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G. Gubbiotti
literature [7, 21, 22]. These approaches relied on searching for difference equations
admitting some extra structures allowing to claim integrability, i.e. invariants of
a fixed form and/or symplectic structures. Higher-order difference equations have
been produced in the literature with different methods, like periodic reductions of
partial difference equations [9, 35–38, 40].
In an upcoming paper [20] we propose a new approach to generate integrable higher-order difference equations through stationary solutions of integrable
differential-difference equations. A differential-difference equation is a function
equation for an unknown sequence {x n (t)} n∈Z of functions a continuous variable,
depending on both shifts and derivatives. A well-known class of differentialdifference equations are the Volterra-like equations:
∂x n
∂t
= f
x n+k , . . . , x n , . . . , x n−k
, k > k
.
(1)
The stationary reduction of a differential-difference equation is obtained by letting
∂x n /∂t ≡ 0 and it is clearly a difference equation, since we suppressed the
dependence on the continuous variable. To be more precise, in [20] we will present
the integrability properties of the stationary solutions for two classes fourth-order
Volterra-like equations, recently classified in [15, 16]. In this short note we present
an interesting example out of this general picture.
The plan of the paper is following: in Sect. 2 we introduce the formal definitions
of integrability we will be using throughout this note. In Sect. 3 we will present our
new example, and show its integrability in the sense of Sect. 2. Finally, in Sect. 4
we give some final comments and an outlook towards the general results.
2 Integrability of Difference Equations
Integrability both for continuous and discrete systems can be defined in several
different ways. In this note we will limit ourselves to two alternative definition,
out of all the possible ones.
Consider an autonomous Nth-order difference equation:
x n+N = Q (x n+N −1 , x n+N −2 , . . . , x n ) .
(2)
A function
I = I (x n+N −1 , x n+N −2 , . . . , x n )
(3)
is called an invariant if:
I (x n+N , x n+N −1 , . . . , x n+1 ) = I (x n+N −1 , x n+N −2 , . . . , x n )
(4)
on the solutions of equation (2).
G. Gubbiotti
literature [7, 21, 22]. These approaches relied on searching for difference equations
admitting some extra structures allowing to claim integrability, i.e. invariants of
a fixed form and/or symplectic structures. Higher-order difference equations have
been produced in the literature with different methods, like periodic reductions of
partial difference equations [9, 35–38, 40].
In an upcoming paper [20] we propose a new approach to generate integrable higher-order difference equations through stationary solutions of integrable
differential-difference equations. A differential-difference equation is a function
equation for an unknown sequence {x n (t)} n∈Z of functions a continuous variable,
depending on both shifts and derivatives. A well-known class of differentialdifference equations are the Volterra-like equations:
∂x n
∂t
= f
x n+k , . . . , x n , . . . , x n−k
, k > k
.
(1)
The stationary reduction of a differential-difference equation is obtained by letting
∂x n /∂t ≡ 0 and it is clearly a difference equation, since we suppressed the
dependence on the continuous variable. To be more precise, in [20] we will present
the integrability properties of the stationary solutions for two classes fourth-order
Volterra-like equations, recently classified in [15, 16]. In this short note we present
an interesting example out of this general picture.
The plan of the paper is following: in Sect. 2 we introduce the formal definitions
of integrability we will be using throughout this note. In Sect. 3 we will present our
new example, and show its integrability in the sense of Sect. 2. Finally, in Sect. 4
we give some final comments and an outlook towards the general results.
2 Integrability of Difference Equations
Integrability both for continuous and discrete systems can be defined in several
different ways. In this note we will limit ourselves to two alternative definition,
out of all the possible ones.
Consider an autonomous Nth-order difference equation:
x n+N = Q (x n+N −1 , x n+N −2 , . . . , x n ) .
(2)
A function
I = I (x n+N −1 , x n+N −2 , . . . , x n )
(3)
is called an invariant if:
I (x n+N , x n+N −1 , . . . , x n+1 ) = I (x n+N −1 , x n+N −2 , . . . , x n )
(4)
on the solutions of equation (2).