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Nonclassical symmetries for a family of Cahn–Hilliard equations
6 December 1999
Physics Letters A 263 Ž1999. 331–337 www.elsevier.nlrlocaterphysleta
Nonclassical symmetries for a family of Cahn–Hilliard equations M.L. Gandarias ) , M.S. Bruzon ´ Departamento Matematicas, UniÕersidad de Cadiz, ´ ´ P.O. Box 40, 11510 Puerto Real, Cadiz, ´ Spain Received 16 August 1999; accepted 20 October 1999 Communicated by A.R. Bishop
Abstract In this Letter we make a full analysis of the symmetry reductions of the family of Cahn–Hilliard equations by using the classical Lie method of infinitesimals, the functional forms of the diffusion coefficients for which the Cahn–Hilliard equations can be fully reduced to ordinary differential equations by classical Lie symmetries are derived. We prove that by using the nonclassical method, we obtain several solutions which are not invariant under any Lie group admitted by the equation and consequently which are not obtainable through the Lie classical method. For this Cahn–Hilliard equation, we obtain nonclassical symmetries that reduce the original equation to ordinary differential equations with the Painleve´ property. We remark that these symmetries have not been derived elsewhere by the singular manifold method. q 1999 Published by Elsevier Science B.V. All rights reserved.
1. Introduction The Cahn–Hilliard equation was introduced to study phase separation in binary alloys glasses and polymers w4x and is a good approach to spinodal decomposition. Based in numerical version of the Fourier transformation approach to the nonlinear Cahn–Hilliard diffusion equation, computer simulations of the spinodal decomposition for a model alloy were carried out by Liu and Haasen w14x. The Cahn–Hilliard diffusion equation is also an equation that serves as a model for many problems in physical chemistry, developmental biology w2x, as well as population movement w8x. The existence of a weak solution for the Cahn–Hilliard equation with degenerate mobility was proved in w9x.
)
Corresponding author.
The Cahn–Hilliard flux equation describing diffusion for decomposition of a one-dimensional binary solution can be written as u t q Ž ku x x x y f Ž u . u x . x s 0,
Ž 1.
which is appropriate to cases where the motion is isotropic. Here u is the solute concentration at point x, t is the time, f X Ž u. is the interdiffusion coefficient of solute, which is concentration dependent and kr2 is the gradient energy coefficient describing the contribution of the diffuse interface to the decomposition. In w8x bifurcations of the equilibrium to nonuniform states have been discussed for f Ž u. s D 0 q D 2 u 2 and in w15x several nonlinear results were derived. Although the direct method of Clarkson and Kruskal is found to more to be more powerful than the classical method w5x., similarity reductions for the Cahn–Hilliard equation with f Ž u. s u and f Ž u. s u 2 ,
0375-9601r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 Ž 9 9 . 0 0 7 5 0 - 1
332
M.L. Gandarias, M.S. Bruzonr ´ Physics Letters A 263 (1999) 331–337
have been obtained in w17,18x, by using classical Lie symmetries as well as the direct method. In w17,18x, the direct method did not yield any reduction that could not be obtained by Lie classical symmetries. In this Letter we solve a complete group classification problem for Eq. Ž1., by studying those diffusion coefficients f Ž u. which admit the classical symmetry group. Both the symmetry group and the diffusion coefficients will be found through consistent application of the Lie-group formalism. Motivated by the fact that symmetry reductions for many partial differential equations ŽPDE’s. are known that are not obtained by using the classical Lie group method, there have been several generalizations of the classical Lie group method for symmetry reductions. The notion of nonclassical symmetries was firstly introduced by Bluman and Cole w3x to study the symmetry reductions of the heat equation. Since then, a great number of papers have been devoted to the study of nonclassical symmetries of nonlinear partial differential equations in both one and several dimensions Žsee, i.e. w6,12,13x.. Clarkson and Mansfield w7x presented an algorithm for calculating the determining equations associated with the nonclassical method. Recently Zhdanov and Lahno w20x have applied the nonclassical Žor conditional. method to the onedimensional porous medium equation u t y Ž uu x . x s 0.
Ž 2.
According to these authors the nonclassical method for Ž2. as well as for the parabolic type PDE’s is inefficient. Once obtained new nonclassical symmetries performing the symmetry reductions gives rise to invariant solutions corresponding to the Lie symmetries of Ž2.. The aim of this Letter is to prove that the nonclassical method applied to the Cahn–Hilliard equation gives rise to new solutions of Ž1. which are not group-invariant and consequently cannot be obtained by Lie classical symmetries. Some of these solutions Žthe characteristic solutions. are solutions of Ž2. which are not invariant under any Lie group admitted by Ž2. and consequently cannot been obtained by Lie classical symmetries this result is a counterexample of the statement done in w20x because according to them all the solutions of Ž2. derived by the nonclassical method are group-invariant.
Recently, the family of Cahn–Hilliard equations has arisen a great interest because of an apparent contradiction between the scope of the singular manifold method ŽSMM. and the nonclassical symmetry reductions w19,11x. In w10x Estevez and Gordoa developed a method ´ for identifying the nonclassical symmetries of PDE’s using the SMM based on the Painleve´ property ŽPP. as a tool. They propose the following conjecture: ‘‘The singular manifold method allows us to identify the nonclassical symmetries that reduce the original equation to an ODE with the Painleve´ property’’. The combination of this statement with the Ablowitz, Ramani and Segur conjecture w1x means that for equations with the PP, the SMM should identify all the nonclassical symmetries. Nevertheless, for equations with the conditional PP, the SMM is only able to identify the symmetries for which the associated reduced ODE’s are of the Painleve´ type. In w11x the authors claim that, for Ž1. with f Ž u. s u an s d f Ž u. s u 2 , besides a trivial symmetry, the SMM allows them to determine two different symmetries and that these symmetries are the only ones in which the associated similarity reduction leads to an ODE of Painleve´ type. Nevertheless, for Ž1. with f Ž u. s u and f Ž u. s u 2 , besides the symmetries derived in w11x, we have derived three new nonclassical symmetries for which the corresponding associated similarity reductions leads to three different ODE’s of Painleve´ type. Consequently, for Ž1. with f Ž u. s u and f Ž u. s u 2 , the nonclassical method is more general than the SMM and that this latter method does not allow us to identify all the nonclassical symmetries that reduce Eq. Ž1., with f Ž u. s u and f Ž u. s u 2 , to ODE’s with the PP.
2. Lie classical classification. To apply the classical method to Ž1. we consider the one-parameter Lie group of infinitesimal transformations in Ž x,t,u. given by x ) s x q ej Ž x ,t ,u . q O Ž e 2 . , t ) s t q et Ž x ,t ,u . q O Ž e 2 . , u ) s u q ef Ž x ,t ,u . q O Ž e 2 . ,
Ž 3.
M.L. Gandarias, M.S. Bruzonr ´ Physics Letters A 263 (1999) 331–337
where e is the group parameter. Then one requires that this transformation leaves invariant the set of solutions of Ž1.. This yields to an overdetermined, linear system of equations for the infinitesimals j Ž x,t,u., t Ž x,t,u. and f Ž x,t,u.. The associated Lie algebra of infinitesimal symmetries is the set of vector fields of the form
E V s j Ž x ,t ,u .
E
Ex
q t Ž x ,t ,u .
Et
q f Ž x ,t ,u .
Eu
.
Having determined the infinitesimals, the symmetry variables are found by solving the invariant surface condition
Eu Ex
qt
y f s 0.
Et
js
Ž 5.
x dt 4 dt
q j 1Ž t . ,
where t , j 1 , f 1 and f 2 are related by the following conditions: 1 dt yf 1 u y f2 y f du du 2 dt d2 f df 1 df dt yŽ f1 u q f 2 . y f1 y du 2 d u d ,t d u2 2 x d t d f Ef 2 d j1 y y2 y 2 4 dt du Ex dt 4 2 d f1 E f2 E f2 Ef 2 uqk yf q 4 2 dt Et Ex Ex df
df
E
E ,
where hŽ z ., after integrating once with respect to z, satisfies khXXX y f Ž h . hX y l h s k 1 .
V2 s
Et
.
Ž 6.
Eq. Ž6. is invariant under translations, this allow us to reduce the order by one. The only functional forms of f Ž u., with f Ž u. / const. for which Eq. Ž1. have extra symmetries are f Ž u. s Ž au q b . n and f Ž u. s de au, and these symmetries are, respectively defined, by the following infinitesimal generators: V31 s x
Ex
E q4t
E V32 s x
Ex
2 y
Et
an
E q4t
E
Ž au q b .
2 E
Et
y
a Eu
Eu
,
.
Without loss of generality we can consider b s 0 and a s d s 1. For the sake of completeness, we provide next, the generators of the nontrivial one-dimensional optimal system which are: Ø For f Ž u. s u n , the set
Ø For f Ž u. s e u , the set
- V1 ) ,- lV1 q V2 ) ,- V32 ) 4 . s 0, s 0, s 0, s 0.
The solutions of this system depends of f Ž u.. For f Ž u. arbitrary, the only symmetries admitted by Ž1. are the group of space and time translations, which are defined by the infinitesimal generators
Ex
u s hŽ z . ,
- V1 ) ,- lV1 q V2 ) ,- V31 ) 4 .
f s f 1 Ž t . u q f 2 Ž x ,t . ,
V1 s
z s x y lt ,
E
Eu
We consider the classical Lie group symmetry analysis of Eq. Ž1.. Invariance of Eq. Ž1. under a Lie group of point transformations with infinitesimal generator Ž4. leads to a set of forty determining equations for the infinitesimals j Ž x,t,u., t Ž x,t,u. and f Ž x,t,u.. Solving this system we obtain
tst Ž t. ,
In this case, we obtain travelling wave reductions
E
Ž 4.
F'j
333
In both sets, l g R is arbitrary. Since Eq. Ž1., with f Ž u. s u n and f Ž u. s e u has additional symmetries and the reductions that correspond to V1 and V2 have already been derived, we must only determine the similarity variables and similarity solutions corresponding to V31 and V32 which are: Ø For V31 : z s xty1 r4 ,
u s ty1 r2 n h Ž z . ,
where hŽ z . satisfies the ODE khXXXX y h n hXX y
z 4
2
hX y nh ny1 Ž hX . y
1 2n
h s 0.
Ž 7.
M.L. Gandarias, M.S. Bruzonr ´ Physics Letters A 263 (1999) 331–337
334
Eq. Ž7. does not admit Lie symmetries. Nevertheless, for n s 2 this equation can be easily integrated once respect to z yielding khXXX y h 2 hX y
z 4
h s k1 .
2
q 3k fu u Ž f x . q 6 k f 2fu u u f x q 12 k ffu u x f x q 10k ffu fu u f x q 6 k fu x x f x q 4 k fu fu x f x
Ø For V32 : z s xty1 r4 ,
u s yln Ž t 1r2 h Ž z . . ,
y 3 f Xff x q k f 4fu u u u q 4 k f 3fu u u x
where hŽ z . satisfies the ODE 4 kh3 hXXXX y 4 h2 hXX y 2 h Ž hX . y 4 kh2 4 hX hXXX q 3 Ž hXX . X 2 XX
Ø In the case t s 0, without loss of generality, we may set j s 1 and the determining equation for the infinitesimal f is k f x x x x q 4 k ffu u f x x q 4 k fu x f x x y ff x x
2
q 6 k f 3fu fu uu q 6 k f 2fu u x x
2
q 12 k f 2fu fu u x q 4 k f 3 Ž fu u . 2
q 12 k f 2fu x fu u q 7k f 2 Ž fu . fu u
y zh3 hX X 4
2
y ff 2fu u q 4 k ffu x x x q 6 k ffu fu x x 4
q 48 kh Ž h . h y 24 k Ž h . q 2 h s 0.
Ž 8.
Eq. Ž8. does not admit Lie symmetries. Due to the fact that Ž1. is only invariant under time and space translations and under a scaling group we have considered nonclassical symmetries.
3. Nonclassical symmetries The basic idea of the method is that the PDE Ž1. is augmented with the invariance surface condition Ž5. which is associated with the vector field Ž4.. By requiring that both Ž1. and Ž5. are invariant under the transformation with infinitesimal generator Ž4. one obtains an overdetermined, nonlinear system of equations for the infinitesimals j Ž x,t,u., t Ž x,t,u. and f Ž x,t,u.. The number of determining equations arising in the nonclassical method is smaller than for the classical method, consequently the set of solutions is in general, larger than for the classical method as in this method one requires only the subset of solutions of Ž1. and Ž5. to be invariant under the infinitesimal generator Ž4.. To obtain nonclassical symmetries of Ž1. we apply the algorithm described in w7x for calculating the determining equations. We can distinguish two different cases: Ø In the case t / 0, without loss of generality, we may set t Ž x,t,u. s 1. The nonclassical method applied to Ž1. gives only rise to the classical symmetries.
2
2
q 8 k f Ž fu x . q 4 k f Ž fu . fu x y 2 fffu x y 2 f Xf 2fu q f t y f XXf 3 s 0
Ž 9.
The complexity of this equation is the reason why we cannot solve Ž9. in general. Thus we proceed, by making ansatz on the form of f Ž x,t,u., to solve Ž9.. In this way we get the following functional forms of f Ž u. and the following similarity reductions which are unobtainable by Lie classical symmetries. Due to the invariance under temporal and spatial translations, we take t 0 s 0 and x 0 s 0 without loss of generality. 1. Choosing f s f Ž x,t ., we find that for any function f Ž u. the infinitesimal generators take the form
j s 1,
t s 0,
f s f Ž x ,t . ,
where f Ž x,t . satisfies the following equation k f x x x x y f 3 f XX Ž u . y 3ff x f X Ž u . q f t s 0. Ž 10 . n Ž . Ž . Setting f u s u and solving 10 we obtain that n s 1 is the only value for which we obtain solutions which are not group-invariant. For f Ž u. s u we get the infinitesimal generators x j s 1, t s 0, f Ž x ,t . s y . Ž 11 . 3t It is easy to check that these generators do not satisfy Lie classical determining equations. Therefore we obtain the nonclassical symmetry reduction zst,
usy
x2 6t
qwŽ t. ,
Ž 12 .
M.L. Gandarias, M.S. Bruzonr ´ Physics Letters A 263 (1999) 331–337
where w Ž t . satisfies the linear ODE X
3tw q w s 0.
Ž 13 .
Consequently, an exact solution of Ž1. is usy
x
2
q 6t
k1
.
t 1r3
k2
'u
Ž 14 .
,
single PDE Ž1. to a system of ODE’s which has the common solution Ž19.. 2.2. For f Ž u . s k 1 u 2r3 q k 2 uy2 r3
We remark that solution Ž14. is not a travelling wave reduction and it is not inÕariant under the scaling group. Solution Ž14. is a characteristic solution of Ž1. consequently Ž14. is a solution of Ž2.. This is in contradiction with the statement done in w20x:‘‘the conditional symmetries of Ž2. with t s 0 yield solutions which are nothing else than group-invariant solutions’’. We point out that in Ž21., the infinitesimals for the independent variables are autonomous with respect to the dependent variable, generating a group of ‘ fibre-preserÕing transformations’. Consequently, the solution Ž14. should also be obtained by the direct method w16x, this solution is missing in w17,18x. 2. Choosing f s Ž x qcuk 1 . besides the classical reductions we obtain: 2.1. For f Ž u. s k1 u q
335
Ž 15 .
Ž 20 .
solving Ž9. we get the infinitesimal generators
j s 1, t s 0,
f Ž x ,u . s
j s 1, t s 0,
f Ž x ,u . s
x
zst,
and we obtain the symmetry reduction zst,
usx2 wŽ t. ,
Ž 17 .
2
w y 6 k 1w s 0.
Ž 18 .
We can choose k 1 s 1 without loss of generality. Consequently an exact solution is usy
x
2
6t
.
Ž 21 .
Ž 22 .
where w Ž t . satisfies wX y 12 k 1w 5r3 s 0.
Ž 23 .
Consequently, an exact solution is us
x3 16'2 Ž yk 1 t .
3r2
.
Ž 24 .
We remark that when f Ž u. adopts the form Ž20. Eq. Ž1. does not admit, besides translations, any classical symmetry. Therefore, this solution cannot be obtained by classical reduction. The scaling reduction can be used to reduce the single PDE Ž1. to a system of ODE’s which has the common solution Ž24.. 3. Choosing f s h Ž t . u, solving Ž9. we obtain k2 u
q k3 ,
and we get the infinitesimal generators u j s 1, t s 0, f Ž t ,u . s y . 2 k1 t
(
Ž 25 .
Ž 26 .
The corresponding symmetry reduction is
where w Ž t . satisfies X
.
usx3wŽ t. ,
f Ž u . s yk 1 Ž log u y 1 . q
Ž 16 .
x
The corresponding symmetry reduction is
we get the infinitesimal generators 2u
3u
Ž 19 .
We must point out that, when f Ž u. adopts the the functional form Ž15., Eq. Ž1. does not admit any classical symmetry but translations, consequently this solution cannot be obtained by Lie classical symmetries. The scaling reduction can be used to reduce the
zst,
ž
u s w Ž t . exp y
x
(2 k t 1
/
,
Ž 27 .
where w Ž t . satisfies 4 k 12 t 2 wX q 2 k 1 tw Ž k 1 log w y k 3 . q kw s 0. We can choose k 1 s 1 and the integrating constant k 4 s 0 without loss of generality. Consequently, an exact solution is
ž
u s exp y
x
'2 t
k q 2t
/
q k3 .
Ž 28 .
M.L. Gandarias, M.S. Bruzonr ´ Physics Letters A 263 (1999) 331–337
336
When f Ž u. adopts the functional form Ž25., Eq. Ž1. does not admit any classical symmetries but translations, consequently solution Ž28. is unobtainable by Lie classical symmetries, we observe that this is a noncharacteristic solution. We remark that most of the solutions obtained are characteristic solutions and so they are new solutions of the diffusion equation obtained when k s 0 in these cases the solutions do not feel the influence of the diffusive interface.
4. Nonclassical symmetries and the singular manifold method The complexity of determining Eq. Ž9., appears for many t s 0 symmetries w6x and one advantage of the SMM is that provides non trivial solutions for Ž9.. In a recent paper w11x Estevez and Gordoa have ´ studied the Cahn–Hilliard equation Ž1. with f Ž u. s u and f Ž u. s u 2 by using the SMM. In the following we compare these results with our results by using the nonclassical method: Ø In w11x they claim that besides the trivial generator
j s 0, t s 1,
fs0
Ž 29 .
Žwhich corresponds to a classical symmetry.. For f Ž u. s u 2 the only infinitesimal generator of the nonclassical symmetries that reduce Ž1. to an ODE with the PP is
j s 1, t s 0,
fsy
1 k1 x q w Ž t .
u
'6 k
,
2
Ž 30 .
Ž 31 .
where w Ž t . satisfies the ODE wX s 0, which satisfies the PP. Nevertheless, it is easy to check that the following symmetry
j s 1, t s 0,
fs
ity1 r2 2
us
ix 2 t 1r2
qwŽ t. ,
Ž 33 .
where w Ž t . satisfies the ODE 2 twX q w s 0, which also satisfies the PP. Ø In w11x, for f Ž u. s u they have got the following symmetry
j s 1, t s 0,
fs
2u x q x0
.
Ž 34 .
They claim that, besides Ž29., the SMM allows to determine Ž34. and that these symmetries are the only ones in which the associated similarity reduction leads to an ODE of Painleve´ type s 2E Nevertheless, it is easy to check that the following symmetry
j s 1, t s 0,
fsy
u 3r2
'3k
,
Ž 35 .
satisfies Eq. Ž9. for the nonclassical symmetries with t s 0 and yields the similarity reduction 12 k
us
Ž xqwŽ t. .
2
,
Ž 36 .
where w Ž t . satisfies the ODE wX s 0,
1
The generator Ž30. yields to the similarity reduction us
satisfies Eq. Ž9. for the nonclassical symmetries with t s 0 and yields the similarity reduction
,
Ž 32 .
which also satisfies the PP. It is also easy to check that the infinitesimals Ž11. satisfy Eq. Ž9. for the nonclassical symmetries with t s 0 s 2E These infinitesimals lead to the similarity reduction Ž12., where w Ž t . satisfies Ž13., which is an ODE of Painleve´ type. Therefore, for the Cahn–Hilliard equation Ž1. with f Ž u. s u and f Ž u. s u 2 by using the nonclassical method we have obtained three different symmetries that are respectively Ž32., Ž35. and Ž11. and that were not obtained in w11x by using the SMM. We remark that although the generators Ž30.,Ž32., Ž34., Ž35. do not satisfy the Lie classical determining equations, the corresponding solutions are group-invariant and can be derived from Lie classical symmetries. However solution Ž14., derived from Ž11. is
M.L. Gandarias, M.S. Bruzonr ´ Physics Letters A 263 (1999) 331–337
not inÕariant under translations nor under the scaling group.
337
than the SMM and the SMM does not identify all the nonclassical symmetries that reduce the equation to ODE’s with the PP.
5. Concluding remarks References In this Letter we have seen a classification of symmetry reductions of a family of Cahn–Hilliard equation Ž1. using the classical Lie method of infinitesimals. We have proved that for the parabolic type equation Ž1. the nonclassical method yields to symmetry reductions which are unobtainable by using the Lie classical method and the exact solutions obtained are not group invariant solutions. Consequently, in contradiction with the statement done in w20x, we have proved that the nonclassical method is efficient for PDE’s of the parabolic type. We have obtained solutions by the nonclassical method that should also be obtainable by the direct method, these solutions were missing in w17,18x. We have discussed the symmetry reductions of this equation by using the nonclassical method with those derived in w11x by using the SMM. For this Cahn–Hilliard equation we have derived three nonclassical symmetries that reduce the equation to ODE’s with the Painleve´ property and were not obtained in w11x by the SMM. Therefore for this equation the nonclassical method is more general
w1x M.J. Ablowitz, A. Ramani, H. Segur, Nuovo Cimento 23 Ž1989. 333. w2x A. Bishop, Fronts, Interfaces and Patterns, Elsevier, Amterdam, 1984. w3x G.W. Bluman, J. Cole, J. Math. Mech. 18 Ž1969. 1025. w4x J.W. Cahn, J.E. Hilliard, J. Chem. Phys. 28 Ž1958. 258. w5x P.A. Clarkson, M.D. Kruskal, J. Math. Phys. 30 Ž1989. 2201. w6x P.A. Clarkson, E.L. Mansfield, Physica D 70 Ž1993. 250. w7x P.A. Clarkson, E.L. Mansfield, SIAM J. Appl. Math. 55 Ž1994. 1693. w8x D.S. Cohen, J.D. Murray, J. Math. Biol. 12 Ž1981. 237. w9x C.M. Elliot, H. Garcke, SIAM J. Math. Anal. 27 Ž1996. 404. w10x P.G. Estevez, P.R. Gordoa, Stud. Appl. Math. 95 Ž1995. 73. ´ w11x P.G. Estevez, P.R. Gordoa, J. Phys. A: Math. Gen. 31 Ž1998. ´ 7511. w12x M.L. Gandarias, J. Phys A: Math. Gen. 30 Ž1997. 6081. w13x M.L. Gandarias, J.L. Romero, J.M. Diaz, J. Phys A: Math. Gen. 32 Ž1999. 1461. w14x J.M. Liu, P. Haasen, Z. Metallkd. 85 Ž1994. 293. w15x A. Novick-Cohn, L.A. Segel, Physica D 10 Ž1984. 277. w16x P.J. Olver, Proc. R. Soc. ŽLondon. A 444 Ž1994. 509. w17x S.R. Choudhury, Can. J. Phys. 70 Ž1992. 273. w18x S.R. Choudhury, Nonlinear Anal. 24 Ž1995. 131. w19x U. Tanriver, S.R. Choudhury, J. Phys. A: Math. Gen. 31 Ž1998. 1487. w20x R.Z. Zhdanov, V.I. Lahno VI, Physica D 122 Ž1998. 178.
Physics Letters A 263 Ž1999. 331–337 www.elsevier.nlrlocaterphysleta
Nonclassical symmetries for a family of Cahn–Hilliard equations M.L. Gandarias ) , M.S. Bruzon ´ Departamento Matematicas, UniÕersidad de Cadiz, ´ ´ P.O. Box 40, 11510 Puerto Real, Cadiz, ´ Spain Received 16 August 1999; accepted 20 October 1999 Communicated by A.R. Bishop
Abstract In this Letter we make a full analysis of the symmetry reductions of the family of Cahn–Hilliard equations by using the classical Lie method of infinitesimals, the functional forms of the diffusion coefficients for which the Cahn–Hilliard equations can be fully reduced to ordinary differential equations by classical Lie symmetries are derived. We prove that by using the nonclassical method, we obtain several solutions which are not invariant under any Lie group admitted by the equation and consequently which are not obtainable through the Lie classical method. For this Cahn–Hilliard equation, we obtain nonclassical symmetries that reduce the original equation to ordinary differential equations with the Painleve´ property. We remark that these symmetries have not been derived elsewhere by the singular manifold method. q 1999 Published by Elsevier Science B.V. All rights reserved.
1. Introduction The Cahn–Hilliard equation was introduced to study phase separation in binary alloys glasses and polymers w4x and is a good approach to spinodal decomposition. Based in numerical version of the Fourier transformation approach to the nonlinear Cahn–Hilliard diffusion equation, computer simulations of the spinodal decomposition for a model alloy were carried out by Liu and Haasen w14x. The Cahn–Hilliard diffusion equation is also an equation that serves as a model for many problems in physical chemistry, developmental biology w2x, as well as population movement w8x. The existence of a weak solution for the Cahn–Hilliard equation with degenerate mobility was proved in w9x.
)
Corresponding author.
The Cahn–Hilliard flux equation describing diffusion for decomposition of a one-dimensional binary solution can be written as u t q Ž ku x x x y f Ž u . u x . x s 0,
Ž 1.
which is appropriate to cases where the motion is isotropic. Here u is the solute concentration at point x, t is the time, f X Ž u. is the interdiffusion coefficient of solute, which is concentration dependent and kr2 is the gradient energy coefficient describing the contribution of the diffuse interface to the decomposition. In w8x bifurcations of the equilibrium to nonuniform states have been discussed for f Ž u. s D 0 q D 2 u 2 and in w15x several nonlinear results were derived. Although the direct method of Clarkson and Kruskal is found to more to be more powerful than the classical method w5x., similarity reductions for the Cahn–Hilliard equation with f Ž u. s u and f Ž u. s u 2 ,
0375-9601r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 Ž 9 9 . 0 0 7 5 0 - 1
332
M.L. Gandarias, M.S. Bruzonr ´ Physics Letters A 263 (1999) 331–337
have been obtained in w17,18x, by using classical Lie symmetries as well as the direct method. In w17,18x, the direct method did not yield any reduction that could not be obtained by Lie classical symmetries. In this Letter we solve a complete group classification problem for Eq. Ž1., by studying those diffusion coefficients f Ž u. which admit the classical symmetry group. Both the symmetry group and the diffusion coefficients will be found through consistent application of the Lie-group formalism. Motivated by the fact that symmetry reductions for many partial differential equations ŽPDE’s. are known that are not obtained by using the classical Lie group method, there have been several generalizations of the classical Lie group method for symmetry reductions. The notion of nonclassical symmetries was firstly introduced by Bluman and Cole w3x to study the symmetry reductions of the heat equation. Since then, a great number of papers have been devoted to the study of nonclassical symmetries of nonlinear partial differential equations in both one and several dimensions Žsee, i.e. w6,12,13x.. Clarkson and Mansfield w7x presented an algorithm for calculating the determining equations associated with the nonclassical method. Recently Zhdanov and Lahno w20x have applied the nonclassical Žor conditional. method to the onedimensional porous medium equation u t y Ž uu x . x s 0.
Ž 2.
According to these authors the nonclassical method for Ž2. as well as for the parabolic type PDE’s is inefficient. Once obtained new nonclassical symmetries performing the symmetry reductions gives rise to invariant solutions corresponding to the Lie symmetries of Ž2.. The aim of this Letter is to prove that the nonclassical method applied to the Cahn–Hilliard equation gives rise to new solutions of Ž1. which are not group-invariant and consequently cannot be obtained by Lie classical symmetries. Some of these solutions Žthe characteristic solutions. are solutions of Ž2. which are not invariant under any Lie group admitted by Ž2. and consequently cannot been obtained by Lie classical symmetries this result is a counterexample of the statement done in w20x because according to them all the solutions of Ž2. derived by the nonclassical method are group-invariant.
Recently, the family of Cahn–Hilliard equations has arisen a great interest because of an apparent contradiction between the scope of the singular manifold method ŽSMM. and the nonclassical symmetry reductions w19,11x. In w10x Estevez and Gordoa developed a method ´ for identifying the nonclassical symmetries of PDE’s using the SMM based on the Painleve´ property ŽPP. as a tool. They propose the following conjecture: ‘‘The singular manifold method allows us to identify the nonclassical symmetries that reduce the original equation to an ODE with the Painleve´ property’’. The combination of this statement with the Ablowitz, Ramani and Segur conjecture w1x means that for equations with the PP, the SMM should identify all the nonclassical symmetries. Nevertheless, for equations with the conditional PP, the SMM is only able to identify the symmetries for which the associated reduced ODE’s are of the Painleve´ type. In w11x the authors claim that, for Ž1. with f Ž u. s u an s d f Ž u. s u 2 , besides a trivial symmetry, the SMM allows them to determine two different symmetries and that these symmetries are the only ones in which the associated similarity reduction leads to an ODE of Painleve´ type. Nevertheless, for Ž1. with f Ž u. s u and f Ž u. s u 2 , besides the symmetries derived in w11x, we have derived three new nonclassical symmetries for which the corresponding associated similarity reductions leads to three different ODE’s of Painleve´ type. Consequently, for Ž1. with f Ž u. s u and f Ž u. s u 2 , the nonclassical method is more general than the SMM and that this latter method does not allow us to identify all the nonclassical symmetries that reduce Eq. Ž1., with f Ž u. s u and f Ž u. s u 2 , to ODE’s with the PP.
2. Lie classical classification. To apply the classical method to Ž1. we consider the one-parameter Lie group of infinitesimal transformations in Ž x,t,u. given by x ) s x q ej Ž x ,t ,u . q O Ž e 2 . , t ) s t q et Ž x ,t ,u . q O Ž e 2 . , u ) s u q ef Ž x ,t ,u . q O Ž e 2 . ,
Ž 3.
M.L. Gandarias, M.S. Bruzonr ´ Physics Letters A 263 (1999) 331–337
where e is the group parameter. Then one requires that this transformation leaves invariant the set of solutions of Ž1.. This yields to an overdetermined, linear system of equations for the infinitesimals j Ž x,t,u., t Ž x,t,u. and f Ž x,t,u.. The associated Lie algebra of infinitesimal symmetries is the set of vector fields of the form
E V s j Ž x ,t ,u .
E
Ex
q t Ž x ,t ,u .
Et
q f Ž x ,t ,u .
Eu
.
Having determined the infinitesimals, the symmetry variables are found by solving the invariant surface condition
Eu Ex
qt
y f s 0.
Et
js
Ž 5.
x dt 4 dt
q j 1Ž t . ,
where t , j 1 , f 1 and f 2 are related by the following conditions: 1 dt yf 1 u y f2 y f du du 2 dt d2 f df 1 df dt yŽ f1 u q f 2 . y f1 y du 2 d u d ,t d u2 2 x d t d f Ef 2 d j1 y y2 y 2 4 dt du Ex dt 4 2 d f1 E f2 E f2 Ef 2 uqk yf q 4 2 dt Et Ex Ex df
df
E
E ,
where hŽ z ., after integrating once with respect to z, satisfies khXXX y f Ž h . hX y l h s k 1 .
V2 s
Et
.
Ž 6.
Eq. Ž6. is invariant under translations, this allow us to reduce the order by one. The only functional forms of f Ž u., with f Ž u. / const. for which Eq. Ž1. have extra symmetries are f Ž u. s Ž au q b . n and f Ž u. s de au, and these symmetries are, respectively defined, by the following infinitesimal generators: V31 s x
Ex
E q4t
E V32 s x
Ex
2 y
Et
an
E q4t
E
Ž au q b .
2 E
Et
y
a Eu
Eu
,
.
Without loss of generality we can consider b s 0 and a s d s 1. For the sake of completeness, we provide next, the generators of the nontrivial one-dimensional optimal system which are: Ø For f Ž u. s u n , the set
Ø For f Ž u. s e u , the set
- V1 ) ,- lV1 q V2 ) ,- V32 ) 4 . s 0, s 0, s 0, s 0.
The solutions of this system depends of f Ž u.. For f Ž u. arbitrary, the only symmetries admitted by Ž1. are the group of space and time translations, which are defined by the infinitesimal generators
Ex
u s hŽ z . ,
- V1 ) ,- lV1 q V2 ) ,- V31 ) 4 .
f s f 1 Ž t . u q f 2 Ž x ,t . ,
V1 s
z s x y lt ,
E
Eu
We consider the classical Lie group symmetry analysis of Eq. Ž1.. Invariance of Eq. Ž1. under a Lie group of point transformations with infinitesimal generator Ž4. leads to a set of forty determining equations for the infinitesimals j Ž x,t,u., t Ž x,t,u. and f Ž x,t,u.. Solving this system we obtain
tst Ž t. ,
In this case, we obtain travelling wave reductions
E
Ž 4.
F'j
333
In both sets, l g R is arbitrary. Since Eq. Ž1., with f Ž u. s u n and f Ž u. s e u has additional symmetries and the reductions that correspond to V1 and V2 have already been derived, we must only determine the similarity variables and similarity solutions corresponding to V31 and V32 which are: Ø For V31 : z s xty1 r4 ,
u s ty1 r2 n h Ž z . ,
where hŽ z . satisfies the ODE khXXXX y h n hXX y
z 4
2
hX y nh ny1 Ž hX . y
1 2n
h s 0.
Ž 7.
M.L. Gandarias, M.S. Bruzonr ´ Physics Letters A 263 (1999) 331–337
334
Eq. Ž7. does not admit Lie symmetries. Nevertheless, for n s 2 this equation can be easily integrated once respect to z yielding khXXX y h 2 hX y
z 4
h s k1 .
2
q 3k fu u Ž f x . q 6 k f 2fu u u f x q 12 k ffu u x f x q 10k ffu fu u f x q 6 k fu x x f x q 4 k fu fu x f x
Ø For V32 : z s xty1 r4 ,
u s yln Ž t 1r2 h Ž z . . ,
y 3 f Xff x q k f 4fu u u u q 4 k f 3fu u u x
where hŽ z . satisfies the ODE 4 kh3 hXXXX y 4 h2 hXX y 2 h Ž hX . y 4 kh2 4 hX hXXX q 3 Ž hXX . X 2 XX
Ø In the case t s 0, without loss of generality, we may set j s 1 and the determining equation for the infinitesimal f is k f x x x x q 4 k ffu u f x x q 4 k fu x f x x y ff x x
2
q 6 k f 3fu fu uu q 6 k f 2fu u x x
2
q 12 k f 2fu fu u x q 4 k f 3 Ž fu u . 2
q 12 k f 2fu x fu u q 7k f 2 Ž fu . fu u
y zh3 hX X 4
2
y ff 2fu u q 4 k ffu x x x q 6 k ffu fu x x 4
q 48 kh Ž h . h y 24 k Ž h . q 2 h s 0.
Ž 8.
Eq. Ž8. does not admit Lie symmetries. Due to the fact that Ž1. is only invariant under time and space translations and under a scaling group we have considered nonclassical symmetries.
3. Nonclassical symmetries The basic idea of the method is that the PDE Ž1. is augmented with the invariance surface condition Ž5. which is associated with the vector field Ž4.. By requiring that both Ž1. and Ž5. are invariant under the transformation with infinitesimal generator Ž4. one obtains an overdetermined, nonlinear system of equations for the infinitesimals j Ž x,t,u., t Ž x,t,u. and f Ž x,t,u.. The number of determining equations arising in the nonclassical method is smaller than for the classical method, consequently the set of solutions is in general, larger than for the classical method as in this method one requires only the subset of solutions of Ž1. and Ž5. to be invariant under the infinitesimal generator Ž4.. To obtain nonclassical symmetries of Ž1. we apply the algorithm described in w7x for calculating the determining equations. We can distinguish two different cases: Ø In the case t / 0, without loss of generality, we may set t Ž x,t,u. s 1. The nonclassical method applied to Ž1. gives only rise to the classical symmetries.
2
2
q 8 k f Ž fu x . q 4 k f Ž fu . fu x y 2 fffu x y 2 f Xf 2fu q f t y f XXf 3 s 0
Ž 9.
The complexity of this equation is the reason why we cannot solve Ž9. in general. Thus we proceed, by making ansatz on the form of f Ž x,t,u., to solve Ž9.. In this way we get the following functional forms of f Ž u. and the following similarity reductions which are unobtainable by Lie classical symmetries. Due to the invariance under temporal and spatial translations, we take t 0 s 0 and x 0 s 0 without loss of generality. 1. Choosing f s f Ž x,t ., we find that for any function f Ž u. the infinitesimal generators take the form
j s 1,
t s 0,
f s f Ž x ,t . ,
where f Ž x,t . satisfies the following equation k f x x x x y f 3 f XX Ž u . y 3ff x f X Ž u . q f t s 0. Ž 10 . n Ž . Ž . Setting f u s u and solving 10 we obtain that n s 1 is the only value for which we obtain solutions which are not group-invariant. For f Ž u. s u we get the infinitesimal generators x j s 1, t s 0, f Ž x ,t . s y . Ž 11 . 3t It is easy to check that these generators do not satisfy Lie classical determining equations. Therefore we obtain the nonclassical symmetry reduction zst,
usy
x2 6t
qwŽ t. ,
Ž 12 .
M.L. Gandarias, M.S. Bruzonr ´ Physics Letters A 263 (1999) 331–337
where w Ž t . satisfies the linear ODE X
3tw q w s 0.
Ž 13 .
Consequently, an exact solution of Ž1. is usy
x
2
q 6t
k1
.
t 1r3
k2
'u
Ž 14 .
,
single PDE Ž1. to a system of ODE’s which has the common solution Ž19.. 2.2. For f Ž u . s k 1 u 2r3 q k 2 uy2 r3
We remark that solution Ž14. is not a travelling wave reduction and it is not inÕariant under the scaling group. Solution Ž14. is a characteristic solution of Ž1. consequently Ž14. is a solution of Ž2.. This is in contradiction with the statement done in w20x:‘‘the conditional symmetries of Ž2. with t s 0 yield solutions which are nothing else than group-invariant solutions’’. We point out that in Ž21., the infinitesimals for the independent variables are autonomous with respect to the dependent variable, generating a group of ‘ fibre-preserÕing transformations’. Consequently, the solution Ž14. should also be obtained by the direct method w16x, this solution is missing in w17,18x. 2. Choosing f s Ž x qcuk 1 . besides the classical reductions we obtain: 2.1. For f Ž u. s k1 u q
335
Ž 15 .
Ž 20 .
solving Ž9. we get the infinitesimal generators
j s 1, t s 0,
f Ž x ,u . s
j s 1, t s 0,
f Ž x ,u . s
x
zst,
and we obtain the symmetry reduction zst,
usx2 wŽ t. ,
Ž 17 .
2
w y 6 k 1w s 0.
Ž 18 .
We can choose k 1 s 1 without loss of generality. Consequently an exact solution is usy
x
2
6t
.
Ž 21 .
Ž 22 .
where w Ž t . satisfies wX y 12 k 1w 5r3 s 0.
Ž 23 .
Consequently, an exact solution is us
x3 16'2 Ž yk 1 t .
3r2
.
Ž 24 .
We remark that when f Ž u. adopts the form Ž20. Eq. Ž1. does not admit, besides translations, any classical symmetry. Therefore, this solution cannot be obtained by classical reduction. The scaling reduction can be used to reduce the single PDE Ž1. to a system of ODE’s which has the common solution Ž24.. 3. Choosing f s h Ž t . u, solving Ž9. we obtain k2 u
q k3 ,
and we get the infinitesimal generators u j s 1, t s 0, f Ž t ,u . s y . 2 k1 t
(
Ž 25 .
Ž 26 .
The corresponding symmetry reduction is
where w Ž t . satisfies X
.
usx3wŽ t. ,
f Ž u . s yk 1 Ž log u y 1 . q
Ž 16 .
x
The corresponding symmetry reduction is
we get the infinitesimal generators 2u
3u
Ž 19 .
We must point out that, when f Ž u. adopts the the functional form Ž15., Eq. Ž1. does not admit any classical symmetry but translations, consequently this solution cannot be obtained by Lie classical symmetries. The scaling reduction can be used to reduce the
zst,
ž
u s w Ž t . exp y
x
(2 k t 1
/
,
Ž 27 .
where w Ž t . satisfies 4 k 12 t 2 wX q 2 k 1 tw Ž k 1 log w y k 3 . q kw s 0. We can choose k 1 s 1 and the integrating constant k 4 s 0 without loss of generality. Consequently, an exact solution is
ž
u s exp y
x
'2 t
k q 2t
/
q k3 .
Ž 28 .
M.L. Gandarias, M.S. Bruzonr ´ Physics Letters A 263 (1999) 331–337
336
When f Ž u. adopts the functional form Ž25., Eq. Ž1. does not admit any classical symmetries but translations, consequently solution Ž28. is unobtainable by Lie classical symmetries, we observe that this is a noncharacteristic solution. We remark that most of the solutions obtained are characteristic solutions and so they are new solutions of the diffusion equation obtained when k s 0 in these cases the solutions do not feel the influence of the diffusive interface.
4. Nonclassical symmetries and the singular manifold method The complexity of determining Eq. Ž9., appears for many t s 0 symmetries w6x and one advantage of the SMM is that provides non trivial solutions for Ž9.. In a recent paper w11x Estevez and Gordoa have ´ studied the Cahn–Hilliard equation Ž1. with f Ž u. s u and f Ž u. s u 2 by using the SMM. In the following we compare these results with our results by using the nonclassical method: Ø In w11x they claim that besides the trivial generator
j s 0, t s 1,
fs0
Ž 29 .
Žwhich corresponds to a classical symmetry.. For f Ž u. s u 2 the only infinitesimal generator of the nonclassical symmetries that reduce Ž1. to an ODE with the PP is
j s 1, t s 0,
fsy
1 k1 x q w Ž t .
u
'6 k
,
2
Ž 30 .
Ž 31 .
where w Ž t . satisfies the ODE wX s 0, which satisfies the PP. Nevertheless, it is easy to check that the following symmetry
j s 1, t s 0,
fs
ity1 r2 2
us
ix 2 t 1r2
qwŽ t. ,
Ž 33 .
where w Ž t . satisfies the ODE 2 twX q w s 0, which also satisfies the PP. Ø In w11x, for f Ž u. s u they have got the following symmetry
j s 1, t s 0,
fs
2u x q x0
.
Ž 34 .
They claim that, besides Ž29., the SMM allows to determine Ž34. and that these symmetries are the only ones in which the associated similarity reduction leads to an ODE of Painleve´ type s 2E Nevertheless, it is easy to check that the following symmetry
j s 1, t s 0,
fsy
u 3r2
'3k
,
Ž 35 .
satisfies Eq. Ž9. for the nonclassical symmetries with t s 0 and yields the similarity reduction 12 k
us
Ž xqwŽ t. .
2
,
Ž 36 .
where w Ž t . satisfies the ODE wX s 0,
1
The generator Ž30. yields to the similarity reduction us
satisfies Eq. Ž9. for the nonclassical symmetries with t s 0 and yields the similarity reduction
,
Ž 32 .
which also satisfies the PP. It is also easy to check that the infinitesimals Ž11. satisfy Eq. Ž9. for the nonclassical symmetries with t s 0 s 2E These infinitesimals lead to the similarity reduction Ž12., where w Ž t . satisfies Ž13., which is an ODE of Painleve´ type. Therefore, for the Cahn–Hilliard equation Ž1. with f Ž u. s u and f Ž u. s u 2 by using the nonclassical method we have obtained three different symmetries that are respectively Ž32., Ž35. and Ž11. and that were not obtained in w11x by using the SMM. We remark that although the generators Ž30.,Ž32., Ž34., Ž35. do not satisfy the Lie classical determining equations, the corresponding solutions are group-invariant and can be derived from Lie classical symmetries. However solution Ž14., derived from Ž11. is
M.L. Gandarias, M.S. Bruzonr ´ Physics Letters A 263 (1999) 331–337
not inÕariant under translations nor under the scaling group.
337
than the SMM and the SMM does not identify all the nonclassical symmetries that reduce the equation to ODE’s with the PP.
5. Concluding remarks References In this Letter we have seen a classification of symmetry reductions of a family of Cahn–Hilliard equation Ž1. using the classical Lie method of infinitesimals. We have proved that for the parabolic type equation Ž1. the nonclassical method yields to symmetry reductions which are unobtainable by using the Lie classical method and the exact solutions obtained are not group invariant solutions. Consequently, in contradiction with the statement done in w20x, we have proved that the nonclassical method is efficient for PDE’s of the parabolic type. We have obtained solutions by the nonclassical method that should also be obtainable by the direct method, these solutions were missing in w17,18x. We have discussed the symmetry reductions of this equation by using the nonclassical method with those derived in w11x by using the SMM. For this Cahn–Hilliard equation we have derived three nonclassical symmetries that reduce the equation to ODE’s with the Painleve´ property and were not obtained in w11x by the SMM. Therefore for this equation the nonclassical method is more general
w1x M.J. Ablowitz, A. Ramani, H. Segur, Nuovo Cimento 23 Ž1989. 333. w2x A. Bishop, Fronts, Interfaces and Patterns, Elsevier, Amterdam, 1984. w3x G.W. Bluman, J. Cole, J. Math. Mech. 18 Ž1969. 1025. w4x J.W. Cahn, J.E. Hilliard, J. Chem. Phys. 28 Ž1958. 258. w5x P.A. Clarkson, M.D. Kruskal, J. Math. Phys. 30 Ž1989. 2201. w6x P.A. Clarkson, E.L. Mansfield, Physica D 70 Ž1993. 250. w7x P.A. Clarkson, E.L. Mansfield, SIAM J. Appl. Math. 55 Ž1994. 1693. w8x D.S. Cohen, J.D. Murray, J. Math. Biol. 12 Ž1981. 237. w9x C.M. Elliot, H. Garcke, SIAM J. Math. Anal. 27 Ž1996. 404. w10x P.G. Estevez, P.R. Gordoa, Stud. Appl. Math. 95 Ž1995. 73. ´ w11x P.G. Estevez, P.R. Gordoa, J. Phys. A: Math. Gen. 31 Ž1998. ´ 7511. w12x M.L. Gandarias, J. Phys A: Math. Gen. 30 Ž1997. 6081. w13x M.L. Gandarias, J.L. Romero, J.M. Diaz, J. Phys A: Math. Gen. 32 Ž1999. 1461. w14x J.M. Liu, P. Haasen, Z. Metallkd. 85 Ž1994. 293. w15x A. Novick-Cohn, L.A. Segel, Physica D 10 Ž1984. 277. w16x P.J. Olver, Proc. R. Soc. ŽLondon. A 444 Ž1994. 509. w17x S.R. Choudhury, Can. J. Phys. 70 Ž1992. 273. w18x S.R. Choudhury, Nonlinear Anal. 24 Ž1995. 131. w19x U. Tanriver, S.R. Choudhury, J. Phys. A: Math. Gen. 31 Ž1998. 1487. w20x R.Z. Zhdanov, V.I. Lahno VI, Physica D 122 Ž1998. 178.