- Email: [email protected]
Symmetry groups of differential–difference equations and their compatibility
J. Math. Anal. Appl. 371 (2010) 355–362
Contents lists available at ScienceDirect
Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa
Symmetry groups of differential–difference equations and their compatibility Shen Shoufeng a,b , Qu Changzheng a,c,∗ a b c
Center for Nonlinear Studies, Northwest University, Xi’an 710069, China Department of Applied Mathematics, Zhejiang University of Technology, Hangzhou 310023, China Department of Mathematics, Northwest University, Xi’an 710069, China
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 28 December 2009 Available online 12 May 2010 Submitted by P. Broadbridge
It is shown that the intrinsic determining equations of a given differential–difference equation (DDE) can be derived by the compatibility between the original equation and the intrinsic invariant surface condition. The (2 + 1)-dimensional Toda lattice, the special Toda lattice and the DD-KP equation serving as examples are used to illustrate this approach. Then, Bäcklund transformations of the (2 + 1)-dimensional DDEs including the special Toda lattice, the modified Toda lattice and the DD-KZ equation are presented by using the nonintrinsic direct method. In addition, the Clarkson–Kruskal direct method is developed to find similarity reductions of the DDEs. © 2010 Elsevier Inc. All rights reserved.
Keywords: Symmetry Differential–difference equation Compatibility Bäcklund transformation
1. Introduction Symmetry group method [1–5] introduced by Sophus Lie in the later part of 19th century has played an important role in the reduction of differential equations and the construction of their exact solutions. Lie’s symmetry method for continuous differential equations has by now been well established, though the same for differential–difference equations (DDEs) or difference equations is much less investigated or understood. However some important results [6–13] have been obtained. For instance, Maeda has proposed a method to obtain the point symmetries of ordinary difference equations in Ref. [6]. More interestingly, the intrinsic symmetry method for computing symmetries of the (1 + 1)-dimensional and (2 + 1)-dimensional Toda lattice was derived by Levi and Winternitz [7,8], where “intrinsic” means that the Lie symmetry operator takes the form
Γ = T (x, t )
∂ ∂ ∂ + X (x, t ) + U x, t , u (m), m . ∂t ∂x ∂ u (m) m∈ Z
The software packages such as MathLie [14] and DESOLV [15] have been explored for computing continuous differential equations with finite variables. However, to the best of our knowledge, there are no similar means for DDEs where the main difficulty in computing symmetries of DDEs is to deal with infinitely many variables [9,10]. In Ref. [16], Clarkson and Kruskal (CK) introduced a direct and effective method to derive similarity reductions of partial differential equations without using any group theoretical techniques. Soon afterwards this method was applied to solve some important DDEs such as the (2 + 1)-dimensional Toda lattice [17,18]. Interestingly, the modified CK direct method [19] can be used to compute symmetry groups for continuous differential equations.
*
Corresponding author at: Department of Mathematics, Northwest University, Xi’an 710069, China. E-mail addresses: [email protected] (S. Shen), [email protected] (C. Qu).
0022-247X/$ – see front matter doi:10.1016/j.jmaa.2010.05.025
©
2010 Elsevier Inc. All rights reserved.
356
S. Shen, C. Qu / J. Math. Anal. Appl. 371 (2010) 355–362
Recently, inspired by Pucci, Saccomandi and Seiler’s related works [20,21], Arrigo et al. [22–24] presented a new method that nonclassical symmetries can be derived from the compatibility between the invariant surface conditions and equations under consideration. Such method has been used to obtain the symmetries of Burgers equation, Klein–Gordon equation, Boussinesq equation and the Nizhnik–Novikov–Veselov equation. Thus, a natural and important question is how this method can be applied to DDEs? The purpose of this paper is to develop this method to solve DDEs. The paper is organized as follows. In Section 2, we show that the intrinsic determining equations of DDEs can be derived by the compatibility between the original equation and the intrinsic invariant surface condition, some DDEs are used to illustrate the approach. Then, the Bäcklund transformations of some DDEs are presented by using the non-intrinsic direct method. In Section 3, the CK direct reductions of the DD-KP equation are obtained. Some conclusions are given in Section 4. 2. The new method and examples Consider the (2 + 1)-dimensional Nth order DDE
( N ) = F t , x, . . . , ut (k−i )x(i ) (n − 1), ut (k−i )x(i ) (n), ut (k−i )x(i ) (n + 1), . . . = 0,
(1)
where F is a smooth function of its arguments, ut (k−i )x(i ) (m) = ∂ k u (m)/∂ t k−i ∂ xi . The intrinsic invariant surface condition with T = 1 (as general) is
1 = ut (m) + X (x, t )u x (m) − U x, t , u (m), m = 0.
(2)
The intrinsic determining equations of Eq. (1) can be obtained by requiring that
Γ ( N ) ( N ) (N ) =0,
1 =0
= 0,
(3)
where the infinitesimal generator Γ is
Γ =T
∂ ∂ ∂ ∂ ∂ ∂ = . +X + U (m) +X + U (m) ∂t ∂x ∂ u (m) ∂ t ∂x ∂ u (m) m∈ Z
(4)
m∈ Z
Its first, second and kth prolongation are given respectively by
Γ (1 ) = Γ +
U t (m)
m∈ Z
Γ (2 ) = Γ (1 ) + ···
m∈ Z
Γ (k) = Γ (k−1) +
∂ ∂ + , U x (m) ∂ ut (m) ∂ u x (m) m∈ Z
∂ ∂ ∂ + + , U tt (m) U tx (m) U xx (m) ∂ utt (m) ∂ utx (m) ∂ u xx (m) m∈ Z
k
U t (k−i )x(i ) (m)
m ∈ Z i =0
m∈ Z
∂ ∂ ut (k−i )x(i ) (m)
.
(5)
The coefficients of the above operators are
U t (k−i )x(i ) (m) = D kt −i D ix U (m) − X u x (m) + X ut (k−i )x(i +1) (m).
(6)
Here D t and D x denote the total derivatives. Thus we have the following intrinsic determining equation
Γ (N ) F = F t + X F x +
k N
U t (k−i )x(i ) (m) F ut (k−i)x(i) (m) = 0.
(7)
m ∈ Z k =0 i =0
Next, we prove Eq. (7) can be derived through the compatibility between the original equation (1) and the intrinsic invariant surface condition (2). Theorem 1. The intrinsic determining equations can be obtained from the compatibility between the original equation (1) and the intrinsic invariant surface condition (2) directly. Proof. Total differentiation D t of Eq. (1) gives
Ft +
k N m ∈ Z k =0 i =0
F ut (k−i)x(i) (m) ut (k−i +1)x(i ) (m) = 0.
S. Shen, C. Qu / J. Math. Anal. Appl. 371 (2010) 355–362
357
Through the compatibility between Eq. (1) and the intrinsic invariant surface condition (2), substituting ut (m) = U (m) − X u x (m) into the above equation gives
Ft +
k N
F ut (k−i)x(i) (m) D kt −i D ix U (m) − X u x (m) = 0.
(8)
m ∈ Z k =0 i =0
Adding
X Fx +
k N
F ut (k−i)x(i) (m) X ut (k−i )x(i +1) (m)
m ∈ Z k =0 i =0
to both sides of the above equation, we have
Ft + X F x +
k N
F ut (k−i)x(i) (m) D kt −i D ix U (m) − X u x (m) + X ut (k−i )x(i +1) (m)
m ∈ Z k =0 i =0 k N
= X Fx +
F ut (k−i)x(i) (m) X ut (k−i )x(i +1) (m).
(9)
m ∈ Z k =0 i =0
By the definition of U t (k−i )x(i ) (m), it follows that
Ft + X F x +
k N
F ut (k−i)x(i) (m) U t (k−i )x(i ) (m)
m ∈ Z k =0 i =0 k N
= X Fx +
F ut (k−i)x(i) (m) ut (k−i )x(i +1) (m) .
(10)
m ∈ Z k =0 i =0
Total differentiation D x of Eq. (1) gives
Fx +
k N
F ut (k−i)x(i) (m) ut (k−i )x(i +1) (m) = 0.
m ∈ Z k =0 i =0
Substituting this equation into the right side of Eq. (10), we arrive at Eq. (7).
2
Similarly, we have the following proposition. Proposition 1. The intrinsic determining equations of the (r + 1 + M )-dimensional differential–difference system
), . . . , u st (k−i )x(i ) (m ), . . . = 0 ( j = 1, . . . , s), (jN ) = F j t , x1 , . . . , xr , . . . , u 1t (k−i )x(i ) (m
(11)
can be obtained from the compatibility between the original equation (11) and the intrinsic invariant surface conditions
) = U j x, t , u (m ), m − u j t (m
r
) ( j = 1, 2, . . . , s), X k (x, t )u j x (m
k =1
k
(12)
) are respectively the infinitesimals corresponding to xk and u j (m ), and the infinitesimal corresponding to t directly. Here X k and U j (m is T = 1. Now we use three examples to show that this new method can yield the intrinsic Lie symmetries of the given DDEs without calculating redundant kth order symmetry prolongation. Examples. Consider the (2 + 1)-dimensional Toda lattice [7,8]
u xt (n) = e u (n+1)−u (n) − e u (n)−u (n−1) .
(13)
Firstly, differentiating Eq. (2) with respect to t gives
utt (n) = U t (n) + U u (n) (n)ut (n) − X t u x (n) − X u xt (n). Thus, from the compatibility condition D x utt (n) = D t u xt (n), we obtain
(14)
358
S. Shen, C. Qu / J. Math. Anal. Appl. 371 (2010) 355–362
U xt (n) + U tu (n) (n)u x (n) + U u (n)x (n) + U u (n)u (n) (n)u x (n) ut (n)
+ U u (n) (n)u xt (n) − X xt u x (n) − X t u xx (n) − X x u xt (n) − X u xxt (n) − e u (n+1)−u (n) ut (n + 1) − ut (n) + e u (n)−u (n−1) ut (n) − ut (n − 1) = 0.
(15)
Furthermore, differentiation x of Eq. (13) gives
u xxt (n) = e u (n+1)−u (n) u x (n + 1) − u x (n) − e u (n)−u (n−1) u x (n) − u x (n − 1) .
(16)
Substituting Eqs. (2), (13) and (16) into Eq. (15), we obtain the following equation
U xt (n) + U tu (n) (n)u x (n) + U u (n)x (n) + U u (n)u (n) (n)u x (n) U (n) − X u x (n)
+ U u (n) (n) − X x e u (n+1)−u (n) − e u (n)−u (n−1) − X xt u x (n) − X t u xx (n) − e u (n+1)−u (n) U (n + 1) − U (n) + e u (n)−u (n−1) U (n) − U (n − 1) = 0.
(17)
Similarly from Eq. (2), we also have
u xt (n) = U x (n) + U u (n) (n)u x (n) − X x u x (n) − X u xx (n).
(18)
It follows from Eq. (7) that
e u (n+1)−u (n) − e u (n)−u (n−1) = U x (n) + U u (n) (n)u x (n) − X x u x (n) − X u xx (n). Solving it, we get
u xx (n) = −
1 u (n+1)−u (n) e − e u (n)−u (n−1) − U x (n) − U u (n) (n)u x (n) + X x u x (n) .
(19)
X
Substituting the above equation into Eq. (17) and collecting homogeneous terms of u 2x (n), u 1x (n) and u 0x (n), we deduce the following determined system
U u (n)u (n) (n) = 0,
(20)
U tu (n) (n) − U u (n)x (n) X + U u (n)u (n) (n)U (n) − X tx −
Xt X
X
U u (n) (n) +
Xt X x X
= 0,
(21)
X t u (n+1)−u (n) e − e u (n)−u (n−1) X
U xt (n) + U u (n)x (n)U (n) + U u (n) (n) − X x +
−
Xt
U x (n) − e u (n+1)−u (n) U (n + 1) − U (n) + e u (n)−u (n−1) U (n) − U (n − 1) = 0.
(22)
Solving this system, we arrive at the following intrinsic Lie symmetries of the Toda lattice (13)
T = 1,
X=
g (x) f (t )
,
U (n) = −
f (t ) + g (x) f (t )
n+
h(t ) + k(x)
(23)
f (t )
and this result is equivalent to the intrinsic Lie symmetries obtained in Ref. [8]. Next, we consider the (2 + 1)-dimensional special Toda lattice [25]
u xt (n) = e u (n+1)−u (n) u x (n + 1) + u x (n) − e u (n)−u (n−1) u x (n) + u x (n − 1) .
(24)
Using the above approach, we obtain the intrinsic Lie symmetries given by
T = 1,
X=
g (x) f (t )
,
U (n) = −
f (t ) f (t )
n + h(t ).
(25)
Furthermore we consider the (2 + 1)-dimensional DD-KP equation
ut (n + 1) − ut (n) = u xx (n + 1) + u xx (n) − 2 1 − u (n + 1) u x (n + 1) + 2 1 − u (n) u x (n),
(26)
which is obtained by using the differential–difference version of Sato’s theory [26]. Here, we have the intrinsic Lie symmetries
T = 1,
X = − v 1 (t )x + v 3 (t ),
U (n) = v 1 (t )u (n) +
v 1 (t ) 2
−
v 21 (t )
x−
v 3 (t ) 2
− v 1 (t ) 1 − v 3 (t ) .
(27)
The symmetries obtained by the intrinsic Lie symmetry method in Ref. [26] are equivalent to Eq. (27) since the corresponding characteristic equations are the same by using the transformation v 1 (t ) = − f (t )/(2 f (t )), v 3 (t ) = g (t )/ f (t ).
S. Shen, C. Qu / J. Math. Anal. Appl. 371 (2010) 355–362
359
Notice that if we replace the intrinsic equation (2) by the non-intrinsic equations such as
1 = ut (m) − U m x, t , . . . , u (n − 1), u (n), u (n + 1), . . . = 0, some new results can be derived. For example, Levi and Winternitz [8] have extended the concept of conditional symmetries from differential equations to DDEs and they proved that the Toda lattice equation (13) admits the Bäcklund transformation
u (n) ut (n) + ut (n) = δ e u (n+1)− − e u (n)− u(n−1) ,
u x (n + 1) − u x (n) =
1 u (n)−u (n) . e u (n+1)−u (n+1) + e
(28)
δ
In the following, we construct Bäcklund transformations of some DDEs by using the non-intrinsic direct method. Theorem 2. The (2 + 1)-dimensional DDE
H u (n), . . . , ut (n), . . .
x
= G u (n), . . . , u x (n), . . . f u (n + 1) − u (n) − G u (n − 1), . . . , u x (n − 1), . . . f u (n) − u (n − 1)
(29)
admits the Bäcklund transformation
H u (n), . . . , ut (n), . . . + H u (n), . . . , ut (n), . . . = δ f u (n + 1) − u (n) − f u (n) − u (n − 1) , u x (n) = u x (n + 1) −
1
(30)
G u (n), . . . , u x (n), . . . f u (n + 1) − u (n)
δ f [u (n + 1) − u (n)] +G u (n), . . . , u x (n), . . . f u (n + 1) − u (n) .
(31)
Namely, if u (n) is a solution, then so is u (n). Here H , G , f are the smooth functions of their arguments and δ is an arbitrary parameter. Proof. Differentiating Eq. (30) with respect to x and substituting Eq. (31) into it, one can prove this theorem immediately. 2 Thus, in terms of the above theorem, the Bäcklund transformation of the special Toda equation (24) reads
u (n) ut (n) + ut (n) = δ e u (n+1)− − e u (n)− u(n−1) ,
u x (n + 1) − u x (n) =
1 u (n+1)−u (n+1) e u (n)−u (n) u x (n + 1) + u x (n) + e u x (n + 1) + u x (n) . δ
Similarly, for the modified Toda lattice [13]
u xt (n) = e u (n+1)−u (n) ut (n) − e u (n)−u (n−1) ut (n),
(32)
we can obtain the following Bäcklund transformation
u (n) ln ut (n) + ln ut (n) = δ e u (n+1)− − e u (n)− u(n−1) ,
u x (n + 1) − u x (n) =
1 u (n)−u (n) . e u (n+1)−u (n+1) + e δ
Furthermore, we can show that the DD-KZ equation [9]
ut (n) + u (n)u x (n)
x
+ u (n + 1) − 2u (n) + u (n − 1) = 0,
(33)
admits the Bäcklund transformation
ut (n) + u (n)u x (n) + ut (n) + u (n) u x (n) = −δ u (n + 1) − u (n) − u (n) + u (n − 1) , u x (n + 1) − u x (n) =
1
δ
u (n + 1) − u (n) + u (n + 1) − u (n) .
360
S. Shen, C. Qu / J. Math. Anal. Appl. 371 (2010) 355–362
3. The intrinsic differential–difference CK direct method In this section, we extend the CK direct method to seek intrinsic symmetry reductions of the DD-KP equation (26). Using the intrinsic ansatz
u (x, t , n) = A (x, t ) + B (x, t ) · V ( z, n),
z = z(x, t ), B (x, t ) = 0,
(34)
where A (x, t ), B (x, t ) and z(x, t ) are to be determined. Substituting this equation into Eq. (26), we have
B z2x V zz (n + 1) + 2B 2 zx V (n + 1) V z (n + 1) + 2B x zx + B zxx − 2(1 − A ) B zx − B zt V z (n + 1)
+ 2B B x V 2 (n + 1) + B xx + 2B A x − 2(1 − A ) B x − B t V (n + 1) + B z2x V zz (n) − 2B 2 zx V (n) V z (n) + 2B x zx + B zxx + 2(1 − A ) B zx + B zt V z (n) + B xx − 2B A x + 2(1 − A ) B x + B t V (n) − 2B B x V 2 (n) + 2 A xx = 0.
(35)
The above equation can be further reduced if we set
2 A xx = B z2x Γ1 ( z), B xx − 2B A x + 2(1 − A ) B x + B t = B z2x Γ2 ( z), B xx + 2B A x − 2(1 − A ) B x − B t = B z2x Γ3 ( z),
−2B B x = B z2x Γ4 ( z), 2B B x = B z2x Γ5 ( z),
−2B 2 zx = B z2x Γ6 ( z), 2B 2 zx = B z2x Γ7 ( z), 2B x zx + B zxx + 2(1 − A ) B zx + B zt = B z2x Γ8 ( z), 2B x zx + B zxx − 2(1 − A ) B zx − B zt = B z2x Γ9 ( z),
(36)
where Γi ( z) (i = 1, . . . , 9) are functions of z to be determined. Analogous to the rules used in the CK direct method for PDEs, the following ones are helpful to solve the above equations. 1. We reserve uppercase Greek letters for undetermined functions of z so that after performing operations (differentiation, integration, exponentiation, rescaling, etc.) the result can be denoted by the same letter (e.g., the derivative of Γ ( z) will be called Γ ( z)). 2. If B (x, t ) is shown to have the form B (x, t ) = Bˆ (x, t )Γ ( z), then we may choose Γ ( z) ≡ 1. ˆ (x, t ) + B (x, t )Γ (z), we can put Γ (z) ≡ 0. 3. If A (x, t ) has the form A (x, t ) = A 4. If z(x, t ) is determined from the implicit relation f ( z) = zˆ (x, t ) where f ( z) is an invertible function, then we may simply put f ( z) ≡ z. It is easy to see that B = zx . Using it and solving (36), we obtain
z = xθ1 (t ) + θ2 (t ),
B = θ1 (t ),
A =1+
θ1 (t )x + θ2 (t ) θ1 (t ) − Γ8 ( z). 2θ1 (t ) 2
By using the Rules 1 and 3, we may put Γ8 ( z) = 0 and in the same way, we can obtain Γ9 ( z) = Γ1 ( z) = Γ2 ( z) = Γ3 ( z) = 0. Namely, we have proved the following theorem. Theorem 3. The DD-KP equation (26) has the similarity reduction
u=1+
θ1 (t )x + θ2 (t ) + θ1 (t ) V ( z, n), 2θ1 (t )
z = θ1 (t )x + θ2 (t ),
and the corresponding reduced equation is
V zz (n + 1) + V zz (n) + 2V (n + 1) V z (n + 1) − 2V (n) V z (n) = 0, which is equivalent to
V z (n + 1) + V z (n) + V 2 (n + 1) − V 2 (n) = a(n), where a(n) is an arbitrary function.
(37)
S. Shen, C. Qu / J. Math. Anal. Appl. 371 (2010) 355–362
361
Remark 1. The reduced equation (37) is just Eq. (4.15) in Ref. [26] and can be translated into the non-autonomous reduced V ( z, n) = V ( z, n) − (a1 /4) z. equation (4.19) by setting In general, to seek for similarity reductions of the DDEs, we can use the simple form (34). However, to find the solution transformations but not similarity reductions, we may use the following expression
u (x, t , n) = F (x, t ) + G (x, t ) · V y (x, t ), z(x, t ), n ,
(38)
where F (x, t ), G (x, t ), y (x, t ) and z(x, t ) are smooth functions of the indicated variables, to be determined. Substituting the above expression into Eq. (26) and requiring that V ( y , z, n) is also a solution of the DD-KP equation, we have
z2x G V zz (n + 1) + V zz (n) + 2G zx y x V yz (n + 1) + V yz (n) + 2G 2 y x − 2G y 2x V (n + 1) V y (n + 1)
+ 2G y 2x − 2G 2 y x V (n) V y (n) + 2G 2 zx V (n + 1) V z (n + 1) − 2G 2 zx V (n) V z (n) − 2G G x V 2 (n) + 2G G x V 2 (n + 1) + 2G x zx + G y 2x + G zxx − 2(1 − F )G zx − G zt V z (n + 1) + 2G x zx − G y 2x + G zxx + 2(1 − F )G zx + G zt V z (n) + 2G x y x + 2G y 2x + G y xx − 2(1 − F )G y x − G yt V y (n + 1) + 2G x y x − 2G y 2x + G y xx + 2(1 − F )G y x + G yt V y (n) + G xx − 2(1 − F )G x + 2G F x − G t V (n + 1) + G xx + 2(1 − F )G x − 2G F x + G t V (n) + 2F xx = 0.
(39)
From Eq. (39), setting the coefficients of the polynomials of V and its derivatives to be zero, we can obtain the following theorem. Theorem 4. If V (x, t , n) is a solution of the DD-KP equation (26) then so is
G (t )x + H (t ) − 2G (t )
u=
2G (t )
t z=
+ 1 + G (t ) V ( y , z, n),
G 2 (tˆ) dtˆ,
y = G (t )x + H (t ).
(40)
4. Conclusion and remarks In summary, it was shown that the intrinsic determining equations of a given DDE can be derived by using the compatibility between the original equation (1) and the intrinsic invariant surface condition (2). Several well-known equations such as the Toda lattice, the special Toda lattice and the DD-KP equation were used to elucidate this method. As the results, the Bäcklund transformations of some DDEs were derived by using the non-intrinsic direct method. Finally, the Clarkson– Kruskal direct method was also extended to find similarity reductions of the DD-KP equation. It is of great importance to explore a software to compute symmetries and perform classification of DDEs [27] based on their symmetries. Acknowledgments We are appreciated to the referee and editor for their constructive suggestions and helpful comments. This work was supported by the China NSF for Distinguished Young Scholars (Grant No. 10925104), the China NSF (Grant No. 10671156) and the NSF of Zhejiang Province (Grant Nos. Y6090383 and Y6090359).
References [1] P.J. Olever, Applications of Lie Groups to Differential Equations, Springer-Verlag, New York, 1986. [2] G.W. Bluman, S. Kumei, Symmetries and Differential Equations, Springer-Verlag, New York, 1989. [3] N.H. Ibragimov, CRC Handbook of Lie Group Analysis of Differential Equations, vol. 1: Symmetries, Exact Solutions and Conservation Laws, CRC Press Inc., Boca Roton, 1994. [4] N.H. Ibragimov, CRC Handbook of Lie Group Analysis of Differential Equations, vol. 2: Symmetries, Exact Solutions and Conservation Laws, CRC Press Inc., Boca Roton, 1995. [5] N.H. Ibragimov, CRC Handbook of Lie Group Analysis of Differential Equations, vol. 3: Symmetries, Exact Solutions and Conservation Laws, CRC Press Inc., Boca Raton, 1996. [6] S. Maeda, The similarity method for difference equations, SIAM J. Appl. Math. 38 (1987) 129–134. [7] D. Levi, P. Winternitz, Continuous symmetries of discrete equations, Phys. Lett. A 152 (1991) 335–338. [8] D. Levi, P. Winternitz, Symmetries and conditional symmetries of differential–difference equations, J. Math. Phys. 34 (1993) 3713–3730. [9] Zhuhan Jiang, Lie symmetries and their local determinacy for a class of differential–difference equations, Phys. Lett. A 240 (1998) 137–143. (k)
[10] Zhuhan Jiang, The intrinsicality of Lie symmetries of un (t ) = F n (t , un+a , . . . , un+b ), J. Math. Anal. Appl. 227 (1998) 396–419. [11] D. Levi, P. Winternitz, Continuous symmetries of difference equations, J. Phys. A 39 (2006) R1–R63. [12] R. Yamilov, Symmetries as integrability criteria for differential difference equations, J. Phys. A 39 (2006) R541–R623.
362
S. Shen, C. Qu / J. Math. Anal. Appl. 371 (2010) 355–362
[13] Shoufeng Shen, Lie symmetry reductions and exact solutions of some differential–difference equations, J. Phys. A 40 (2007) 1775–1783. [14] G. Baumann, MathLie a program of doing symmetry analysis, Math. Comput. Simulation 48 (1998) 205–223. [15] J. Buthcher, J. Carminati, K.T. Vu, A comparative study of some computer algebra packages which determine the Lie point symmetries of differential equations, Comput. Phys. Comm. 155 (2003) 92–114. [16] P.A. Clarkson, M.D. Kruskal, New similarity solutions of the Boussinesq equation, J. Math. Phys. 30 (1989) 2201–2213. [17] Shoufeng Shen, Clarkson–Kruskal direct similarity approach for differential–difference equations, Commun. Theor. Phys. 44 (2005) 964–966. [18] N. Joshi, Direct delay reductions of the Toda equation, J. Phys. A 42 (2009) 022001 (8 pp.). [19] Senyue Lou, Hongcai Ma, Non-Lie symmetry groups of (2 + 1)-dimensional nonlinear systems obtained from a simple direct method, J. Phys. A 38 (2005) L129–L137. [20] E. Pucci, G. Saccomandi, On the weak symmetry groups of partial differential equations, J. Math. Anal. Appl. 163 (1992) 588–598. [21] W. Seiler, Involution and symmetry reductions, Math. Comput. Modelling 25 (8–9) (1997) 63–73. [22] D.J. Arrigo, J.R. Beckham, Nonclassical symmetries of evolutionary partial differential equations and compatibility, J. Math. Anal. Appl. 289 (2004) 55–65. [23] Xiaohua Niu, Zuliang Pan, Nonclassical symmetries of a class of nonlinear partial differential equations with arbitrary order and compatibility, J. Math. Anal. Appl. 311 (2005) 479–488. [24] Xiaohua Niu, Lidu Huang, Zuliang Pan, The determining equations for the nonclassical method of the nonlinear differential equation(s) with arbitrary order can be obtained through the compatibility, J. Math. Anal. Appl. 320 (2006) 499–509. [25] Cewen Cao, Xianguo Geng, Yongtang Wu, From the special 2 + 1 Toda lattice to the Kadomtsev–Petviashvili equation, J. Phys. A 32 (1999) 8059–8078. [26] K.M. Tamizhmani, S. Kanagavel, B. Grammaticos, A. Ramani, Singularity structure and algebraic properties of the differential–difference Kadomtsev– Petviashvili equation, Chaos Solitons Fractals 11 (2000) 1423–1431. [27] F. Güngör, V.I. Lahno, R.Z. Zhdanov, Symmetry classification of KdV-type nonlinear evolution equations, J. Math. Phys. 45 (2004) 2280–2313.
Contents lists available at ScienceDirect
Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa
Symmetry groups of differential–difference equations and their compatibility Shen Shoufeng a,b , Qu Changzheng a,c,∗ a b c
Center for Nonlinear Studies, Northwest University, Xi’an 710069, China Department of Applied Mathematics, Zhejiang University of Technology, Hangzhou 310023, China Department of Mathematics, Northwest University, Xi’an 710069, China
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 28 December 2009 Available online 12 May 2010 Submitted by P. Broadbridge
It is shown that the intrinsic determining equations of a given differential–difference equation (DDE) can be derived by the compatibility between the original equation and the intrinsic invariant surface condition. The (2 + 1)-dimensional Toda lattice, the special Toda lattice and the DD-KP equation serving as examples are used to illustrate this approach. Then, Bäcklund transformations of the (2 + 1)-dimensional DDEs including the special Toda lattice, the modified Toda lattice and the DD-KZ equation are presented by using the nonintrinsic direct method. In addition, the Clarkson–Kruskal direct method is developed to find similarity reductions of the DDEs. © 2010 Elsevier Inc. All rights reserved.
Keywords: Symmetry Differential–difference equation Compatibility Bäcklund transformation
1. Introduction Symmetry group method [1–5] introduced by Sophus Lie in the later part of 19th century has played an important role in the reduction of differential equations and the construction of their exact solutions. Lie’s symmetry method for continuous differential equations has by now been well established, though the same for differential–difference equations (DDEs) or difference equations is much less investigated or understood. However some important results [6–13] have been obtained. For instance, Maeda has proposed a method to obtain the point symmetries of ordinary difference equations in Ref. [6]. More interestingly, the intrinsic symmetry method for computing symmetries of the (1 + 1)-dimensional and (2 + 1)-dimensional Toda lattice was derived by Levi and Winternitz [7,8], where “intrinsic” means that the Lie symmetry operator takes the form
Γ = T (x, t )
∂ ∂ ∂ + X (x, t ) + U x, t , u (m), m . ∂t ∂x ∂ u (m) m∈ Z
The software packages such as MathLie [14] and DESOLV [15] have been explored for computing continuous differential equations with finite variables. However, to the best of our knowledge, there are no similar means for DDEs where the main difficulty in computing symmetries of DDEs is to deal with infinitely many variables [9,10]. In Ref. [16], Clarkson and Kruskal (CK) introduced a direct and effective method to derive similarity reductions of partial differential equations without using any group theoretical techniques. Soon afterwards this method was applied to solve some important DDEs such as the (2 + 1)-dimensional Toda lattice [17,18]. Interestingly, the modified CK direct method [19] can be used to compute symmetry groups for continuous differential equations.
*
Corresponding author at: Department of Mathematics, Northwest University, Xi’an 710069, China. E-mail addresses: [email protected] (S. Shen), [email protected] (C. Qu).
0022-247X/$ – see front matter doi:10.1016/j.jmaa.2010.05.025
©
2010 Elsevier Inc. All rights reserved.
356
S. Shen, C. Qu / J. Math. Anal. Appl. 371 (2010) 355–362
Recently, inspired by Pucci, Saccomandi and Seiler’s related works [20,21], Arrigo et al. [22–24] presented a new method that nonclassical symmetries can be derived from the compatibility between the invariant surface conditions and equations under consideration. Such method has been used to obtain the symmetries of Burgers equation, Klein–Gordon equation, Boussinesq equation and the Nizhnik–Novikov–Veselov equation. Thus, a natural and important question is how this method can be applied to DDEs? The purpose of this paper is to develop this method to solve DDEs. The paper is organized as follows. In Section 2, we show that the intrinsic determining equations of DDEs can be derived by the compatibility between the original equation and the intrinsic invariant surface condition, some DDEs are used to illustrate the approach. Then, the Bäcklund transformations of some DDEs are presented by using the non-intrinsic direct method. In Section 3, the CK direct reductions of the DD-KP equation are obtained. Some conclusions are given in Section 4. 2. The new method and examples Consider the (2 + 1)-dimensional Nth order DDE
( N ) = F t , x, . . . , ut (k−i )x(i ) (n − 1), ut (k−i )x(i ) (n), ut (k−i )x(i ) (n + 1), . . . = 0,
(1)
where F is a smooth function of its arguments, ut (k−i )x(i ) (m) = ∂ k u (m)/∂ t k−i ∂ xi . The intrinsic invariant surface condition with T = 1 (as general) is
1 = ut (m) + X (x, t )u x (m) − U x, t , u (m), m = 0.
(2)
The intrinsic determining equations of Eq. (1) can be obtained by requiring that
Γ ( N ) ( N ) (N ) =0,
1 =0
= 0,
(3)
where the infinitesimal generator Γ is
Γ =T
∂ ∂ ∂ ∂ ∂ ∂ = . +X + U (m) +X + U (m) ∂t ∂x ∂ u (m) ∂ t ∂x ∂ u (m) m∈ Z
(4)
m∈ Z
Its first, second and kth prolongation are given respectively by
Γ (1 ) = Γ +
U t (m)
m∈ Z
Γ (2 ) = Γ (1 ) + ···
m∈ Z
Γ (k) = Γ (k−1) +
∂ ∂ + , U x (m) ∂ ut (m) ∂ u x (m) m∈ Z
∂ ∂ ∂ + + , U tt (m) U tx (m) U xx (m) ∂ utt (m) ∂ utx (m) ∂ u xx (m) m∈ Z
k
U t (k−i )x(i ) (m)
m ∈ Z i =0
m∈ Z
∂ ∂ ut (k−i )x(i ) (m)
.
(5)
The coefficients of the above operators are
U t (k−i )x(i ) (m) = D kt −i D ix U (m) − X u x (m) + X ut (k−i )x(i +1) (m).
(6)
Here D t and D x denote the total derivatives. Thus we have the following intrinsic determining equation
Γ (N ) F = F t + X F x +
k N
U t (k−i )x(i ) (m) F ut (k−i)x(i) (m) = 0.
(7)
m ∈ Z k =0 i =0
Next, we prove Eq. (7) can be derived through the compatibility between the original equation (1) and the intrinsic invariant surface condition (2). Theorem 1. The intrinsic determining equations can be obtained from the compatibility between the original equation (1) and the intrinsic invariant surface condition (2) directly. Proof. Total differentiation D t of Eq. (1) gives
Ft +
k N m ∈ Z k =0 i =0
F ut (k−i)x(i) (m) ut (k−i +1)x(i ) (m) = 0.
S. Shen, C. Qu / J. Math. Anal. Appl. 371 (2010) 355–362
357
Through the compatibility between Eq. (1) and the intrinsic invariant surface condition (2), substituting ut (m) = U (m) − X u x (m) into the above equation gives
Ft +
k N
F ut (k−i)x(i) (m) D kt −i D ix U (m) − X u x (m) = 0.
(8)
m ∈ Z k =0 i =0
Adding
X Fx +
k N
F ut (k−i)x(i) (m) X ut (k−i )x(i +1) (m)
m ∈ Z k =0 i =0
to both sides of the above equation, we have
Ft + X F x +
k N
F ut (k−i)x(i) (m) D kt −i D ix U (m) − X u x (m) + X ut (k−i )x(i +1) (m)
m ∈ Z k =0 i =0 k N
= X Fx +
F ut (k−i)x(i) (m) X ut (k−i )x(i +1) (m).
(9)
m ∈ Z k =0 i =0
By the definition of U t (k−i )x(i ) (m), it follows that
Ft + X F x +
k N
F ut (k−i)x(i) (m) U t (k−i )x(i ) (m)
m ∈ Z k =0 i =0 k N
= X Fx +
F ut (k−i)x(i) (m) ut (k−i )x(i +1) (m) .
(10)
m ∈ Z k =0 i =0
Total differentiation D x of Eq. (1) gives
Fx +
k N
F ut (k−i)x(i) (m) ut (k−i )x(i +1) (m) = 0.
m ∈ Z k =0 i =0
Substituting this equation into the right side of Eq. (10), we arrive at Eq. (7).
2
Similarly, we have the following proposition. Proposition 1. The intrinsic determining equations of the (r + 1 + M )-dimensional differential–difference system
), . . . , u st (k−i )x(i ) (m ), . . . = 0 ( j = 1, . . . , s), (jN ) = F j t , x1 , . . . , xr , . . . , u 1t (k−i )x(i ) (m
(11)
can be obtained from the compatibility between the original equation (11) and the intrinsic invariant surface conditions
) = U j x, t , u (m ), m − u j t (m
r
) ( j = 1, 2, . . . , s), X k (x, t )u j x (m
k =1
k
(12)
) are respectively the infinitesimals corresponding to xk and u j (m ), and the infinitesimal corresponding to t directly. Here X k and U j (m is T = 1. Now we use three examples to show that this new method can yield the intrinsic Lie symmetries of the given DDEs without calculating redundant kth order symmetry prolongation. Examples. Consider the (2 + 1)-dimensional Toda lattice [7,8]
u xt (n) = e u (n+1)−u (n) − e u (n)−u (n−1) .
(13)
Firstly, differentiating Eq. (2) with respect to t gives
utt (n) = U t (n) + U u (n) (n)ut (n) − X t u x (n) − X u xt (n). Thus, from the compatibility condition D x utt (n) = D t u xt (n), we obtain
(14)
358
S. Shen, C. Qu / J. Math. Anal. Appl. 371 (2010) 355–362
U xt (n) + U tu (n) (n)u x (n) + U u (n)x (n) + U u (n)u (n) (n)u x (n) ut (n)
+ U u (n) (n)u xt (n) − X xt u x (n) − X t u xx (n) − X x u xt (n) − X u xxt (n) − e u (n+1)−u (n) ut (n + 1) − ut (n) + e u (n)−u (n−1) ut (n) − ut (n − 1) = 0.
(15)
Furthermore, differentiation x of Eq. (13) gives
u xxt (n) = e u (n+1)−u (n) u x (n + 1) − u x (n) − e u (n)−u (n−1) u x (n) − u x (n − 1) .
(16)
Substituting Eqs. (2), (13) and (16) into Eq. (15), we obtain the following equation
U xt (n) + U tu (n) (n)u x (n) + U u (n)x (n) + U u (n)u (n) (n)u x (n) U (n) − X u x (n)
+ U u (n) (n) − X x e u (n+1)−u (n) − e u (n)−u (n−1) − X xt u x (n) − X t u xx (n) − e u (n+1)−u (n) U (n + 1) − U (n) + e u (n)−u (n−1) U (n) − U (n − 1) = 0.
(17)
Similarly from Eq. (2), we also have
u xt (n) = U x (n) + U u (n) (n)u x (n) − X x u x (n) − X u xx (n).
(18)
It follows from Eq. (7) that
e u (n+1)−u (n) − e u (n)−u (n−1) = U x (n) + U u (n) (n)u x (n) − X x u x (n) − X u xx (n). Solving it, we get
u xx (n) = −
1 u (n+1)−u (n) e − e u (n)−u (n−1) − U x (n) − U u (n) (n)u x (n) + X x u x (n) .
(19)
X
Substituting the above equation into Eq. (17) and collecting homogeneous terms of u 2x (n), u 1x (n) and u 0x (n), we deduce the following determined system
U u (n)u (n) (n) = 0,
(20)
U tu (n) (n) − U u (n)x (n) X + U u (n)u (n) (n)U (n) − X tx −
Xt X
X
U u (n) (n) +
Xt X x X
= 0,
(21)
X t u (n+1)−u (n) e − e u (n)−u (n−1) X
U xt (n) + U u (n)x (n)U (n) + U u (n) (n) − X x +
−
Xt
U x (n) − e u (n+1)−u (n) U (n + 1) − U (n) + e u (n)−u (n−1) U (n) − U (n − 1) = 0.
(22)
Solving this system, we arrive at the following intrinsic Lie symmetries of the Toda lattice (13)
T = 1,
X=
g (x) f (t )
,
U (n) = −
f (t ) + g (x) f (t )
n+
h(t ) + k(x)
(23)
f (t )
and this result is equivalent to the intrinsic Lie symmetries obtained in Ref. [8]. Next, we consider the (2 + 1)-dimensional special Toda lattice [25]
u xt (n) = e u (n+1)−u (n) u x (n + 1) + u x (n) − e u (n)−u (n−1) u x (n) + u x (n − 1) .
(24)
Using the above approach, we obtain the intrinsic Lie symmetries given by
T = 1,
X=
g (x) f (t )
,
U (n) = −
f (t ) f (t )
n + h(t ).
(25)
Furthermore we consider the (2 + 1)-dimensional DD-KP equation
ut (n + 1) − ut (n) = u xx (n + 1) + u xx (n) − 2 1 − u (n + 1) u x (n + 1) + 2 1 − u (n) u x (n),
(26)
which is obtained by using the differential–difference version of Sato’s theory [26]. Here, we have the intrinsic Lie symmetries
T = 1,
X = − v 1 (t )x + v 3 (t ),
U (n) = v 1 (t )u (n) +
v 1 (t ) 2
−
v 21 (t )
x−
v 3 (t ) 2
− v 1 (t ) 1 − v 3 (t ) .
(27)
The symmetries obtained by the intrinsic Lie symmetry method in Ref. [26] are equivalent to Eq. (27) since the corresponding characteristic equations are the same by using the transformation v 1 (t ) = − f (t )/(2 f (t )), v 3 (t ) = g (t )/ f (t ).
S. Shen, C. Qu / J. Math. Anal. Appl. 371 (2010) 355–362
359
Notice that if we replace the intrinsic equation (2) by the non-intrinsic equations such as
1 = ut (m) − U m x, t , . . . , u (n − 1), u (n), u (n + 1), . . . = 0, some new results can be derived. For example, Levi and Winternitz [8] have extended the concept of conditional symmetries from differential equations to DDEs and they proved that the Toda lattice equation (13) admits the Bäcklund transformation
u (n) ut (n) + ut (n) = δ e u (n+1)− − e u (n)− u(n−1) ,
u x (n + 1) − u x (n) =
1 u (n)−u (n) . e u (n+1)−u (n+1) + e
(28)
δ
In the following, we construct Bäcklund transformations of some DDEs by using the non-intrinsic direct method. Theorem 2. The (2 + 1)-dimensional DDE
H u (n), . . . , ut (n), . . .
x
= G u (n), . . . , u x (n), . . . f u (n + 1) − u (n) − G u (n − 1), . . . , u x (n − 1), . . . f u (n) − u (n − 1)
(29)
admits the Bäcklund transformation
H u (n), . . . , ut (n), . . . + H u (n), . . . , ut (n), . . . = δ f u (n + 1) − u (n) − f u (n) − u (n − 1) , u x (n) = u x (n + 1) −
1
(30)
G u (n), . . . , u x (n), . . . f u (n + 1) − u (n)
δ f [u (n + 1) − u (n)] +G u (n), . . . , u x (n), . . . f u (n + 1) − u (n) .
(31)
Namely, if u (n) is a solution, then so is u (n). Here H , G , f are the smooth functions of their arguments and δ is an arbitrary parameter. Proof. Differentiating Eq. (30) with respect to x and substituting Eq. (31) into it, one can prove this theorem immediately. 2 Thus, in terms of the above theorem, the Bäcklund transformation of the special Toda equation (24) reads
u (n) ut (n) + ut (n) = δ e u (n+1)− − e u (n)− u(n−1) ,
u x (n + 1) − u x (n) =
1 u (n+1)−u (n+1) e u (n)−u (n) u x (n + 1) + u x (n) + e u x (n + 1) + u x (n) . δ
Similarly, for the modified Toda lattice [13]
u xt (n) = e u (n+1)−u (n) ut (n) − e u (n)−u (n−1) ut (n),
(32)
we can obtain the following Bäcklund transformation
u (n) ln ut (n) + ln ut (n) = δ e u (n+1)− − e u (n)− u(n−1) ,
u x (n + 1) − u x (n) =
1 u (n)−u (n) . e u (n+1)−u (n+1) + e δ
Furthermore, we can show that the DD-KZ equation [9]
ut (n) + u (n)u x (n)
x
+ u (n + 1) − 2u (n) + u (n − 1) = 0,
(33)
admits the Bäcklund transformation
ut (n) + u (n)u x (n) + ut (n) + u (n) u x (n) = −δ u (n + 1) − u (n) − u (n) + u (n − 1) , u x (n + 1) − u x (n) =
1
δ
u (n + 1) − u (n) + u (n + 1) − u (n) .
360
S. Shen, C. Qu / J. Math. Anal. Appl. 371 (2010) 355–362
3. The intrinsic differential–difference CK direct method In this section, we extend the CK direct method to seek intrinsic symmetry reductions of the DD-KP equation (26). Using the intrinsic ansatz
u (x, t , n) = A (x, t ) + B (x, t ) · V ( z, n),
z = z(x, t ), B (x, t ) = 0,
(34)
where A (x, t ), B (x, t ) and z(x, t ) are to be determined. Substituting this equation into Eq. (26), we have
B z2x V zz (n + 1) + 2B 2 zx V (n + 1) V z (n + 1) + 2B x zx + B zxx − 2(1 − A ) B zx − B zt V z (n + 1)
+ 2B B x V 2 (n + 1) + B xx + 2B A x − 2(1 − A ) B x − B t V (n + 1) + B z2x V zz (n) − 2B 2 zx V (n) V z (n) + 2B x zx + B zxx + 2(1 − A ) B zx + B zt V z (n) + B xx − 2B A x + 2(1 − A ) B x + B t V (n) − 2B B x V 2 (n) + 2 A xx = 0.
(35)
The above equation can be further reduced if we set
2 A xx = B z2x Γ1 ( z), B xx − 2B A x + 2(1 − A ) B x + B t = B z2x Γ2 ( z), B xx + 2B A x − 2(1 − A ) B x − B t = B z2x Γ3 ( z),
−2B B x = B z2x Γ4 ( z), 2B B x = B z2x Γ5 ( z),
−2B 2 zx = B z2x Γ6 ( z), 2B 2 zx = B z2x Γ7 ( z), 2B x zx + B zxx + 2(1 − A ) B zx + B zt = B z2x Γ8 ( z), 2B x zx + B zxx − 2(1 − A ) B zx − B zt = B z2x Γ9 ( z),
(36)
where Γi ( z) (i = 1, . . . , 9) are functions of z to be determined. Analogous to the rules used in the CK direct method for PDEs, the following ones are helpful to solve the above equations. 1. We reserve uppercase Greek letters for undetermined functions of z so that after performing operations (differentiation, integration, exponentiation, rescaling, etc.) the result can be denoted by the same letter (e.g., the derivative of Γ ( z) will be called Γ ( z)). 2. If B (x, t ) is shown to have the form B (x, t ) = Bˆ (x, t )Γ ( z), then we may choose Γ ( z) ≡ 1. ˆ (x, t ) + B (x, t )Γ (z), we can put Γ (z) ≡ 0. 3. If A (x, t ) has the form A (x, t ) = A 4. If z(x, t ) is determined from the implicit relation f ( z) = zˆ (x, t ) where f ( z) is an invertible function, then we may simply put f ( z) ≡ z. It is easy to see that B = zx . Using it and solving (36), we obtain
z = xθ1 (t ) + θ2 (t ),
B = θ1 (t ),
A =1+
θ1 (t )x + θ2 (t ) θ1 (t ) − Γ8 ( z). 2θ1 (t ) 2
By using the Rules 1 and 3, we may put Γ8 ( z) = 0 and in the same way, we can obtain Γ9 ( z) = Γ1 ( z) = Γ2 ( z) = Γ3 ( z) = 0. Namely, we have proved the following theorem. Theorem 3. The DD-KP equation (26) has the similarity reduction
u=1+
θ1 (t )x + θ2 (t ) + θ1 (t ) V ( z, n), 2θ1 (t )
z = θ1 (t )x + θ2 (t ),
and the corresponding reduced equation is
V zz (n + 1) + V zz (n) + 2V (n + 1) V z (n + 1) − 2V (n) V z (n) = 0, which is equivalent to
V z (n + 1) + V z (n) + V 2 (n + 1) − V 2 (n) = a(n), where a(n) is an arbitrary function.
(37)
S. Shen, C. Qu / J. Math. Anal. Appl. 371 (2010) 355–362
361
Remark 1. The reduced equation (37) is just Eq. (4.15) in Ref. [26] and can be translated into the non-autonomous reduced V ( z, n) = V ( z, n) − (a1 /4) z. equation (4.19) by setting In general, to seek for similarity reductions of the DDEs, we can use the simple form (34). However, to find the solution transformations but not similarity reductions, we may use the following expression
u (x, t , n) = F (x, t ) + G (x, t ) · V y (x, t ), z(x, t ), n ,
(38)
where F (x, t ), G (x, t ), y (x, t ) and z(x, t ) are smooth functions of the indicated variables, to be determined. Substituting the above expression into Eq. (26) and requiring that V ( y , z, n) is also a solution of the DD-KP equation, we have
z2x G V zz (n + 1) + V zz (n) + 2G zx y x V yz (n + 1) + V yz (n) + 2G 2 y x − 2G y 2x V (n + 1) V y (n + 1)
+ 2G y 2x − 2G 2 y x V (n) V y (n) + 2G 2 zx V (n + 1) V z (n + 1) − 2G 2 zx V (n) V z (n) − 2G G x V 2 (n) + 2G G x V 2 (n + 1) + 2G x zx + G y 2x + G zxx − 2(1 − F )G zx − G zt V z (n + 1) + 2G x zx − G y 2x + G zxx + 2(1 − F )G zx + G zt V z (n) + 2G x y x + 2G y 2x + G y xx − 2(1 − F )G y x − G yt V y (n + 1) + 2G x y x − 2G y 2x + G y xx + 2(1 − F )G y x + G yt V y (n) + G xx − 2(1 − F )G x + 2G F x − G t V (n + 1) + G xx + 2(1 − F )G x − 2G F x + G t V (n) + 2F xx = 0.
(39)
From Eq. (39), setting the coefficients of the polynomials of V and its derivatives to be zero, we can obtain the following theorem. Theorem 4. If V (x, t , n) is a solution of the DD-KP equation (26) then so is
G (t )x + H (t ) − 2G (t )
u=
2G (t )
t z=
+ 1 + G (t ) V ( y , z, n),
G 2 (tˆ) dtˆ,
y = G (t )x + H (t ).
(40)
4. Conclusion and remarks In summary, it was shown that the intrinsic determining equations of a given DDE can be derived by using the compatibility between the original equation (1) and the intrinsic invariant surface condition (2). Several well-known equations such as the Toda lattice, the special Toda lattice and the DD-KP equation were used to elucidate this method. As the results, the Bäcklund transformations of some DDEs were derived by using the non-intrinsic direct method. Finally, the Clarkson– Kruskal direct method was also extended to find similarity reductions of the DD-KP equation. It is of great importance to explore a software to compute symmetries and perform classification of DDEs [27] based on their symmetries. Acknowledgments We are appreciated to the referee and editor for their constructive suggestions and helpful comments. This work was supported by the China NSF for Distinguished Young Scholars (Grant No. 10925104), the China NSF (Grant No. 10671156) and the NSF of Zhejiang Province (Grant Nos. Y6090383 and Y6090359).
References [1] P.J. Olever, Applications of Lie Groups to Differential Equations, Springer-Verlag, New York, 1986. [2] G.W. Bluman, S. Kumei, Symmetries and Differential Equations, Springer-Verlag, New York, 1989. [3] N.H. Ibragimov, CRC Handbook of Lie Group Analysis of Differential Equations, vol. 1: Symmetries, Exact Solutions and Conservation Laws, CRC Press Inc., Boca Roton, 1994. [4] N.H. Ibragimov, CRC Handbook of Lie Group Analysis of Differential Equations, vol. 2: Symmetries, Exact Solutions and Conservation Laws, CRC Press Inc., Boca Roton, 1995. [5] N.H. Ibragimov, CRC Handbook of Lie Group Analysis of Differential Equations, vol. 3: Symmetries, Exact Solutions and Conservation Laws, CRC Press Inc., Boca Raton, 1996. [6] S. Maeda, The similarity method for difference equations, SIAM J. Appl. Math. 38 (1987) 129–134. [7] D. Levi, P. Winternitz, Continuous symmetries of discrete equations, Phys. Lett. A 152 (1991) 335–338. [8] D. Levi, P. Winternitz, Symmetries and conditional symmetries of differential–difference equations, J. Math. Phys. 34 (1993) 3713–3730. [9] Zhuhan Jiang, Lie symmetries and their local determinacy for a class of differential–difference equations, Phys. Lett. A 240 (1998) 137–143. (k)
[10] Zhuhan Jiang, The intrinsicality of Lie symmetries of un (t ) = F n (t , un+a , . . . , un+b ), J. Math. Anal. Appl. 227 (1998) 396–419. [11] D. Levi, P. Winternitz, Continuous symmetries of difference equations, J. Phys. A 39 (2006) R1–R63. [12] R. Yamilov, Symmetries as integrability criteria for differential difference equations, J. Phys. A 39 (2006) R541–R623.
362
S. Shen, C. Qu / J. Math. Anal. Appl. 371 (2010) 355–362
[13] Shoufeng Shen, Lie symmetry reductions and exact solutions of some differential–difference equations, J. Phys. A 40 (2007) 1775–1783. [14] G. Baumann, MathLie a program of doing symmetry analysis, Math. Comput. Simulation 48 (1998) 205–223. [15] J. Buthcher, J. Carminati, K.T. Vu, A comparative study of some computer algebra packages which determine the Lie point symmetries of differential equations, Comput. Phys. Comm. 155 (2003) 92–114. [16] P.A. Clarkson, M.D. Kruskal, New similarity solutions of the Boussinesq equation, J. Math. Phys. 30 (1989) 2201–2213. [17] Shoufeng Shen, Clarkson–Kruskal direct similarity approach for differential–difference equations, Commun. Theor. Phys. 44 (2005) 964–966. [18] N. Joshi, Direct delay reductions of the Toda equation, J. Phys. A 42 (2009) 022001 (8 pp.). [19] Senyue Lou, Hongcai Ma, Non-Lie symmetry groups of (2 + 1)-dimensional nonlinear systems obtained from a simple direct method, J. Phys. A 38 (2005) L129–L137. [20] E. Pucci, G. Saccomandi, On the weak symmetry groups of partial differential equations, J. Math. Anal. Appl. 163 (1992) 588–598. [21] W. Seiler, Involution and symmetry reductions, Math. Comput. Modelling 25 (8–9) (1997) 63–73. [22] D.J. Arrigo, J.R. Beckham, Nonclassical symmetries of evolutionary partial differential equations and compatibility, J. Math. Anal. Appl. 289 (2004) 55–65. [23] Xiaohua Niu, Zuliang Pan, Nonclassical symmetries of a class of nonlinear partial differential equations with arbitrary order and compatibility, J. Math. Anal. Appl. 311 (2005) 479–488. [24] Xiaohua Niu, Lidu Huang, Zuliang Pan, The determining equations for the nonclassical method of the nonlinear differential equation(s) with arbitrary order can be obtained through the compatibility, J. Math. Anal. Appl. 320 (2006) 499–509. [25] Cewen Cao, Xianguo Geng, Yongtang Wu, From the special 2 + 1 Toda lattice to the Kadomtsev–Petviashvili equation, J. Phys. A 32 (1999) 8059–8078. [26] K.M. Tamizhmani, S. Kanagavel, B. Grammaticos, A. Ramani, Singularity structure and algebraic properties of the differential–difference Kadomtsev– Petviashvili equation, Chaos Solitons Fractals 11 (2000) 1423–1431. [27] F. Güngör, V.I. Lahno, R.Z. Zhdanov, Symmetry classification of KdV-type nonlinear evolution equations, J. Math. Phys. 45 (2004) 2280–2313.