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On double reductions from symmetries and conservation laws
Nonlinear Analysis: Real World Applications 10 (2009) 3472–3477
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Nonlinear Analysis: Real World Applications journal homepage: www.elsevier.com/locate/nonrwa
On double reductions from symmetries and conservation laws A. Sjöberg Department of Applied Mathematics, University of Johannesburg (APK Campus), PO Box 524, Auckland Park, 2006, South Africa
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info
Article history: Received 30 April 2008 Accepted 30 September 2008 Keywords: Lie point symmetries Conservation law Double reduction Visco-elastic liquid Nonlinear PDE
abstract We present the theory of double reductions of PDEs with two independent variables that admit a Lie point symmetry and a conserved vector invariant under the symmetry. The theory is applied to a third order nonlinear partial differential equation which describes the filtration of a visco-elastic liquid with relaxation through a porous medium. © 2009 Elsevier Ltd. All rights reserved.
1. Introduction When a Lie point or Lie–Bäcklund symmetry generator is applied to a conserved vector, one of two things may happen. We may find that the conservation law is associated with the symmetry or we may find another conservation law which could either be trivial, known or new. The papers [1,2] deal with the case when we expect to generate new conservation laws from the action of a symmetry on a conserved vector. We have shown in [3] that when the generated conserved vector is the null vector, i.e. the symmetry is associated with the conserved vector (association defined as in [1]), a double reduction is possible for PDEs with two independent variables. In this double reduction the PDE of order q is reduced to an ODE of order q − 1. Thus the use of one symmetry which is associated with a conservation law leads to two reductions, the first being a reduction of the number of independent variables and the second being a reduction of the order of the DE. In this article we discuss a slight generalization of the theory given in [3] to include more cases where a double reduction of PDEs with two independent variables is possible. 2. Theory The notation of [3] is used and some definitions are repeated here to make the article self-contained. The notation of summation over repeated indices is followed. The generation theorem as given in [1,2] states that the action of a Lie point (or Lie–Bäcklund) symmetry X on a conserved vector T with components T i generates a conserved vector T˜ ∗ : XT i + T i Dj ξ j − T j Dj ξ i = T˜ ∗i ,
i = 1, 2.
(1)
(An earlier form of the generation theorem for symmetries in their vertical form [i.e. a symmetry transformation where transformation is confined to the dependent variables uα and the operator is of the form X = W α ∂∂uα with W = ηα − uαi ξ i
E-mail address: [email protected]. 1468-1218/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.nonrwa.2008.09.029
A. Sjöberg / Nonlinear Analysis: Real World Applications 10 (2009) 3472–3477
3473
the characteristic function, see page 125 of [4]] was given in [5] and the references therein.) Furthermore we can write the components of T˜ ∗ in terms of a trivial part (or divergence term), and a nontrivial part, T˜ ∗i = Dj (−1)j g − Di (−1)i g + T ∗i . ∗i Here both the trivial components Ttriv = Dj (−1)j g − Di (−1)i g and the nontrivial components T i∗ depend upon derivatives of the dependent variable u of up to order q − 1 (one order less than the PDE/system that is studied) [6]. Note that ∗i Di Ttriv = Di Dj (−1)j g − Di Di (−1)i g = Di Dj6=i (−1)j g = D1 D2 g − D2 D1 g = 0. is a trivial conservation law ∗i If T = 0 then (1) becomes
XT i + T i Dj ξ j − T j Dj ξ i = Dj (−1)j g − Di (−1)i g ,
i, j = 1, 2.
(2)
If this criteria is met then we say T is associated with X . This happens when the conserved form of the conservation law is invariant under the symmetry X . (In the article [1] association is defined as when the right-hand side of Eq. (1) is zero. This definition was augmented by the authors Kara and Mahomed to include the divergence term. Here we define strict association as when g = 0 and weak association if g 6= 0.) The double reduction of PDEs that admit Lie point symmetries that are strictly associated with conservation laws was presented in [3]. We consider a Lie point symmetry admitted by a PDE of order q with two independent variables F (x1 , x2 , u, u(1) , . . . , u(q) ) = 0.
(3)
To affect a reduction of this PDE to an ODE we transform X into its canonical form X = ∂∂s so that new coordinates are r , s and w . In these new coordinates invariant solutions of (3) solve an ODE G(r , w(r ), w 0 , . . . , w (q) ) = 0 of the same order as (3). This is the first reduction as described in all standard texts on symmetry methods for PDEs such as [7,6,8,4]. The conservation laws Di T i = 0 and Di T i∗ = 0 can be transformed into conservation laws Dr T r + Ds T s = 0
and Dr T ∗r + Ds T ∗s = 0
by the formulae (see [3]) Ts = Tr =
T t Dt (s) + T x Dx (s) Dt (r )Dx (s) − Dx (r )Dt (s)
,
(4)
T t Dt (r ) + T x Dx (r )
(5)
Dt (r )Dx (s) − Dx (r )Dt (s)
and for T ∗r and T ∗s by changing T to T ∗ in the above. The denominator Dt (r )Dx (s) − Dx (r )Dt (s) in the above equations is the Jacobian which is nonzero for transformations that are at least locally invertible as in our case. If we consider invariant solutions of (3) then T s , T r , T ∗s , T ∗r are functions of (s, r , w, . . . , w (q−1) ). This means that s ∗s Ds T s = ∂∂Ts and Ds T ∗s = ∂ ∂T s . Thus the two conservation laws read
∂T s ∂ T ∗s + Dr T r = 0 and + Dr T ∗r = 0. ∂s ∂s
(6)
Now we consider (2) in canonical coordinates
∂T i = Dj6=i (−1)j g + T ∗i for i = r , s, ∂s or
∂T r = Ds g + T ∗r , ∂s
∂T s = −Dr g + T ∗s . ∂s
(7)
Substitute (7) into (6):
− Dr g + T ∗s + Dr T r = 0.
(8)
We define two nonlocal (or potential) variables v and v such that T = vr , T = −vs , T = vr , T Thus (8) becomes −Dr g + vr∗ + Dr T r = 0 which can be integrated with respect to r to yield ∗
−g + v ∗ + T r = g˜ (s). To find g˜ (s) we take the derivative with respect to s:
−Ds g + vs∗ + Ds T r =
dg˜ ds
.
s
r
∗s
∗
∗r
∗
= −vs .
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Note that from (7) Ds g =
∂T r ∂T r − T ∗r = + vs∗ . ∂s ∂s
dg˜
Thus ds = 0 or g˜ is a constant. Solutions of a PDE invariant under X therefore solves T r − g + v∗ = k
(9)
where k is a constant, i.e. g˜ = k. Now note the following: If T ∗ is a nontrivial conserved vector then v ∗ is a nonlocal variable. This means that (9) is an ordinary integro differential equation with highest derivative of order (q − 1). In this case we have made no progress. On the other hand if T ∗ is zero then (9) is the ordinary DE of order q − 1, T r − g = k,
(10)
and we have accomplished a double reduction. We have therefore proved the following theorem: Theorem 2.1. If a PDE of order q admits a Lie point (or contact) symmetry with generator X which is associated with a conserved vector T of the PDE then the PDE can be reduced to an ODE of order q − 1 given by (10). This theorem is a slight generalization (or an alternative form) of the theorem on double reduction recently given in [3]. r Another interesting case is when T ∗ = kT . We consider the case where g = 0. In this case Eq. (8) becomes ∂∂Ts = kT r
and ∂∂Ts = kT s and therefore T r = eks Ar (r , w, wr , . . . , wq−1 ) and T s = eks As (r , w, wr , . . . , wq−1 ). This means that the conservation law Dr T r + Ds T s = 0 becomes eks (Dr Ar + kAs ) = 0. The reduced equation can therefore be written in the form Dr Ar + kAs = 0. Although it is not possible to integrate this equation in general, this form may have advantages when considering numerical schemes. We should now consider whether or not the ODE (10) retains any of the symmetries of the PDE. It is easy to see that when the conserved vector T is strictly associated with X then the ODE (10) should retain all other symmetries that are also strictly associated with T because for strict association the components of T are in fact invariants of the symmetries it is associated with in canonical coordinates. This seems to be quite a strong restriction. We conjecture that in weak association the ODE retains only symmetries that are weakly associated with T with identical function g in the divergence term. Are these the only symmetries of the reduced equation? This is still an open problem. In the following section we demonstrate the theory by applications to a scalar PDE. s
3. Application As an example we consider the third order PDE pt = [|px |n−1 px + f (|px |n−1 px )t ]x
(11)
which describes the filtration of a visco-elastic liquid with relaxation through a porous medium [9]. 3.1. Symmetries A group classification of Lie point symmetries of this equation was done in [10,11]. Eq. (11) admits the following Lie point symmetries with f (z ) an arbitrary function of its argument z = |px |n−1 px and n an arbitrary constant: X1 =
∂ , ∂t
X2 =
∂ ∂ and X3 = . ∂x ∂p
The Lie algebra is extended in the following cases: Case 1: f = Az 1/n + B, n arbitrary, X4 = (1 − n)t
∂ ∂ +p . ∂t ∂p
Case 1.1: n = 1, Xg = g (t , x)
∂ , ∂p
where g (x, t ) is a solution of gt = gxx + Agxxt (which is Eq. (11) with f = Az + B and n = 1).
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Table 1 Commutator table of symmetry operators of (11).
X1 X2 X3
X1
X2
X3
0 0 0
0 0 0
0 0 0
X4 Case 1
Case 2
Case 3
Case 4
Case 5
(1 − n)X1
(n + 1)X1
0 X3
X2 0
2(A − 1)X1 (A − 1)X2 + 2CX3 (A + 1)X3
2nX1 X2 −X3
2X1 X2 − 2CX3 −X3
Table 2 Conserved vectors T ∗ generated by the symmetries of (11) applied to T given by (12). X1
T∗
X2
(0, 0)
Case 2: f =
B z 1/n
X3
(0, 0)
(1, 0)
X4 Case 1
Case 2
Case 3
Case 4
Case 5
T
T
2AT + (2Cx, −2C )
(0, 0)
(−2Cx, 2C )
+ D, n arbitrary,
X4 = (n + 1)t
∂ ∂ +x . ∂t ∂x
Case 3: f = B(z + C )A , n = 1, X4 = 2(A − 1)t
∂ ∂ ∂ + (A − 1)x + [p(A + 1) + 2Cx] . ∂t ∂x ∂p
Case 4: f = A ln |z 1/n | + B, n arbitrary, X4 = 2nt
∂ ∂ ∂ +x −p . ∂t ∂x ∂p
Case 5: f = A ln |z + C | + B, n = 1, X4 = 2t
∂ ∂ ∂ + x − (p + 2Cx) . ∂t ∂x ∂p
The authors also established that (11) has one nontrivial conserved vector for f and n arbitrary namely T = (T t , T x ) = (p, −|px |n−1 px − f (|px |n−1 px )t ).
(12)
The commutator table of the symmetries are given in Table 1. 3.2. Double reductions The conserved vectors T ∗ generated by the symmetries of Eq. (11) and the only nontrivial conserved vector for this equation, (12), are given in Table 2. From these results we see that a double reduction is possible in the case where f and n are arbitrary as well as in cases 4 and 5. For the case where f and n are arbitrary, Eq. (11) admits the symmetry operator X =
∂ ∂ ∂ + c2 + c1 . ∂t ∂x ∂p
(13)
This symmetry is associated with T since XT t + T t Dx ξ x − T x Dx ξ t = c1 ,
(14)
XT + T Dt ξ − T Dt ξ = 0.
(15)
x
x
t
t
x
Thus from (2) g = c1 x. To perform a double reduction of (11) we determine the canonical coordinates for X . These variables satisfy Xs = 1, Xr = X w = 0. We make the choice r = c2 t − x, s = t , w = c1 t − p. Solutions of (11) invariant under (13) are of the
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A. Sjöberg / Nonlinear Analysis: Real World Applications 10 (2009) 3472–3477
form w(r ). From Theorem 2.1 w(r ) satisfies Dr (T r − g ) = 0 (which is equivalent to (11)) or T r − g = c3 . We calculate the conserved component T r from (5): Tr =
T t Dt (r ) + T x Dx (r )
(16)
Dx (s)Dt (r ) − Dt (s)Dx (r )
= c2 p + |px |n−1 px + n|px |n−1 ptx f 0 (|px |n−1 px ) = c2 (c1 s − w) + |wr |
n−1
wr + c2 n|wr |
n −1
(17)
wrr f (|wr | 0
n −1
wr ).
(18)
The function g = c1 x = c1 (c2 s − r ) in the canonical coordinates. Thus T r − g = c1 r − c2 w + |wr |n−1 wr + c2 n|wr |n−1 wrr f 0 (|wr |n−1 wr ). This means that w(r ) satisfies c1 r − c2 w + |wr |n−1 wr + c2 n|wr |n−1 wrr f 0 (|wr |n−1 wr ) = c3 or c2 f (|wr |n−1 wr )r = c3 + c2 w − c1 r − |wr |n−1 wr .
(19)
Eq. (19) is a second order ordinary differential equation which is a double reduction of the third order PDE (11). In case 4 Eq. (11) with f = A ln |z 1/n | + B admits X = (c3 + x) ∂∂x + (c2 + 2nt ) ∂∂ t + (c1 − p) ∂∂p . Here canonical variables
are r = (c3 + x)(c2 + 2nt )−1/2n , s = ln(x + c3 ) and w = (p − c1 )(c2 + 2nt )1/2n so that
T r = nA 2 +
r wrr
wr
− |wr |n−1 wr + r w + c1 (c2 + x)
and g = c1 x. Thus solutions of (11) invariant under X have the form p = c1 + (c2 + 2nt )−1/2n w(s) where w(r ) solves the second order ODE r wrr − |wr |n−1 wr + r w = k. nA
wr
The last case where a double reduction is possible is case 5. Here Eq. (11) with n = 1 and f = A ln |z + C | + B admits X = (c3 + x)
∂ ∂ ∂ + (c2 + 2t ) − (p + 2Cx − c1 ) . ∂x ∂t ∂p
The conserved component T r is given by T = w + c1 x − Cx + r
2
1 r
2r wr − w − Cx − 2Cc3 x + c1 c3 + A 2
2r 2 wr + 4r 3 wrr
2r wr − w + c1 c3 + Cc32
(20)
(x+c3 )2 ,s 2t +c2
= ln |x + c3 | and w = (x + c3 )p + Cx2 − c1 x. The function g is defined by Dx g = c1 − 2Cx, Dt g = −2C and therefore g = c1 x − Cx2 − 2Ct. Thus the double reduction of (11) in this case is given by in terms of canonical coordinates r =
w + 2wr +
Cc32 + c1 c3 − w r
+A
2r wr + 4r 2 wrr 2r wr − w + c1 c3 + Cc32
Solutions of the PDE invariant under (20) are thus given by p =
= c4 . w(
(x+c3 )2 )+c1 x−Cx2 2t +c2 x + c3
(21)
where w(r ) solves (21).
4. Discussion We considered PDEs with two independent variables of order q which admit a Lie point symmetry with generator X and have a conservation law with conserved vector T . We have shown that an ordinary integro-differential equation of order q − 1 can be constructed for the solutions of the PDE invariant under X . That is, solutions of the PDE invariant under X also solve this ordinary integro-differential equation. Furthermore, if X is associated with T by the relation (2) then the invariant solutions of the PDE satisfy an ODE of order (at most) (q − 1). Thus we have affected a double reduction of the PDE. Examples of such a double reduction have been given to illustrate the theory. Much work is needed to generalize (if possible) the theory to PDEs with more than two independent variables. Acknowledgement This material is based upon the work financially supported by the National Research Foundation, grant number TTK2006060700006.
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References [1] A. Kara, F. Mahomed, The relationship between symmetries and conservation laws, Int. J. Theor. Phys. 39 (1) (2000) 23–40. [2] A. Kara, F. Mahomed, A basis of conservation laws for partial differential equations, J. Nonlinear Math. Phys 9 (Suppl. 2) (2002) 60–72. [3] A. Sjöberg, Double reduction of PDEs from the association of symmetries with conservation laws with applications, Appl. Math. Comput. 184 (2007) 608–616. [4] N. Euler, W.-H. Steeb, Continuous Symmetries, Lie Algebras and Differential Equations, B.I. Wissenschaftsverlag, Mannheim, Leipzig, Wien, Zürich, 1992. [5] W.-H. Steeb, Continuous Symmetries, Lie Algebras, Differential Equations and Computer Algebra, World Scientific Publishing Co. Pty. Ltd., Singapore, 1996. [6] P. Olver, Applications of Lie Groups to Differential Equations, in: Graduate Texts in Mathematics, vol. 107, Springer Verlag, New York, 1986. [7] H. Stephani, Differential Equations: Their Solutions Using Symmetries, Cambridge University Press, Cambridge, 1989. [8] G. Bluman, S. Kumei, Symmetries and Differential Equations, in: Graduate Texts in Mathematics, vol. 81, Springer Verlag, New York, 1989. [9] V. Baikov, Filtration of a non-Newtonian liquid in porous media: Models, symmetries and solutions, in: Interdisciplinary Workshop on Symmetry Analysis and Mathematical Modelling, University of the North-West, Mmabatho, 1988. [10] Ö. Kartal, Visco-elastic liquid with relaxation: Symmetries, conservation laws and solutions, Dissertation, University of Johannesburg, Auckland Park Campus, Gauteng, 2004. [11] A. Sjöberg, Ö. Kartal, Filtration of a visco-elastic liquid with relaxation: A note on Lie point symmetries and reductions, J. Nonlinear Math. Phys. 15 (Suppl. 1) (2008) 203–210.
Contents lists available at ScienceDirect
Nonlinear Analysis: Real World Applications journal homepage: www.elsevier.com/locate/nonrwa
On double reductions from symmetries and conservation laws A. Sjöberg Department of Applied Mathematics, University of Johannesburg (APK Campus), PO Box 524, Auckland Park, 2006, South Africa
article
info
Article history: Received 30 April 2008 Accepted 30 September 2008 Keywords: Lie point symmetries Conservation law Double reduction Visco-elastic liquid Nonlinear PDE
abstract We present the theory of double reductions of PDEs with two independent variables that admit a Lie point symmetry and a conserved vector invariant under the symmetry. The theory is applied to a third order nonlinear partial differential equation which describes the filtration of a visco-elastic liquid with relaxation through a porous medium. © 2009 Elsevier Ltd. All rights reserved.
1. Introduction When a Lie point or Lie–Bäcklund symmetry generator is applied to a conserved vector, one of two things may happen. We may find that the conservation law is associated with the symmetry or we may find another conservation law which could either be trivial, known or new. The papers [1,2] deal with the case when we expect to generate new conservation laws from the action of a symmetry on a conserved vector. We have shown in [3] that when the generated conserved vector is the null vector, i.e. the symmetry is associated with the conserved vector (association defined as in [1]), a double reduction is possible for PDEs with two independent variables. In this double reduction the PDE of order q is reduced to an ODE of order q − 1. Thus the use of one symmetry which is associated with a conservation law leads to two reductions, the first being a reduction of the number of independent variables and the second being a reduction of the order of the DE. In this article we discuss a slight generalization of the theory given in [3] to include more cases where a double reduction of PDEs with two independent variables is possible. 2. Theory The notation of [3] is used and some definitions are repeated here to make the article self-contained. The notation of summation over repeated indices is followed. The generation theorem as given in [1,2] states that the action of a Lie point (or Lie–Bäcklund) symmetry X on a conserved vector T with components T i generates a conserved vector T˜ ∗ : XT i + T i Dj ξ j − T j Dj ξ i = T˜ ∗i ,
i = 1, 2.
(1)
(An earlier form of the generation theorem for symmetries in their vertical form [i.e. a symmetry transformation where transformation is confined to the dependent variables uα and the operator is of the form X = W α ∂∂uα with W = ηα − uαi ξ i
E-mail address: [email protected]. 1468-1218/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.nonrwa.2008.09.029
A. Sjöberg / Nonlinear Analysis: Real World Applications 10 (2009) 3472–3477
3473
the characteristic function, see page 125 of [4]] was given in [5] and the references therein.) Furthermore we can write the components of T˜ ∗ in terms of a trivial part (or divergence term), and a nontrivial part, T˜ ∗i = Dj (−1)j g − Di (−1)i g + T ∗i . ∗i Here both the trivial components Ttriv = Dj (−1)j g − Di (−1)i g and the nontrivial components T i∗ depend upon derivatives of the dependent variable u of up to order q − 1 (one order less than the PDE/system that is studied) [6]. Note that ∗i Di Ttriv = Di Dj (−1)j g − Di Di (−1)i g = Di Dj6=i (−1)j g = D1 D2 g − D2 D1 g = 0. is a trivial conservation law ∗i If T = 0 then (1) becomes
XT i + T i Dj ξ j − T j Dj ξ i = Dj (−1)j g − Di (−1)i g ,
i, j = 1, 2.
(2)
If this criteria is met then we say T is associated with X . This happens when the conserved form of the conservation law is invariant under the symmetry X . (In the article [1] association is defined as when the right-hand side of Eq. (1) is zero. This definition was augmented by the authors Kara and Mahomed to include the divergence term. Here we define strict association as when g = 0 and weak association if g 6= 0.) The double reduction of PDEs that admit Lie point symmetries that are strictly associated with conservation laws was presented in [3]. We consider a Lie point symmetry admitted by a PDE of order q with two independent variables F (x1 , x2 , u, u(1) , . . . , u(q) ) = 0.
(3)
To affect a reduction of this PDE to an ODE we transform X into its canonical form X = ∂∂s so that new coordinates are r , s and w . In these new coordinates invariant solutions of (3) solve an ODE G(r , w(r ), w 0 , . . . , w (q) ) = 0 of the same order as (3). This is the first reduction as described in all standard texts on symmetry methods for PDEs such as [7,6,8,4]. The conservation laws Di T i = 0 and Di T i∗ = 0 can be transformed into conservation laws Dr T r + Ds T s = 0
and Dr T ∗r + Ds T ∗s = 0
by the formulae (see [3]) Ts = Tr =
T t Dt (s) + T x Dx (s) Dt (r )Dx (s) − Dx (r )Dt (s)
,
(4)
T t Dt (r ) + T x Dx (r )
(5)
Dt (r )Dx (s) − Dx (r )Dt (s)
and for T ∗r and T ∗s by changing T to T ∗ in the above. The denominator Dt (r )Dx (s) − Dx (r )Dt (s) in the above equations is the Jacobian which is nonzero for transformations that are at least locally invertible as in our case. If we consider invariant solutions of (3) then T s , T r , T ∗s , T ∗r are functions of (s, r , w, . . . , w (q−1) ). This means that s ∗s Ds T s = ∂∂Ts and Ds T ∗s = ∂ ∂T s . Thus the two conservation laws read
∂T s ∂ T ∗s + Dr T r = 0 and + Dr T ∗r = 0. ∂s ∂s
(6)
Now we consider (2) in canonical coordinates
∂T i = Dj6=i (−1)j g + T ∗i for i = r , s, ∂s or
∂T r = Ds g + T ∗r , ∂s
∂T s = −Dr g + T ∗s . ∂s
(7)
Substitute (7) into (6):
− Dr g + T ∗s + Dr T r = 0.
(8)
We define two nonlocal (or potential) variables v and v such that T = vr , T = −vs , T = vr , T Thus (8) becomes −Dr g + vr∗ + Dr T r = 0 which can be integrated with respect to r to yield ∗
−g + v ∗ + T r = g˜ (s). To find g˜ (s) we take the derivative with respect to s:
−Ds g + vs∗ + Ds T r =
dg˜ ds
.
s
r
∗s
∗
∗r
∗
= −vs .
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Note that from (7) Ds g =
∂T r ∂T r − T ∗r = + vs∗ . ∂s ∂s
dg˜
Thus ds = 0 or g˜ is a constant. Solutions of a PDE invariant under X therefore solves T r − g + v∗ = k
(9)
where k is a constant, i.e. g˜ = k. Now note the following: If T ∗ is a nontrivial conserved vector then v ∗ is a nonlocal variable. This means that (9) is an ordinary integro differential equation with highest derivative of order (q − 1). In this case we have made no progress. On the other hand if T ∗ is zero then (9) is the ordinary DE of order q − 1, T r − g = k,
(10)
and we have accomplished a double reduction. We have therefore proved the following theorem: Theorem 2.1. If a PDE of order q admits a Lie point (or contact) symmetry with generator X which is associated with a conserved vector T of the PDE then the PDE can be reduced to an ODE of order q − 1 given by (10). This theorem is a slight generalization (or an alternative form) of the theorem on double reduction recently given in [3]. r Another interesting case is when T ∗ = kT . We consider the case where g = 0. In this case Eq. (8) becomes ∂∂Ts = kT r
and ∂∂Ts = kT s and therefore T r = eks Ar (r , w, wr , . . . , wq−1 ) and T s = eks As (r , w, wr , . . . , wq−1 ). This means that the conservation law Dr T r + Ds T s = 0 becomes eks (Dr Ar + kAs ) = 0. The reduced equation can therefore be written in the form Dr Ar + kAs = 0. Although it is not possible to integrate this equation in general, this form may have advantages when considering numerical schemes. We should now consider whether or not the ODE (10) retains any of the symmetries of the PDE. It is easy to see that when the conserved vector T is strictly associated with X then the ODE (10) should retain all other symmetries that are also strictly associated with T because for strict association the components of T are in fact invariants of the symmetries it is associated with in canonical coordinates. This seems to be quite a strong restriction. We conjecture that in weak association the ODE retains only symmetries that are weakly associated with T with identical function g in the divergence term. Are these the only symmetries of the reduced equation? This is still an open problem. In the following section we demonstrate the theory by applications to a scalar PDE. s
3. Application As an example we consider the third order PDE pt = [|px |n−1 px + f (|px |n−1 px )t ]x
(11)
which describes the filtration of a visco-elastic liquid with relaxation through a porous medium [9]. 3.1. Symmetries A group classification of Lie point symmetries of this equation was done in [10,11]. Eq. (11) admits the following Lie point symmetries with f (z ) an arbitrary function of its argument z = |px |n−1 px and n an arbitrary constant: X1 =
∂ , ∂t
X2 =
∂ ∂ and X3 = . ∂x ∂p
The Lie algebra is extended in the following cases: Case 1: f = Az 1/n + B, n arbitrary, X4 = (1 − n)t
∂ ∂ +p . ∂t ∂p
Case 1.1: n = 1, Xg = g (t , x)
∂ , ∂p
where g (x, t ) is a solution of gt = gxx + Agxxt (which is Eq. (11) with f = Az + B and n = 1).
A. Sjöberg / Nonlinear Analysis: Real World Applications 10 (2009) 3472–3477
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Table 1 Commutator table of symmetry operators of (11).
X1 X2 X3
X1
X2
X3
0 0 0
0 0 0
0 0 0
X4 Case 1
Case 2
Case 3
Case 4
Case 5
(1 − n)X1
(n + 1)X1
0 X3
X2 0
2(A − 1)X1 (A − 1)X2 + 2CX3 (A + 1)X3
2nX1 X2 −X3
2X1 X2 − 2CX3 −X3
Table 2 Conserved vectors T ∗ generated by the symmetries of (11) applied to T given by (12). X1
T∗
X2
(0, 0)
Case 2: f =
B z 1/n
X3
(0, 0)
(1, 0)
X4 Case 1
Case 2
Case 3
Case 4
Case 5
T
T
2AT + (2Cx, −2C )
(0, 0)
(−2Cx, 2C )
+ D, n arbitrary,
X4 = (n + 1)t
∂ ∂ +x . ∂t ∂x
Case 3: f = B(z + C )A , n = 1, X4 = 2(A − 1)t
∂ ∂ ∂ + (A − 1)x + [p(A + 1) + 2Cx] . ∂t ∂x ∂p
Case 4: f = A ln |z 1/n | + B, n arbitrary, X4 = 2nt
∂ ∂ ∂ +x −p . ∂t ∂x ∂p
Case 5: f = A ln |z + C | + B, n = 1, X4 = 2t
∂ ∂ ∂ + x − (p + 2Cx) . ∂t ∂x ∂p
The authors also established that (11) has one nontrivial conserved vector for f and n arbitrary namely T = (T t , T x ) = (p, −|px |n−1 px − f (|px |n−1 px )t ).
(12)
The commutator table of the symmetries are given in Table 1. 3.2. Double reductions The conserved vectors T ∗ generated by the symmetries of Eq. (11) and the only nontrivial conserved vector for this equation, (12), are given in Table 2. From these results we see that a double reduction is possible in the case where f and n are arbitrary as well as in cases 4 and 5. For the case where f and n are arbitrary, Eq. (11) admits the symmetry operator X =
∂ ∂ ∂ + c2 + c1 . ∂t ∂x ∂p
(13)
This symmetry is associated with T since XT t + T t Dx ξ x − T x Dx ξ t = c1 ,
(14)
XT + T Dt ξ − T Dt ξ = 0.
(15)
x
x
t
t
x
Thus from (2) g = c1 x. To perform a double reduction of (11) we determine the canonical coordinates for X . These variables satisfy Xs = 1, Xr = X w = 0. We make the choice r = c2 t − x, s = t , w = c1 t − p. Solutions of (11) invariant under (13) are of the
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A. Sjöberg / Nonlinear Analysis: Real World Applications 10 (2009) 3472–3477
form w(r ). From Theorem 2.1 w(r ) satisfies Dr (T r − g ) = 0 (which is equivalent to (11)) or T r − g = c3 . We calculate the conserved component T r from (5): Tr =
T t Dt (r ) + T x Dx (r )
(16)
Dx (s)Dt (r ) − Dt (s)Dx (r )
= c2 p + |px |n−1 px + n|px |n−1 ptx f 0 (|px |n−1 px ) = c2 (c1 s − w) + |wr |
n−1
wr + c2 n|wr |
n −1
(17)
wrr f (|wr | 0
n −1
wr ).
(18)
The function g = c1 x = c1 (c2 s − r ) in the canonical coordinates. Thus T r − g = c1 r − c2 w + |wr |n−1 wr + c2 n|wr |n−1 wrr f 0 (|wr |n−1 wr ). This means that w(r ) satisfies c1 r − c2 w + |wr |n−1 wr + c2 n|wr |n−1 wrr f 0 (|wr |n−1 wr ) = c3 or c2 f (|wr |n−1 wr )r = c3 + c2 w − c1 r − |wr |n−1 wr .
(19)
Eq. (19) is a second order ordinary differential equation which is a double reduction of the third order PDE (11). In case 4 Eq. (11) with f = A ln |z 1/n | + B admits X = (c3 + x) ∂∂x + (c2 + 2nt ) ∂∂ t + (c1 − p) ∂∂p . Here canonical variables
are r = (c3 + x)(c2 + 2nt )−1/2n , s = ln(x + c3 ) and w = (p − c1 )(c2 + 2nt )1/2n so that
T r = nA 2 +
r wrr
wr
− |wr |n−1 wr + r w + c1 (c2 + x)
and g = c1 x. Thus solutions of (11) invariant under X have the form p = c1 + (c2 + 2nt )−1/2n w(s) where w(r ) solves the second order ODE r wrr − |wr |n−1 wr + r w = k. nA
wr
The last case where a double reduction is possible is case 5. Here Eq. (11) with n = 1 and f = A ln |z + C | + B admits X = (c3 + x)
∂ ∂ ∂ + (c2 + 2t ) − (p + 2Cx − c1 ) . ∂x ∂t ∂p
The conserved component T r is given by T = w + c1 x − Cx + r
2
1 r
2r wr − w − Cx − 2Cc3 x + c1 c3 + A 2
2r 2 wr + 4r 3 wrr
2r wr − w + c1 c3 + Cc32
(20)
(x+c3 )2 ,s 2t +c2
= ln |x + c3 | and w = (x + c3 )p + Cx2 − c1 x. The function g is defined by Dx g = c1 − 2Cx, Dt g = −2C and therefore g = c1 x − Cx2 − 2Ct. Thus the double reduction of (11) in this case is given by in terms of canonical coordinates r =
w + 2wr +
Cc32 + c1 c3 − w r
+A
2r wr + 4r 2 wrr 2r wr − w + c1 c3 + Cc32
Solutions of the PDE invariant under (20) are thus given by p =
= c4 . w(
(x+c3 )2 )+c1 x−Cx2 2t +c2 x + c3
(21)
where w(r ) solves (21).
4. Discussion We considered PDEs with two independent variables of order q which admit a Lie point symmetry with generator X and have a conservation law with conserved vector T . We have shown that an ordinary integro-differential equation of order q − 1 can be constructed for the solutions of the PDE invariant under X . That is, solutions of the PDE invariant under X also solve this ordinary integro-differential equation. Furthermore, if X is associated with T by the relation (2) then the invariant solutions of the PDE satisfy an ODE of order (at most) (q − 1). Thus we have affected a double reduction of the PDE. Examples of such a double reduction have been given to illustrate the theory. Much work is needed to generalize (if possible) the theory to PDEs with more than two independent variables. Acknowledgement This material is based upon the work financially supported by the National Research Foundation, grant number TTK2006060700006.
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