Applications of Nonclassical  Symmetry Reductions of Nonlinear Reaction-Diffusion Equations

Philip Broadbridge

Acknowledgements to Australian Research Council,James Hill, Daniel Arrigo,

Peter Tritscher, Joanna Goard, Bronwyn Hajek, Peter Clarkson, Elizabeth Mansfield, Clara Nucci

and George Bluman.

Tuesday, 13 May 14

L. Ovsiannikov 1966: Partial InvarianceG. Bluman & J. Cole 1969: Nonclassical SymmetriesP. Clarkson & M. Kruskal 1989: Direct Method for ReductionA. Fokas & Q. M. Liu 1994: Generalized Conditional SymmetriesP. Olver 1994: Direct Reduction and Differential Constraints

Personal list of influential publications on nonclassical reductions

Tuesday, 13 May 14

Nonclassical Symmetry AnalysisG. W. Bluman and J. D. Cole (1969), ‘The general similarity solution of the heat equation’, J. Math. Mech. 18, 1025-42. What is the Lie point symmetry group

that leaves invariant a system consisting of governing n’th order scalar PDE A[xj, u, u(1),...,u(n)]=0 plus invariant surface condition

?

d

d✏

[u� f(xj)] = 0 () X

ju,j = U.

xk = xk + ✏Xk(xj , u) +O(✏2),

u = u+ ✏U(xj , u) +O(✏2)

Tuesday, 13 May 14

Reaction-diffusion equations.

✓t = r · [D(✓)r✓] +R(✓)

have applications in combustion theory, population genetics, population dynamics, tissue growth, irreversible thermodynamics,microwave heating, construction of Lyapunov functionals, etc.

u =

ZD(✓)d✓

F (u)ut =r2u+Q(u) ; F (u) = 1/D(✓), Q(u) = R(✓).

Simplify by Kirchhoff transformation (1891)

Heat flux is �ru.

Tuesday, 13 May 14

In 1+1-D, with ut = uxx+Q(u),full nonclassical symmetry classification has been completed byMansfield and Clarkson Physica D (1993) (rec. 26 Mar, acc 12 Jul 1993)Arrigo, Hill & Broadbridge IMA J App Math (1994) (rec. Feb 3, acc 29 July 1993)

Various types of cubic polynomial Q(u) (but not all cubics)are the only sources that admit genuine nonclassical symmetries.

Xuu

= 0,

Uuu

� 2Xxu

+ 2XXu

= 0,

Xt

+ 2XXx

�Xxx

+ 2Uxu

� 2UXu

+ 3Xu

Q(u) = 0,

Ut

+ 2UXx

� Uxx

+ (Utt

� 2Xx

)Q(u)� UQ0(u) = 0.

Nonclassical determining relations:

Tuesday, 13 May 14

Fisher (1937): advantageous new allele ‘a’ in diploid Mendelian species, total population approx. const, propagates as

pt

= Dpxx

+ p(1� p).

as is well-known but incorrect!: These assumptions lead to cubic source term.

Skellam 1973, using simple DE continuum models.Broadbridge et al 2002, taking limits of system with discrete breeding cycles.

Tuesday, 13 May 14

Diploid population, two possible alleles A1,A2 at one locus, with respective genotype population densities ⇢11, ⇢12, ⇢22.

⇢11 + ⇢12 + ⇢22 = ⇢.Total population density is

p =2⇢11 + ⇢12

2⇢

g1 = �12 � �22, g2 = 2�12 � �11 � �22.

@p

@t

=@

2p

@x

2+

2

⇢

@⇢

@x

@p

@x

+ p(1� p)(g1 � g2p).

@⇢11

@t

=@

2⇢11

@x

2� µ⇢11 + �11p

2⇢

@⇢12

@t

=@

2⇢12

@x

2� µ⇢12 + 2�12p(1� p)⇢

@⇢22

@t

=@

2⇢22

@x

2� µ⇢22 + �22(1� p)2⇢

Tuesday, 13 May 14

If A1 is advantageous but fully recessive,�11 > �12 = �22,

source term in p-equation is (�11 � �22)p2(1� p)

as in Huxley equation.If fitnesses occur in arithmetic progression, get quadratic source term0.5(�11 � �22)p(1� p)

as in Fisher equation.

pt

= Dpxx

+ p(1� p)(p� r1)

Otherwise we have Fitzhugh-Nagumo equation

Huxley eq. has nonclassical infinitesimal symmetry generator

� =

r

2(3p� 1)@

x

+3

2p2(1� p)@

p

.

Tuesday, 13 May 14

Tuesday, 13 May 14

Diploid population, three possible alleles A1,A2, A3 at one locus, with respective genotype population densities

⇢11, ⇢12, ⇢13, ⇢22, ⇢23, ⇢33.

@p1

@t

=@

2p1

@x

2+

2

⇢

@⇢

@x

@p1

@x

+ �(p1, p2).

@p2

@t

=@

2p2

@x

2+

2

⇢

@⇢

@x

@p2

@x

+ (p1, p2).

Cubic source terms in p1, p2 . Bradshaw-Hajek 2004.

Tuesday, 13 May 14

u = �(x, y)eAt ; r2�+K

2� = 0.

In 2+1-D, PDE has simple nonclassical symmetry

u = eA✏u, t = t+ ✏ whenever

Q=Au F(u) + K2 u. (1)

This gives a reduction to linear Helmholtz equation

This is true also in 3+1-D.Sub (2) in governing PDE, get (1), equivalent to ODE

(2)

(3)D(✓) = u0(✓) =Au

R(✓)�K2u

Nonlinear reaction-diffusion in 2+1-D

Nonclassical symmetry analysis- Goard & Broadbridge 1996.

Tuesday, 13 May 14

0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

kT/E

R

Arrhenius reaction rate

0.5

R = e�E/kT

Tuesday, 13 May 14

u = exp

A

Z ✓

✓0

e

1/sds

!.

Then

⇠ Ae

A✓, ✓ ! 1

⇠ exp[A Ei(1/✓0)� ✓0e1/✓0 ] ✓

�2 [A✓2e1/✓]

exp[A✓

2e

1/✓]! 0, ✓ ! 0.

In particular we can get exact solutions with Arrhenius reaction term that is related to Gibbs probability for a single activation energy jump

R = e�1/✓ =1

Ze��E/kT

.

With K=0, the ODE (3) directly integrates to

D(✓) = u0(✓) = Ae1/✓ exp

⇣A[✓e1/✓ � Ei(1/✓)]

⌘

Tuesday, 13 May 14

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

200

400

600

800

1000

1200

1400

D

✓

Tuesday, 13 May 14

With K=0, with an arbitrary solution of One instructive solution would be the radial solution

u = �(x, y)eAt

� r2� = 0.

However, the rate of heat supply from singular point source is

This solution has an unrealistic point heat source apart from combustion and it has a rapidly increasing heat diffusivity that manages to bound temperature increase to an exponential.

(in 2D) � 2⇡rur = 2⇡↵eAt, (in 3D) � 4⇡r2ur = 4⇡↵eAt.

(in 2D) u = �↵ log r + �, (in 3D) u = �↵r�1+ �.

Tuesday, 13 May 14

Now consider u = �(x, y)eAt ; r2�+K

2� = 0. (2)

With K2>0, the Helmholtz equation has positive radial solutions J0(Kr) between successive zeros of the Bessel function. Solution (2) then satisfies meaningful boundary conditions

ur = 0, r = 0,

u = 0, r = r1; J0(Kr1) = 0.

In terms of inverse temperature ODE (3) to determine D=u’ is

� = 1/✓,

��2u0(�) =Au

e�� �K2u.

� ! 1, K2u >> e�� , =) D ⇡ �A

K2

� ! 0 (✓ ! 1), K2u >> e�� , =) D ⇡ �A

K2.

Tuesday, 13 May 14

Now require A (=-K2D(0) ) <0.

��2u0(�) =Au

e�� �K2u.

With these boundary conditions of ideal heat extraction at circular boundary,

In 3D, J0(Kr) is replaced by spherical Bessel j0(Kr).

u = J0(Kr)e�|A|t ! 0, t ! 1.

Tuesday, 13 May 14

0 1 2 3 4 5 60

0.5

1

1.5

temp kT/E

diffu

sivi

ty

1=-A/K2

Tuesday, 13 May 14

0 0.5 1 1.5 2 2.5−0.2

0

0.2

0.4

0.6

0.8

1

1.2

Kr

u ex

p(−A

t)

Tuesday, 13 May 14

Application of Bluman’s notion of potential symmetries.

Fujita 1952-54 presented exact similarity solutions for u(x,t) with

D(u) =1

au + b, D(u) =

1(au + b)2

, D(u) =1

au + b + cu2

Bluman, Kumei and Reid 1988: these come from “potential” symmetries, or from point symmetries of the system

yx = u ; yt = D(u)ux

ut

= [D(u)ux

]x

Every nonlinear diffusion eq. has Boltzmann scaling symmetry

x = e�x ; t = e2�t ; u = u.

Invariant solution u=f(z) (z=xt -1/2) satisfies ODE

2d

dz

�D(u)

du

dz

⇥+ z

du

dz = 0

Tuesday, 13 May 14

which has a Lie point symmetry, allowing reduction of order, iff D(u) is power-law or exponential. Bluman & Reid 1988: the sufficient system

dw/dz = u ; 2D

�dw

dz

⇥d2w

dz2+ z

dw

dz� w = 0

has an additional symmetry iff

D(u) =1

au2 + bu + cexp

��

⇤1

au2 + bu + cdu

⇥

Reduced 1st order ODE has yet another symmetry, allowing separable form and solution by quadrature, when i.e. Fujita cases.

� = 0,

Tuesday, 13 May 14

Mullins 1957 theory of thermal grooving. e.g. for Mg, evaporation-condensation is the dominant mechanism. This leads to evolution by mean curvature.

�y

�t= B

yxx

1 + y2x

yx = u ; yt = D(u)ux

“ curve-shortening equation”

ut

=

B

1 + u2ux

�

x

Tuesday, 13 May 14

�y�N

cos(�)⇥y

⇥t=

1(1 + y2

x)1/2B

yxx

1 + y2x

�y

�t= B

yxx

1 + y2x

Tuesday, 13 May 14

HRTEM image. Z. Zhang et al, Science 302, 846-49 (2003)

Tuesday, 13 May 14

−3 −2 −1 0 1 2 3−1

−0.9

−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

�b

�s �s

�

m = tan(⇥) ; �b = 2�s sin(⇥)

yx(0, t) = m; y(x, t)� 0, x�⇥; y(x, 0) = 0

Tuesday, 13 May 14

Broadbridge 1989: Exact CSE parametric similarity solution for grain boundary groove � ⇥� (�, ⇥) = (x/2

⇤t, y/2

⇤t)

⇥ = ��1/2⇤sin[F (⇤; �)] + (1� ⇤2 � �ln⇤)1/2cos[F ] ; 0 ⇥ � ⇥ �⇥

⇥ = ��1/2⇤sin[G(⇤; �)] + (1� ⇤2 � �ln⇤)1/2cos[G] ; �⇥ ⇥ � ⇥ 1

⌅ = m[⇥��H(⇤; �)]

F (⇥; �) =� �

0(1� q2 � �lnq)�1/2dq

G(⇥; �) = tan�1m� F (⇥; �) + F (⇥m; �)

H(⇥; �) = ��1/2m�1⇥ sec[F (⇥; �)] ; 0 � � � �⇥

H(⇥; �) = ��1/2m�1⇥ sec[G(⇥; �)] ; �⇥ � � � 1

�⇥ = m�1tan[F (1; �)] ; � =⇥2

m � 1�m2

⇥2mln ⇥m

2F (1; �)� F (⇥m; �) = tan�1m

� = m�1tan[F (⇥; �)] ; 0 � � � �⇥

� = m�1tan[G(⇥; �)] ; �⇥ � � � 1

Tuesday, 13 May 14

slope m

opening angle = π-2 arctan m

Ishimura 1995 “opening angle” solutions.

Closely related to the grain boundary solutions- prescribed slope at x/Bt1/2 = a (const). a=0 for grain boundary.Mg hinges are self-welding !

Tuesday, 13 May 14

Arrigo et al 1997: for classical diffusion, groove depth F(0) proportional to m but for CSE with m sufficiently large,

�ln(

m

2⌅

�) +

14

⇥1/2

� 12⇥ |y(0, t)|⇤

(Bt)⇥

�2ln(

m⌅�

)⇥1/2

Use upper and lower bounds for D(yx), using linearisable model of the form D =

ai

(bi

+ yx

)2

Tuesday, 13 May 14

Tritscher & Broadbridge, 1995

yt = �B �x

⇤�1 + y2

x

⇥�1/2�x

yxx

(1 + y2x)3/2

⌅

Analytic solution has finite groove depth in limit of infinite groove slope.

Tuesday, 13 May 14

Classical Lie point symmetry assumption leads to system of linear determining PDE for coefficients Xj, U of infinitesimal symmetry generator Nonclassical symmetry assumption leads to system of nonlinear determining relations.Some target PDEs have classical symmetries but no additional nonclassical symmetries.Some target PDEs have additional nonclassical symmetries but all invariant solutions are invariant under some classical symmetry.

� = Xj@j + U@u.

**Some PDEs have nonclassically invariant solutions that are not invariant under any classical symmetry.

Tuesday, 13 May 14

Theorem (Arrigo & Broadbridge 1998): Every solution of any linear PDE that has an independent-variable-deforming classical Lie point symmetry, is invariant under some classical symmetry.Choose a classical symmetry

� = X

i(xj, u)

@

@x

i+ U(xj

, u)@

@u

; U = f(xj)u+ g(xj).

Let be any solution of linear Lu=0.u = (xj)

Then u is invariant under �+ g1(xj)

@

@u

,

where g1 = X

k @

@x

k� f(xj) � g.

Tuesday, 13 May 14

e.g. A solution of the heat equation ut=uxx that is well known not tobe invariant under any any classical symmetry (in the finite part of the Lie algebra) is

Choose

g1 = x

@

@x

+ 2t@

@t

= 3x3 + 18xt.

u = x

3 + 6xt+ c = (x, t)

�1 = x@

x

+ 2t@t

.

�2 = �1 + g1@u.

This is a classical symmetry with invariant surface condition

xu

x

+ 2tut

= g1.

The ISC is clearly satisfied by the solution (1) above.

(1)

xu

x

+ 2tut

= 3(u� c)

u� c = x

3 + 6xt = t

3/2(x3t

�3/2 + 6xt�1/2)

Tuesday, 13 May 14

Initial data (and presumably boundary data) do not have to be invariant in order to solve IVPs systematically by symmetry reductions.J. M. Goard, “Finding symmetries by incorporating initial conditions as side conditions.”, E.J.A.M. (2008).

e.g. ut

= �uxxxx

� uxx

+4

5u2x

+u2xx

5= 0

with “side” (initial) cond. u(x,0)=cos(2x),

has conditional symmetry

@

t

+ 6 cot(2x)@

x

+ (�24u+

16

5

)@

u

and invariant solution

2

15

[1� e

�24t] + e

�12tcos(2x)

even though init cond is not invariant.

Tuesday, 13 May 14

ut

=

au

x

(b+ u)2

�

x

v = b+ u; v

t

=v

xx

v

2

v = Y

x

; Y

t

=Y

xx

Y

2x

,

x

t

= ax

Y Y

.

Same applies to xt = a@

nY/@x

n,

e.g. Bluman and Kumei 1980 gave symmetry recursion operator for integrable hierarchy of equation for u(x,t),

Tuesday, 13 May 14

⇤t = �B1⌅2x[

�

⇥ + ⇤⌅x{(

�

⇥ + ⇤)3⇤x}]

yt = �B �x

⇤�1 + y2

x

⇥�1/2�x

yxx

(1 + y2x)3/2

⌅

gives (with parameters piecewise-constant) gives integrable approximation for

yx

= ✓

Tuesday, 13 May 14

f(u)=1/(1+u2)1/2

f(u)=ai/(bi+u); mi≤u≤mi+1

Best fit by Chebychev norm

Tuesday, 13 May 14