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.
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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⇢
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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
.
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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.
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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/✓)]
⌘
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0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
200
400
600
800
1000
1200
1400
D
✓
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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+ �.
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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.
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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.
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0 1 2 3 4 5 60
0.5
1
1.5
temp kT/E
diffu
sivi
ty
1=-A/K2
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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)
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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,
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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
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�y�N
cos(�)⇥y
⇥t=
1(1 + y2
x)1/2B
yxx
1 + y2x
�y
�t= B
yxx
1 + y2x
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HRTEM image. Z. Zhang et al, Science 302, 846-49 (2003)
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−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
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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
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