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Generalization of the double reduction theory
Nonlinear Analysis: Real World Applications 11 (2010) 3763–3769
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Nonlinear Analysis: Real World Applications journal homepage: www.elsevier.com/locate/nonrwa
Generalization of the double reduction theory Ashfaque H. Bokhari a , Ahmad Y. Al-Dweik a,∗ , F.D. Zaman a , A.H. Kara b , F.M. Mahomed c a
Department of Mathematical Sciences, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia
b
School of Mathematics, University of the Witwatersrand, Wits 2050, South Africa
c
School of Computation and Applied Mathematics, Center for Differential Equations, Continuum Mechanics and Applications, University of the Witwatersrand, Wits 2050, South Africa
article
info
Article history: Received 4 September 2009 Accepted 10 February 2010 Keywords: Double reduction theory Conservation laws Associated symmetry Invariant solutions
abstract In a recent work Sjöberg (2007, 2008) [1,2] remarked that generalization of the double reduction theory to partial differential equations of higher dimensions is still an open problem. In this note we have attempted to provide this generalization to find invariant solution for a non linear system of qth order partial differential equations with n independent and m dependent variables provided that the non linear system of partial differential equations admits a nontrivial conserved form which has at least one associated symmetry in every reduction. In order to give an application of the procedure we apply it to the nonlinear (2 + 1) wave equation for arbitrary function f (u) and g (u). © 2010 Elsevier Ltd. All rights reserved.
1. Introduction Applying a Lie point or Lie–Bäcklund symmetry generator to a conserved vector provides either (1) conservation law associated with that symmetry or (2) conservation law that may be trivial, known already or new. A pioneering work in this direction was published by Kara et al. [3,4]. Sjöberg later showed that [1,2], when the generated conserved vector is null, i.e. the symmetry is associated with the conserved vector (association defined as in [3]), 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 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. Sjöberg also constructed the reduction formula for PDEs with two independent variables which transform the conserved form of the PDE to a reduced conserved form via an associated symmetry. Application of this method to the linear heat, the BBM and the sine-Gordon equation and a system of differential equations from one dimensional gas dynamics are given [1]. The double reduction theory says that a PDE of order q with two independent and m dependent variables, which admits a nontrivial conserved form that has at least one associated symmetry, can be reduced to an ODE of order (q − 1). In her papers [1,2] Sjöberg opines that generalizing the double reduction theory to PDEs of higher dimensions is still an open problem and it is not clear how to overcome the problem when not all derivatives of non-local variables are known explicitly. Further calculations for higher dimensions are quite tedious and cumbersome. There do not exist enough examples of potential symmetries and symmetries with associated conservation laws for higher dimensional PDEs so that the complexity of this problem can be demonstrated. And much work is needed to generalize (if possible) the theory to PDEs with more than two independent variables. In this article we discuss a generalization of the double reduction theory showing that a non linear system of qth order PDEs with n independent and m dependent variables can be reduced to a non linear system of (q − 1)th order ODEs. It is shown
∗
Corresponding author. Tel.: +966 555786178. E-mail address: [email protected] (A.Y. Al-Dweik).
1468-1218/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.nonrwa.2010.02.006
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that these reductions are possible provided the system admits a nontrivial conserved form with at least one associated symmetry in every reduction. In order to solve this we use two main steps: (a) generalize the reduction formula of Sjöberg in [1] from two independent variables to n independent variables and (b) prove that the conserved form of PDEs with n independent variables can be transformed to a reduced conserved form via an associated symmetry. Finally we apply the generalized double reduction to the nonlinear (2 + 1) wave equation for arbitrary function f (u) and g (u) to obtain invariant solution. 2. The fundamental theorem of double reduction Consider the qth-order system of partial differential equations (PDEs) of n independent variables x = (x1 , x2 , . . . , xn ) and m dependent variables u = (u1 , u2 , . . . , um ) E α (x, u, u(1) , . . . , u(q) ) = 0,
α = 1, . . . , m,
(2.1) α
where u(1) , u(2) , . . . , u(q) denote the collections of all first, second, . . . , qth-order partial derivatives, i.e., ui = Di (u ), uαij = α
Dj Di (uα ), . . . respectively, with the total differentiation operator with respect to xi given by Di =
∂ ∂ ∂ + uαi α + uαij α + · · · , ∂ xi ∂u ∂ uj
i = 1 , . . . , n,
(2.2)
in which the summation convention is used. The following definitions are well-known (see, e.g. [5,6,3]). The Lie–Bäcklund operator is X = ξi
∂ ∂ + ηα α ∂ xi ∂u
ξ i , ηα ∈ A,
(2.3)
where A is the space of differential functions. The operator (2.3) is an abbreviated form of the infinite formal sum X = ξi
X ∂ ∂ ∂ ζiα1 i2 ...is α + ηα α + , i ∂x ∂u ∂ ui1 i2 ...is s≥1
(2.4)
where the additional coefficients are determined uniquely by the prolongation formulae,
ζiα = Di (W α ) + ξ j uαij ζiα1 ...is = Di1 . . . Dis (W α ) + ξ j uαji1 ...is ,
(2.5)
s > 1,
in which W α is the Lie characteristic function, W α = ηα − ξ j uαj .
(2.6)
The n-tuple vector T = (T 1 , T 2 , . . . , T n ), T j ∈ A, j = 1, . . . , n is a conserved vector of (2.1) if T i satisfies Di T i |(2.1) = 0.
(2.7)
A Lie–Bäcklund symmetry generator X of the form (2.4) is associated with a conserved vector T of the system (2.1) if X and T satisfy the relations
[T i , X ] = X (T i ) + T i Dk (ξ k ) − T k Dk (ξ i ) = 0,
i = 1, . . . , n.
(2.8)
Theorem 2.1 ([5,6]). Suppose that X is any Lie–Bäcklund symmetry of (2.1) and T i , i = 1, . . . , n are the components of conserved vector of (2.1). Then T ∗ = [T i , X ] = X (T i ) + T i Dj ξ j − T j Dj ξ i , i
i = 1, . . . , n.
(2.9)
∗i
constitute the components of a conserved vector of (2.1), i.e. Di T |(2.1) = 0. Theorem 2.2 ([7]). Suppose Di T i = 0 is a conservation law of PDE system (2.1). Under the contact transformation, there exist ˜ i T˜ i , where T˜ i is given explicitly in terms of the determinant obtained through replacing the ith functions T˜ i such that J Di T i = D row of the Jacobian determinant by [T 1 , T 2 , . . . , T n ], where
D˜ 1 x1 D˜ 2 x1 J = . .. D˜ x n 1
˜ 1 x2 D ˜ 2 x2 D .. .
˜ n x2 D
... ... .. . ...
˜ 1 xn D ˜ 2 xn D
.. . . ˜ n xn D
(2.10)
Theorem 2.3. Suppose Di T i = 0 is a conservation law of PDE system (2.1). Under the contact transformation, there exist functions ˜ i T˜ i where T˜ i is given explicitly in terms of T˜ i such that J Di T i = D
A.H. Bokhari et al. / Nonlinear Analysis: Real World Applications 11 (2010) 3763–3769
1 1 T˜ T T˜ 2 T 2 . = J (A−1 )T . , .. .. ˜T n Tn
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1 T˜ T T 2 T˜ 2 J . = AT . , .. .. Tn T˜ n 1
(2.11)
where
˜ 1 x1 D D˜ 2 x1
˜ 1 x2 D ˜ 2 x2 D
A= . .
.. .
. ˜ n x1 D
˜ n x2 D
... ... .. .
...
˜ 1 xn D ˜ 2 xn D
.. , .
A −1 = ..
˜ n xn D
... ... .. .
D1 x˜2 D2 x˜2
D1 x˜1 D2 x˜1
.
.. .
Dn x˜1
Dn x˜2
...
D1 x˜n D2 x˜n
.. . Dn x˜n
(2.12)
and J = det (A). Proof. Using Theorem 2.2 we can write
T˜ 1
T˜ 2
T˜ n
T1 D˜ 2 x1 = .. . D˜ n x1 D˜ 1 x1 T1 = .. . D˜ n x1 D˜ 1 x1 D˜ 2 x1 = . .. T1
T2 ˜ 2 x2 D
.. .
... ... .. .
˜ n x2 D
...
˜ 1 x2 D T2
... ... .. .
.. .
˜ n x2 D
...
˜ 1 x2 D ˜ 2 x2 D
... ... .. .
.. .
T2
...
J T1 D˜ 2 x1 ˜ 2 x2 1 J T2 D .. = .. .. J . . . J T D˜ x ˜ n xn D n 2 n D˜ 1 x1 J T1 ˜ 1 xn D ˜ 1 x2 J T2 Tn 1 D .. = .. .. J . . . D˜ x J T ˜ n xn D 1 n n D˜ 1 x1 D˜ 2 x1 ˜ 1 xn D ˜D2 xn 1 D˜ 1 x2 D˜ 2 x2 .. .. = J .. . . . D˜ x D˜ x Tn 1 n 2 n
Tn ˜ 2 xn D
... ... .. .
˜ n x1 D ˜ n x2 D
.. , . . . . D˜ n xn . . . D˜ n x1 . . . D˜ n x2 .. .. , . . . . . D˜ n xn . . . J T1 . . . J T2 .. .. . . . . . . J Tn
(2.13)
(2.14)
(2.15)
Since
D˜ 1 x1 D˜ 2 x1 J = . .. D˜ x n 1
˜ 1 x2 D ˜D2 x2 .. .
˜ n x2 D
... ... .. . ...
D˜ 1 x1 D˜ 1 x2 .. = .. . . ˜ n xn D˜ 1 xn D
˜ 2 x1 D ˜ 2 x2 D
˜ 1 xn D ˜D2 xn
.. .
˜ 2 xn D
... ... .. . ...
˜ n x1 D ˜ n x2 D
T .. = A , . ˜ x D
(2.16)
n n
one can use Cramer’s rule to find that T˜ , T˜ , . . . , T˜ can be written as follows: 1
2
n
1 T˜ T˜ 2 . = AT . . .. .. n JT T˜ n
J T1 J T 2
(2.17)
Lastly, one can easily see that AA−1 = I .
(2.18)
Lemma 2.1. Consider n independent variables x = (x1 , x2 , . . . , xn ), m dependent variables u = (u1 , u2 , . . . , um ) and the change of independent variables x˜ = (˜x1 , x˜ 2 , . . . , x˜ n ), then any vector f (x, u, u1 ) = (f 1 , f 2 , . . . , f n ) must satisfy the following identity
˜1 D D˜ 2
. .. ˜n D
˜1 D ˜2 D .. . ˜n D
... ... ... ...
˜1 D f1 1 ˜2 D f
.. .. . . ˜Dn f1
f2 f2
... ...
f2
... ...
.. .
fn f n
D1 D2
.. = A .. . .
fn
Dn
D1 D2
... ...
Dn
... ...
.. .
f1 D1 D2 f 1
f2 f2
... ...
f1
f2
... ...
.. . . ..
Dn
.. .
fn f n
.. , .
fn
(2.19)
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where
˜ 1 x1 D D˜ 2 x1
... ... .. .
˜ 1 x2 D ˜D2 x2
A= . .
.. .
. ˜ n x1 D
˜ 1 xn D ˜D2 xn
.. .
...
˜ n x2 D
(2.20)
˜ n xn D
Proof. Since
˜ i f j = D˜ i xk Dk f j , D
i, j = 1, . . . , n,
(2.21)
then
˜ 1f 1 D D˜ 2 f 1
. .. ˜ nf 1 D
˜ 1f 2 D ˜ 2f 2 D
... ...
˜ nf 2 D
... ...
.. .
˜ 1f n D ˜ 2f n D
D1 f 1 D2 f 1
D1 f 2 D2 f 2
... ...
Dn f 1
Dn f 2
... ...
.. = A .. . .
˜ nf n D
.. .
D1 f n D2 f n
.. . .
(2.22)
Dn f n
Theorem 2.4 (Fundamental Theorem of Double Reduction). Suppose Di T i = 0 is a conservation law of PDE system (2.1). Under the similarity transformation of a symmetry X of the form (2.4) for the PDE, there exist functions T˜ i such that X is still a symmetry ˜ i T˜ i = 0 and for the PDE D X T˜ 1 X T˜ 2
[T 1 , X ] [T 2 , X ] . = J (A−1 )T . , .. .. [T n , X ] X T˜ n
(2.23)
where
˜ 1 x1 D D˜ 2 x1
A= . .
. ˜ n x1 D
... ... .. .
˜ 1 x2 D ˜ 2 x2 D .. .
˜ 1 xn D ˜ 2 xn D
.. , .
...
˜ n x2 D
D1 x˜1 D2 x˜1
A−1 = ..
˜ n xn D
D1 x˜2 D2 x˜2
.
.. .
Dn x˜1
Dn x˜2
... ... .. . ...
D1 x˜n D2 x˜n
.. . Dn x˜n
(2.24)
and J = det (A).
˜ i T˜ i and Proof. By the above theorem there exist functions T˜ i such that J Di T i = D 1 1 T˜ T T˜ 2 T 2 . = J (A−1 )T . , .. .. n Tn T˜
1 T˜ T˜ 2 J . = AT . . .. .. n T T˜ n 1 T T 2
(2.25)
˜ i T˜ i = 0, because X (J ) Di T i + J X (Di T i ) = X (D˜ i T˜ i ) and Then X is a symmetry for the PDE D X T˜ 1 X T˜ 2
1
XT 1 XT 2
1
T T 2
. = J (A−1 )T . + JX ((A−1 )T ) . .. .. .. n n ˜ XT Tn XT
T
T 2 + X (J )(A−1 )T . . ..
(2.26)
Tn
Since J = det (A), then
D˜ 1 ξ 1 D˜ 2 x1 X (J ) = . .. D˜ x n 1
˜ 1ξ 2 D ˜ 2 x2 D .. .
˜ n x2 D
... ... .. .
...
˜ 1ξ n D ˜ 2 xn D
D˜ 1 x1 D˜ 2 ξ 1 + .. .. . . ˜ n xn D˜ n x1 D
˜ 1 x2 D ˜ 2ξ 2 D .. .
˜ n x2 D
... ... .. .
...
˜ 1 xn D ˜ 2ξ n D
D˜ 1 x1 D˜ 2 x1 + · · · + . .. .. . D˜ ξ 1 ˜Dn xn n
˜ 1 x2 D ˜ 2 x2 D
.. . ˜ nξ 2 D
... ... .. . ...
˜ 1 xn D ˜ 2 xn D
.. . . ˜ nξ n D
(2.27)
A.H. Bokhari et al. / Nonlinear Analysis: Real World Applications 11 (2010) 3763–3769
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˜ i ξ j , then it is the cofactor of D˜ i xj for the matrix A. Thus Let ζij denote the cofactor of D ˜ i ξ j ζij = Dk ξ j D˜ i xk ζij = Dk ξ j δjk J . X (J ) = D
(2.28)
˜ i xk ζij = δjk J for every fixed j, where δjk is the Kronecker delta, then Since D X (J ) = J (D1 ξ 1 + D2 ξ 2 + · · · + Dn ξ n ).
(2.29)
Now using the previous lemma one gets,
˜1 D D˜ 2
. .. ˜n D
... ...
˜1 D ˜2 D .. . ˜n D
... ...
˜1 D ξ1 ˜ 2 ξ 1 D
ξ2 ξ2 .. .
.. .. . . ˜Dn ξ1
... ... ... ...
ξ2
ξn D1 n ξ D2 .. = A .. . .
D1 D2
... ...
Dn
Dn
... ...
ξn
.. .
ξ D1 1 D2 ξ
1
.. . . .. Dn ξ1
ξ2 ξ2 .. . ξ2
... ... ... ...
ξn ξ n .. . .
(2.30)
ξn
Now transposing both sides gives, D1 ξ 1 D1 ξ 2
X (AT ) = . .
. D1 ξ n
D2 ξ 1 D2 ξ 2
.. . D2 ξ n
... ... .. .
...
Dn ξ 1 Dn ξ 2
T .. A . . Dn ξ n
(2.31)
Since AT (A−1 )T = I, then X (AT )(A−1 )T + AT X ((A−1 )T ) = 0, thus X ((A−1 )T ) = −(AT )−1 X (AT )(A−1 )T = −(A−1 )T X (AT )(AT )−1 D1 ξ 1 D1 ξ 2
= −(A−1 )T . .. D1 ξ n
D2 ξ 1 D2 ξ 2
.. . D2 ξ n
... ... .. .
...
Dn ξ 1 Dn ξ 2
.. . . Dn ξ n
(2.32)
Lastly we get the result X T˜ 1 X T˜ 2
XT 1 XT 2
D1 ξ 1 D1 ξ 2
. = J (A−1 )T . − . .. .. .. n ˜ D1 ξ n XT n XT
D2 ξ 1 D2 ξ 2
.. . D2 ξ n
... ... .. .
...
T1 Dn ξ 1 T 2 Dn ξ 2
1 T
T 2 i .. .. + Di ξ .. . . . . n n Tn T Dn ξ
(2.33)
Corollary 2.1 (The Necessary and Sufficient Condition to Get Reduced Conserved Form). The conserved form Di T i = 0 of PDE ˜ i T˜ i = 0 if and system (2.1) can be reduced under the similarity transformation of a symmetry X to a reduced conserved form D only if X is associated with the conservation law T , i.e. [T , X ]|(2.1) = 0. Corollary 2.2 (The Generalized Double Reduction Theory). A non linear system of qth order PDEs with n independent and m dependent variables, which admits a nontrivial conserved form that has at least one associated symmetry in every reduction from the n reductions (the first step of double reduction) can be reduced to a non linear system (q − 1)th order of ODEs. Remark. According to the procedure of Sjöberg [1,2], one can arrive from a PDE of order q to an ODE of order q provided there exists at least one associated symmetry in every reduction. This follows directly from the above theorem by the invariance of the fluxes and using canonical coordinates. Lastly, the qth order ODE (written in the conserved form) is reduced to an ODE of order (q − 1). Corollary 2.3 (The Inherited Symmetries). Any symmetry Y for the conserved form Di T i = 0 of PDE system (2.1) can be ˜ i T˜ i = 0. transformed under the similarity transformation of a symmetry X for the PDE to the symmetry Y˜ for the PDE D Remark. There is a possibility to get an associated symmetry with a reduced conserved form by inheriting the non associated symmetry with the original conserved form. So there is an important use of the non associated symmetry also in Double reduction. Finally, we conjecture that the reduction under a combination of associated and non associated symmetries will give us two PDEs, one of them being a reduced conserved form and the second being a non reduced conserved form, and we can ˜ i T˜ i ) = 0 such that the solution of a reduced conserved form is also a solution of the non separate them via the condition X (D reduced conserved form.
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3. Application of the generalized double reduction theory to the nonlinear (2 + 1) wave equation The nonlinear (2 + 1) wave equation for arbitrary function f (u) and g (u) utt − (f (u)ux )x − (g (u)uy )y = 0,
(3.1)
has the obvious conservation law T = (−ut , f (u) ux , g (u) uy ).
(3.2)
It admits the following four symmetries:
∂ , ∂t ∂ X3 = ∂y
∂ ∂x ∂ ∂ ∂ X4 = t +x +y . ∂t ∂x ∂y
X1 =
X2 =
(3.3)
We can get a reduced conserved form for the PDE by the associated symmetry which satisfies the following formula Tt Tx Ty
X
Dt ξ t Dt ξ x Dt ξ y
! −
Dx ξ t Dx ξ x Dx ξ y
Dy ξ t Dy ξ x Dy ξ y
Tt Tx Ty
!
!
Tt + (Dt ξ + Dx ξ + Dy ξ ) T x Ty t
x
y
! = 0.
(3.4)
Then the only associated symmetries are X1 , X2 and X3 , so we can get a reduced conserved form by the combination of them X = ∂∂t + c1 ∂∂x + c2 ∂∂y , where the generator X has a canonical form X = ∂∂q when dt 1
=
dx c1
=
dy c2
=
du
=
0
dr
=
0
ds 0
=
dq 1
=
dw 0
,
(3.5)
or r = y − c2 t ,
s = x − c1 t ,
q = t,
w(r , s) = u.
(3.6)
Using the following formula, we can get the reduced conserved form Tr Ts Tq
! = J (A−1 )T
Tt Tx Ty
! ,
(3.7)
where −1
A
Dt r Dx r Dy r
=
Dt s Dx s Dy s
Dt q Dx q , Dy q
!
J = det (A).
(3.8)
Then the reduced conserved form is Dr T r + Ds T s = 0,
(3.9)
where T r = c22 wr + c2 c1 ws − g (w)wr , T s = c1 c2 wr + c12 ws − f (w)ws , T q = −c2 wr − c1 ws .
(3.10)
The reduced conserved form admits the inherited symmetry: X˜ 4 = r
∂ ∂ +s . ∂r ∂s
(3.11)
Similarly we can get a reduced conserved form for the PDE by the associated symmetry which satisfies the following formula
r X
T Ts
−
Dr ξ r Dr ξ s
Ds ξ r Ds ξ s
r T Ts
+ (Dr ξ + Ds ξ ) r
s
r T Ts
= 0.
(3.12)
One can see that X˜ 4 is an associated symmetry, so we can get a reduced conserved form by Y = r ∂∂r + s ∂∂s , where the generator Y has a canonical form Y = ∂∂m when dr r
=
ds s
=
dw 0
=
dn 0
=
dm 1
=
dv 0
,
(3.13)
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3769
or n=
s r
,
m = lnr ,
v(n) = w.
(3.14)
So by using the following formula, we can get the reduced conserved form
Tn Tm
= J (A−1 )T
r T Ts
,
(3.15)
where A −1 =
Dr n Ds n
Dr m , Ds m
J = det (A).
(3.16)
Then the reduced conserved form is: Dn T n = 0 ,
(3.17)
where T n = vn (−c22 n2 + 2c2 c1 n + n2 g (v) − c12 + f (v)), T m = −vn (−c22 n + c2 c1 + ng (v)).
(3.18)
The second step of double reduction can be given as
vn (−c22 n2 + 2c2 c1 n + n2 g (v) − c12 + f (v)) = C ,
(3.19)
where C is a constant, n = y−c1 t and v = u. 2 x −c t
4. Conclusion In order to find invariant solutions for a nonlinear system of qth order PDEs with n independent and m dependent variables, a generalization of the double reduction theory due to Sjöberg [1,2] is proposed. This generalization allows one to reduce the PDEs of order q to an ODE of (q − 1) from the association of the symmetry with its conserved form via the new generalized formula (2.11). The procedure is applied to a nonlinear (2 + 1) wave equation by showing, in one case, how reduction to an ODE of order one is achieved. Acknowledgements Ashfaque H. Bokhari, Ahmad Y. Al-Dweik and F.-D. Zaman would like to thank the King Fahd University of Petroleum and Minerals for providing research support through FT100001. References [1] A. Sjöberg, Double reduction of PDEs from the association of symmetries with conservation laws with applications, Appl. Math. Comput. 184 (2007) 608–616. [2] A. Sjöberg, On double reductions from symmetries and conservation laws, Nonlinear Anal. RWA 10 (6) (2009) 3472–3477. [3] A. Kara, F. Mahomed, The relationship between symmetries and conservation laws, Int. J. Theor. Phys. 39 (1) (2000) 23–40. [4] A. Kara, F. Mahomed, A basis of conservation laws for partial differential equations, J. Nonlinear Math. Phys. 9 (Suppl. 2) (2002) 60–72. [5] W.H. Steeb, W. Strampp, Diffusion equations and Lie and Lie-Backlund transformation groups, Physica 114A (1982) 95–99. [6] A. Kara, F. Mahomed, Action of Lie-Backlund symmetries on conservation laws, in: Modern Group Analysis, vol. VII, Norway, 1997. [7] S.C. Anco, G.W. Bluman, New conservation laws obtained directly from symmetry action on a known conservation law, J. Math. Anal. Appl. 322 (2006) 233–250.
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Nonlinear Analysis: Real World Applications journal homepage: www.elsevier.com/locate/nonrwa
Generalization of the double reduction theory Ashfaque H. Bokhari a , Ahmad Y. Al-Dweik a,∗ , F.D. Zaman a , A.H. Kara b , F.M. Mahomed c a
Department of Mathematical Sciences, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia
b
School of Mathematics, University of the Witwatersrand, Wits 2050, South Africa
c
School of Computation and Applied Mathematics, Center for Differential Equations, Continuum Mechanics and Applications, University of the Witwatersrand, Wits 2050, South Africa
article
info
Article history: Received 4 September 2009 Accepted 10 February 2010 Keywords: Double reduction theory Conservation laws Associated symmetry Invariant solutions
abstract In a recent work Sjöberg (2007, 2008) [1,2] remarked that generalization of the double reduction theory to partial differential equations of higher dimensions is still an open problem. In this note we have attempted to provide this generalization to find invariant solution for a non linear system of qth order partial differential equations with n independent and m dependent variables provided that the non linear system of partial differential equations admits a nontrivial conserved form which has at least one associated symmetry in every reduction. In order to give an application of the procedure we apply it to the nonlinear (2 + 1) wave equation for arbitrary function f (u) and g (u). © 2010 Elsevier Ltd. All rights reserved.
1. Introduction Applying a Lie point or Lie–Bäcklund symmetry generator to a conserved vector provides either (1) conservation law associated with that symmetry or (2) conservation law that may be trivial, known already or new. A pioneering work in this direction was published by Kara et al. [3,4]. Sjöberg later showed that [1,2], when the generated conserved vector is null, i.e. the symmetry is associated with the conserved vector (association defined as in [3]), 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 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. Sjöberg also constructed the reduction formula for PDEs with two independent variables which transform the conserved form of the PDE to a reduced conserved form via an associated symmetry. Application of this method to the linear heat, the BBM and the sine-Gordon equation and a system of differential equations from one dimensional gas dynamics are given [1]. The double reduction theory says that a PDE of order q with two independent and m dependent variables, which admits a nontrivial conserved form that has at least one associated symmetry, can be reduced to an ODE of order (q − 1). In her papers [1,2] Sjöberg opines that generalizing the double reduction theory to PDEs of higher dimensions is still an open problem and it is not clear how to overcome the problem when not all derivatives of non-local variables are known explicitly. Further calculations for higher dimensions are quite tedious and cumbersome. There do not exist enough examples of potential symmetries and symmetries with associated conservation laws for higher dimensional PDEs so that the complexity of this problem can be demonstrated. And much work is needed to generalize (if possible) the theory to PDEs with more than two independent variables. In this article we discuss a generalization of the double reduction theory showing that a non linear system of qth order PDEs with n independent and m dependent variables can be reduced to a non linear system of (q − 1)th order ODEs. It is shown
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Corresponding author. Tel.: +966 555786178. E-mail address: [email protected] (A.Y. Al-Dweik).
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that these reductions are possible provided the system admits a nontrivial conserved form with at least one associated symmetry in every reduction. In order to solve this we use two main steps: (a) generalize the reduction formula of Sjöberg in [1] from two independent variables to n independent variables and (b) prove that the conserved form of PDEs with n independent variables can be transformed to a reduced conserved form via an associated symmetry. Finally we apply the generalized double reduction to the nonlinear (2 + 1) wave equation for arbitrary function f (u) and g (u) to obtain invariant solution. 2. The fundamental theorem of double reduction Consider the qth-order system of partial differential equations (PDEs) of n independent variables x = (x1 , x2 , . . . , xn ) and m dependent variables u = (u1 , u2 , . . . , um ) E α (x, u, u(1) , . . . , u(q) ) = 0,
α = 1, . . . , m,
(2.1) α
where u(1) , u(2) , . . . , u(q) denote the collections of all first, second, . . . , qth-order partial derivatives, i.e., ui = Di (u ), uαij = α
Dj Di (uα ), . . . respectively, with the total differentiation operator with respect to xi given by Di =
∂ ∂ ∂ + uαi α + uαij α + · · · , ∂ xi ∂u ∂ uj
i = 1 , . . . , n,
(2.2)
in which the summation convention is used. The following definitions are well-known (see, e.g. [5,6,3]). The Lie–Bäcklund operator is X = ξi
∂ ∂ + ηα α ∂ xi ∂u
ξ i , ηα ∈ A,
(2.3)
where A is the space of differential functions. The operator (2.3) is an abbreviated form of the infinite formal sum X = ξi
X ∂ ∂ ∂ ζiα1 i2 ...is α + ηα α + , i ∂x ∂u ∂ ui1 i2 ...is s≥1
(2.4)
where the additional coefficients are determined uniquely by the prolongation formulae,
ζiα = Di (W α ) + ξ j uαij ζiα1 ...is = Di1 . . . Dis (W α ) + ξ j uαji1 ...is ,
(2.5)
s > 1,
in which W α is the Lie characteristic function, W α = ηα − ξ j uαj .
(2.6)
The n-tuple vector T = (T 1 , T 2 , . . . , T n ), T j ∈ A, j = 1, . . . , n is a conserved vector of (2.1) if T i satisfies Di T i |(2.1) = 0.
(2.7)
A Lie–Bäcklund symmetry generator X of the form (2.4) is associated with a conserved vector T of the system (2.1) if X and T satisfy the relations
[T i , X ] = X (T i ) + T i Dk (ξ k ) − T k Dk (ξ i ) = 0,
i = 1, . . . , n.
(2.8)
Theorem 2.1 ([5,6]). Suppose that X is any Lie–Bäcklund symmetry of (2.1) and T i , i = 1, . . . , n are the components of conserved vector of (2.1). Then T ∗ = [T i , X ] = X (T i ) + T i Dj ξ j − T j Dj ξ i , i
i = 1, . . . , n.
(2.9)
∗i
constitute the components of a conserved vector of (2.1), i.e. Di T |(2.1) = 0. Theorem 2.2 ([7]). Suppose Di T i = 0 is a conservation law of PDE system (2.1). Under the contact transformation, there exist ˜ i T˜ i , where T˜ i is given explicitly in terms of the determinant obtained through replacing the ith functions T˜ i such that J Di T i = D row of the Jacobian determinant by [T 1 , T 2 , . . . , T n ], where
D˜ 1 x1 D˜ 2 x1 J = . .. D˜ x n 1
˜ 1 x2 D ˜ 2 x2 D .. .
˜ n x2 D
... ... .. . ...
˜ 1 xn D ˜ 2 xn D
.. . . ˜ n xn D
(2.10)
Theorem 2.3. Suppose Di T i = 0 is a conservation law of PDE system (2.1). Under the contact transformation, there exist functions ˜ i T˜ i where T˜ i is given explicitly in terms of T˜ i such that J Di T i = D
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1 1 T˜ T T˜ 2 T 2 . = J (A−1 )T . , .. .. ˜T n Tn
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1 T˜ T T 2 T˜ 2 J . = AT . , .. .. Tn T˜ n 1
(2.11)
where
˜ 1 x1 D D˜ 2 x1
˜ 1 x2 D ˜ 2 x2 D
A= . .
.. .
. ˜ n x1 D
˜ n x2 D
... ... .. .
...
˜ 1 xn D ˜ 2 xn D
.. , .
A −1 = ..
˜ n xn D
... ... .. .
D1 x˜2 D2 x˜2
D1 x˜1 D2 x˜1
.
.. .
Dn x˜1
Dn x˜2
...
D1 x˜n D2 x˜n
.. . Dn x˜n
(2.12)
and J = det (A). Proof. Using Theorem 2.2 we can write
T˜ 1
T˜ 2
T˜ n
T1 D˜ 2 x1 = .. . D˜ n x1 D˜ 1 x1 T1 = .. . D˜ n x1 D˜ 1 x1 D˜ 2 x1 = . .. T1
T2 ˜ 2 x2 D
.. .
... ... .. .
˜ n x2 D
...
˜ 1 x2 D T2
... ... .. .
.. .
˜ n x2 D
...
˜ 1 x2 D ˜ 2 x2 D
... ... .. .
.. .
T2
...
J T1 D˜ 2 x1 ˜ 2 x2 1 J T2 D .. = .. .. J . . . J T D˜ x ˜ n xn D n 2 n D˜ 1 x1 J T1 ˜ 1 xn D ˜ 1 x2 J T2 Tn 1 D .. = .. .. J . . . D˜ x J T ˜ n xn D 1 n n D˜ 1 x1 D˜ 2 x1 ˜ 1 xn D ˜D2 xn 1 D˜ 1 x2 D˜ 2 x2 .. .. = J .. . . . D˜ x D˜ x Tn 1 n 2 n
Tn ˜ 2 xn D
... ... .. .
˜ n x1 D ˜ n x2 D
.. , . . . . D˜ n xn . . . D˜ n x1 . . . D˜ n x2 .. .. , . . . . . D˜ n xn . . . J T1 . . . J T2 .. .. . . . . . . J Tn
(2.13)
(2.14)
(2.15)
Since
D˜ 1 x1 D˜ 2 x1 J = . .. D˜ x n 1
˜ 1 x2 D ˜D2 x2 .. .
˜ n x2 D
... ... .. . ...
D˜ 1 x1 D˜ 1 x2 .. = .. . . ˜ n xn D˜ 1 xn D
˜ 2 x1 D ˜ 2 x2 D
˜ 1 xn D ˜D2 xn
.. .
˜ 2 xn D
... ... .. . ...
˜ n x1 D ˜ n x2 D
T .. = A , . ˜ x D
(2.16)
n n
one can use Cramer’s rule to find that T˜ , T˜ , . . . , T˜ can be written as follows: 1
2
n
1 T˜ T˜ 2 . = AT . . .. .. n JT T˜ n
J T1 J T 2
(2.17)
Lastly, one can easily see that AA−1 = I .
(2.18)
Lemma 2.1. Consider n independent variables x = (x1 , x2 , . . . , xn ), m dependent variables u = (u1 , u2 , . . . , um ) and the change of independent variables x˜ = (˜x1 , x˜ 2 , . . . , x˜ n ), then any vector f (x, u, u1 ) = (f 1 , f 2 , . . . , f n ) must satisfy the following identity
˜1 D D˜ 2
. .. ˜n D
˜1 D ˜2 D .. . ˜n D
... ... ... ...
˜1 D f1 1 ˜2 D f
.. .. . . ˜Dn f1
f2 f2
... ...
f2
... ...
.. .
fn f n
D1 D2
.. = A .. . .
fn
Dn
D1 D2
... ...
Dn
... ...
.. .
f1 D1 D2 f 1
f2 f2
... ...
f1
f2
... ...
.. . . ..
Dn
.. .
fn f n
.. , .
fn
(2.19)
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where
˜ 1 x1 D D˜ 2 x1
... ... .. .
˜ 1 x2 D ˜D2 x2
A= . .
.. .
. ˜ n x1 D
˜ 1 xn D ˜D2 xn
.. .
...
˜ n x2 D
(2.20)
˜ n xn D
Proof. Since
˜ i f j = D˜ i xk Dk f j , D
i, j = 1, . . . , n,
(2.21)
then
˜ 1f 1 D D˜ 2 f 1
. .. ˜ nf 1 D
˜ 1f 2 D ˜ 2f 2 D
... ...
˜ nf 2 D
... ...
.. .
˜ 1f n D ˜ 2f n D
D1 f 1 D2 f 1
D1 f 2 D2 f 2
... ...
Dn f 1
Dn f 2
... ...
.. = A .. . .
˜ nf n D
.. .
D1 f n D2 f n
.. . .
(2.22)
Dn f n
Theorem 2.4 (Fundamental Theorem of Double Reduction). Suppose Di T i = 0 is a conservation law of PDE system (2.1). Under the similarity transformation of a symmetry X of the form (2.4) for the PDE, there exist functions T˜ i such that X is still a symmetry ˜ i T˜ i = 0 and for the PDE D X T˜ 1 X T˜ 2
[T 1 , X ] [T 2 , X ] . = J (A−1 )T . , .. .. [T n , X ] X T˜ n
(2.23)
where
˜ 1 x1 D D˜ 2 x1
A= . .
. ˜ n x1 D
... ... .. .
˜ 1 x2 D ˜ 2 x2 D .. .
˜ 1 xn D ˜ 2 xn D
.. , .
...
˜ n x2 D
D1 x˜1 D2 x˜1
A−1 = ..
˜ n xn D
D1 x˜2 D2 x˜2
.
.. .
Dn x˜1
Dn x˜2
... ... .. . ...
D1 x˜n D2 x˜n
.. . Dn x˜n
(2.24)
and J = det (A).
˜ i T˜ i and Proof. By the above theorem there exist functions T˜ i such that J Di T i = D 1 1 T˜ T T˜ 2 T 2 . = J (A−1 )T . , .. .. n Tn T˜
1 T˜ T˜ 2 J . = AT . . .. .. n T T˜ n 1 T T 2
(2.25)
˜ i T˜ i = 0, because X (J ) Di T i + J X (Di T i ) = X (D˜ i T˜ i ) and Then X is a symmetry for the PDE D X T˜ 1 X T˜ 2
1
XT 1 XT 2
1
T T 2
. = J (A−1 )T . + JX ((A−1 )T ) . .. .. .. n n ˜ XT Tn XT
T
T 2 + X (J )(A−1 )T . . ..
(2.26)
Tn
Since J = det (A), then
D˜ 1 ξ 1 D˜ 2 x1 X (J ) = . .. D˜ x n 1
˜ 1ξ 2 D ˜ 2 x2 D .. .
˜ n x2 D
... ... .. .
...
˜ 1ξ n D ˜ 2 xn D
D˜ 1 x1 D˜ 2 ξ 1 + .. .. . . ˜ n xn D˜ n x1 D
˜ 1 x2 D ˜ 2ξ 2 D .. .
˜ n x2 D
... ... .. .
...
˜ 1 xn D ˜ 2ξ n D
D˜ 1 x1 D˜ 2 x1 + · · · + . .. .. . D˜ ξ 1 ˜Dn xn n
˜ 1 x2 D ˜ 2 x2 D
.. . ˜ nξ 2 D
... ... .. . ...
˜ 1 xn D ˜ 2 xn D
.. . . ˜ nξ n D
(2.27)
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˜ i ξ j , then it is the cofactor of D˜ i xj for the matrix A. Thus Let ζij denote the cofactor of D ˜ i ξ j ζij = Dk ξ j D˜ i xk ζij = Dk ξ j δjk J . X (J ) = D
(2.28)
˜ i xk ζij = δjk J for every fixed j, where δjk is the Kronecker delta, then Since D X (J ) = J (D1 ξ 1 + D2 ξ 2 + · · · + Dn ξ n ).
(2.29)
Now using the previous lemma one gets,
˜1 D D˜ 2
. .. ˜n D
... ...
˜1 D ˜2 D .. . ˜n D
... ...
˜1 D ξ1 ˜ 2 ξ 1 D
ξ2 ξ2 .. .
.. .. . . ˜Dn ξ1
... ... ... ...
ξ2
ξn D1 n ξ D2 .. = A .. . .
D1 D2
... ...
Dn
Dn
... ...
ξn
.. .
ξ D1 1 D2 ξ
1
.. . . .. Dn ξ1
ξ2 ξ2 .. . ξ2
... ... ... ...
ξn ξ n .. . .
(2.30)
ξn
Now transposing both sides gives, D1 ξ 1 D1 ξ 2
X (AT ) = . .
. D1 ξ n
D2 ξ 1 D2 ξ 2
.. . D2 ξ n
... ... .. .
...
Dn ξ 1 Dn ξ 2
T .. A . . Dn ξ n
(2.31)
Since AT (A−1 )T = I, then X (AT )(A−1 )T + AT X ((A−1 )T ) = 0, thus X ((A−1 )T ) = −(AT )−1 X (AT )(A−1 )T = −(A−1 )T X (AT )(AT )−1 D1 ξ 1 D1 ξ 2
= −(A−1 )T . .. D1 ξ n
D2 ξ 1 D2 ξ 2
.. . D2 ξ n
... ... .. .
...
Dn ξ 1 Dn ξ 2
.. . . Dn ξ n
(2.32)
Lastly we get the result X T˜ 1 X T˜ 2
XT 1 XT 2
D1 ξ 1 D1 ξ 2
. = J (A−1 )T . − . .. .. .. n ˜ D1 ξ n XT n XT
D2 ξ 1 D2 ξ 2
.. . D2 ξ n
... ... .. .
...
T1 Dn ξ 1 T 2 Dn ξ 2
1 T
T 2 i .. .. + Di ξ .. . . . . n n Tn T Dn ξ
(2.33)
Corollary 2.1 (The Necessary and Sufficient Condition to Get Reduced Conserved Form). The conserved form Di T i = 0 of PDE ˜ i T˜ i = 0 if and system (2.1) can be reduced under the similarity transformation of a symmetry X to a reduced conserved form D only if X is associated with the conservation law T , i.e. [T , X ]|(2.1) = 0. Corollary 2.2 (The Generalized Double Reduction Theory). A non linear system of qth order PDEs with n independent and m dependent variables, which admits a nontrivial conserved form that has at least one associated symmetry in every reduction from the n reductions (the first step of double reduction) can be reduced to a non linear system (q − 1)th order of ODEs. Remark. According to the procedure of Sjöberg [1,2], one can arrive from a PDE of order q to an ODE of order q provided there exists at least one associated symmetry in every reduction. This follows directly from the above theorem by the invariance of the fluxes and using canonical coordinates. Lastly, the qth order ODE (written in the conserved form) is reduced to an ODE of order (q − 1). Corollary 2.3 (The Inherited Symmetries). Any symmetry Y for the conserved form Di T i = 0 of PDE system (2.1) can be ˜ i T˜ i = 0. transformed under the similarity transformation of a symmetry X for the PDE to the symmetry Y˜ for the PDE D Remark. There is a possibility to get an associated symmetry with a reduced conserved form by inheriting the non associated symmetry with the original conserved form. So there is an important use of the non associated symmetry also in Double reduction. Finally, we conjecture that the reduction under a combination of associated and non associated symmetries will give us two PDEs, one of them being a reduced conserved form and the second being a non reduced conserved form, and we can ˜ i T˜ i ) = 0 such that the solution of a reduced conserved form is also a solution of the non separate them via the condition X (D reduced conserved form.
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3. Application of the generalized double reduction theory to the nonlinear (2 + 1) wave equation The nonlinear (2 + 1) wave equation for arbitrary function f (u) and g (u) utt − (f (u)ux )x − (g (u)uy )y = 0,
(3.1)
has the obvious conservation law T = (−ut , f (u) ux , g (u) uy ).
(3.2)
It admits the following four symmetries:
∂ , ∂t ∂ X3 = ∂y
∂ ∂x ∂ ∂ ∂ X4 = t +x +y . ∂t ∂x ∂y
X1 =
X2 =
(3.3)
We can get a reduced conserved form for the PDE by the associated symmetry which satisfies the following formula Tt Tx Ty
X
Dt ξ t Dt ξ x Dt ξ y
! −
Dx ξ t Dx ξ x Dx ξ y
Dy ξ t Dy ξ x Dy ξ y
Tt Tx Ty
!
!
Tt + (Dt ξ + Dx ξ + Dy ξ ) T x Ty t
x
y
! = 0.
(3.4)
Then the only associated symmetries are X1 , X2 and X3 , so we can get a reduced conserved form by the combination of them X = ∂∂t + c1 ∂∂x + c2 ∂∂y , where the generator X has a canonical form X = ∂∂q when dt 1
=
dx c1
=
dy c2
=
du
=
0
dr
=
0
ds 0
=
dq 1
=
dw 0
,
(3.5)
or r = y − c2 t ,
s = x − c1 t ,
q = t,
w(r , s) = u.
(3.6)
Using the following formula, we can get the reduced conserved form Tr Ts Tq
! = J (A−1 )T
Tt Tx Ty
! ,
(3.7)
where −1
A
Dt r Dx r Dy r
=
Dt s Dx s Dy s
Dt q Dx q , Dy q
!
J = det (A).
(3.8)
Then the reduced conserved form is Dr T r + Ds T s = 0,
(3.9)
where T r = c22 wr + c2 c1 ws − g (w)wr , T s = c1 c2 wr + c12 ws − f (w)ws , T q = −c2 wr − c1 ws .
(3.10)
The reduced conserved form admits the inherited symmetry: X˜ 4 = r
∂ ∂ +s . ∂r ∂s
(3.11)
Similarly we can get a reduced conserved form for the PDE by the associated symmetry which satisfies the following formula
r X
T Ts
−
Dr ξ r Dr ξ s
Ds ξ r Ds ξ s
r T Ts
+ (Dr ξ + Ds ξ ) r
s
r T Ts
= 0.
(3.12)
One can see that X˜ 4 is an associated symmetry, so we can get a reduced conserved form by Y = r ∂∂r + s ∂∂s , where the generator Y has a canonical form Y = ∂∂m when dr r
=
ds s
=
dw 0
=
dn 0
=
dm 1
=
dv 0
,
(3.13)
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3769
or n=
s r
,
m = lnr ,
v(n) = w.
(3.14)
So by using the following formula, we can get the reduced conserved form
Tn Tm
= J (A−1 )T
r T Ts
,
(3.15)
where A −1 =
Dr n Ds n
Dr m , Ds m
J = det (A).
(3.16)
Then the reduced conserved form is: Dn T n = 0 ,
(3.17)
where T n = vn (−c22 n2 + 2c2 c1 n + n2 g (v) − c12 + f (v)), T m = −vn (−c22 n + c2 c1 + ng (v)).
(3.18)
The second step of double reduction can be given as
vn (−c22 n2 + 2c2 c1 n + n2 g (v) − c12 + f (v)) = C ,
(3.19)
where C is a constant, n = y−c1 t and v = u. 2 x −c t
4. Conclusion In order to find invariant solutions for a nonlinear system of qth order PDEs with n independent and m dependent variables, a generalization of the double reduction theory due to Sjöberg [1,2] is proposed. This generalization allows one to reduce the PDEs of order q to an ODE of (q − 1) from the association of the symmetry with its conserved form via the new generalized formula (2.11). The procedure is applied to a nonlinear (2 + 1) wave equation by showing, in one case, how reduction to an ODE of order one is achieved. Acknowledgements Ashfaque H. Bokhari, Ahmad Y. Al-Dweik and F.-D. Zaman would like to thank the King Fahd University of Petroleum and Minerals for providing research support through FT100001. References [1] A. Sjöberg, Double reduction of PDEs from the association of symmetries with conservation laws with applications, Appl. Math. Comput. 184 (2007) 608–616. [2] A. Sjöberg, On double reductions from symmetries and conservation laws, Nonlinear Anal. RWA 10 (6) (2009) 3472–3477. [3] A. Kara, F. Mahomed, The relationship between symmetries and conservation laws, Int. J. Theor. Phys. 39 (1) (2000) 23–40. [4] A. Kara, F. Mahomed, A basis of conservation laws for partial differential equations, J. Nonlinear Math. Phys. 9 (Suppl. 2) (2002) 60–72. [5] W.H. Steeb, W. Strampp, Diffusion equations and Lie and Lie-Backlund transformation groups, Physica 114A (1982) 95–99. [6] A. Kara, F. Mahomed, Action of Lie-Backlund symmetries on conservation laws, in: Modern Group Analysis, vol. VII, Norway, 1997. [7] S.C. Anco, G.W. Bluman, New conservation laws obtained directly from symmetry action on a known conservation law, J. Math. Anal. Appl. 322 (2006) 233–250.