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Continuous symmetries of certain nonlinear partial difference equations and their reductions
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Physics Letters A ••• (••••) •••–•••
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Physics Letters A www.elsevier.com/locate/pla
Continuous symmetries of certain nonlinear partial difference equations and their reductions R. Sahadevan ∗ , G. Nagavigneshwari Ramanujan Institute for Advanced Study in Mathematics, University of Madras, Chepauk, Chennai 600 005, Tamilnadu, India
a r t i c l e
i n f o
Article history: Received 3 July 2014 Received in revised form 12 September 2014 Accepted 12 September 2014 Available online xxxx Communicated by C.R. Doering Keywords: Lie point symmetries Integrability Partial difference equation Ordinary difference equation
a b s t r a c t In this article, Quispel, Roberts and Thompson type of nonlinear partial difference equation with two independent variables is considered and identified five distinct nonlinear partial difference equations admitting continuous point symmetries quadratic in the dependent variable. Using the degree growth of iterates the integrability nature of the obtained nonlinear partial difference equations is discussed. It is also shown how to derive higher order ordinary difference equations from the periodic reduction of the identified nonlinear partial difference equations. The integrability nature of the obtained ordinary difference equations is investigated wherever possible. © 2014 Elsevier B.V. All rights reserved.
1. Introduction Lie groups and Lie algebras are mathematical objects which have originated in the seminal work of Sophus Lie (1842–1899) on solving differential equations by quadrature, using symmetry methods. Lie symmetries approach originally introduced by Sophus Lie has been used as a tool to unify various integration techniques for ordinary differential equations (ODEs) and has played a very important role in the study of both ODEs and partial differential equations (PDEs). Symmetry groups are invariant transformations which do not alter the structural form of the equation under investigation. Once the symmetry group of a system of differential equations is known, it can be used to generate new solutions from the old ones, often interesting ones from trivial ones [20]. It can be used to classify solutions into conjugacy classes and to classify and simplify differential equations. An important application of the symmetry approach is the reduction of an ODE to a lower order one, the reduction of a PDE to one with fewer independent variables. The usefulness of Lie symmetry approach has been widely illustrated for a variety of dynamical systems governed by both nonlinear ODEs and PDEs which arise in different contexts [3,8, 9,19,20,27] during the past several decades [3,8,9,19,20,27]. During 1980s Maeda [15–18] has extended Lie symmetries approach to discrete systems governed by nonlinear mappings or ordinary
*
Corresponding author. E-mail addresses: [email protected] (R. Sahadevan), [email protected] (G. Nagavigneshwari). http://dx.doi.org/10.1016/j.physleta.2014.09.021 0375-9601/© 2014 Elsevier B.V. All rights reserved.
difference equations (OEs) and demonstrated how it provides an effective tool to derive their continuous point symmetries. Later on the Lie symmetry approach has been further developed by Levi and Winternitz [10,12], Quispel et al. [22], Levi et al. [11,13] and others [4,7,23,24] to nonlinear discrete systems governed by partial differential–difference equations (PDEs) and OEs. Also, several groups have profitably exploited to derive mathematical structures related with integrability of different nonlinear PEs possessing solitons [1,6,25]. Though the Lie symmetry approach has been extended to discrete nonlinear systems governed by lattice equations or nonlinear partial difference equations (PEs), its effectiveness has not yet been demonstrated widely. The objective of this article is to illustrate its usefulness on other nonlinear PEs. More specifically a scalar nonlinear PE of Quispel, Roberts and Thompson (QRT) type [21]
v (l + 1, m + 1) f 1 ( v (l, m + 1), v (l + 1, m)) − v (l, m) f 2 ( v (l, m + 1), v (l + 1, m)) = f 3 ( v (l, m + 1), v (l + 1, m)) − v (l, m) f 4 ( v (l, m + 1), v (l + 1, m)) is considered and under what conditions on f i it possesses continuous point symmetries quadratic in the dependent variable is investigated. The integrability nature of obtained PEs is analyzed using the degree growth of iterates [26], another characteristic of integrable discrete systems. Also, it is shown how higher order OEs are explicitly derived. The paper is organized as follows. In Section 2, QRT type of nonlinear PE is considered as mentioned above and under what conditions on f i it possesses continuous point symmetries
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2
quadratic in the dependent variable is investigated. In Section 3, the integrability nature of the obtained PEs is analyzed through the degree growth of iterates. In Section 4, higher order OEs are derived from the periodic reduction of the obtained PEs. Also, it is shown that the derived OEs are measure preserving and they admit sufficient number of integrals, if exist, leading to their integrability. In Section 5, a brief summary of the obtained results and concluding remarks are provided. 2. Lie point symmetries of partial difference equations Consider a scalar nonlinear PE of the form
v (l + 1, m + 1) = F v (l, m), v (l, m + 1), v (l + 1, m) ,
(2.1)
where F ( v (l, m), v (l, m + 1), v (l + 1, m)) is an arbitrary function. Let us assume that (2.1) is invariant under a one-parameter ( ) continuous point transformations ∗
∗
l = l,
m = m,
(2.2)
v = v (l, m) + η l, m, v (l, m) + O 2 ∗
(2.3)
with infinitesimal generator
X = η l, m, v (l, m)
2 + c 1 + c 2 v lm+1 + c 3 v lm+1 ∂ f4 ∂ f3 + f 22 l+1 × f 12 l+1 f1 f2 ∂ vm ∂ vm + 2(c 1 f 3 f 4 + c 2 f 1 f 4 + c 3 f 1 f 2 ) = 0, l 2 ∂ f3 2 l f 1 c 1 + c 2 v m +1 + c 3 v m +1 ∂ v lm+1 f 1 2 ∂ f3 + f 12 c 1 + c 2 v lm+1 + c 3 v lm+1 l +1 f1 ∂ vm 2 + c 1 f 3 + f 2 f 3 − f 1 f 4 + c 2 f 1 f 3 + c 3 f 12 = 0.
Case 1: c 1 = 0, c 2 = 0, c 3 = 0
(2.4) f 1 = 1,
≡ v (l, m + 1), ≡ v (l + 1, m + 1)
f3 =
v lm ≡ v (l, m),
F ≡ F v (l, m), v (l, m + 1), v (l + 1, m) , v lm++11
≡ v (l + 1, m),
f2 =
η(l + 1, m + 1, F ) = η l, m
+ 1, v lm+1
v lm++11 =
∂F
∂ v lm+1 ∂F l +1
+ η l + 1, m , v m
+ η l, m, v lm
∂F ∂ v lm
f1 − f3 −
(2.5)
.
v lm f 2 , v lm f 4
(2.6)
2
(2.7)
,
where A (l, m) is an arbitrary function and c i , i = 1, 2, 3 are arbitrary parameters. To start with let A (l, m) = 1. Using (2.7) along with their shifts in Eq. (2.5) and equating powers of ( v lm ) j , j = 0, 1, 2 to zero the following three equations are obtained:
f 22 c 1 + c 2 v lm+1 + c 3 v lm+1
+ +
f 22
c 1 + c 2 v lm+1
c 1 f 42
2
∂ v lm+1
2 + c 3 v lm+1
+ c 2 f 2 f 4 + c 3 f 22
∂ ∂
∂ v lm+1
f4
f2
v lm+1
v lm+1 f4 =
,
, 1
v lm+1 v lm+1
(2.11)
,
f4 f2
+ f2 f3 − f1 f4
m∗ = m,
X = c 1 + c 2 v lm + c 3 v lm
X1 =
2 ∂ ∂ v lm
∂ ∂ v lm
X 2 = v lm
,
∂ ∂ v lm
X 3 = v lm
,
2 ∂ ∂ v lm
.
It is straight forward to check that the above generators satisfy
[ X1, X2] = X1,
[ X1, X3] = 2 X2,
[ X2, X3] = X3
indicating that the underlying Lie algebra of PE (2.12) is not solvable [3]. Case 2: c 1 = 0, c 2 = 0, c 3 = 0 Proceeding as before, we find that Eqs. (2.8)–(2.10) satisfy provided
f3 = (2.8)
+ O 2
leading to the following generators:
= 0,
2
with infinitesimal generator
f 1 = 1,
(2.12)
[av lm+1 + (1 − a) v lm+1 ] − v lm
v ∗ = v lm + c 1 + c 2 v lm + c 3 v lm
where f 1 , f 2 , f 3 and f 4 are arbitrary functions of v lm+1 and v lm+1 . In order to solve the invariance equation (2.5) with (2.6), assume that
(1 − a)
(1 − a)
v lm+1 v lm+1 − v lm [av lm+1 + (1 − a) v lm+1 ]
l∗ = l,
∂ v lm+1
η l, m, v lm = A (l, m) c1 + c2 v lm + c3 v lm
+
+
which is invariant under
Eq. (2.5) is a functional difference equation and there is no known method to solve it. In the present work, the investigation is restricted to QRT type nonlinear PE having the form
F=
a v lm+1
a v lm+1
where a is an arbitrary parameter and so the QRT PEs (2.6) becomes
unless otherwise specified. Then the invariant equation reads
(2.10)
Then there exist different possibilities which will be discussed below separately.
provided any solution v (l, m) satisfies (2.1). Hereafter denoting
v lm+1
(2.9)
After a detailed calculation it is found that Eqs. (2.8)–(2.10) satisfy identically provided
∂ ∂ v (l, m)
v lm+1
2
c 1 + c 2 v lm+1 + c 3 v lm+1 ∂ f4 ∂ f3 2 2 + f2 l × f1 l f ∂ v m +1 ∂ v m +1 f 2 1
a v lm+1
f2 =
+
a v lm+1
(1 − a) v lm+1
,
+
(1 − a) v lm+1 f4 =
, 1
v lm+1 v lm+1
+b
v lm+1 − v lm+1
2
v lm+1 v lm+1
(2.13)
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where a and b are arbitrary parameters and so the nonlinear PE (2.6) becomes v lm+1 v lm+1 [ v lm+1 v lm+1 − v lm [av lm+1 + (1 − a) v lm+1 ]] l +1 l v m+1 v m [av lm+1 + (1 − a) v lm+1 ] − v lm [ v lm+1 v lm+1 + b( v lm+1 −
v lm++11 =
v lm+1 )2 ]
.
proceeding as before it is found that the invariance equation (2.5) is satisfied when f 2 = 1, f 3 = 1 and f 4 = − f 1 and
f1 =
(2.14)
∂
∂ ∂ v lm
∂ v lm
X 2 = v lm
,
1 − v lm+1 v lm+1
v lm++11 =
and so the following generators are obtained
X 1 = v lm
v lm+1 + v lm+1
(2.19)
.
Hence Eq. (2.6) becomes
Here the infinitesimal generator X reads
X = v lm c 2 + c 3 v lm
3
( v lm+1 + v lm+1 ) − v lm (1 − v lm+1 v lm+1 ) (1 − v lm+1 v lm+1 ) + v lm ( v lm+1 + v lm+1 )
(2.20)
with infinitesimal generator
2 ∂
X=
∂ v lm
α + β(l + m) 1 + v lm
2 ∂ ∂ v lm
.
satisfying [ X 1 , X 2 ] = X 2 indicating that the underlying Lie algebra of Eq. (2.14) is solvable [3].
It is crucial to mention here that for other choices of A (l, m) different nonlinear PEs can be derived. For more details see [4].
Case 3: c 1 = 0, c 2 = 0, c 3 = 1
3. Degree growth of iterates of nonlinear PEs
In this case the invariance equation (2.5) is satisfied when
f 1 = 1, f3 = f4 =
f2 =
a v lm+1
+
a v lm+1
(1 − a) v lm+1
1 v lm+1 v lm+1
+
(1 − a) v lm+1
,
,
+b
v lm+1 − v lm+1
+ c,
v lm+1 v lm+1
(2.15)
where a, b and c are arbitrary parameters. Now Eq. (2.6) on simplifying becomes v lm+1 v lm+1 − v lm (av lm+1 + (1 − a) v lm+1 )
v lm++11 =
(av lm+1
+ (1 − a) v lm+1 ) − v lm [1 + b( v lm+1 − v lm+1 ) + cv lm+1 v lm+1 ]
.
(2.16) The infinitesimal generator of Eq. (2.16) is given by
X = v lm
2 ∂ ∂ v lm
.
Case 4: c 1 = 0, c 2 = 1, c 3 = 0
v l0 =
In a similar manner the invariance equation (2.5) is satisfied when
f 1 = 1, f3 =
f2 =
a v lm+1
+
a v lm+1
(1 − a) v lm+1
,
+
(1 − a) v lm+1 f4 =
, b
( v lm+1 )2
+
c v lm+1 v lm+1
,
(2.17)
v lm+1 [ v lm+1 v lm+1 − v lm (av lm+1 + (1 − a) v lm+1 )]
v lm+1 [(av lm+1 + (1 − a) v lm+1 )] − v lm [bv lm+1 + cv lm+1 ]
,
(2.18) with infinitesimal generator
X = v lm
∂ ∂ v lm
pl q
,
0 vm =
rm q
,
l , m = 0, 1 , 2 . . .
(3.1)
provided p 0 = r0 . Here the homogeneous coordinates q, pl and rm , for l, m = 0, 1, 2, . . . have the same homogeneity degree. For l = 0 and m = 0, Eq. (2.12) becomes
v 11 =
v 01 v 10 − v 00 [av 10 + (1 − a) v 01 ]
[av 01 + (1 − a) v 10 ] − v 00
.
(3.2)
Using Eq. (3.1) in Eq. (3.2), it becomes
where a, b and c are arbitrary parameters. In this case, Eq. (2.6) becomes
v lm++11 =
In this section, the integrability nature of the obtained PEs (2.12), (2.14), (2.16), (2.18) and (2.20) is investigated through the degree growth of iterates. It is widely believed that there exists a deep connection between the degree growth of iterates of nonlinear PEs and their integrability [26]. More specifically if the degree growth of iterates of a PE is polynomial then it may be integrable. If the degree growth of iterates is exponential, then it is nonintegrable. In the case of linearizable equations, the degree growth is linear and hence is slower than that of the integrable equations [26]. Note that the term degree is nothing but the common homogeneity degree of the numerators and denominators of their irreducible forms, which is obtained by introducing homogeneous coordinates. Once the degree of the first few iterates is computed, the underlying pattern is conjectured and written in the form of an appropriate analytical expression, which can be further confirmed by computing the degree of subsequent iterates. The computational details of the degree growth of iterates of the obtained nonlinear PEs are presented below. It is explained briefly how to evaluate the degree growth of iterates of nonlinear PE (2.12). Let the initial conditions along the lines l = 0 and m = 0 be respectively denoted by
.
Next assuming that A (l, m) given in Eq. (2.7) is linear, that is A (l, m) = α + β(l + m), where α and β are arbitrary parameters,
v 11 =
r1 p 1 q q
−
r0 [a pq1 q
+ (1 − a) rq1 ]
[a rq1 + (1 − a) pq1 ] −
r0 q
which can be rewritten as
v 11 =
r1 p 1 − r0 [ap 1 + (1 − a)r1 ]
(3.3)
q[ar1 + (1 − a) p 1 ] − r0 q
indicating that the degree of v 11 is 2. Next for l = 0 and m = 1 Eq. (2.12) becomes
v 12 =
v 02 v 11 − v 01 [av 11 + (1 − a) v 02 ]
[av 02 + (1 − a) v 11 ] − v 01
.
(3.4)
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dlm = 2 max(l, m) min(l, m) + 1,
Using (3.1)–(3.3) in (3.4) and rearranging
v 12 =
(r2 − ar1 )[r1 p 1 − r0 [ap 1 + (1 − a)r1 ]] − (1 − a)r1 r2 [ar1 + (1 − a) p 1 − r0 ] . q[(ar2 − r1 )[ar1 + (1 − a) p 1 − r0 ] − (1 − a)[r1 p 1 − r0 [ap 1 + (1 − a)r1 ]]]
(3.5)
a polynomial one. At the moment we are unable to present the growth of iterations of PE (2.18) in an analytic expression. The degree growth of iterates of PE (2.20) is the following
v 12
It is clear that degree of is 3. In a similar way, for different values of l and m, the degrees of various iterates are computed as mentioned hereunder:
.. .
⏐ m⏐ ⏐
1 1
3 5 2 3
7 4
1
1
1
1
⏐ m⏐ ⏐
dlm = lm + 1, which is a polynomial. Here dlm denotes the degree of the (l, m)-entry in the lattice. Next the degree growth of iterates of Eq. (2.14) is calculated below. For l = 0 and m = 0 Eq. (2.14) becomes
v 01 v 10 [ v 01 v 10 − v 00 [av 10 + (1 − a) v 01 ]] v 01 v 10 [av 01 + (1 − a) v 10 ] − v 00 [ v 01 v 10 + b( v 10 − v 01 )2 ]
.
(3.6)
r1 p 1 [r1 p 1 − r0 [ap 1 + (1 − a)r1 ]] q[r1 p 1 [ar1 + (1 − a) p 1 ] − r0 [r1 p 1 + b( p 1 − r1 )2 ]]
(3.7)
9 27 4 9
1
1
1
1
dlm =
1, ml = 0 3, ml = 0
Hence Eq. (2.20) exhibits a linear growth indicating that it is linearizable. It is appropriate to mention here that Levi and Scimiterna [14] have derived a set of necessary conditions for a nonlinear P E to be linearizable and Eq. (2.20) satisfies those conditions [14]. 4. Reductions to ordinary difference equations Consider a solution v lm of the nonlinear PE (2.1) satisfying the periodicity property
(4.1)
v lm+1 = v n+z1 ,
v lm+1 = v n+z2 ,
v lm++11 = v n+z1 +z2 .
As a consequence Eq. (2.1) can be transformed into a ( z1 + z2 ) order OE, that is
v n+z1 +z2 = F ( v n , v n+z1 , v n+z2 ).
(4.2)
Hereafter z1 = 1 and z2 = z unless or otherwise specified. So Eq. (2.12) on reduction yields a ( z + 1)th order OE
19 1
−→ l From the above, the degree growth of iterates of PE (2.14) is
dlm =
1
where gcd( z1 , z2 ) = 1, z1 , z2 ∈ Z. Here n = mz1 + lz2 and so
,
.. .
1
1
l− z
and so degree of v 11 is 4. Proceeding further along the lines described earlier the degree growth of v lm , l, m = 0, 1, 2, . . . is computed as stated below:
⏐ m⏐ ⏐
3 3 3
v m+2z1 = v lm = v n ,
Substituting Eq. (3.1) in Eq. (3.6),
1 1
3 3 3
and it is inferred that
From the above mentioned degree growth of iterates it is conjectured that the degree growth of iterates of nonlinear PE (2.12) is
v 11 =
3 3 3
l
l
=
1 1 1
−→
−→
v 11
.. .
⎧ ⎨ 1,
ml = 0 m = l = 0 l 2 + m2 l 2 + m2 , ⎩ 4 l + 2l2 + 2l − 1, m = l = 0
v n + z +1 =
v n+1 v n+z − v n [av n+z + (1 − a) v n+1 ]
[av n+1 + (1 − a) v n+z ] − v n
.
(4.3)
Schematic reduction of the above equation with solutions satisfying the periodicity property v lm−+z 1 = v lm = v n is given in Fig. 1. For z = 1 Eq. (4.3) becomes trivial. It is imperative to mention here that higher order OEs can be derived when z ≥ 2. For example if z = 2, a third order OE given by
v n +3 =
v n+1 v n+2 − v n [av n+2 + (1 − a) v n+1 ]
[av n+1 + (1 − a) v n+2 ] − v n
(4.4)
which is a polynomial. Next calculating the degree growth of Eq. (2.16)
.. .
⏐
m⏐ ⏐
1 1
5 9 3 5 7
1
1 1 1
−→ l which can be written as
Fig. 1. Schematic representation of the ( z + 1)th order mapping with solutions satisfying the periodicity property v lm−+z 1 = v lm = v n .
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+ (3a − 4) v n2 v n+1 + (2 − a) v n2 v n+2
is obtained. To derive integrals consider I (n) to take the form
I (n) =
A 1 ( v n , v n+1 ) v n2+2 + A 2 ( v n , v n+1 ) v n+2 + A 3 ( v n , v n+1 ) B 1 ( v n , v n+1 ) v n2+2 + B 2 ( v n , v n+1 ) v n+2 + B 3 ( v n , v n+1 )
+ (2 − a) v n v n2+1 + (4 − 5a) v n v n2+2
,
+ 6(a − 1) v n v n+1 v n+2 + (4 − 5a) v n2+1 v n+2 + (3a − 2) v n+1 v n2+2 v n+3 + (3a − 4) v n2 v n2+2
(4.5) where A i ’s and B i ’s are arbitrary functions. Next for what forms of A i ’s and B i ’s, I (n + 1) − I (n) = 0 holds along with Eq. (4.4) is investigated. After detailed calculation it is found that Eq. (4.4) possesses two independent integrals
I 1 (n) = I 2 (n) =
P 1 (n) P 2 (n) P 3 (n) P 2 (n)
P 1 (n) = (a − 1) v n2+2 − av n+1 + (a − 2) v n v n+2 + av n2+1
+ (a − 1) v n2 − av n v n+1 , P 2 (n) = ( v n − v n+1 ) v n2+2 − ( v n + v n+1 ) v n+2 + v n v n+1 , P 3 (n) = (3a − 2) v n − av n+1 v n2+2 + (2 − a) v n2 − 2(3a − 2) v n v n+1 + (3a − 2) v n2+1 v n+2 + (3a − 4) v n2 v n+1 + (2 − a) v n v n2+1 . The above integrals have also been identified in [5] using Adjoint equation method. It is straightforward to check that Eq. (4.4) is measure preserving with measure 1
[ a−a 1 ( v n+2 + v n )( v n+2 − v n+1 )( v n+1 − v n ) − 2v n+1 ( v n+2 − v n )2 ]
.
Next for z = 3 a fourth order OE given by
v n+1 v n+3 − v n [av n+3 + (1 − a) v n+1 ]
(4.6)
[av n+1 + (1 − a) v n+3 ] − v n
is obtained. Here again integrals having the form
I (n) =
A 1 ( v n , v n+1 , v n+2 ) v n2+3 + A 2 ( v n , v n+1 , v n+2 ) v n+3 + A 3 ( v n , v n+1 , v n+2 ) B 1 ( v n , v n+1 , v n+2 ) v n2+3 + B 2 ( v n , v n+1 , v n+2 ) v n+3 + B 3 ( v n , v n+1 , v n+2 )
,
(4.7) where A i ’s and B i ’s are arbitrary functions, are considered. Proceeding as above and after a detailed calculation, it is clear that OE (4.6) possesses two independent integrals
J 1 (n) = J 2 (n) =
Q 1 (n) Q 2 (n) Q 3 (n)
,
It is easy to show that the OE (4.6) is measure preserving with measure
− 1) v n2+2
In a similar manner, higher order OEs can be derived from the periodic reductions of the other obtained nonlinear PEs namely (2.14), (2.16), (2.18) and (2.20), which are under investigation and will be reported elsewhere. 5. Summary and concluding remarks In this article QRT type of nonlinear PE (2.6) is considered and identified five distinct nonlinear PEs namely (2.12), (2.14), (2.16), (2.18) and (2.20) admitting continuous point symmetries quadratic in the dependent variable. Since there exists no direct connection between the Lie point symmetries and the concept of complete integrability of differential equations or difference equations, the integrability nature of the above-mentioned PEs cannot be ascertained at this point. The nature of integrability of the identified PEs is analyzed through the degree growth of iterates and reported that the degree growth of iterates of each of the PEs (2.12), (2.14) and (2.16) is a polynomial one and hence they are expected to be integrable. It is clear that the above PEs do not fall into the well-known ABS classification [2] which may be derived from some reductions of Hirota–Miwa equation or BKP equation which is under investigation. Also it would be of interest to investigate whether any further mathematical structures of PEs (2.12), (2.14) and (2.16) exist related with the integrability. The integrability nature of the identified PE (2.18) is under investigation. Next the degree growth of iterates of PE (2.20) is linear. It is checked that Eq. (2.20) satisfies a set of necessary conditions for a nonlinear P E to be linearizable [14]. Also, Eq. (2.20) possesses a Lax pair
⎜
Llm = ⎝
where
Q 1 (n) = (a − 1)( v n+2 −
a ( v n+3 + v n )( v n+3 − v n+2 )( v n+2 − v n+1 )( v n+1 − v n ) a−1 − 2v n+2 ( v n+3 − v n )( v n+2 − v n+1 )( v n+1 − v n )
⎛
Q 2 (n)
− (a
+ (2 − a) v n v n+1 v n2+2 .
−1 − 2v n+1 ( v n+3 − v n )( v n+3 − v n+2 )( v n+2 − v n ) .
where
v n +4 =
+ (6 − 5a) v n2 v n+1 v n+2 + (3a − 4) v n v n2+1 v n+2
,
v n ) v n2+3
+ (a
⎛ − 1) v n2
+
v n2+1
− v n v n +1 + v n v n +2 − v n +1 v n +2 v n +3
− (a − 1) v n v n+1 ( v n − v n+1 ) − v n v n2+2 − av n+1 v n+2 ( v n+1 − v n+2 ) + v n v n+1 v n+2 , Q 2 (n) = ( v n+2 − v n+1 )( v n − v n+1 )
5
× v n2+3 − ( v n + v n+2 ) v n+3 + v n v n+2 , Q 3 (n) = (4 − 5a) v n v n+1 + (3a − 2) v n v n+2 − av n+1 v n+2 + (3a − 2) v n2+1 v n2+3
⎜
l Mm =⎝
v lm v lm+1 +(i +k) v lm+1 +ik
⎞
ik( v lm+1 − v lm )
v lm v lm+1 +i ( v lm +k)+kv lm+1
v lm v lm+1 +i ( v lm +k)+kv lm+1
v lm v lm+1 +i ( v lm +k)+kv lm+1
v lm v lm+1 +i ( v lm +k)+kv lm+1
v lm − v lm+1
v lm v lm+1 +(k−i ) v lm+1 −ik
v lm v lm+1 −i ( v lm +k)+kv lm+1 v lm − v lm+1 v lm v lm+1 −i ( v lm +k)+kv lm+1
v lm v lm+1 +(i +k) v lm +ik
⎟ ⎠,
−ik( v lm+1 − v lm ) l l v m v m+1 −i ( v lm +k)+kv lm+1 v lm v lm+1 +(k−i ) v lm −ik v lm v lm+1 −i ( v lm +k)+kv lm+1
(5.1)
⎞ ⎟ ⎠ , (5.2)
where i 2 = −1 and k is the spectral parameter, satisfying L lm+1 ·
l l+1 l Mm − Mm L m = 0. Hence Eq. (2.20) is linearizable and so integrable. It is appropriate to report here that the following nonlinear PE
v lm++11 =
˜ 1 ( v lm+1 , v l ) − v lm H˜ 2 ( v lm+1 , v l ) H m +1 m +1 ˜ 2 ( v lm+1 , v l ) + v lm H˜ 1 ( v lm+1 , v l ) H m +1 m +1
,
(5.3)
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6
with
2
2
˜ 1 = 1 − v lm+1 H ˜ 2 = 1 − v lm+1 H H1 =
| r −1 |
(−1)
H 1 + 2v lm+1 H 2 ,
No: 09/115(0729)/2010-EMR-I dated 8th November 2010), Government of India, New Delhi for providing financial support in the form of Senior Research Fellowship.
H 2 − 2v lm+1 H 1 ,
References
j −1 |r | 2
j =1 j -odd
j
j |r |− j 1 + v lm+1 v lm+1 , × sgn(r ) v lm+1 − v lm+1 | r| j −1 |r | H2 = (−1) 2
j =0 j -even
j
j |r |− j × sgn(r ) v lm+1 − v lm+1 1 + v lm+1 v lm+1 and r ∈ 2Z\{0, 2} satisfies the invariance condition (2.5) where η(l, m, v lm ) = (α + β(l + m))[1 + ( v lm )2 ]. The above equation can also be transformed into a linear PE
θml++11 − r θml+1 + (r − 2)θml +1 + θml = p π , θml = tan−1 v lm and p ∈ Z. Similar conclusions can be arrived at for r-odd. Also nonlinear PE (5.3) possesses the following Lax representation
⎛ ⎜
Llm = ⎝
( v lm +i )2 ( v lm+1 −i )r +1 (1+i v lm )2 (−1+i v lm+1 )r +1 k( v lm +i )2 ( v lm+1 −i )r +1 (1+i v lm )2 (−1+i v lm+1 )r +1
⎛ ⎜
l =⎝ Mm
(1−i v lm )( v lm+1 −i )r
(i − v lm )(i + v lm+1 )r k(1−i v lm )( v lm+1 −i )r (i − v lm )(i + v lm+1 )r
( v lm +i )2 ( v lm+1 −i )r +1 k(1+i v lm )2 (−1+i v lm+1 )r +1 ( v lm +i )2 ( v lm+1 −i )r +1 (1+i v lm )2 (−1+i v lm+1 )r +1 (1−i v lm )( v lm+1 −i )r k(i − v lm )(i + v lm+1 )r (1−i v lm )( v lm+1 −i )r (i − v lm )(i + v lm+1 )r
⎞ ⎟ ⎠,
(5.4)
⎞
⎟ ⎠,
(5.5)
where i 2 = −1 and k is the spectral parameter, indicating that it is Lax integrable. In Section 4, it is explained that how higher order OEs can be derived through periodic reduction. This has been illustrated for the identified PE (2.12) and derived its reductions i.e., ( z + 1)th order OE (4.3). Furthermore the explicit forms of third order OE (4.4) and fourth order OE (4.6) are given. Each of the OEs (4.4) and (4.6) admits two independent integrals and is measure preserving, therefore integrable. At the moment it is not clear whether the fourth order OE (4.6) is symplectic, which is under investigation. In a similar manner, higher order OEs can be derived from the periodic reduction of the remaining identified PEs (2.14), (2.16), (2.18) and (2.20). Acknowledgements The authors wish to thank the anonymous referees for their helpful and critical comments. One of the authors (G.N.) wishes to thank the Council of Scientific and Industrial Research (CSIR) (File
[1] V.E. Adler, A.B. Shabat, R.I. Yamilov, Symmetry approach to the integrability problem, Theor. Math. Phys. 125 (2000) 1603–1661. [2] V.E. Adler, A.I. Bobenko, Yu.B. Suris, Classification of integrable equations on quad-graphs. The consistency approach, Commun. Math. Phys. 233 (2003) 513–543. [3] G. Bluman, S. Anco, Symmetry and Integration Methods for Differential Equations, Springer-Verlag, New York, 2002. [4] G.B. Byrnes, R. Sahadevan, G.R.W. Quispel, Factorizable Lie symmetries and the linearization of difference equations, Nonlinearity 8 (1995) 443–459. [5] V. Dorodnitsyn, E. Kaptsov, R. Kozlov, P. Winternitz, First integrals of ordinary difference equations beyond Lagrangian methods, arXiv:1311.1597v1, 2013. [6] Ü. Göktas, ¸ W. Hereman, Algorithmic computation of generalized symmetries of nonlinear evolution and lattice equations, Adv. Comput. Math. 11 (1999) 55–80. [7] P.E. Hydon, Symmetries and first integrals of ordinary difference equations, Proc. R. Soc. Lond. Ser. A, Math. Phys. Sci. 456 (2000) 2835–2855. [8] N.H. Ibragimov, Transformation Groups Applied to Mathematical Physics, Reidel, Boston, 1985. [9] M. Lakshmanan, P. Kaliappan, Lie transformations, nonlinear evolution equations, and Painlevé forms, J. Math. Phys. 24 (1983) 795–806. [10] D. Levi, P. Winternitz, Continuous symmetries of discrete equations, Phys. Lett. A 152 (1991) 335–338. [11] D. Levi, S. Trembley, P. Winternitz, Lie point symmetries of difference equations and lattices, J. Phys. A, Math. Gen. 33 (2000) 8507–8523. [12] D. Levi, P. Winternitz, Continuous symmetries of difference equations, J. Phys. A, Math. Gen. 39 (2006) R1–R63. [13] D. Levi, P. Winternitz, R.I. Yamilov, Lie point symmetries of differential– difference equations, J. Phys. A, Math. Theor. 43 (2010) 292002. [14] D. Levi, C. Scimiterna, Linearizability of nonlinear equations on a quad-graph by a point, two points and generalized Hopf–Cole transformations, SIGMA 7 (2011) 079. [15] S. Maeda, Canonical structure and symmetries for discrete systems, Math. Jpn. 25 (1980) 405–420. [16] S. Maeda, Extension of discrete Noether theorem, Math. Jpn. 26 (1981) 85–90. [17] S. Maeda, Lagrangian formulation of discrete systems and concept of difference space, Math. Jpn. 27 (1982) 345–356. [18] S. Maeda, The similarity method for difference equations, IMA J. Appl. Math. 38 (1987) 129–134. [19] P.J. Olver, Applications of Lie Groups to Differential Equations, Springer, Berlin, 1993. [20] L.V. Ovsiannikov, Group Analysis of Differential Equations, Academic Press, New York, 1982. [21] G.R.W. Quispel, J.A.G. Roberts, C.J. Thompson, Integrable mappings and soliton equations II, Physica D 34 (1989) 183–192. [22] G.R.W. Quispel, H.W. Capel, R. Sahadevan, Continuous symmetries of differential–difference equations: the Kac–van Moerbeke equation and Painlevé reduction, Phys. Lett. A 170 (1992) 379–383. [23] G.R.W. Quispel, R. Sahadevan, Lie symmetries and integration of difference equations, Phys. Lett. A 184 (1993) 64–70. [24] R. Sahadevan, G.B. Byrnes, G.R.W. Quispel, Linearisation of difference equations using factorisable Lie symmetries, in: Proc. Workshop on Symmetries and Integrability of Difference Equation 1994, Estérel, in: D. Levi, L. Vinet, P. Winternitz (Eds.), CRM Proceedings and Lecture Notes Series, Am. Math. Soc., RI, Providence, 1994. [25] R. Sahadevan, S. Khousalya, Belov–Chaltikian and Blaszak–Marciniak lattice equations: recursion operators and factorization, J. Math. Phys. 44 (2003) 882–898. [26] S. Tremblay, B. Grammaticos, A. Ramani, Integrable lattice equations and their growth properties, Phys. Lett. A 278 (2001) 319–324. [27] P. Winternitz, Lie groups and solutions of nonlinear partial differential equations, in: Integrable Systems, Quantum Groups, and Quantum Field Theories NATO ASI Series, vol. 409, 1993, pp. 429–495.
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Physics Letters A ••• (••••) •••–•••
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Continuous symmetries of certain nonlinear partial difference equations and their reductions R. Sahadevan ∗ , G. Nagavigneshwari Ramanujan Institute for Advanced Study in Mathematics, University of Madras, Chepauk, Chennai 600 005, Tamilnadu, India
a r t i c l e
i n f o
Article history: Received 3 July 2014 Received in revised form 12 September 2014 Accepted 12 September 2014 Available online xxxx Communicated by C.R. Doering Keywords: Lie point symmetries Integrability Partial difference equation Ordinary difference equation
a b s t r a c t In this article, Quispel, Roberts and Thompson type of nonlinear partial difference equation with two independent variables is considered and identified five distinct nonlinear partial difference equations admitting continuous point symmetries quadratic in the dependent variable. Using the degree growth of iterates the integrability nature of the obtained nonlinear partial difference equations is discussed. It is also shown how to derive higher order ordinary difference equations from the periodic reduction of the identified nonlinear partial difference equations. The integrability nature of the obtained ordinary difference equations is investigated wherever possible. © 2014 Elsevier B.V. All rights reserved.
1. Introduction Lie groups and Lie algebras are mathematical objects which have originated in the seminal work of Sophus Lie (1842–1899) on solving differential equations by quadrature, using symmetry methods. Lie symmetries approach originally introduced by Sophus Lie has been used as a tool to unify various integration techniques for ordinary differential equations (ODEs) and has played a very important role in the study of both ODEs and partial differential equations (PDEs). Symmetry groups are invariant transformations which do not alter the structural form of the equation under investigation. Once the symmetry group of a system of differential equations is known, it can be used to generate new solutions from the old ones, often interesting ones from trivial ones [20]. It can be used to classify solutions into conjugacy classes and to classify and simplify differential equations. An important application of the symmetry approach is the reduction of an ODE to a lower order one, the reduction of a PDE to one with fewer independent variables. The usefulness of Lie symmetry approach has been widely illustrated for a variety of dynamical systems governed by both nonlinear ODEs and PDEs which arise in different contexts [3,8, 9,19,20,27] during the past several decades [3,8,9,19,20,27]. During 1980s Maeda [15–18] has extended Lie symmetries approach to discrete systems governed by nonlinear mappings or ordinary
*
Corresponding author. E-mail addresses: [email protected] (R. Sahadevan), [email protected] (G. Nagavigneshwari). http://dx.doi.org/10.1016/j.physleta.2014.09.021 0375-9601/© 2014 Elsevier B.V. All rights reserved.
difference equations (OEs) and demonstrated how it provides an effective tool to derive their continuous point symmetries. Later on the Lie symmetry approach has been further developed by Levi and Winternitz [10,12], Quispel et al. [22], Levi et al. [11,13] and others [4,7,23,24] to nonlinear discrete systems governed by partial differential–difference equations (PDEs) and OEs. Also, several groups have profitably exploited to derive mathematical structures related with integrability of different nonlinear PEs possessing solitons [1,6,25]. Though the Lie symmetry approach has been extended to discrete nonlinear systems governed by lattice equations or nonlinear partial difference equations (PEs), its effectiveness has not yet been demonstrated widely. The objective of this article is to illustrate its usefulness on other nonlinear PEs. More specifically a scalar nonlinear PE of Quispel, Roberts and Thompson (QRT) type [21]
v (l + 1, m + 1) f 1 ( v (l, m + 1), v (l + 1, m)) − v (l, m) f 2 ( v (l, m + 1), v (l + 1, m)) = f 3 ( v (l, m + 1), v (l + 1, m)) − v (l, m) f 4 ( v (l, m + 1), v (l + 1, m)) is considered and under what conditions on f i it possesses continuous point symmetries quadratic in the dependent variable is investigated. The integrability nature of obtained PEs is analyzed using the degree growth of iterates [26], another characteristic of integrable discrete systems. Also, it is shown how higher order OEs are explicitly derived. The paper is organized as follows. In Section 2, QRT type of nonlinear PE is considered as mentioned above and under what conditions on f i it possesses continuous point symmetries
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2
quadratic in the dependent variable is investigated. In Section 3, the integrability nature of the obtained PEs is analyzed through the degree growth of iterates. In Section 4, higher order OEs are derived from the periodic reduction of the obtained PEs. Also, it is shown that the derived OEs are measure preserving and they admit sufficient number of integrals, if exist, leading to their integrability. In Section 5, a brief summary of the obtained results and concluding remarks are provided. 2. Lie point symmetries of partial difference equations Consider a scalar nonlinear PE of the form
v (l + 1, m + 1) = F v (l, m), v (l, m + 1), v (l + 1, m) ,
(2.1)
where F ( v (l, m), v (l, m + 1), v (l + 1, m)) is an arbitrary function. Let us assume that (2.1) is invariant under a one-parameter ( ) continuous point transformations ∗
∗
l = l,
m = m,
(2.2)
v = v (l, m) + η l, m, v (l, m) + O 2 ∗
(2.3)
with infinitesimal generator
X = η l, m, v (l, m)
2 + c 1 + c 2 v lm+1 + c 3 v lm+1 ∂ f4 ∂ f3 + f 22 l+1 × f 12 l+1 f1 f2 ∂ vm ∂ vm + 2(c 1 f 3 f 4 + c 2 f 1 f 4 + c 3 f 1 f 2 ) = 0, l 2 ∂ f3 2 l f 1 c 1 + c 2 v m +1 + c 3 v m +1 ∂ v lm+1 f 1 2 ∂ f3 + f 12 c 1 + c 2 v lm+1 + c 3 v lm+1 l +1 f1 ∂ vm 2 + c 1 f 3 + f 2 f 3 − f 1 f 4 + c 2 f 1 f 3 + c 3 f 12 = 0.
Case 1: c 1 = 0, c 2 = 0, c 3 = 0
(2.4) f 1 = 1,
≡ v (l, m + 1), ≡ v (l + 1, m + 1)
f3 =
v lm ≡ v (l, m),
F ≡ F v (l, m), v (l, m + 1), v (l + 1, m) , v lm++11
≡ v (l + 1, m),
f2 =
η(l + 1, m + 1, F ) = η l, m
+ 1, v lm+1
v lm++11 =
∂F
∂ v lm+1 ∂F l +1
+ η l + 1, m , v m
+ η l, m, v lm
∂F ∂ v lm
f1 − f3 −
(2.5)
.
v lm f 2 , v lm f 4
(2.6)
2
(2.7)
,
where A (l, m) is an arbitrary function and c i , i = 1, 2, 3 are arbitrary parameters. To start with let A (l, m) = 1. Using (2.7) along with their shifts in Eq. (2.5) and equating powers of ( v lm ) j , j = 0, 1, 2 to zero the following three equations are obtained:
f 22 c 1 + c 2 v lm+1 + c 3 v lm+1
+ +
f 22
c 1 + c 2 v lm+1
c 1 f 42
2
∂ v lm+1
2 + c 3 v lm+1
+ c 2 f 2 f 4 + c 3 f 22
∂ ∂
∂ v lm+1
f4
f2
v lm+1
v lm+1 f4 =
,
, 1
v lm+1 v lm+1
(2.11)
,
f4 f2
+ f2 f3 − f1 f4
m∗ = m,
X = c 1 + c 2 v lm + c 3 v lm
X1 =
2 ∂ ∂ v lm
∂ ∂ v lm
X 2 = v lm
,
∂ ∂ v lm
X 3 = v lm
,
2 ∂ ∂ v lm
.
It is straight forward to check that the above generators satisfy
[ X1, X2] = X1,
[ X1, X3] = 2 X2,
[ X2, X3] = X3
indicating that the underlying Lie algebra of PE (2.12) is not solvable [3]. Case 2: c 1 = 0, c 2 = 0, c 3 = 0 Proceeding as before, we find that Eqs. (2.8)–(2.10) satisfy provided
f3 = (2.8)
+ O 2
leading to the following generators:
= 0,
2
with infinitesimal generator
f 1 = 1,
(2.12)
[av lm+1 + (1 − a) v lm+1 ] − v lm
v ∗ = v lm + c 1 + c 2 v lm + c 3 v lm
where f 1 , f 2 , f 3 and f 4 are arbitrary functions of v lm+1 and v lm+1 . In order to solve the invariance equation (2.5) with (2.6), assume that
(1 − a)
(1 − a)
v lm+1 v lm+1 − v lm [av lm+1 + (1 − a) v lm+1 ]
l∗ = l,
∂ v lm+1
η l, m, v lm = A (l, m) c1 + c2 v lm + c3 v lm
+
+
which is invariant under
Eq. (2.5) is a functional difference equation and there is no known method to solve it. In the present work, the investigation is restricted to QRT type nonlinear PE having the form
F=
a v lm+1
a v lm+1
where a is an arbitrary parameter and so the QRT PEs (2.6) becomes
unless otherwise specified. Then the invariant equation reads
(2.10)
Then there exist different possibilities which will be discussed below separately.
provided any solution v (l, m) satisfies (2.1). Hereafter denoting
v lm+1
(2.9)
After a detailed calculation it is found that Eqs. (2.8)–(2.10) satisfy identically provided
∂ ∂ v (l, m)
v lm+1
2
c 1 + c 2 v lm+1 + c 3 v lm+1 ∂ f4 ∂ f3 2 2 + f2 l × f1 l f ∂ v m +1 ∂ v m +1 f 2 1
a v lm+1
f2 =
+
a v lm+1
(1 − a) v lm+1
,
+
(1 − a) v lm+1 f4 =
, 1
v lm+1 v lm+1
+b
v lm+1 − v lm+1
2
v lm+1 v lm+1
(2.13)
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where a and b are arbitrary parameters and so the nonlinear PE (2.6) becomes v lm+1 v lm+1 [ v lm+1 v lm+1 − v lm [av lm+1 + (1 − a) v lm+1 ]] l +1 l v m+1 v m [av lm+1 + (1 − a) v lm+1 ] − v lm [ v lm+1 v lm+1 + b( v lm+1 −
v lm++11 =
v lm+1 )2 ]
.
proceeding as before it is found that the invariance equation (2.5) is satisfied when f 2 = 1, f 3 = 1 and f 4 = − f 1 and
f1 =
(2.14)
∂
∂ ∂ v lm
∂ v lm
X 2 = v lm
,
1 − v lm+1 v lm+1
v lm++11 =
and so the following generators are obtained
X 1 = v lm
v lm+1 + v lm+1
(2.19)
.
Hence Eq. (2.6) becomes
Here the infinitesimal generator X reads
X = v lm c 2 + c 3 v lm
3
( v lm+1 + v lm+1 ) − v lm (1 − v lm+1 v lm+1 ) (1 − v lm+1 v lm+1 ) + v lm ( v lm+1 + v lm+1 )
(2.20)
with infinitesimal generator
2 ∂
X=
∂ v lm
α + β(l + m) 1 + v lm
2 ∂ ∂ v lm
.
satisfying [ X 1 , X 2 ] = X 2 indicating that the underlying Lie algebra of Eq. (2.14) is solvable [3].
It is crucial to mention here that for other choices of A (l, m) different nonlinear PEs can be derived. For more details see [4].
Case 3: c 1 = 0, c 2 = 0, c 3 = 1
3. Degree growth of iterates of nonlinear PEs
In this case the invariance equation (2.5) is satisfied when
f 1 = 1, f3 = f4 =
f2 =
a v lm+1
+
a v lm+1
(1 − a) v lm+1
1 v lm+1 v lm+1
+
(1 − a) v lm+1
,
,
+b
v lm+1 − v lm+1
+ c,
v lm+1 v lm+1
(2.15)
where a, b and c are arbitrary parameters. Now Eq. (2.6) on simplifying becomes v lm+1 v lm+1 − v lm (av lm+1 + (1 − a) v lm+1 )
v lm++11 =
(av lm+1
+ (1 − a) v lm+1 ) − v lm [1 + b( v lm+1 − v lm+1 ) + cv lm+1 v lm+1 ]
.
(2.16) The infinitesimal generator of Eq. (2.16) is given by
X = v lm
2 ∂ ∂ v lm
.
Case 4: c 1 = 0, c 2 = 1, c 3 = 0
v l0 =
In a similar manner the invariance equation (2.5) is satisfied when
f 1 = 1, f3 =
f2 =
a v lm+1
+
a v lm+1
(1 − a) v lm+1
,
+
(1 − a) v lm+1 f4 =
, b
( v lm+1 )2
+
c v lm+1 v lm+1
,
(2.17)
v lm+1 [ v lm+1 v lm+1 − v lm (av lm+1 + (1 − a) v lm+1 )]
v lm+1 [(av lm+1 + (1 − a) v lm+1 )] − v lm [bv lm+1 + cv lm+1 ]
,
(2.18) with infinitesimal generator
X = v lm
∂ ∂ v lm
pl q
,
0 vm =
rm q
,
l , m = 0, 1 , 2 . . .
(3.1)
provided p 0 = r0 . Here the homogeneous coordinates q, pl and rm , for l, m = 0, 1, 2, . . . have the same homogeneity degree. For l = 0 and m = 0, Eq. (2.12) becomes
v 11 =
v 01 v 10 − v 00 [av 10 + (1 − a) v 01 ]
[av 01 + (1 − a) v 10 ] − v 00
.
(3.2)
Using Eq. (3.1) in Eq. (3.2), it becomes
where a, b and c are arbitrary parameters. In this case, Eq. (2.6) becomes
v lm++11 =
In this section, the integrability nature of the obtained PEs (2.12), (2.14), (2.16), (2.18) and (2.20) is investigated through the degree growth of iterates. It is widely believed that there exists a deep connection between the degree growth of iterates of nonlinear PEs and their integrability [26]. More specifically if the degree growth of iterates of a PE is polynomial then it may be integrable. If the degree growth of iterates is exponential, then it is nonintegrable. In the case of linearizable equations, the degree growth is linear and hence is slower than that of the integrable equations [26]. Note that the term degree is nothing but the common homogeneity degree of the numerators and denominators of their irreducible forms, which is obtained by introducing homogeneous coordinates. Once the degree of the first few iterates is computed, the underlying pattern is conjectured and written in the form of an appropriate analytical expression, which can be further confirmed by computing the degree of subsequent iterates. The computational details of the degree growth of iterates of the obtained nonlinear PEs are presented below. It is explained briefly how to evaluate the degree growth of iterates of nonlinear PE (2.12). Let the initial conditions along the lines l = 0 and m = 0 be respectively denoted by
.
Next assuming that A (l, m) given in Eq. (2.7) is linear, that is A (l, m) = α + β(l + m), where α and β are arbitrary parameters,
v 11 =
r1 p 1 q q
−
r0 [a pq1 q
+ (1 − a) rq1 ]
[a rq1 + (1 − a) pq1 ] −
r0 q
which can be rewritten as
v 11 =
r1 p 1 − r0 [ap 1 + (1 − a)r1 ]
(3.3)
q[ar1 + (1 − a) p 1 ] − r0 q
indicating that the degree of v 11 is 2. Next for l = 0 and m = 1 Eq. (2.12) becomes
v 12 =
v 02 v 11 − v 01 [av 11 + (1 − a) v 02 ]
[av 02 + (1 − a) v 11 ] − v 01
.
(3.4)
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4
dlm = 2 max(l, m) min(l, m) + 1,
Using (3.1)–(3.3) in (3.4) and rearranging
v 12 =
(r2 − ar1 )[r1 p 1 − r0 [ap 1 + (1 − a)r1 ]] − (1 − a)r1 r2 [ar1 + (1 − a) p 1 − r0 ] . q[(ar2 − r1 )[ar1 + (1 − a) p 1 − r0 ] − (1 − a)[r1 p 1 − r0 [ap 1 + (1 − a)r1 ]]]
(3.5)
a polynomial one. At the moment we are unable to present the growth of iterations of PE (2.18) in an analytic expression. The degree growth of iterates of PE (2.20) is the following
v 12
It is clear that degree of is 3. In a similar way, for different values of l and m, the degrees of various iterates are computed as mentioned hereunder:
.. .
⏐ m⏐ ⏐
1 1
3 5 2 3
7 4
1
1
1
1
⏐ m⏐ ⏐
dlm = lm + 1, which is a polynomial. Here dlm denotes the degree of the (l, m)-entry in the lattice. Next the degree growth of iterates of Eq. (2.14) is calculated below. For l = 0 and m = 0 Eq. (2.14) becomes
v 01 v 10 [ v 01 v 10 − v 00 [av 10 + (1 − a) v 01 ]] v 01 v 10 [av 01 + (1 − a) v 10 ] − v 00 [ v 01 v 10 + b( v 10 − v 01 )2 ]
.
(3.6)
r1 p 1 [r1 p 1 − r0 [ap 1 + (1 − a)r1 ]] q[r1 p 1 [ar1 + (1 − a) p 1 ] − r0 [r1 p 1 + b( p 1 − r1 )2 ]]
(3.7)
9 27 4 9
1
1
1
1
dlm =
1, ml = 0 3, ml = 0
Hence Eq. (2.20) exhibits a linear growth indicating that it is linearizable. It is appropriate to mention here that Levi and Scimiterna [14] have derived a set of necessary conditions for a nonlinear P E to be linearizable and Eq. (2.20) satisfies those conditions [14]. 4. Reductions to ordinary difference equations Consider a solution v lm of the nonlinear PE (2.1) satisfying the periodicity property
(4.1)
v lm+1 = v n+z1 ,
v lm+1 = v n+z2 ,
v lm++11 = v n+z1 +z2 .
As a consequence Eq. (2.1) can be transformed into a ( z1 + z2 ) order OE, that is
v n+z1 +z2 = F ( v n , v n+z1 , v n+z2 ).
(4.2)
Hereafter z1 = 1 and z2 = z unless or otherwise specified. So Eq. (2.12) on reduction yields a ( z + 1)th order OE
19 1
−→ l From the above, the degree growth of iterates of PE (2.14) is
dlm =
1
where gcd( z1 , z2 ) = 1, z1 , z2 ∈ Z. Here n = mz1 + lz2 and so
,
.. .
1
1
l− z
and so degree of v 11 is 4. Proceeding further along the lines described earlier the degree growth of v lm , l, m = 0, 1, 2, . . . is computed as stated below:
⏐ m⏐ ⏐
3 3 3
v m+2z1 = v lm = v n ,
Substituting Eq. (3.1) in Eq. (3.6),
1 1
3 3 3
and it is inferred that
From the above mentioned degree growth of iterates it is conjectured that the degree growth of iterates of nonlinear PE (2.12) is
v 11 =
3 3 3
l
l
=
1 1 1
−→
−→
v 11
.. .
⎧ ⎨ 1,
ml = 0 m = l = 0 l 2 + m2 l 2 + m2 , ⎩ 4 l + 2l2 + 2l − 1, m = l = 0
v n + z +1 =
v n+1 v n+z − v n [av n+z + (1 − a) v n+1 ]
[av n+1 + (1 − a) v n+z ] − v n
.
(4.3)
Schematic reduction of the above equation with solutions satisfying the periodicity property v lm−+z 1 = v lm = v n is given in Fig. 1. For z = 1 Eq. (4.3) becomes trivial. It is imperative to mention here that higher order OEs can be derived when z ≥ 2. For example if z = 2, a third order OE given by
v n +3 =
v n+1 v n+2 − v n [av n+2 + (1 − a) v n+1 ]
[av n+1 + (1 − a) v n+2 ] − v n
(4.4)
which is a polynomial. Next calculating the degree growth of Eq. (2.16)
.. .
⏐
m⏐ ⏐
1 1
5 9 3 5 7
1
1 1 1
−→ l which can be written as
Fig. 1. Schematic representation of the ( z + 1)th order mapping with solutions satisfying the periodicity property v lm−+z 1 = v lm = v n .
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+ (3a − 4) v n2 v n+1 + (2 − a) v n2 v n+2
is obtained. To derive integrals consider I (n) to take the form
I (n) =
A 1 ( v n , v n+1 ) v n2+2 + A 2 ( v n , v n+1 ) v n+2 + A 3 ( v n , v n+1 ) B 1 ( v n , v n+1 ) v n2+2 + B 2 ( v n , v n+1 ) v n+2 + B 3 ( v n , v n+1 )
+ (2 − a) v n v n2+1 + (4 − 5a) v n v n2+2
,
+ 6(a − 1) v n v n+1 v n+2 + (4 − 5a) v n2+1 v n+2 + (3a − 2) v n+1 v n2+2 v n+3 + (3a − 4) v n2 v n2+2
(4.5) where A i ’s and B i ’s are arbitrary functions. Next for what forms of A i ’s and B i ’s, I (n + 1) − I (n) = 0 holds along with Eq. (4.4) is investigated. After detailed calculation it is found that Eq. (4.4) possesses two independent integrals
I 1 (n) = I 2 (n) =
P 1 (n) P 2 (n) P 3 (n) P 2 (n)
P 1 (n) = (a − 1) v n2+2 − av n+1 + (a − 2) v n v n+2 + av n2+1
+ (a − 1) v n2 − av n v n+1 , P 2 (n) = ( v n − v n+1 ) v n2+2 − ( v n + v n+1 ) v n+2 + v n v n+1 , P 3 (n) = (3a − 2) v n − av n+1 v n2+2 + (2 − a) v n2 − 2(3a − 2) v n v n+1 + (3a − 2) v n2+1 v n+2 + (3a − 4) v n2 v n+1 + (2 − a) v n v n2+1 . The above integrals have also been identified in [5] using Adjoint equation method. It is straightforward to check that Eq. (4.4) is measure preserving with measure 1
[ a−a 1 ( v n+2 + v n )( v n+2 − v n+1 )( v n+1 − v n ) − 2v n+1 ( v n+2 − v n )2 ]
.
Next for z = 3 a fourth order OE given by
v n+1 v n+3 − v n [av n+3 + (1 − a) v n+1 ]
(4.6)
[av n+1 + (1 − a) v n+3 ] − v n
is obtained. Here again integrals having the form
I (n) =
A 1 ( v n , v n+1 , v n+2 ) v n2+3 + A 2 ( v n , v n+1 , v n+2 ) v n+3 + A 3 ( v n , v n+1 , v n+2 ) B 1 ( v n , v n+1 , v n+2 ) v n2+3 + B 2 ( v n , v n+1 , v n+2 ) v n+3 + B 3 ( v n , v n+1 , v n+2 )
,
(4.7) where A i ’s and B i ’s are arbitrary functions, are considered. Proceeding as above and after a detailed calculation, it is clear that OE (4.6) possesses two independent integrals
J 1 (n) = J 2 (n) =
Q 1 (n) Q 2 (n) Q 3 (n)
,
It is easy to show that the OE (4.6) is measure preserving with measure
− 1) v n2+2
In a similar manner, higher order OEs can be derived from the periodic reductions of the other obtained nonlinear PEs namely (2.14), (2.16), (2.18) and (2.20), which are under investigation and will be reported elsewhere. 5. Summary and concluding remarks In this article QRT type of nonlinear PE (2.6) is considered and identified five distinct nonlinear PEs namely (2.12), (2.14), (2.16), (2.18) and (2.20) admitting continuous point symmetries quadratic in the dependent variable. Since there exists no direct connection between the Lie point symmetries and the concept of complete integrability of differential equations or difference equations, the integrability nature of the above-mentioned PEs cannot be ascertained at this point. The nature of integrability of the identified PEs is analyzed through the degree growth of iterates and reported that the degree growth of iterates of each of the PEs (2.12), (2.14) and (2.16) is a polynomial one and hence they are expected to be integrable. It is clear that the above PEs do not fall into the well-known ABS classification [2] which may be derived from some reductions of Hirota–Miwa equation or BKP equation which is under investigation. Also it would be of interest to investigate whether any further mathematical structures of PEs (2.12), (2.14) and (2.16) exist related with the integrability. The integrability nature of the identified PE (2.18) is under investigation. Next the degree growth of iterates of PE (2.20) is linear. It is checked that Eq. (2.20) satisfies a set of necessary conditions for a nonlinear P E to be linearizable [14]. Also, Eq. (2.20) possesses a Lax pair
⎜
Llm = ⎝
where
Q 1 (n) = (a − 1)( v n+2 −
a ( v n+3 + v n )( v n+3 − v n+2 )( v n+2 − v n+1 )( v n+1 − v n ) a−1 − 2v n+2 ( v n+3 − v n )( v n+2 − v n+1 )( v n+1 − v n )
⎛
Q 2 (n)
− (a
+ (2 − a) v n v n+1 v n2+2 .
−1 − 2v n+1 ( v n+3 − v n )( v n+3 − v n+2 )( v n+2 − v n ) .
where
v n +4 =
+ (6 − 5a) v n2 v n+1 v n+2 + (3a − 4) v n v n2+1 v n+2
,
v n ) v n2+3
+ (a
⎛ − 1) v n2
+
v n2+1
− v n v n +1 + v n v n +2 − v n +1 v n +2 v n +3
− (a − 1) v n v n+1 ( v n − v n+1 ) − v n v n2+2 − av n+1 v n+2 ( v n+1 − v n+2 ) + v n v n+1 v n+2 , Q 2 (n) = ( v n+2 − v n+1 )( v n − v n+1 )
5
× v n2+3 − ( v n + v n+2 ) v n+3 + v n v n+2 , Q 3 (n) = (4 − 5a) v n v n+1 + (3a − 2) v n v n+2 − av n+1 v n+2 + (3a − 2) v n2+1 v n2+3
⎜
l Mm =⎝
v lm v lm+1 +(i +k) v lm+1 +ik
⎞
ik( v lm+1 − v lm )
v lm v lm+1 +i ( v lm +k)+kv lm+1
v lm v lm+1 +i ( v lm +k)+kv lm+1
v lm v lm+1 +i ( v lm +k)+kv lm+1
v lm v lm+1 +i ( v lm +k)+kv lm+1
v lm − v lm+1
v lm v lm+1 +(k−i ) v lm+1 −ik
v lm v lm+1 −i ( v lm +k)+kv lm+1 v lm − v lm+1 v lm v lm+1 −i ( v lm +k)+kv lm+1
v lm v lm+1 +(i +k) v lm +ik
⎟ ⎠,
−ik( v lm+1 − v lm ) l l v m v m+1 −i ( v lm +k)+kv lm+1 v lm v lm+1 +(k−i ) v lm −ik v lm v lm+1 −i ( v lm +k)+kv lm+1
(5.1)
⎞ ⎟ ⎠ , (5.2)
where i 2 = −1 and k is the spectral parameter, satisfying L lm+1 ·
l l+1 l Mm − Mm L m = 0. Hence Eq. (2.20) is linearizable and so integrable. It is appropriate to report here that the following nonlinear PE
v lm++11 =
˜ 1 ( v lm+1 , v l ) − v lm H˜ 2 ( v lm+1 , v l ) H m +1 m +1 ˜ 2 ( v lm+1 , v l ) + v lm H˜ 1 ( v lm+1 , v l ) H m +1 m +1
,
(5.3)
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R. Sahadevan, G. Nagavigneshwari / Physics Letters A ••• (••••) •••–•••
6
with
2
2
˜ 1 = 1 − v lm+1 H ˜ 2 = 1 − v lm+1 H H1 =
| r −1 |
(−1)
H 1 + 2v lm+1 H 2 ,
No: 09/115(0729)/2010-EMR-I dated 8th November 2010), Government of India, New Delhi for providing financial support in the form of Senior Research Fellowship.
H 2 − 2v lm+1 H 1 ,
References
j −1 |r | 2
j =1 j -odd
j
j |r |− j 1 + v lm+1 v lm+1 , × sgn(r ) v lm+1 − v lm+1 | r| j −1 |r | H2 = (−1) 2
j =0 j -even
j
j |r |− j × sgn(r ) v lm+1 − v lm+1 1 + v lm+1 v lm+1 and r ∈ 2Z\{0, 2} satisfies the invariance condition (2.5) where η(l, m, v lm ) = (α + β(l + m))[1 + ( v lm )2 ]. The above equation can also be transformed into a linear PE
θml++11 − r θml+1 + (r − 2)θml +1 + θml = p π , θml = tan−1 v lm and p ∈ Z. Similar conclusions can be arrived at for r-odd. Also nonlinear PE (5.3) possesses the following Lax representation
⎛ ⎜
Llm = ⎝
( v lm +i )2 ( v lm+1 −i )r +1 (1+i v lm )2 (−1+i v lm+1 )r +1 k( v lm +i )2 ( v lm+1 −i )r +1 (1+i v lm )2 (−1+i v lm+1 )r +1
⎛ ⎜
l =⎝ Mm
(1−i v lm )( v lm+1 −i )r
(i − v lm )(i + v lm+1 )r k(1−i v lm )( v lm+1 −i )r (i − v lm )(i + v lm+1 )r
( v lm +i )2 ( v lm+1 −i )r +1 k(1+i v lm )2 (−1+i v lm+1 )r +1 ( v lm +i )2 ( v lm+1 −i )r +1 (1+i v lm )2 (−1+i v lm+1 )r +1 (1−i v lm )( v lm+1 −i )r k(i − v lm )(i + v lm+1 )r (1−i v lm )( v lm+1 −i )r (i − v lm )(i + v lm+1 )r
⎞ ⎟ ⎠,
(5.4)
⎞
⎟ ⎠,
(5.5)
where i 2 = −1 and k is the spectral parameter, indicating that it is Lax integrable. In Section 4, it is explained that how higher order OEs can be derived through periodic reduction. This has been illustrated for the identified PE (2.12) and derived its reductions i.e., ( z + 1)th order OE (4.3). Furthermore the explicit forms of third order OE (4.4) and fourth order OE (4.6) are given. Each of the OEs (4.4) and (4.6) admits two independent integrals and is measure preserving, therefore integrable. At the moment it is not clear whether the fourth order OE (4.6) is symplectic, which is under investigation. In a similar manner, higher order OEs can be derived from the periodic reduction of the remaining identified PEs (2.14), (2.16), (2.18) and (2.20). Acknowledgements The authors wish to thank the anonymous referees for their helpful and critical comments. One of the authors (G.N.) wishes to thank the Council of Scientific and Industrial Research (CSIR) (File
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