Stability analysis of travelling wave solutions using the ...jitse/durham04.pdf · solutions using...

Stability analysis of travelling wavesolutions using the Evans function

Jitse Niesenin collaboration with Simon Malham

Heriot-Watt University, Edinburgh

e-Science Institute, Durham, February 2004

Stability analysis of travelling wave solutions using the Evans function – p. 1/25

Overview

the Gray–Scott equation. . . is a reaction-diffusion equation.

travelling wave solutions. . . are solutions which travel as a whole. We want to study theirstability by computing the spectrum of the correspondingdifferential operator.

the Evans function. . . is a tool for finding eigenvalues of a differential operator.Evaluating the Evans functions entails solving an ordinarydifferential equation.

the Magnus method. . . is used to solve ordinary differential equations.


Acknowledgments

This research is based on the following paper:

Nairo D. Aparicio (Oxford Brookes),Simon J. A. Malham (Heriot-Watt),Marcel Oliver (Internat. Univ. Bremen)

Numerical evaluation of the Evans function byMagnus integration.

The project is being funded by the EPSRC.


The Gray–Scott equation

The Gray–Scott equation describes a reaction of twochemicals U and V with concentrations � ��

and � ��

.

��

��

The three terms on the right-hand side describe:

diffusion of U and V;�

= (inverse) Lewis number.

autocatalytic reaction U � �

V � �

V.

in-flow of U with rate

�

and out-flow of V with rate

�

;(

��

describes additional reaction V � inert product).


History

��

Originally proposed by Gray and Scott in 1983 withoutthe

� �

terms to model autocatalysis in a well-stirredreactor.

Simulations by Pearson in 1993 show a surprisingvariety of patterns.

At the same time, Lee, McCormick, Ouyang andSwinning obtained similar patterns experimentally. Theystudied a ferrocyanide-iodate-sulphide reaction in acontinuously-fed gel reactor.


Simulations in 2D

� ��

,

� �� ,

� ��

� ��

,� ��

,

� ��

All simulations have

� � � and are due to Roy Williams.


History — II

Boissonade writes in Nature about:spots that replicate, grow and die in uncannyresemblance to living things.

The Gray–Scott equation isalso used to model combus-tion (U fuel, V heat) andfungal mycelia.(Davidson, Sleeman,

Crawford, 1997)

The case

� � �

was considered by Aparicio,Malham and Oliver.


Simulations in 1D

No-flow BCs,

� ��

,

� ��

,

� � � .Computed with moving-grid code of Blom and Zegeling.


Simulations in 1D — II

No-flow BCs,

� ��

,

� ��

,



Simulations in 1D — III

Dirichlet BCs,

� ��

,

� ��

,



Travelling wave solutions

Travelling waves are solutions to the equation that movewith constant speed � while maintaining their shape.

The Gray–Scott equation in

� � � dimensions is

��

We write this in short form as

��

where

� � ��

is the diffusion matrix and � � � � � � � �

.

In the moving frame� � � �, this is �� .

Travelling wave solutions are of the form � � � � � � � � � � �

.Hence, they satisfy

� � � � � � � � � � � � � �

.


Stability of travelling waves

Given a travelling wave

� �, what is the fate of solutionswhose initial conditions are small perturbations of

� �? If anysuch solution stay close to

� �, we say that the travelling waveis stable.

Technical detail: If

� � � � �

is a travelling wave solution, then sois its translate

� � � � ��

. So, we require for stability thatsmall perturbations of

� � � � �

stay close to the family

� � � � � ��

��

.

A natural approach is to linearize the equation about thetravelling wave.


Linear stability analysis

Travelling wave

� � � � �

solves

� � � � � � � � � � � � � �

.

Suppose that � � � � � � � � � � � � � � � � � � �

is a perturbation of thetravelling wave at

� �

.

� � � � � �

solves the full PDE �� , so

� ��

� � � � � � � � � ��

� � � � � � ��

Separation of variables yields� � � � � � � ��

with �

satisfying

� � � � � � ��

� � � � � � � �.

So, we need to study the spectrum of the differentialoperator � � � � � � ��

�

� � � � � . If has an eigenvalue

�

with

� � �

, then the wave is unstable.


The spectrum of

The spectrum of can be decomposed in two parts.

The point spectrum �

pt consists of eigenvalues�

, for which

� � � has a solution. We have

� �

pt.

The essential spectrum �

ess typically contains open sets,and is asymptotically (as

� � �) inside the parabolic region

� ��

��

.

�

ess

Im

�

Re

�

�

pt


Nonlinear stability vs linear stability

If the spectrum of is contained in the left half-plane andbounded away from the imaginary axis, except for a simpleeigenvalue at the origin, then the travelling wave is(nonlinearly) stable. (Henri ’81)

If the essential spectrum touches theimaginary axis at the origin, then one typicallyhas to introduce weighted norms to provestability.

For the Gray–Scott equation, the essential spectrum iseasily computed. Hence, we now assume that �

ess is in theleft half-plane, and we concentrate on the point spectrum.


Eigenvalues of

Recall:

� �

pt if there is a � with � � � and � ��

.

This is a second order ODE, so we can rewrite it as a firstorder equation �� ( �)

Here,

�

is a

��

matrix and linear in�

. If

�

is to the right ofthe �

ess, then

� �� has two stable and two unstableeigenvalues.

Denote the two stable eigenvalues of

� � � � � by � �� and � ��

with corresponding eigenvectors

�� and

�� .Then, every solution of ( �) with

� � � � � �

is of the form

� � � � � � � � �� as

� � � � with � � � � � �

.


Eigenvalues of — II

Every solution of

��

with

� � � � � �satisfies

� � � � � � � � � �� as

� � � �.Let

� �� denote the solution with

� �� .

Then

� � ��

is a basis for the linear space of solutionswith

� � � � � �

.

Similarly, we find a basis

� � ��

for the space ofsolutions with

� � � � �

.

If there is a solution which vanishes at both ends, then it isin both spaces. So,

�is an eigenvalue if ��

� � � ��

and

��

have a nontrivial intersection.


The Evans function

The Evans function is defined by

� � � � � � � � � ��

� � � � ��

Properties of the Evans function:

� � � � �

iff

�

is an eigenvalue of . (Evans, 1972)

If the eigenvectors

��

are chosen to be analytic, thenthe Evans function is analytic to the right of �

ess.(Alexander, Gardner, Jones, 1990)

Hence, we can count the eigenvalues inthe right half-plane by looking at thechange of ��

as�

traverses thepurple loop.

Im

�

Re

�


Computing the Evans function

To evaluate the Evans function at

�

, we need to

compute the eigenvectors

�� of

� � � � ,solve

�� with

� � � � �� (where

�

is

some large number) to obtain

� �� , and

compute

�� and

� ��

, and calculate the determinant.

Some points to consider while implementing this procedure:

The eigenvectors

�� should be analytic functions of

�

.

The solutions of the ODE are growing exponentially.

� �� grows faster than

� �� if

� � �� , so round-offerrors in the

� �� direction will dominate the

� �� solution.


Exterior algebra spaces

The idea is to track the two-dimensional subspace

��

under the flow of

��

.(Bridges, Derks, 1999)

Let � � � � � � � �� be an orthonormal basis of� �

. Define the� � � � �

to be the linear space with basis

� � � � � � � � �� with

� � � � � � � � ��

where the wedge product is bilinear and anti-symmetric.

We can identify the subspace ��

with theelement � � ��

.

This allows us to extend

��

to

��

with

� � � � � � � �.


Computing the Evans function – II

Recall: the old procedure was to compute the eivecs �� ,

solve

�� with

� � � � �� to obtain

� ��

,do the same from � �

, and calculate the determinant.

Now, we take as initial value

� � � � � ��

and solve the extended equation

�� toobtain

� � � � �

. Compute

� � � � �

in a similar way, and theEvans function is

� � � � � � � � � � � � � � �

.

Even better is to solve

�� , where

� � �

is the most unstable eigenvalue of

� � � � � � � �

, to getrid of the exponential grow.

The only remaining task is to solve the ODE.


The Magnus expansion

The solution of the scalar equation � � � � � � � with � � � � �

is

� � � � ��

The solution of the vector equation � � � � with � � � � ��

is

� � � � �� However, the solution of � � � � � � � with � � � � ��

is not

� � � � ��

� ��

So, what is the solution? Write � � � � ��

, then

� � � �

� � � � � ��

where

� are the Bernouilli numbers and � � � is defined by

� � � � � � � � � � � � � � �

. (Hausdorff, 1906)


The Magnus expansion — II

We can solve

� � � �

� � � � � ��

by Picard iteration

��

� �

� � � � � ��

� ��

which yields the Magnus expansion: (Magnus, 1954)

� � � � ��

� ��

� ��

�

� �

� � ��

� ��

��

� �

� �

� � ��

��

� ��

��

� �

� �

� � � ��

��

This expansion is also known as the Feynman-Dyson pathordered exponential or the Chen-Fliess series. It convergesfor sufficiently small

�. (Blanes et al. & Moan, 1998)


The Magnus expansion — III

With some tricks, we find

� � � � � � � � � � � � � � � ��

where

� � � � � � � ��

� � � � � � � � � � � � � ��

� � � � �

This forms the basis of a fourth-order numerical method forsolving

��

.

Advantages of Magnus methods above classical methods:

Superior performance in highly oscillatory regimes,e.g., if

�

approaches the essential spectrum.

If, as in our case,� � � � � �

is linear in

�

, we can split thecomputation in a

�-dependent and a

�

-dependent part.(Moan, 1998)


Putting everything together

We are analysing a reaction-diffusion system, like theGray–Scott model. Suppose that we have found atravelling wave solution, and that we wish to study its(linear) stability.

We need to find out whether part of the spectrum of thecorresponding differential operator is in the right half-plane.Any eigenvalues can be found by evaluating the Evansfunction along the imaginary axis.

This means we have to solve an ODE, which we lift to theexterior algebra space for stability reasons. Finally, wesolve the lifted equation with the Magnus method. Thebulk of the computation needs to be done only once.


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