A Novel Integrable Fourth-Order Difference Equation Admitting Three Invariants
73
This proves that Eq. (13) is an integrable equation according to both definitions
stated in Sect. 2. Moreover, the number and shape of the invariants proves that
this equation is outside the known classifications given in [7, 22]. For instance,
the invariants are rational in the affine coordinate, differently from [22], where
the invariants are polynomials in affine coordinates. Furthermore, these invariants
are not ratios of biquadratic polynomials, therefore are outside the class considered
in [7].
From the invariants (16) it is possible to construct a dual map [32]. Differently
from the dual maps appearing in [7, 26, 32] this dual map are not integrable
according to the algebraic entropy test. We underline that dual maps with such
features already appeared in [22].
Remark 2 The invariant I 1 is linear in x n+1 and x n−2 . This implies that T n I 1 − I 1
is actually equivalent to Eq. (13). That is, Eq. (13) is resummable to an autonomous
third-order difference equation.
4 Final Remarks
In this short note we showed that the stationary reduction of equation (12), namely
Eq. (13), is integrable in the sense of algebraic entropy and in the sense of the
existence of invariants. The first properties follow from our previous work [19], yet
we showed that the quadratic growth is preserved.
In our upcoming paper [20] we will consider all the stationary reductions of
the fourth-order Volterra-like differential-difference equations classified in [15, 16].
The application of the algebraic entropy test will give rise to a vast “zoology” of
possibilities, consisting in periodic equations, explicitly linear equations, linearizable equations (linear growth), and integrable equations (quadratic growth). We will
explain these growth properties using the following notions:
1. correspondence to idempotent maps,
2. explicit linearization,
3. resummation, in the sense of [3], to integrable second-order non-autonomous
difference equations,
4. deflation, in the sense of [26], to integrable third-order difference equations,
5. existence of three invariants.
Genuinely new integrable fourth-order difference equations belong to the last
class. With these new examples we aim to broaden our knowledge of higher-order
integrable difference equations and give the foundation for a new algorithmic search
method based on hierarchies of differential-difference equations.
Acknowledgments We thank Dr. D. T. Tran for the helpful discussions during the preparation of
this paper.
GG is supported through Prof. N. Joshi’s Australian Laureate Fellowship #FL120100094.
73
This proves that Eq. (13) is an integrable equation according to both definitions
stated in Sect. 2. Moreover, the number and shape of the invariants proves that
this equation is outside the known classifications given in [7, 22]. For instance,
the invariants are rational in the affine coordinate, differently from [22], where
the invariants are polynomials in affine coordinates. Furthermore, these invariants
are not ratios of biquadratic polynomials, therefore are outside the class considered
in [7].
From the invariants (16) it is possible to construct a dual map [32]. Differently
from the dual maps appearing in [7, 26, 32] this dual map are not integrable
according to the algebraic entropy test. We underline that dual maps with such
features already appeared in [22].
Remark 2 The invariant I 1 is linear in x n+1 and x n−2 . This implies that T n I 1 − I 1
is actually equivalent to Eq. (13). That is, Eq. (13) is resummable to an autonomous
third-order difference equation.
4 Final Remarks
In this short note we showed that the stationary reduction of equation (12), namely
Eq. (13), is integrable in the sense of algebraic entropy and in the sense of the
existence of invariants. The first properties follow from our previous work [19], yet
we showed that the quadratic growth is preserved.
In our upcoming paper [20] we will consider all the stationary reductions of
the fourth-order Volterra-like differential-difference equations classified in [15, 16].
The application of the algebraic entropy test will give rise to a vast “zoology” of
possibilities, consisting in periodic equations, explicitly linear equations, linearizable equations (linear growth), and integrable equations (quadratic growth). We will
explain these growth properties using the following notions:
1. correspondence to idempotent maps,
2. explicit linearization,
3. resummation, in the sense of [3], to integrable second-order non-autonomous
difference equations,
4. deflation, in the sense of [26], to integrable third-order difference equations,
5. existence of three invariants.
Genuinely new integrable fourth-order difference equations belong to the last
class. With these new examples we aim to broaden our knowledge of higher-order
integrable difference equations and give the foundation for a new algorithmic search
method based on hierarchies of differential-difference equations.
Acknowledgments We thank Dr. D. T. Tran for the helpful discussions during the preparation of
this paper.
GG is supported through Prof. N. Joshi’s Australian Laureate Fellowship #FL120100094.