2500 3000−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

Vel

ocity

t

Velocities

mu1mu2

0 500 1000 1500 2000 2500 3000−50

0

50

Pos

ition

t

Positions

g1g2




0 500 1000 1500 2000 2500 300010

−12

10−10

10−8

10−6

10−4

Convergence error

t

||vt|| L2

−50 −40 −30 −20 −10

−0.01

−0.005

0

0.005

0.01

Eigenvalues

ReIm

• Traveling 3-front (traveling multifront):I. Nonfrozen solution: Consider the nonfrozen system


= 0 , x ∈ ∂BR(0), t ∈ [0,∞[u(0) = u0 , x ∈ BR(0), t = 0


α2 = 12 , α3 = 7

10 , R = 250 and initial data u0(x) = 14 tanh (x+ 50) + 1

4 tanh (x)− 14 tanh (x− 50) + 1

4 .Moreover, for the spatial discretization we use FEM (continuous piecewise linear finite elements) with△x = 0.4. For the temporal discretization we use BDF(2) with △t = 1.0, rtol = 10−3, atol = 10−4

and intermediate timesteps.Nonfrozen solution:


II. Frozen system (with freezing method ([11])): Consider the associated frozen system (j = 1, 2, 3)



3

k=1ϕ(·−gk+gj)

· f(

∑3k=1 vk(· − gk + gj , ·)

)

+µjvj,x , x ∈ BR(0), t ∈ [0,∞[∂vj∂n



16 , α2 = 12 , α3 = 7

10 , R = 200, initial datau10(x) = 1

4 tanh(

x5

)

+ 14 , u20(x) = u10(x), u30(x) = −1

4 tanh(

x5

)

− 14 , initial positions γ10 = −50,

γ20 = 0, γ30 = 50, reference functions v̂1(x) = u10(x), v̂2(x) = u20(x), v̂3(x) = u30(x) and bumpfunction ϕ(x) = 2

exp( 1

2x)+exp(− 1

2x)


piecewise linear finite elements) with △x = 0.4. For the temporal discretization we use BDF(2) with△t = 0.3, rtol = 10−3, atol = 10−4 and intermediate timesteps.Frozen solutions:

Velocities, positions and Spatial-temporal patterns:

0 200 400 600 800 1000−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

Vel

ocity

t

Velocities

mu1mu2mu3

0 200 400 600 800 1000−200

−150

−100

−50

0

50

100

150

Pos

ition

t

Positions

g1g2g3




0 200 400 600 800 100010

−100

10−80

10−60

10−40

10−20

100

Convergence error

t

||vt|| L2

−1 −0.8 −0.6 −0.4 −0.2 0

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

Eigenvalues

ReIm

• Traveling 4-front (traveling multifront):I. Nonfrozen solution: Consider the nonfrozen system


= 0 , x ∈ ∂BR(0), t ∈ [0,∞[u(0) = u0 , x ∈ BR(0), t = 0


α2 = 12 , α3 = 7


(

x+1005

)

+ 14 tanh

(

x+505

)

− 14 tanh

(

x−505

)

−14 tanh

(

x−1005

)

. Moreover, for the spatial discretization we use FEM (continuous piecewise linear finite

elements) with △x = 0.3. For the temporal discretization we use BDF(2) with △t = 0.5, rtol = 10−3,atol = 10−4 and intermediate timesteps.Nonfrozen solution:


II. Frozen system (with freezing method ([11])): Consider the associated frozen system (j = 1, 2, 3, 4)



4

k=1ϕ(·−gk+gj)

· f(

∑4k=1 vk(· − gk + gj , ·)

)

+µjvj,x , x ∈ BR(0), t ∈ [0,∞[∂vj∂n



16 , α2 = 12 , α3 = 7


4 tanh(

x5

)

+ 14 , u20(x) = u10(x), u30(x) = −1

4 tanh(

x5

)

− 14 , u40(x) = u30(x), initial positions

γ10 = −100, γ20 = −50, γ30 = 50, γ40 = 100, reference functions v̂1(x) = u10(x), v̂2(x) = u20(x),v̂3(x) = u30(x), v̂4(x) = u40(x) and bump function ϕ(x) = 2

exp( 1

2x)+exp(− 1

2x)

. Moreover, for the

spatial discretization we use FEM (continuous piecewise linear finite elements) with △x = 0.5. Forthe temporal discretization we use BDF(2) with △t = 1.0, rtol = 10−2, atol = 10−3 and intermediatetimesteps.Frozen solutions, velocities and positions:

0 500 1000 1500 2000 2500 3000

−0.2

−0.1

0

0.1

0.2

Vel

ocity

t

Velocities

mu1mu2mu3mu4

0 500 1000 1500 2000 2500 3000−800

−600

−400

−200

0

200

400

600

800

Pos

ition

t

Positions

g1g2g3g4




0 500 1000 1500 2000 2500 300010

−12

10−10

10−8

10−6

10−4

10−2

Convergence error

t

||vt|| L2

−10 −8 −6 −4 −2−0.15

−0.1

−0.05

0

0.05

0.1

0.15

Eigenvalues

Re

Im

• Front interaction (4-front double collision):I. Nonfrozen solution: Consider the nonfrozen system


= 0 , x ∈ ∂BR(0), t ∈ [0,∞[u(0) = u0 , x ∈ BR(0), t = 0


α2 = 12 , α3 = 3


(

x+1005

)

+ 14 tanh

(

x+205

)

− 14 tanh

(

x−205

)

−14 tanh

(

x−1005

)

. Moreover, for the spatial discretization we use FEM (continuous piecewise linear finite

elements) with △x = 0.3. For the temporal discretization we use BDF(2) with △t = 1.0, rtol = 10−3,atol = 10−4 and intermediate timesteps.Nonfrozen solution:



II. Frozen system (with freezing method ([11])): Consider the associated frozen system (j = 1, 2, 3, 4)


4

k=1ϕ(·−gk+gj)

· f(

∑4k=1 vk(· − gk + gj , ·)

)

+µjvj,x , x ∈ BR(0), t ∈ [0,∞[∂vj∂n



4 , α2 = 12 , α3 = 3


4 tanh(

x5

)

+ 14 , u20(x) = u10(x), u30(x) = −1

4 tanh(

x5

)

− 14 , u40(x) = u30(x), initial positions

γ10 = −100, γ20 = −20, γ30 = 20, γ40 = 100, reference functions v̂1(x) = u10(x), v̂2(x) = u20(x),v̂3(x) = u30(x), v̂4(x) = u40(x) and bump function ϕ(x) = 2

exp( 1

2x)+exp(− 1

2x)

. Moreover, for the

spatial discretization we use FEM (continuous piecewise linear finite elements) with △x = 0.4. Forthe temporal discretization we use BDF(2) with △t = 1.0, rtol = 10−2, atol = 10−3 and intermediatetimesteps.Frozen solutions, velocities and positions:

0 500 1000 1500 2000 2500 3000−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

Vel

ocity

t

Velocities

mu1mu2mu3mu4

0 500 1000 1500 2000 2500 3000−100

−50

0

50

100

Pos

ition

t

Positions

g1g2g3g4




0 500 1000 1500 2000 2500 300010

−12

10−10

10−8

10−6

10−4

Convergence error

t

||vt|| L2

−10 −8 −6 −4 −2

−0.015

−0.01

−0.005

0

0.005

0.01

0.015

Eigenvalues

Re

Im

Explicit solutions: not available

Literature: [11], [73]

1.4 FitzHugh-Nagumo model 27

1.4 FitzHugh-Nagumo model

Name: FitzHugh-Nagumo model (FHN) (sometimes called Bonhoeffer-van der Pol model)

Equations:

(

u1

u2

)

t

=

(

△u1 + f (u1)− u2 + αD△u2 + β (γu1 − δu2 + ǫ)

)

u1 = u1(x, t) ∈ ℝ, u2 = u2(x, t) ∈ ℝ, u = (u1, u2)T , x ∈ ℝd, d ∈ {1, 2, 3}, D,α, β, γ, δ, ǫ ∈ ℝ

and f : ℝ → ℝ is a (cubic nonlinearity) polynomial of third degree (i. e. f(w) = w − ζw3 orf(w) = w(1− w)(w − ζ) with 0 6= ζ ∈ ℝ, cf. Nagumo equation).

Notations:

u1 : membran potential (measure the potential difference across the cell membrane)u2 : recovery variable (measure transmembrane currents which affect the ability of

the cell to recover before being able to fire again)D : diffusion coefficient (usually D = 0)α : magnitude of stimulus currentβ, γ, δ, ǫ, ζ : constant system parameters

Short description: The FitzHugh-Nagumo model ([33],[62]), named after Richard FitzHugh (1922-2007) and Jin-Ichi Nagumo (1926-1999), discribes nerve conduction ([83]), propagation of waves andnerve pulses in excitable media (e.g. heart tissue or nerve fiber) and spike generation in squid giantaxons. This model exhibits traveling pulses, traveling fronts, traveling multipulses ([64]) in 1D, spiralwaves, spiral breakups ([39]), spiral turbulences ([40]), labyrinthine patterns ([40]), spot splittings([39]) and rotating vortices ([17],[15]) in 2D as well as other phenomena in 3D.

Phenomena:


• Traveling 1-pulse (traveling pulse)

• Traveling 2-pulse (traveling multipulse)


d D f(w) α β γ δ ǫ ζ R △x △t Boundary Phenomena

1 110 w − ζw3 0 2

25 1 8 710

13 60 0.1 0.5 ∂u


1 110 w − ζw3 0 2

25 1 45

710

13 60 0.1 0.1 ∂u

∂n= 0 on ∂BR(0) 1-pulse

1 110 w − ζw3 0 2

25 1 45

710

13 100 0.1 0.1 ∂u

∂n= 0 on ∂BR(0) 2-pulse


(

u1

u2

)

t

=

(

u1,xx + f (u1)− u2 + αDu2,xx + β (γu1 − δu2 + ǫ)

)

, x ∈ BR(0), t ∈ [0,∞[

∂u∂n

= 0 , x ∈ ∂BR(0), t ∈ [0,∞[u(0) = u0 , x ∈ BR(0), t = 0


where f(w) = w − ζw3 and d = 1 (i. e. BR(0) = [−R,R]). For the numerical computations weuse parameters D = 1

10 , α = 0, β = 225 , γ = 1, δ = 8, ǫ = 7

10 , ζ = 13 , R = 60 and initial data

u0(x) = (tanh(x), 1− tanh(x))T . Moreover, for the spatial discretization we use FEM (continuouspiecewise linear finite elements) with △x = 0.1. For the temporal discretization we use BDF(2) with△t = 0.5, rtol = 10−2, atol = 10−5 and intermediate timesteps ([59]).Nonfrozen solution:



(

v1v2

)

t

=

(

v1,xx + f (v1)− v2 + α+ λ1v1,xDv2,xx + β (γv1 − δv2 + ǫ) + λ1v2,x

)

, x ∈ BR(0), t ∈ [0,∞[

∂v∂n

= 0 , x ∈ ∂BR(0), t ∈ [0,∞[v(0) = u0 , x ∈ BR(0), t = 0γt = λ1 , t ∈ [0,∞[γ(0) = 0 , t = 0

0 = 〈v̂x, v − v̂〉L2(BR(0),ℝ) , t ∈ [0,∞[

where f(w) = w − ζw3 and d = 1 (i. e. BR(0) = [−R,R]). For the numerical computations we useparameters D = 1

10 , α = 0, β = 225 , γ = 1, δ = 8, ǫ = 7

10 , ζ = 13 , R = 60, initial data u0(x) =


(tanh(x), 1− tanh(x))T and reference function v̂(x) =?. Moreover, for the spatial discretization weuse FEM (continuous piecewise linear finite elements) with △x = 0.1. For the temporal discretizationwe use BDF(2) with △t = 0.1, rtol = 10−2, atol = 10−5 and intermediate timesteps.Frozen solution and velocity:

Spatial-temporal pattern, convergence error of freezing method and eigenvalues of the linearizationabout the frozen solution:

• Traveling 1-pulse (traveling pulse):I. Nonfrozen solution: Consider the nonfrozen system

(

u1

u2

)

t

=

(


)

, x ∈ BR(0), t ∈ [0,∞[

∂u∂n

= 0 , x ∈ ∂BR(0), t ∈ [0,∞[u(0) = u0 , x ∈ BR(0), t = 0


10 , α = 0, β = 225 , γ = 1, δ = 4

5 , ǫ = 710 , ζ = 1

3 , R = 60 and initial data

u0(x) = (tanh(x),−0.6)T . Moreover, for the spatial discretization we use FEM (continuous piecewiselinear finite elements) with △x = 0.1. For the temporal discretization we use BDF(2) with △t = 0.1,rtol = 10−2, atol = 10−5 and intermediate timesteps ([59]).Nonfrozen solution:




(

v1v2

)

t

=

(

v1,xx + f (v1)− v2 + α+ λ1v1,xDv2,xx + β (γv1 − δv2 + ǫ) + λ1v2,x

)

, x ∈ BR(0), t ∈ [0,∞[

∂v∂n

= 0 , x ∈ ∂BR(0), t ∈ [0,∞[v(0) = u0 , x ∈ BR(0), t = 0γt = λ1 , t ∈ [0,∞[γ(0) = 0 , t = 0

0 = 〈v̂x, v − v̂〉L2(BR(0),ℝ) , t ∈ [0,∞[

where f(w) = w − ζw3 and d = 1 (i. e. BR(0) = [−R,R]). For the numerical computationswe use parameters D = 1

10 , α = 0, β = 225 , γ = 1, δ = 4

5 , ǫ = 710 , ζ = 1

3 , R = 60, initial

data u0(x) = (tanh(x), 1− tanh(x))T and reference function v̂(x) =?. Moreover, for the spatialdiscretization we use FEM (continuous piecewise linear finite elements) with △x = 0.1. For thetemporal discretization we use BDF(2) with △t = 0.1, rtol = 10−2, atol = 10−5 and intermediatetimesteps.Frozen solution and velocity:

Spatial-temporal pattern, convergence error of freezing method and eigenvalues of the linearizationabout the frozen solution:

• Traveling 2-pulse (traveling multipulse):I. Nonfrozen solution: Consider the nonfrozen system

(

u1

u2

)

t

=

(


)

, x ∈ BR(0), t ∈ [0,∞[

∂u∂n

= 0 , x ∈ ∂BR(0), t ∈ [0,∞[u(0) = u0 , x ∈ BR(0), t = 0


10 , α = 0, β = 225 , γ = 1, δ = 4

5 , ǫ = 710 , ζ = 1

3 , R = 100 and initial data

u0(x) =

(

−1.2 + 2.5

1+(x3 )2 ,−0.6

)T


piecewise linear finite elements) with △x = 0.1. For the temporal discretization we use BDF(2) with△t = 0.1, rtol = 10−2, atol = 10−5 and intermediate timesteps ([12],[11]).Nonfrozen solution:




(

v1,jv2,j

)

t

=

(

v1,j,xxDv2,j,xx

)

+ ϕ(·)∑

2

k=1ϕ(·−gk+gj)

· F(

∑2k=1 vk(· − gk + gj , ·)

)

+µj

(

v1,j,xv2,j,x

)

, x ∈ BR(0), t ∈ [0,∞[

∂vj∂n

= 0 , x ∈ ∂BR(0), t ∈ [0,∞[vj(0) = uj0 , x ∈ BR(0), t = 0gj,t = µj , t ∈ [0,∞[gj(0) = gj0 , t = 0

0 = 〈(v̂j)x, vj − v̂j〉L2(BR(0),ℝ) , t ∈ [0,∞[

with

F

(

w1

w2

)

=

(

f(w1)− w2 + αβ (γw1 − δw2 + ǫ)

)


where vj = (v1,j , v2,j)T , f(w) = w − ζw3 and d = 1 (i. e. BR(0) = [−R,R]). For the numerical

computations we use parameters D = 110 , α = 0, β = 2

25 , γ = 1, δ = 45 , ǫ = 7

10 , ζ = 13 , R =

50, initial data u0(x) =

(

−1.2 + 2.5

1+(x3 )2 ,−0.6

)T

, initial positions γ10 = −50, γ20 = 50, reference

functions v̂1(x) =?, v̂2(x) =? and bump function ϕ(x) = 2exp( 1

2x)+exp(− 1

2x)

. Moreover, for the spatial

discretization we use FEM (continuous piecewise linear finite elements) with △x = 0.1. For thetemporal discretization we use BDF(2) with △t = 0.1, rtol = 10−2, atol = 10−5 and intermediatetimesteps.Frozen solutions:





Literature: [33], [62], [32], [35], [34], [68], [12], [11], [40], [59], [64], [45], [17], [15]

1.5 Purwins model 33

1.5 Purwins model

Name: Purwins model

Equations:

ut = Du△u+ f(u)− κ1v − κ2w + κ3

τvt = Dv△v + u− vθwt = Dw△v + u− w

u = u(x, t) ∈ ℝ, v = v(x, t) ∈ ℝ, w = w(x, t) ∈ ℝ, x ∈ ℝd, d ∈ {1, 2, 3}, Du, Dv, Dw, α, β, γ, τ, θ ∈ ℝ

and f : ℝ→ ℝ is a (cubic nonlinearity) polynomial of third degree (i.e. f(u) = λu− u3).

Notations:

u : activatorv, w : inhibitorDu, Dv, Dw : diffusion constants with slow diffusion Dv and fast diffusion Dw, i.e. Dv << Dwκ1, κ2, κ3, λ, τ, θ : some additional constants

Short description: not available

Set of parameter values: not available

Phenomena: not available


Literature: not available


1.6 Barkley model

Name: Barkley model

Equations:

(

u1

u2

)

t

=

(

△u1 + 1ε· u1 · (1− u1) ·

(

u1 − u2+ba

)

D△u2 + g(u1)− u2

)

u1 = u1(x, t) ∈ ℝ, u2 = u2(x, t) ∈ ℝ, u = (u1, u2)T , x ∈ ℝd, d ∈ {1, 2, 3}, D, a, b, ε ∈ ℝ, g(w) = w ([7])

(or g(w) = w3, g(w) =

0 , w ∈ [0, 13 [

1− 6.75w(w − 1)2 , w ∈ [13 , 1]

1 , w ∈]1,∞[

([20])).

Notations:

D : diffusion constant for the slow species u2 (0 ⩽ D << 1)g(u1) : reaction kineticsε : sets the timescale separation between the fast u1- and the slow u2-equation (0 < ε << 1)a, b : other system parameters

Short description: The Barkley model ([7],[8]), named after Dwight Barkley, discribes excitablemedia, oscillatory media ([7]), catalytic surface reactions ([20],[21]), the interaction of a fast activatoru and a slow inhibitor v (in this case g(u) discribes a delayed production of the inhibitor) and is oftenused as a qualitative model in pattern forming systems (i.e. Belousov-Zhabotinsky reaction). Thismodel exhibits spiral wave ([78], [12]), rotating spiral and scroll wave solutions. Larger a gives a longerexcitation duration and increasing b

agives a larger excitability threshold (or equivalently decreasing b

a

produce a spiral with many windings). The standard reaction kinetics g(u) = u can also be replacedby one of two above mentioned possibilities. In both of these nonstandard cases the model can exhibitspiral breakups followed by spiral turbulences ([21],[70]).

Phenomena:

• Rotating spiral wave (rigidly rotating spiral)

• Meandering spiral wave (meandering spiral)


d D a b ε g(w) R △x △t Boundary Phenomena

2 110

34

1100

150 w 40 0.7 0.1 ∂u

∂n= 0 on ∂BR(0) Rotating spiral wave

Numerical results:• Rotating spiral wave (rigidly rotating spiral):I. Nonfrozen solution: Consider the nonfrozen system

(

u1

u2

)

t

=

(

△u1 + 1ε· u1 · (1− u1) ·

(

u1 − u2+ba

)

D△u2 + g(u1)− u2

)

, x ∈ BR(0), t ∈ [0,∞[

∂u∂n

= 0 , x ∈ ∂BR(0), t ∈ [0,∞[u(0) = u0 , x ∈ BR(0), t = 0

where g(w) = w and d = 2 (i. e. BR(0) = {x ∈ ℝ2 | ‖x‖

ℝ2 ⩽ R}). For the numerical computations we

use parameters D = 110 , a = 3

4 , b = 1100 , ε = 1

50 , R = 40 and initial data u0(x) = (u1(x, 0), u2(x, 0))T

1.6 Barkley model 35

with u1(x, 0) =

{

0 x ⩽ 0

1 x > 0and u2(x, 0) =

{

0 y ⩽ 0a2 y > 0

. Moreover, for the spatial discretization we use

FEM (continuous piecewise linear finite elements) with △x = 0.7. For the temporal discretization weuse BDF(2) with △t = 0.1, rtol = 10−3, atol = 10−4 and intermediate timesteps ([78],[12]).Nonfrozen solution:

Spatial-temporal pattern: (with x2 = 0, x1 ∈ [−40, 40])

• Rotating spiral wave (rigidly rotating spiral): d = 2, Dv = 110 , a = 3

4 , b = 1100 , ε = 1

50 , g(u) = u,

R = 40, △x = 0.7, △t = 0.1,

(

u

v

)

= 0 on ∂BR(0) (Dirichlet boundary), u0 =

{

0 x ⩽ 0

1 x > 0,

v0 =

{

0 y ⩽ 0a2 y > 0

([78],[12]). I. Nonfrozen solution:

Spatial-temporal pattern: (with y = 0, x ∈ [−40, 40])


• Meandering rotating spiral wave (Drifting rotating spiral wave): d = 2, Dv = 0, a = 35 , b = 1

20 ,

ε = 150 , g(u) = u, R = 20, △x = 0.5, △t = 0.1, ∂

∂n

(

u

v

)

= 0 on ∂BR(0) (Neumann boundary),

u0 =

{

0 x ⩽ 0

1 x > 0, v0 =

{

0 y ⩽ 0a2 y > 0

([8]). I. Nonfrozen solution:

Spatial-temporal pattern: (with y = 0, x ∈ [−20, 20])

• Core breakup (Spiral breakup followed by spiral turbulence): d = 2, Dv = 0, a = 0.75, b = 0.07,

ε = 0.08, R = 50, △x = 1.25, △t = 0.1, g(u) = u3, ∂∂n

(

u

v

)


u0 and v0 spiral patterns (i.e. solution at time t = 15 with parameters Dv = 0.001, a = 0.75, b = 0.01,

ε = 0.02, R = 50, △x = 1.25, △t = 0.1, g(u) = u, ∂∂n

(

u

v

)


u0 =

{

0 x ⩽ 0

1 x > 0, v0 =

{

0 y ⩽ 0a2 y > 0

) ([70]). I. Nonfrozen solution:

1.6 Barkley model 37

• Core breakup (Spiral breakup followed by spiral turbulence): d = 2, Dv = 0, a = 0.84, b = 0.07,

ε = 0.08, R = 50, △x = 1.25, △t = 0.1, g(u) =

0 u ∈ [0, 13 [

1− 6.75u(u− 1)2 u ∈ [13 , 1]

1 u ∈]1,∞[

, ∂∂n

(

u

v

)

= 0 on

∂BR(0) (Neumann boundary), u0 and v0 spiral patterns (i.e. solution at time t = 15 with parameters

Dv = 0.001, a = 0.75, b = 0.01, ε = 0.02, R = 50, △x = 1.25, △t = 0.1, g(u) = u, ∂∂n

(

u

v

)

= 0 on

∂BR(0) (Neumann boundary), u0 =

{

0 x ⩽ 0

1 x > 0, v0 =

{

0 y ⩽ 0a2 y > 0

) ([70]). I. Nonfrozen solution:

• Far field breakup (Spiral breakup): d = 2, Dv = 0.001, a = 0.75, b = 0.01, ε = 0.0752, R = 50,

△x = 1.25, △t = 0.1, g(u) =

0 u ∈ [0, 13 [

1− 6.75u(u− 1)2 u ∈ [13 , 1]

1 u ∈]1,∞[

, ∂∂n

(

u

v

)

= 0 on ∂BR(0) (Neumann

boundary), u0 and v0 spiral patterns (i.e. solution at time t = 15 with parameters Dv = 0.001,

a = 0.75, b = 0.01, ε = 0.02, R = 50, △x = 1.25, △t = 0.1, g(u) = u, ∂∂n

(

u

v

)

= 0 on ∂BR(0)

(Neumann boundary), u0 =

{

0 x ⩽ 0

1 x > 0, v0 =

{

0 y ⩽ 0a2 y > 0

) ([21],[70]). I. Nonfrozen solution:


• Scroll wave: d = 3, Dv = 0, a = 45 , b = 1

100 , ε = 150 , g(u) = u, R = 15, △x = 0.5, △t = 0.1,

∂∂n

(

u

v

)

= 0 on ∂BR(0) (Neumann boundary), u0 =

{

0 x ⩽ 0

1 x > 0, v0 =

{

0 y ⩽ 0a2 y > 0

.


Additional informations: For Dv = 0 and ε = 0.02 fixed the following figure shows the dynamics ofa single spiral wave as a function of parameters a and b as well as various types of meander. Top: Theyellow region denotes periodically rotating spirals and the cyan region denotes meandering spirals.Bottom: Cut through the parameter space at b = 0.05 with the different states illustrated by tippaths ([7],[9]).

For their numerical approximation see [44] and [16].

Literature: [78], [20], [7], [19], [9], [70], [21], [8], [44], [16]

1.7 Schrödinger equation 39

1.7 Schrödinger equation

Name: Schrödinger equation (sometimes called Nonlinear Schrödinger equation (NLS,NLSE))

Equations: ut = α△u+ β |u|p u

u = u(x, t) ∈ ℂ, u1 := Re(u), u2 := Im(u), x ∈ ℝd, d ∈ {1, 2, 3}, t ∈ [0,∞[, α, β ∈ ℂ, p ∈ ℕ0.


Short description: The Schrödinger equation ([71])

Set of parameter values: not available

Phenomena:• Stationary oscillon (standing oscillating pulse, stable rotating pulse, solitary wave): d = 1, α = i,β = i, p = 2, R = 25, △x = 0.1, △t = 0.1, ∂u

∂n= 0 on ∂BR(0) (Neumann boundary), u10 =

C√

2Im(β)

cos(D)cosh(Cx+E) , u20 = C

√

2Im(β)

sin(D)cosh(Cx+E) with C = 1

2 , D = 1 and E = 0. I. Nonfrozen solution:

Explicit solutions:• 1d: For α = i, β = ki (with k ∈ ℝ) and p = 2 we have

u(x, t) = C exp(

i(

Dx+(

kC2 −D2)

t+ E))

u(x, t) = ±C√

2

k

exp(

i(

C2t+D))

cosh (Cx+ E), if k > 0

u(x, t) = ±A√

2

k

exp(

iBx+ i(A2 +B2)t+ iC)

cosh (Ax− 2ABt+D), if k > 0

u(x, t) =C√t

exp

(

i(x+D)2

4t+ i

(

kC2 ln(t) + E)

)

where A,B,C,D,E are arbitrary real constants. For a collection of solutions see [29].

Additional informations: With respect to p ∈ ℕ0 in the literature there are special nomenclature

ut = α△u+ βu : linear Schrödinger equation

ut = α△u+ β |u|u : quadratic nonlinear Schrödinger equation

ut = α△u+ β |u|2 u : cubic nonlinear Schrödinger equation

ut = α△u+ β |u|3 u : quartic nonlinear Schrödinger equation

ut = α△u+ β |u|4 u : quintic nonlinear Schrödinger equation

ut = α△u+ β |u|6 u : septic nonlinear Schrödinger equation

1.8 Gross-Pitaevskii equation 41

1.8 Gross-Pitaevskii equation

Name: Gross-Pitaevskii equation (GPE)

Equations: ut = α△u+ βV (x)u+ γ |u|2 u

u = u(x, t) ∈ ℂ, x ∈ ℝd, d ∈ {1, 2, 3}, t ∈ [0,∞[, α, β, γ ∈ ℂ, V : ℝd → ℝ.


Short description: The Gross-Pitaevskii equation ([37],[67]), named after Eugene P. Gross and LevPetrovich Pitaevskii, describes Bose-Einstein condensates (BEC) at zero or very low temperature andthe ground states of a quantum system of identical bosons. This model exhibits standing solitaryoscillons.

Phenomena:

• Standing solitary oscillon (rotating pulse, stable localized oscillating pulse)


d α β γ V (x) R △x △t Boundary Phenomena

1 i2 −i −i x2

2 10 0.1 0.5 ∂u∂n

= 0 on ∂BR(0) rotating pulse

Numerical results:• Standing solitary oscillon (rotating pulse, stable localized oscillating pulse):I. Nonfrozen solution: Consider the nonfrozen system

ut = α△u+ βV (x)u+ γ |u|2 u , x ∈ BR(0), t ∈ [0,∞[∂u∂n

= 0 , x ∈ ∂BR(0), t ∈ [0,∞[u(0) = u0 , x ∈ BR(0), t = 0

where d = 1 (i. e. BR(0) = [−R,R]). For the numerical computations we use parameters α = i2 ,

β = −i, γ = −i, V (x) = x2

2 , R = 10 and initial data Reu0 = π−1

4 exp(

−x2

2

)

, Im u0 = 0. Moreover, for

the spatial discretization we use FEM (continuous piecewise linear finite elements) with △x = 0.1. Forthe temporal discretization we use BDF(2) with △t = 0.5, rtol = 10−2, atol = 10−4 and intermediatetimesteps.Nonfrozen solution:




vt = α△v + βV (x)v + γ |v|2 v + iλ1v , x ∈ BR(0), t ∈ [0,∞[∂v∂n

= 0 , x ∈ ∂BR(0), t ∈ [0,∞[v(0) = u0 , x ∈ BR(0), t = 0(γ1)t = λ1 , t ∈ [0,∞[γ1(0) = 0 , t = 00 = 〈iv̂, v − v̂〉L2(BR(0),ℂ) , t ∈ [0,∞[

where d = 1 (i. e. BR(0) = [−R,R]). For the numerical computations we use parameters α = i2 ,

β = −i, γ = −i, V (x) = x2

2 , R = 10 and initial data Reu0 = π−1

4 exp(

−x2

2

)

, Im u0 = 0. As reference

functions Rev̂(x) and Imv̂(x) we choose the real and imaginary part of the nonfrozen solution attime t = 1, respectively, with the parameters mentioned above with △x = 0.05 and △t = 0.1.Moreover, for the spatial discretization we use FEM (continuous piecewise linear finite elements) with△x = 0.05. For the temporal discretization we use BDF(2) with △t = 0.1, rtol = 10−2, atol = 10−4



1.8 Gross-Pitaevskii equation 43

Velocities, convergence error of freezing method and eigenvalues of the linearization about the frozensolution:

0 50 100 1500.84

0.85

0.86

0.87

0.88

0.89

0.9

Vel

ocity

t

Velocities

lm1

0 50 100 15010

−35

10−30

10−25

10−20

10−15

Convergence error

t

||vt|| L2

−5 0 5

x 10−14

−800

−600

−400

−200

0

200

400

600

800Eigenvalues

Re

Im


Additional informations: If β = 0 then we obtain the cubic nonlinear Schrödinger equation.



1.9 Complex Ginzburg-Landau equation (CGL)

Name: Complex Ginzburg-Landau equation (CGL, CGLE) (sometimes called Cubic Com-

plex Ginzburg-Landau)

Equations: ut = α△u+ u(

µ+ β |u|2)

+ γu

u = u(x, t) ∈ ℂ, u1 := Re(u), u2 := Im(u), x ∈ ℝd, d ∈ {1, 2, 3}, t ∈ [0,∞[, α, β, γ, µ ∈ ℂ with γ = 0,

u the complex conjugate of u.

Notations: ([41])

u(x, t) : complex-valued amplitude, slowly varying in space x and time tIm(α) : dispersionRe(µ) : distance from the Hopf bifurcationIm(µ) : detuningIm(β) : nonlinear frequency correctionγ : weak periodic forcing term, forcing amplitude

Short description: The complex Ginzburg-Landau equation, named after Vitaly Lazarevich Ginzburg(1916-2009) and Lev Landau (1908-1968), discribes nonlinear waves, second-order phase transitions,Rayleigh-Bénard convection, superconductivity, superfluidity and is used in the study of fiber optics([3]). The equation describes the evolution of amplitudes of unstable modes for any process exhibitinga Hopf bifurcation, for which a continuous spectrum of unstable wavenumbers is taken into account.It can be viewed as a highly general normal form for a large class of bifurcations and nonlinear wavephenomena in spatially extended systems.


d α β γ µ R △x △t Boundary

2 1 + 12 i −1 2.02 1 + 2i 50 1.25 0.1 ∂u

∂n= 0 or u = 0 on ∂BR(0)

2 1 + 12 i −1 1.98 1 + 2i 50 1.25 0.1 u = 0 on ∂BR(0)

Phenomena:• Labyrinthine pattern: d = 2, α = 1 + 1

2 i, β = −1, γ = 2.02, µ = 1 + 2i, R = 50, △x = 1.25,

△t = 0.1, ∂u∂n

= 0 on ∂BR(0) (Neumann boundary), u10 = tanh(x), u20 = 1 − u10 ([41],[38]). I.Nonfrozen solution:

• Labyrinthine pattern: d = 2, α = 1+ 12 i, β = −1, γ = 2.02, µ = 1+2i, R = 50, △x = 1.25, △t = 0.1,

u = 0 on ∂BR(0) (Dirichlet boundary), u10 = tanh(x), u20 = 1−u10 ([41],[38]). I. Nonfrozen solution:

1.9 Complex Ginzburg-Landau equation (CGL) 45

• Scroll wave: d = 3, µ = 1 (or µ = −1), α = 1, β = 1 + i, γ = 0 ([87],[50],[14])• Spiral-vortex nucleation (formation of Bloch-front turbulence): d =?, α = 1 + 3

10 i, β = −1, γ = 15 ,

µ = 12 + 3

20 i, R = 256, △x =?, △t =?, ∂u∂n

= 0 on ∂BR(0) (Neumann boundary), u10 = front,u20 = front ([41]).


Additional informations: (1): (Bloch-)spiralincreasing γ−→ spiral-vortex nucleation

increasing γ−→ labyrinthine,where ([41])

|Im(µ) + Re(µ) · Im(β)|√

1 + (Im(β))2=: γb < γ << γNIB :=

√

(Im(µ))2 +1

9(Re(µ))2 (for spiral)

In case of Re(α) = Re(β) = µ = 0 there exists soliton solutions ([80]). If µ = γ = 0 then we obtainthe cubic nonlinear Schrödinger equation.(2): If γ 6= 0 this equation is called Forced Complex Ginzburg-Landau equation. In case ofIm(α) = Im(β) = 0, γ = 0 and µ ∈ ℝ the equation is called real Ginzburg-Landau equation.

Literature: [42], [83], [6], [41], [38], [87], [50], [14], [3]


1.10 Quintic Complex Ginzburg-Landau equation (QCGL)

Name: Quintic Complex Ginzburg-Landau equation (QCGL, QCGLE) (sometimes calledCubic-Quintic Complex Ginzburg-Landau equation (CQCGL))

Equations: ut = α△u+ u(

µ+ β |u|2 + γ |u|4)

u = u(x, t) ∈ ℂ, x ∈ ℝd, d ∈ {1, 2, 3}, t ∈ [0,∞[, α, β, γ, µ ∈ ℂ.

Notations: ([80])

x : normalized transversal spatial coordinate (or retarded time, in temporal problemswith d = 1)

t : normalized propagation distance or the normalized number of round tipsu : complex anvelope of the electric fieldRe(α) : spatial or temporal spectral filtering, diffusion coefficient, Re(α) > 0Im(α) : determines the lowest order diffraction, i.e. Im(α) > 0 corresponds to anomalous

dispersion and Im(α) < 0 corresponds to normal dispersionµ : linear gain (or loss) at the (spatial or temporal) control frequency, µ ∈ ℝ

Re(β) : nonlinear gain (absorption processes)Re(γ) : a higher-order correction term to the nonlinear amplification (absorption)Im(γ) : a higher-order correction term to the nonlinear refractive index

Short description: The quintic complex Ginzburg-Landau equation ([36]), named after VitalyLazarevich Ginzburg (1916-2009) and Lev Landau (1908-1968), describes different aspects of signalpropagation in heart tissue, superconductivity, superfluidity, nonlinear optical systems ([60]), photon-ics, plasmas, physics of lasers, Bose-Einstein condensation, liquid crystals, fluid dynamics, chemicalwaves, quantum field theory, granular media and is used in the study of hydrodynamic instabilities([58]). This model shows a variety of coherent structures like stable and unstable pulses, fronts, sourcesand sinks in 1D ([47],[79],[2],[80]), vortex solitons ([24]), spinning solitons ([25]), rotating spiral waves,propagating clusters ([69]) and exploding dissipative solitons ([75]) in 2D as well as scroll waves andspinning solitons ([26]) in 3D.

Phenomena:

• Standing solitary oscillon (rotating pulse, stable localized oscillating pulse)

• Traveling oscillating front (stable rotating front)

• Pulsating soliton

• Creeping soliton

• Rotating 2-pulse (rotating multipulse)

• Rotating pulse combined with a traveling rotating front (traveling rotating multistructure)

• Traveling rotating 2-front (traveling rotating multifront)

• Spinning soliton (spinning solitary wave, localized vortex solution)


• Standing solitary oscillon (rotating pulse, stable localized oscillating pulse, merger of 3 collidingsolitons into a single stable pulse)

1.10 Quintic Complex Ginzburg-Landau equation (QCGL) 47

• Spinning 2-soliton (spinning multisoliton, localized 2-vortex solution):


d µ α β γ R △x △t Boundary Phenomena

1 − 110 1 3 + i −11

4 + i 20 0.1 0.1 ∂u∂n

= 0 on ∂BR(0) rotating pulse

1 − 110 1 3 + i −11

4 + i 20 0.1 0.1 ∂u∂n

= 0 on ∂BR(0) rotating front

1 − 110

810 + 1

2 i66100 + i − 1

10 − 110 i 20 0.1 0.1 ∂u

∂n= 0 on ∂BR(0) pulsating soliton

1 − 110

1011000 + 1

2 i1310 + i − 3

10 − 1011000 i 20 0.1 0.3 ∂u

∂n= 0 on ∂BR(0) creeping soliton

2 −12

12 + 1

2 i52 + i −1− 1

10 i 20 0.5 0.1 ∂u∂n

= 0 on ∂BR(0) spinning soliton

2 −12

12 + 1

2 i135 + i −1− 1

10 i 20 0.5 0.1 ∂u∂n

= 0 on ∂BR(0) rotating spiral wave

Numerical results:• Standing solitary oscillon (rotating pulse, stable localized oscillating pulse):I. Nonfrozen solution: Consider the nonfrozen system

ut = α△u+ u(

µ+ β |u|2 + γ |u|4)

, x ∈ BR(0), t ∈ [0,∞[∂u∂n

= 0 , x ∈ ∂BR(0), t ∈ [0,∞[u(0) = u0 , x ∈ BR(0), t = 0

where d = 1 (i. e. BR(0) = [−R,R]). For the numerical computations we use parameters µ = − 110 ,

α = 1, β = 3 + i, γ = −114 + i, R = 20 and initial data Reu0 = 5

2

(

1+(x5 )2) , Im u0 = 0. Moreover, for

the spatial discretization we use FEM (continuous piecewise linear finite elements) with △x = 0.1. Forthe temporal discretization we use BDF(2) with △t = 0.1, rtol = 10−3, atol = 10−4 and intermediatetimesteps ([79],[47],[78],[76]).Nonfrozen solution:




vt = α△v + v(

µ+ β |v|2 + γ |v|4)

+ λ1vx + iλ2v , x ∈ BR(0), t ∈ [0,∞[∂v∂n

= 0 , x ∈ ∂BR(0), t ∈ [0,∞[v(0) = u0 , x ∈ BR(0), t = 0(γ1)t = λ1 , t ∈ [0,∞[γ1(0) = 0 , t = 0(γ2)t = λ2 , t ∈ [0,∞[γ2(0) = 0 , t = 00 = 〈v̂x, v − v̂〉L2(BR(0),ℂ) , t ∈ [0,∞[

0 = 〈iv̂, v − v̂〉L2(BR(0),ℂ) , t ∈ [0,∞[



2

(

1+(x5 )2) , Im u0 = 0. As reference

functions Re v̂(x) and Im v̂(x) we choose the real and imaginary part of the nonfrozen solution attime t = 70, respectively, with the parameters mentioned above and atol = 10−5. Moreover, for thespatial discretization we use FEM (continuous piecewise linear finite elements) with △x = 0.1. Forthe temporal discretization we use BDF(2) with △t = 0.1, rtol = 10−3, atol = 10−5 and intermediatetimesteps ([79],[47],[78]).Frozen solutions:




0 50 100 150−25

−20

−15

−10

−5

0

5

Vel

ocity

t

Velocities

lm1lm2

0 50 100 15010

−8

10−6

10−4

10−2

100

Convergence error

t

||vt|| L2

−250 −200 −150 −100 −50 0

−1

−0.5

0

0.5

1

Eigenvalues

Re

Im

• Traveling oscillating front (stable rotating front):I. Nonfrozen solution: Consider the nonfrozen system

ut = α△u+ u(

µ+ β |u|2 + γ |u|4)

, x ∈ BR(0), t ∈ [0,∞[∂u∂n

= 0 , x ∈ ∂BR(0), t ∈ [0,∞[u(0) = u0 , x ∈ BR(0), t = 0


α = 1, β = 3+i, γ = −114 +i, R = 20 and initial data Reu0 =

2

11·(3+√

9+11µ)

1+exp

(

− x√2

) , Im u0 = 0. Moreover, for

the spatial discretization we use FEM (continuous piecewise linear finite elements) with △x = 0.1. Forthe temporal discretization we use BDF(2) with △t = 0.1, rtol = 10−3, atol = 10−4 and intermediatetimesteps ([79],[47],[78],[76]).Nonfrozen solution:




vt = α△v + v(

µ+ β |v|2 + γ |v|4)

+ λ1vx + iλ2v , x ∈ BR(0), t ∈ [0,∞[∂v∂n


0 = 〈iv̂, v − v̂〉L2(BR(0),ℂ) , t ∈ [0,∞[


α = 1, β = 3 + i, γ = −114 + i, R = 20 and initial data Reu0 = 2.5

1+(x5 )2 , Im u0 = 0. As reference

functions Re v̂(x) and Im v̂(x) we choose the real and imaginary part of the nonfrozen solution attime t = 4, respectively, with the parameters mentioned above and atol = 10−5. Moreover, for thespatial discretization we use FEM (continuous piecewise linear finite elements) with △x = 0.1. Forthe temporal discretization we use BDF(2) with △t = 0.1, rtol = 10−3, atol = 10−5 and intermediatetimesteps ([79],[47],[78]).Frozen solutions:




0 20 40 60 80 100−3

−2

−1

0

1

2

Vel

ocity

t

Velocities

lm1lm2

0 20 40 60 80 10010

−15

10−10

10−5

100

Convergence error

t

||vt|| L2

−250 −200 −150 −100 −50 0−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

Eigenvalues

Re

Im

• Pulsating soliton:I. Nonfrozen solution: Consider the nonfrozen system

ut = α△u+ u(

µ+ β |u|2 + γ |u|4)

, x ∈ BR(0), t ∈ [0,∞[∂u∂n

= 0 , x ∈ ∂BR(0), t ∈ [0,∞[u(0) = u0 , x ∈ BR(0), t = 0


α = 810 + 1

2 i, β = 66100 + i, γ = − 1

10 − 110 i, R = 20 and initial data Reu0 = sech(x), Im u0 = 0.


and intermediate timesteps ([5]).Nonfrozen solution:


II. Frozen system (freezing method still not developed): Consider the associated frozen system


vt = α△v + v(

µ+ β |v|2 + γ |v|4)

+ λ1vx + iλ2v , x ∈ BR(0), t ∈ [0,∞[∂v∂n


0 = 〈iv̂, v − v̂〉L2(BR(0),ℂ) , t ∈ [0,∞[


α = 8100 + 1

2 i, β = 66100 + i, γ = − 1


As reference functions Re v̂(x) and Im v̂(x) we choose the real and imaginary part of the nonfrozensolution at time t = 10, respectively, with the parameters mentioned above. Moreover, for the spatialdiscretization we use FEM (continuous piecewise linear finite elements) with △x = 0.1. For thetemporal discretization we use BDF(2) with △t = 0.1, rtol = 10−2, atol = 10−5 and intermediatetimesteps.Frozen solutions:




0 20 40 60 80 100−2.5

−2

−1.5

−1

−0.5

0

0.5

Vel

ocity

t

Velocities

lm1lm2

0 20 40 60 80 10010

−10

10−8

10−6

10−4

10−2

Convergence error

t

||vt|| L2

−20 −15 −10 −5 0−150

−100

−50

0

50

100

150Eigenvalues

Re

Im

• Creeping soliton:I. Nonfrozen solution: Consider the nonfrozen system

ut = α△u+ u(

µ+ β |u|2 + γ |u|4)

, x ∈ BR(0), t ∈ [0,∞[∂u∂n

= 0 , x ∈ ∂BR(0), t ∈ [0,∞[u(0) = u0 , x ∈ BR(0), t = 0


α = 1011000 + 1

2 i, β = 1310 + i, γ = − 3



and intermediate timesteps ([5]).Nonfrozen solution:


II. Frozen system (freezing method still not developed).• Rotating 2-pulse (rotating multipulse):I. Nonfrozen solution: Consider the nonfrozen system


ut = α△u+ u(

µ+ β |u|2 + γ |u|4)

, x ∈ BR(0), t ∈ [0,∞[∂u∂n

= 0 , x ∈ ∂BR(0), t ∈ [0,∞[u(0) = u0 , x ∈ BR(0), t = 0


α = 1, β = 3+i, γ = −114 +i, R = 50 and initial data Reu0 = 5

2

(

1+(x+20

2 )2)+ 5

2

(

1+(x−20

2 )2) , Im u0 = 0.






α = 1, β = 3+i, γ = −114 +i, R = 50 and initial data Re v10 = 5

2

(

1+(x2 )2) , Im v10 = 0, Re v20 = Re v10,

Im v20 = 0. As reference functions and bump function we use Re v̂1(x) = Re v10, Im v̂1(x) = Im v10,Re v̂2(x) = Re v20, Im v̂2(x) = Im v20 and ϕ(x) = sech(0.5 · x), respectively. Moreover, for the spatialdiscretization we use FEM (continuous piecewise linear finite elements) with △x = 0.1. For thetemporal discretization we use BDF(2) with △t = 0.2, rtol = 10−2, atol = 10−6 and intermediatetimesteps.Frozen solutions:


0 50 100 150 200 250−7

−6

−5

−4

−3

−2

−1

0

1

Vel

ocity

t

Velocities

tau1theta1tau2theta2

0 50 100 150 200 25010

−15

10−10

10−5

100

105

t

Convergence error

||µt||

L2

||vt||

L2

−10 −8 −6 −4 −2 0

−1

−0.5

0

0.5

1

Eigenvalues

Re

Im

• Rotating pulse combined with a traveling rotating front (traveling rotating multistructure):I. Nonfrozen solution: Consider the nonfrozen system

ut = α△u+ u(

µ+ β |u|2 + γ |u|4)

, x ∈ BR(0), t ∈ [0,∞[∂u∂n

= 0 , x ∈ ∂BR(0), t ∈ [0,∞[u(0) = u0 , x ∈ BR(0), t = 0


α = 1, β = 3+i, γ = −114 +i, R = 50 and initial data Reu0 = 5

2

(

1+(x+20

2 )2)+

2

11 (3+√

9+11µ)

1+exp

(

−x−20√

2

) , Im u0 = 0.







α = 1, β = 3 + i, γ = −114 + i, R = 50 and initial data Re v10 = 5

2

(

1+(x2 )2) , Im v10 = 0, Re v20 =

2

11 (3+√

9+11µ)

1+exp

(

− x√2

) , Im v20 = 0. As reference functions and bump function we use Re v̂1(x) = Re v10,

Im v̂1(x) = Im v10, Re v̂2(x) = Re v20, Im v̂2(x) = Im v20 and ϕ(x) = sech(0.5 · x), respectively.Moreover, for the spatial discretization we use FEM (continuous piecewise linear finite elements) with△x = 0.1. For the temporal discretization we use BDF(2) with △t = 0.1, rtol = 10−2, atol = 10−3





0 50 100 150 200 250−6

−5

−4

−3

−2

−1

0

1

2

Vel

ocity

t

Velocities


0 50 100 150 200 25010

−15

10−10

10−5

100

105

t

Convergence error

||µt||

L2

||vt||

L2

−10 −8 −6 −4 −2 0

−2

−1

0

1

2

Eigenvalues

Re

Im

• Traveling rotating 2-front (traveling rotating multifront):I. Nonfrozen solution: Consider the nonfrozen system

ut = α△u+ u(

µ+ β |u|2 + γ |u|4)

, x ∈ BR(0), t ∈ [0,∞[∂u∂n

= 0 , x ∈ ∂BR(0), t ∈ [0,∞[u(0) = u0 , x ∈ BR(0), t = 0



11 (3 +√

9 + 11µ)−2

11 (3+√

9+11µ)

1+exp

(

−x+20√

2

) +

2

11 (3+√

9+11µ)

1+exp

(

−x−20√

2

) , Im u0 = 0. Moreover, for the spatial discretization we use FEM (continuous piecewise






α = 1, β = 3 + i, γ = −114 + i, R = 50 and initial data Re v10 = 2

11 (3 +√

9 + 11µ) −2

11 (3+√

9+11µ)

1+exp

(

− x√2

) ,

Im v10 = 0, Re v20 =2

11 (3+√

9+11µ)

1+exp

(

− x√2

) , Im v20 = 0. As reference functions and bump function we use

Re v̂1(x) = Re v10, Im v̂1(x) = Im v10, Re v̂2(x) = Re v20, Im v̂2(x) = Im v20 and ϕ(x) = sech(0.5 · x),respectively. Moreover, for the spatial discretization we use FEM (continuous piecewise linear finiteelements) with △x = 0.1. For the temporal discretization we use BDF(2) with △t = 0.2, rtol = 10−2,atol = 10−4 and intermediate timesteps.Frozen solutions:




0 50 100 150 200 250−3

−2

−1

0

1

2

Vel

ocity

t

Velocities


0 50 100 150 200 25010

−6

10−4

10−2

100

102

t

Convergence error

||µt||

L2

||vt||

L2

−12 −10 −8 −6 −4 −2

−4

−3

−2

−1

0

1

2

3

4

Eigenvalues

Re

Im

• Spinning soliton (spinning solitary wave, localized vortex solution):I. Nonfrozen solution: Consider the nonfrozen system

ut = α△u+ u(

µ+ β |u|2 + γ |u|4)

, x ∈ BR(0), t ∈ [0,∞[∂u∂n

= 0 , x ∈ ∂BR(0), t ∈ [0,∞[u(0) = u0 , x ∈ BR(0), t = 0

where d = 2 (i. e. BR(0) = {x ∈ ℝ2 | ‖x‖

ℝ2 ⩽ R}). For the numerical computations we use parameters

µ = −12 , α = 1

2 + 12 i, β = 5

2 + i, γ = −1 − 110 i, R = 20 and initial data Reu0 = x1

5 exp(

−x21+x2

2

49

)

,

Im u0 = x2

5 exp(

−x21+x2

2

49

)

(or using polar coordinates u0(r, φ) = r5 exp (iφ) exp

(

− r272

)

). Moreover, for

the spatial discretization we use FEM (continuous piecewise linear finite elements) with △x = 0.5. Forthe temporal discretization we use BDF(2) with △t = 0.1, rtol = 10−4, atol = 10−5 and intermediatetimesteps ([25],[24],[78],[14],[12]).Nonfrozen solution:

Spatial-temporal pattern: (with x1 ∈ [−20, 20] and x2 = 0)



vt = α△v + v(

µ+ β |v|2 + γ |v|4)

+ λ1vx1+ λ2vx2

+ iλ3v , x ∈ BR(0), t ∈ [0,∞[∂v∂n

= 0 , x ∈ ∂BR(0), t ∈ [0,∞[v(0) = u0 , x ∈ BR(0), t = 0(γ1)t = λ1 , t ∈ [0,∞[γ1(0) = 0 , t = 0(γ2)t = λ2 , t ∈ [0,∞[γ2(0) = 0 , t = 0(γ3)t = λ3 , t ∈ [0,∞[γ3(0) = 0 , t = 00 = 〈v̂x1

, v − v̂〉L2(BR(0),ℂ) , t ∈ [0,∞[

0 = 〈v̂x2, v − v̂〉L2(BR(0),ℂ) , t ∈ [0,∞[

0 = 〈iv̂, v − v̂〉L2(BR(0),ℂ) , t ∈ [0,∞[

where d = 2 (i. e. BR(0) = {x ∈ ℝ2 | ‖x‖


µ = −12 , α = 1

2 + 12 i, β = 5


5 exp(

−x21+x2

2

49

)

,

Im u0 = x2

5 exp(

−x21+x2

2

49

)


(

− r272

)

). As reference

functions Re v̂(x) and Im v̂(x) we choose the real and imaginary part of the nonfrozen solution at timet = 150, respectively, with the parameters mentioned above replacing rtol = 10−2 and atol = 10−4.Moreover, for the spatial discretization we use FEM (continuous piecewise linear finite elements) with△x = 0.5. For the temporal discretization we use BDF(2) with △t = 0.05, rtol = 10−2, atol = 10−7





0 50 100 150−1.4

−1.2

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

Vel

ocity

t

Velocities

lm1lm2lm3

0 50 100 15010

−10

10−5

100

105

Convergence error

t

||vt|| L2

−6 −5 −4 −3 −2 −1 0 1−8

−6

−4

−2

0

2

4

6

8Eigenvalues

Re

ImReal parts of the eigenfunctions:


• Rotating spiral wave (rigidly rotating spiral):I. Nonfrozen solution: Consider the nonfrozen system

ut = α△u+ u(

µ+ β |u|2 + γ |u|4)

, x ∈ BR(0), t ∈ [0,∞[∂u∂n

= 0 , x ∈ ∂BR(0), t ∈ [0,∞[u(0) = u0 , x ∈ BR(0), t = 0

where d = 2 (i. e. BR(0) = {x ∈ ℝ2 | ‖x‖


µ = −12 , α = 1

2 + 12 i, β = 13


5 exp(

−x21+x2

2

49

)

,

Im u0 = x2

5 exp(

−x21+x2

2

49

)


(

− r272

)

). Moreover, for

the spatial discretization we use FEM (continuous piecewise linear finite elements) with △x = 0.5. Forthe temporal discretization we use BDF(2) with △t = 0.1, rtol = 10−4, atol = 10−5 and intermediatetimesteps ([78]).Nonfrozen solution:




vt = α△v + v(

µ+ β |v|2 + γ |v|4)

+ λ1vx1+ λ2vx2

+ iλ3v , x ∈ BR(0), t ∈ [0,∞[∂v∂n


, v − v̂〉L2(BR(0),ℂ) , t ∈ [0,∞[

0 = 〈v̂x2, v − v̂〉L2(BR(0),ℂ) , t ∈ [0,∞[

0 = 〈iv̂, v − v̂〉L2(BR(0),ℂ) , t ∈ [0,∞[

where d = 2 (i. e. BR(0) = {x ∈ ℝ2 | ‖x‖


µ = −12 , α = 1

2 + 12 i, β = 13


5 exp(

−x21+x2

2

49

)

,

Im u0 = x2

5 exp(

−x21+x2

2

49

)


(

− r272

)

). As reference

functions Re v̂(x) and Im v̂(x) we choose the real and imaginary part of the nonfrozen solution at timet = 150, respectively, with the parameters mentioned with rtol = 10−2 and atol = 10−4. Moreover, forthe spatial discretization we use FEM (continuous piecewise linear finite elements) with △x = 0.5. Forthe temporal discretization we use BDF(2) with △t = 0.1, rtol = 10−2, atol = 10−5 and intermediatetimesteps.Frozen solutions:




0 50 100 150−1.4

−1.2

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

Vel

ocity

t

Velocities

lm1lm2lm3

0 50 100 15010

−4

10−3

10−2

10−1

100

Convergence error

t

||vt|| L2

−8 −6 −4 −2 0

−10

−5

0

5

10

Eigenvalues

Re

Im

Real parts of the eigenfunctions:

• Spinning 2-soliton (spinning multisoliton, localized 2-vortex solution):I. Nonfrozen solution: Consider the nonfrozen system

ut = α△u+ u(

µ+ β |u|2 + γ |u|4)

, x ∈ BR(0), t ∈ [0,∞[∂u∂n

= 0 , x ∈ ∂BR(0), t ∈ [0,∞[u(0) = u0 , x ∈ BR(0), t = 0

where d = 2 (i. e. BR(0) = {x ∈ ℝ2 | ‖x‖


µ = −12 , α = 1

2 + 12 i, β = 5


5 exp(

−x21+x2

2

49

)

,


Im u0 = x2

5 exp(

−x21+x2

2

49

)


(

− r272

)

). Moreover, for

the spatial discretization we use FEM (continuous piecewise linear finite elements) with △x = 0.5. Forthe temporal discretization we use BDF(2) with △t = 0.1, rtol = 10−4, atol = 10−5 and intermediatetimesteps.Nonfrozen solution:


Explicit solutions: ([80],[56],[57])

Additional informations: In case of Im(α) = Im(β) = Im(γ) = Im(µ) = 0 (i.e. α, β, γ, µ ∈ ℝ) theequation is called quintic real Ginzburg-Landau equation. If γ = 0 then we obtain the cubic

Ginzburg-Landau equation. In the following cases there exists soliton solutions ([80])

1. Re(α) < 2 · Re(β), µ ∈ ℝ with µ < 0, Re(γ) < 0

2. Re(α) > 2 · Re(β), µ ∈ ℝ with µ > 0, Re(γ) < 0

3. Re(α) = Re(β) = Re(γ) = µ = 0

In case of Re(β) > 0 there exists soliton-like solutions ([80]). Multi-armed spirals are everywhereunstable, moreover the only stable rotating spirals are the one-armed. The QCGL equation describesa subcritical Hopf bifurcation at Re(µ) = 0 with three branches:

|u| = 0 and |u|± =

√

−Re(β)±√

(Re(β))2 − 4Re(γ)Re(µ)

2Re(γ)

The first branch is stable if Re(µ) < 0 and unstable if Re(µ) > 0. The second exists only for(Re(β))2

4Re(γ) < Re(µ) < 0 and is always unstable. The third exists only for (Re(β))2

4Re(γ) < Re(µ) and is always

stable ([85]).

Literature: [77], [54], [82], [74], [84], [58], [47], [69], [2], [24], [75], [25], [26], [79], [14], [87], [56], [57],[40], [81], [80], [5], [76]

1.11 λ-ω system 67

1.11 λ-ω system

Name: λ-ω system

Equations: ut = D△u+ (λ(|u|) + iω(|u|)) · u

u = u(x, t) ∈ ℂ, x ∈ ℝd, d ∈ {1, 2, 3}, t ∈ [0,∞[, D ∈ ℂ, λ : ℝ → ℝ with λ = λ(|u|) (i.e.

λ(|u|) = 1− |u|2), ω : ℝ→ ℝ with ω = ω(|u|) (i.e. ω(|u|) = −α|u|2 with α ∈ ℝ).

Notations:

Reu(x, t) : concentration at position x and time t of the first reactantIm u(x, t) : concentration at position x and time t of the second reactantD : diffusion coefficientRe f(u), Im f(u) : f(u) = (λ(|u|) + iω(|u|)) · u, reaction kineticsα : reaction parameter (positive constant)

Short description: The λ-ω system ([53],[61]) describes chemical reaction processes ([53],[52]), phys-iological processes in the study of cardiac arrhythmias ([66]), time evolution of biological systems ([61])and is often used to analyse the mechanism of pattern formation ([46]) as well as to study the onesetof turbulent behavior ([55]). An example of an emerging technological application based on patternforming systems is given by memory devices using magnetic domain patterns ([27]). This modelexhibits rotating spirals as well as scroll wave and scroll ring solutions ([10],[30]).

Phenomena:



d D λ(|u|) ω(|u|) R △x △t Boundary Phenomena

2 1 1− |u|2 −|u|2 50 1.0 0.1 ∂u∂n

= 0 on ∂BR(0) rotating spiral wave

Numerical results:• Rotating spiral wave (rigidly rotating spiral):I. Nonfrozen solution: Consider the nonfrozen system

ut = D△u+ (λ(|u|) + iω(|u|)) · u , x ∈ BR(0), t ∈ [0,∞[∂u∂n

= 0 , x ∈ ∂BR(0), t ∈ [0,∞[u(0) = u0 , x ∈ BR(0), t = 0

where d = 2 (i. e. BR(0) = {x ∈ ℝ2 | ‖x‖

ℝ2 ⩽ R}). For the numerical computations we useparameters D = 1, λ(|u|) = 1 − |u|2, ω(|u|) = −|u|2, R = 50 and initial data u10 = x

20 , u20 = y20 (or

using polar coordinates u0(r, φ) = rR

(cos(ϕ) + i sin(ϕ))). Moreover, for the spatial discretization weuse FEM (continuous piecewise linear finite elements) with △x = 1.0. For the temporal discretizationwe use BDF(2) with △t = 0.1, rtol = 10−3, atol = 10−4 and intermediate timesteps ([13],[53],[12]).Nonfrozen solution:




vt = D△v + (λ(|v|) + iω(|v|)) · v + λ1vx1+ λ2vx2

+ iλ3v , x ∈ BR(0), t ∈ [0,∞[∂v∂n


, v − v̂〉L2(BR(0),ℂ) , t ∈ [0,∞[

0 = 〈v̂x2, v − v̂〉L2(BR(0),ℂ) , t ∈ [0,∞[

0 = 〈iv̂, v − v̂〉L2(BR(0),ℂ) , t ∈ [0,∞[

where d = 2 (i. e. BR(0) = {x ∈ ℝ2 | ‖x‖

ℝ2 ⩽ R}). For the numerical computations we useparameters D = 1, λ(|v|) = 1 − |v|2, ω(|v|) = −|v|2, R = 50, and initial data u10 = x

20 , u20 = y20 (or

using polar coordinates u0(r, φ) = rR

(cos(ϕ) + i sin(ϕ))). As reference functions Re v̂(x) and Im v̂(x)we choose the real and imaginary part of the nonfrozen solution at time t = 150, respectively, withthe parameters mentioned above and △x = 2.0. Moreover, for the spatial discretization we use FEM(continuous piecewise linear finite elements) with △x = 2.0. For the temporal discretization we useBDF(2) with △t = 1.0, rtol = 10−3, atol = 10−4 and intermediate timesteps.Frozen solutions:

1.11 λ-ω system 69



0 100 200 300 400−0.2

0

0.2

0.4

0.6

0.8

1

1.2

Vel

ocity

t

Velocities

lm1lm2lm3

0 100 200 300 40010

−10

10−8

10−6

10−4

10−2

100

Convergence error

t

||vt|| L2

−5 −4 −3 −2 −1 0 1−5

0

5Eigenvalues

Re

Im

Real parts of the eigenfunctions:



Literature: [12], [61], [18], [52], [46], [55], [27], [66], [53], [10], [13], [30]

1.12 Autocatalysis model 71

1.12 Autocatalysis model

Name: Autocatalysis model

Equations:ut = Du△u− uvm

vt = Dv△v + uvm

u = u(x, t) ∈ ℝ, v = v(x, t) ∈ ℝ, x ∈ ℝd, d ∈ {1, 2, 3}, t ∈ [0,∞[, Du, Dv ∈ ℝ, m ∈ ℕ.

Notations:

u(x, t) : concentration of the reactant at position x and time tv(x, t) : concentration of the autocatalyst at position x and time tDu, Dv : diffusion coefficients (positive constant)m : other system parameter (positve integer)

Short description: not available

Set of parameters:

d Du Dv m R △x △t Boundary

1 0.1 1 9 180 0.1 0.1 ∂u∂n

= 0 on ∂[−180, 180]

2 2 1 2 200 5.0 0.1 ∂u∂n

= 0 on ∂BR(0)

Phenomena:• Traveling wave front (traveling wave, traveling front): d = 1, Du = 0.1, Dv = 1, m = 9, R = 180,

△x = 0.1, △t = 0.1, ∂u∂n

= 0 on ∂[−180, 180] (Neumann boundary), u0 =

{

1 x > 0

0 x ⩽ 0, v0 = 1− u0.

• Traveling wave front (traveling wave, traveling front): d = 2, Du = 2, Dv = 1, m = 2, R = 200,△x = 5.0, △t = 0.1, ∂u

∂n= 0 on ∂BR(0) (Neumann boundary), u0 = 1

1+exp

(

−y−50−5 cos(πx20 )

4

) , v0 =

1− u0.



Literature: not available

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Index

A

Allen, Samuel M. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

B

Barkley, Dwight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Belousov, Boris Pavlovich (1893-1970) . . . . . . . .

C

Cahn, John W. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Chafee, Nathaniel L. . . . . . . . . . . . . . . . . . . . . . . . . .

F

Fisher, Ronald Aylmer (1890-1962) . . . . . . . . . . .FitzHugh, Richard Dick (1922-2007) . . . . . . . . . .Frank-Kamenetskii, David Albertovich (1910-

1970) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

G

Ginzburg, Vitaly Lazarevich (1916-2009) . . . . .

H

Hodgkin, Alan Lloyd (1914-1998) . . . . . . . . . . . . .Huxley, Andrew Fielding . . . . . . . . . . . . . . . . . . . . .

I

Infante, Ettore F. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

K

Kolmogorov, Andrei Nikolajewitsch (1903-1987)

L

Landau, Lev Davidovich (1908-1968) . . . . . . . . .

N

Nagumo, Jin-Ichi (1926-1999). . . . . . . . . . . . . . . . .Newell, Alan C. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

P

Petrovsky, Ivan Georgievich (1901-1973) . . . . . .Piskounov, Nicolai . . . . . . . . . . . . . . . . . . . . . . . . . . . .

W

Whitehead, John A. . . . . . . . . . . . . . . . . . . . . . . . . . .

Z

Zel’dovich, Yakov Borisovich (1914-1987) . . . . .Zhabotinsky, Anytol Markovich . . . . . . . . . . . . . . .

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