A Novel Integrable Fourth-Order
Difference Equation Admitting Three
Invariants
Giorgio Gubbiotti
Abstract In this short note we present a novel integrable fourth-order difference
equation. This equation is obtained as a stationary reduction from a known
integrable differential-difference equation. The novelty of the equation is inferred
from the number and shape of its invariants.
Keywords Difference equations · Integrability · Algebraic entropy
1 Introduction
The interest in discrete systems, that is, of systems whose independent degrees of
freedom take values in a discrete set, grew enormously during the past decades,
for reasons which span from very philosophical [23] to purely practical ones
[29]. Nowadays, discrete systems are studied from different points of view and
perspective, see [12, 24].
In this short note we will present a new integrable fourth-order difference
equation. An Nth-order difference equation is a functional equation for an unknown
sequence {x n } n∈Z where the x n+N element is expressible in terms of the previous
x n+i , i = 0, 1, . . . , N − 1. Such kind of functional equations are also called
recurrence relations. Without entering in the details, which will be given in Sect. 2,
we say that an Nth-order difference equation is integrable when its dynamics is
sufficiently regular and predictable.
The integrability of second-order difference equations is a well understood topic,
as it is known that most of the integrable second-order difference equations belong to
the QRT class [30, 31], even though there are some notable exceptions [11, 33, 44].
In higher dimension an analogous general framework does not exist, whereas
some approaches similar to the one of QRT [30, 31] have been pursued in the
G. Gubbiotti ()
School of Mathematics and Statistics, The University of Sydney, Sydney, NSW, Australia
e-mail: giorgio.gubbiotti@sydney.edu.au
© Springer Nature Switzerland AG 2021
M. B. Paranjape et al. (eds.), Quantum Theory and Symmetries, CRM Series in
Mathematical Physics, https://doi.org/10.1007/978-3-030-55777-5_6
67
Difference Equation Admitting Three
Invariants
Giorgio Gubbiotti
Abstract In this short note we present a novel integrable fourth-order difference
equation. This equation is obtained as a stationary reduction from a known
integrable differential-difference equation. The novelty of the equation is inferred
from the number and shape of its invariants.
Keywords Difference equations · Integrability · Algebraic entropy
1 Introduction
The interest in discrete systems, that is, of systems whose independent degrees of
freedom take values in a discrete set, grew enormously during the past decades,
for reasons which span from very philosophical [23] to purely practical ones
[29]. Nowadays, discrete systems are studied from different points of view and
perspective, see [12, 24].
In this short note we will present a new integrable fourth-order difference
equation. An Nth-order difference equation is a functional equation for an unknown
sequence {x n } n∈Z where the x n+N element is expressible in terms of the previous
x n+i , i = 0, 1, . . . , N − 1. Such kind of functional equations are also called
recurrence relations. Without entering in the details, which will be given in Sect. 2,
we say that an Nth-order difference equation is integrable when its dynamics is
sufficiently regular and predictable.
The integrability of second-order difference equations is a well understood topic,
as it is known that most of the integrable second-order difference equations belong to
the QRT class [30, 31], even though there are some notable exceptions [11, 33, 44].
In higher dimension an analogous general framework does not exist, whereas
some approaches similar to the one of QRT [30, 31] have been pursued in the
G. Gubbiotti ()
School of Mathematics and Statistics, The University of Sydney, Sydney, NSW, Australia
e-mail: giorgio.gubbiotti@sydney.edu.au
© Springer Nature Switzerland AG 2021
M. B. Paranjape et al. (eds.), Quantum Theory and Symmetries, CRM Series in
Mathematical Physics, https://doi.org/10.1007/978-3-030-55777-5_6
67