Non-Existence of S-Integrable Three-Point Partial Difference Equations in the Lattice Plane
Determining if an (1+1)-differential-difference equation is integrable or not (in the sense of possessing an infinite number of symmetries) can be reduced to the study of the dependence of the equation on the lattice points, according to Yamilov's theorem. We shall apply this result to a class...
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| description | Determining if an (1+1)-differential-difference equation is integrable or not (in the sense of possessing an infinite number of symmetries) can be reduced to the study of the dependence of the equation on the lattice points, according to Yamilov's theorem. We shall apply this result to a class of differential-difference equations obtained as partial continuous limits of 3-point difference equations in the plane and conclude that they cannot be integrable.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 19 (2023), 084, 7 pages
Non-Existence of S-Integrable Three-Point Partial
Difference Equations in the Lattice Plane
Decio LEVI a and Miguel A. RODRÍGUEZ b
a) Mathematical and Physical Department, Roma Tre University,
Via della Vasca Navale, 84, I00146 Roma, Italy
b) Departamento de F́ısica Teórica, Universidad Complutense de Madrid,
Pza. de las Ciencias, 1, 28040 Madrid, Spain
E-mail: rodrigue@ucm.es
Received April 17, 2023, in final form October 23, 2023; Published online November 01, 2023
https://doi.org/10.3842/SIGMA.2023.084
Abstract. Determining if an (1 + 1)-differential-difference equation is integrable or not
(in the sense of possessing an infinite number of symmetries) can be reduced to the study
of the dependence of the equation on the lattice points, according to Yamilov’s theorem.
We shall apply this result to a class of differential-difference equations obtained as partial
continuous limits of 3-points difference equations in the plane and conclude that they cannot
be integrable.
Key words: difference equations; integrability; Yamilov’s theorem
2020 Mathematics Subject Classification: 39A14; 39A36
This article is our contribution to the celebration
of Peter Olver’s 70th birthday. We want to ex-
press our admiration and gratitude for his work
and friendship.
1 Introduction
Partial difference equations have always played an important role in physics and this has been
noticeable the last decades. From one side, discrete systems seem to be at the base of many
important laws of physics (as in quantum gravity [23]) and on the other side, with the increasing
use of computers, discretizations are playing a growing role in physical applications to numeri-
cally solve differential equations [9] preserving some of the main properties of these equations,
in particular their symmetries [14].
The number of points involved in the equation characterizes these equations, for instance,
partial difference equations on four points in a plane have been studied at length [3, 8, 13, 15],
being the simplest class such that the evolution can be invertible. The continuous limit corre-
sponds to hyperbolic partial differential equations. We can distinguish two classes of integrable
equations, those which are linearizable, C-integrable equations in Calogero’s terminology [4], and
equations whose integrability requires the solution of a scattering problem (characterized by the
compatibility of two linear problems for an auxiliary wave function), S-integrable equations in
Calogero’s terminology.
Partial difference equations defined on three points, which appear as particular situations in
triangular lattices, have not received such detailed attention (see [20] for a thoroughly study
on the linearizable case) although triangularization is a key tool in differential geometry and
This paper is a contribution to the Special Issue on Symmetry, Invariants, and their Applications in honor of
Peter J. Olver. The full collection is available at https://www.emis.de/journals/SIGMA/Olver.html
mailto:rodrigue@ucm.es
https://doi.org/10.3842/SIGMA.2023.084
https://www.emis.de/journals/SIGMA/Olver.html
2 D. Levi and M.A. Rodŕıguez
they allow to define many discrete models [2, 7, 17, 18, 19]. Often, however, in the references
mentioned before, the partial difference equations defined on a triangular lattice involve more
than three lattice points.
Boundary value problems can be defined for equations involving only three lattice points.
In Figure 1, we present just an example of the initial data and the corresponding set of points
(a more complete set of possible configurations is given in [20]).
r1
un+1,m
un,m+1
un,m
r2
un+1,m
un,m+1
un,m
r3
un+1,m
un,m+1
un,m
Figure 1. An example of points related by an equation defined on three points and the choice of initial
conditions (see the main text for a discussion of these graphics).
We consider an equation relating the values of the field u at the points, (m,n), (m,n + 1),
and (m+1, n) (given three points there is only one relative position, disregarding orthogonality
or step sizes). Generically, if we write the equation as
f(un,m, un+1,m, un,m+1) = 0,
we can define the value in a point as a function of the values in the other two points, for instance,
un,m+1 = g1(un,m, un+1,m),
that is, if we give the initial values along r1 (see Figure 1), we could compute (solving the
equation) the values of the field in the upper half plane. We can (again generically) isolate
un+1,m:
un+1,m = g2(un,m, un,m+1).
Then, giving the initial values along the line r2 we obtain the values of the field in the right half
plane. Finally, we can also obtain un,m as a function of the values in the two other points:
un,m = g3(un,m+1, un+1,m).
If we provide the initial values along the line r3, the solution of the equation will be obtained
in a left descending staircase. Other possible schemes can be used for this kind of lattices and
equations (see [22] for a description of different initial-boundary value problems).
Multilinear equations can also be considered as particular cases. In fact, nonlinear equations
defined on three lattice points are simpler than those on four points. The number of parameters
in these equations is less than in the four points case, since the nonlinearities are at most
cubic. As shown in [20], the classification problem for linearizable multilinear partial difference
equations can be carried out up to the end and provides many examples of nonlinear three-point
partial difference equations.
Let us write a generic equation on three lattice points as
En,m(un,m, un+1,m, un,m+1) = 0. (1.1)
Non-Existence of S-Integrable Three-Point Partial Difference Equations in the Lattice Plane 3
If the independent variables are written as xn = nhn and tm = mhm, in terms of the discrete
indices n and m and of the lattice parameters hn and hm, we can get from (1.1) in the continuous
limit, when hn, hm → 0 and n,m → ∞ in the usual way, equations of the form
ut = f(u)ux + g(u)uxx, u = u(x, t),
as, for instance, the continuous Burgers equation [11].
The analysis of the C-integrable nonlinear partial differential equations (linearizable via
a transformation) can be found in [5, 6]. In the continuous case the class of possible trans-
formations is richer than in the discrete one as we can have transformations of the independent
and dependent variables. In the discrete case we have to restrict ourselves to just the transforma-
tions of the dependent variable unless we consider transformable lattices, i.e., partial difference
schemes (see [14] and references quoted therein).
Contrary to the case of C-integrable equations, the situation in the case of S-integrable three-
point partial difference equation is not known, up to our knowledge.
In this sense, Yamilov’s theorem constitutes a useful tool in the discussion of the integrability
of differential-difference equations (see, for instance, [12] for an example of its application in
a particular case).
In fact, the classification of differential, difference or differential-difference equations with
higher order symmetries, the symmetry approach to integrable equations, is a well known field
with many important results (see [1, 16, 21] as interesting examples of these results for differential
evolution equations). In this article, we will only focus on a negative result on the existence of
integrable systems of a particular type to show the role of Yamilov’s theorem in this field.
The content of this article is as follows. In Section 2, we review the main results on the lin-
earization of partial difference equations defined on three lattice points either by point transfor-
mations or by Hopf–Cole transformations [20]. Then in Section 3, by using a theorem introduced
by Yamilov (see [14]), we show that a partial difference equation defined on three points cannot
be S-integrable. The general result is then confirmed by considering multilinear equations where
the calculations are explicit. Section 4 is devoted to some concluding remarks and a proposal of
future researches.
2 C-integrable three-point partial difference equations
Here we review the results on linearizable three-point partial difference equations presented
in [20]. To simplify the presentation, as we will limit ourselves to autonomous equations, we will
not need to index the complex dependent variable by its lattice point but just by its relative
position with respect to the reference point n,m, i.e. un,m = u00. Then, equation (1.1) can be
written as
E (u00, u10, u01) = 0. (2.1)
In some cases, we can use an autonomous point transformation (for a nonconstant function f)
ũ00 = f(u00) (2.2)
to linearize (2.1), providing the linear equation
aũ00 + bũ10 + cũ01 + d = 0,
with a, b, c and d being complex coefficients.
4 D. Levi and M.A. Rodŕıguez
2.1 Linearizability
The following theorem, see [20], provides a solution to this problem:
Theorem 2.1 ([20, Theorem 2]). Necessary and sufficient condition for the linearizability by
a point transformation (2.2) of an equation belonging to the class (2.1) is that the equation can
be written in the form
u00 = H−1(−βH(u10)− αH(u01)− γ),
where H(x) is an arbitrary function of its argument, H−1 is its inverse function and α, β, γ
are arbitrary integration constants. This equation is linearizable by the point transformation
ũ00 = H(u00) to the equation
ũ00 + βũ10 + γũ01 = 0.
We can also linearize (2.1) by the Cole–Hopf transformation:
ũ01 = f (u00) ũ00, (2.3)
which transforms the linear equation
aũ00 + bũ10 + cũ01 = 0,
into (2.1), where a, b and c are complex coefficients with (a, b, c) ̸= (0, 0, 0). We refer to [20] for
details on the conditions over the equation to be linearizable under this transformation.
2.2 Classification of linearizable multilinear equations
We can use the above results to classify multilinear difference equations depending on three
points in a two-dimensional lattice:
au00 + bu01 + cu10 + du00u10 + eu00u01 + fu10u01 + gu00u10u01 + h = 0, (2.4)
where a, . . . , h are arbitrary complex parameters. We can state the following theorem [20]:
Theorem 2.2 ([20, Theorem 3]). Apart from the equations which are Möbius equivalent to a lin-
ear equation, the only equations belonging to the class (2.4) which are linearizable by a point
transformation are, up to a Möbius transformation of the dependent variable (eventually com-
posed with an exchange of the independent variables n ↔ m), the following three
u00u10u01 − 1 = 0, (2.5a)
u00u01 − u10 = 0, (2.5b)
u10u01 − u00 = 0. (2.5c)
Equations (2.5) linearize by the transformation ũ00 = log u00 with log always standing for the
principal branch of the complex logarithmic function, respectively, to the equations (z ∈ Z):
ũ00 + ũ10 + ũ01 = 2πiz,
ũ00 − ũ10 + ũ01 = 2πiz,
−ũ00 + ũ10 + ũ01 = 2πiz.
Let us now consider the linearization via Hopf–Cole transformations (2.3). As we classify
up to a Möbius transformation, we can be always set g00(u00) = u00. We can now state the
following theorem:
Non-Existence of S-Integrable Three-Point Partial Difference Equations in the Lattice Plane 5
Theorem 2.3 ([20, Theorem 5]). The class of complex autonomous multilinear discrete equa-
tions defined on three points which is linearizable to a homogeneous linear equation by a Cole–
Hopf transformation ũ01 = u00ũ00, is given, up to a Möbius transformation of the dependent
variable (eventually composed with an exchange of the independent variables n ↔ m), by the
following equation:
1 + u00
u00
− 1 + u01
u10
= 0.
This equation linearizes to the equation
ũ00 + ũ10 + ũ01 = 0.
3 Yamilov’s theorem and the S-integrability
of three-point partial difference equations
In [14] (see section “Why the shape of integrable equations on the lattice is symmetric”), the
following theorem is proposed and proved.
Theorem 3.1 (Yamilov). If an equation of the form
u̇n = fn = f(un+N , un+N−1, . . . , un+M ), N ≥ M,
∂fn
∂un+N
∂fn
∂un+M
̸= 0,
where f is a smooth enough function of its variables, possesses a conservation law of order m,
such that m > min(|N |, |M |), then
N = −M, (3.1)
and N ≥ 0.
This theorem states a necessary condition for a differential-difference equation to be S-
integrable based on the fact that while a differential-difference equation with an infinity of
symmetries is either S- or C-integrable, if it has not a sufficiently high conservation law it can-
not be S-integrable. As the content of this theorem is a necessary condition, an equation which
satisfies it may still be not integrable. However, a S-integrable equation has to satisfy necessarily
this theorem.
Here in the following we will use this theorem to show that, while as we saw in the previous
Section partial difference equations defined on three lattice points can be C-integrable, they may
not be S-integrable.
To do so we carry out the partial continuous limit of (1.1). Equation (1.1) is symmetric
in the exchange of n and m so in the whole generality we can do the continuous limit when
hm → 0 and m → ∞ so that t = mhm remains finite. For convenience, we will call hm = ε and
un,m+1 = un(t+ hm) = un(t+ ε). Assuming that un(t) is an entire function of t, we can write
un(t+ ε) = un(t) + εu̇n(t) +
1
2
ε2ün(t) +O
(
ε3
)
.
So (1.1) becomes
En,m(un,m, un+1,m, un,m+1) = En(ε, un(t), un+1(t), un(t+ ε)). (3.2)
Expanding the last result in (3.2) in ε, assuming that En(ε, un(t), un+1(t), un(t+ ε)) is an entire
function, we have
En(ε, un(t), un+1(t), un(t+ ε)) = E(0)
n (un(t), un+1(t))+εE(1)
n (un(t), un+1(t), u̇n(t))+O
(
ε2
)
.
6 D. Levi and M.A. Rodŕıguez
By a proper choice of the dependence of (1.1) from um,n, um,n+1 and um+1,n, we can make
E(0)
n = 0 and then its semi-continuous limit becomes
E(1)
n (un(t), un+1(t), u̇n(t)) = 0. (3.3)
Equation (3.3) is not in the form of Yamilov’s theorem, since there is no un−1 dependence in the
equation, required in the theorem, and cannot be reduced to it by any lattice re-parametrization
as it depends just on two points.
As an example let us consider the semi-continuous limit of (2.4) when the complex parameters
a, . . . , h are taken to be entire functions of ε, the lattice spacing in the m direction which we
can assume to be a constant along the lattice. We can expand the parameters of (2.4) in powers
of ε and we have
a = a(0) + εa(1) + ε2a(2) + · · ·
and similar expressions for the other parameters in the equation. The request that the zero
order in ε of (2.4) be zero implies
a(0) = −b(0), c(0) = e(0) = g(0) = h(0) = 0, d(0) = −f (0), (3.4)
and at first order in ε we get(
b(0) + f (0)u1
)
u̇0 + (a(1) + b(1))u0 + c(1)u1 + e(1)u20 + (d(1) + f (1))u0u1
+ g(1)u20u1 + h(1) = 0. (3.5)
Naturally (3.5) is of the form (3.3) and, by a proper choice of the parameters (3.4), we can
always make E(0)
n = 0, as required in the general case. In the notation of Yamilov’s theorem,
(3.5) has N = 1 and M = 0 and consequently N ≥ 0 but the condition (3.1) is not satisfied.
Thus (3.5) cannot be S-integrable as it does not satisfy the necessary S-integrability conditions
implied by Theorem 3.1. As (3.5) is not S-integrable for any choice of the parameters so it will
be also the multilinear partial difference equation (2.4).
4 Conclusions
The application of Yamilov’s theorem to the three-points partial difference equation yields the
result that no integrable differential-difference equation can be constructed by taking the limit
of this class of equations.
We have just discussed the limits in the direct approach, taking the continuous limit in one
of the two indices of the equation. We have also considered skew-limits (through a combination
of both indices n, m), since, as it is known [10, 12], in some cases one can obtain differential-
difference equations satisfying Yamilov’s theorem using this approach. However, this is not
possible in the three-points case.
We plan to extend the results presented here to the case of four-points partial difference
equations on the lattice plane. In this case we know there are many S-integrable and C-integrable
results. Since the classification of C-integrable equations in the multilinear case is not complete,
it would be interesting to prove that, at least in this multilinear case, there is a privileged shape
of the equation which might contain S-integrable equations.
Acknowledgements
We thank the anonymous referees for corrections, useful suggestions, and constructive criticism
that helped improve this article. MAR acknowledges the support of Universidad Complutense
de Madrid (Spain), under grant G/6400100/3000.
Non-Existence of S-Integrable Three-Point Partial Difference Equations in the Lattice Plane 7
Professor Decio Levi passed away at the time we were working on this article. I will always
miss him as my colleague and dearest friend.
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1 Introduction
2 C-integrable three-point partial difference equations
2.1 Linearizability
2.2 Classification of linearizable multilinear equations
3 Yamilov's theorem and the S-integrability of three-point partial difference equations
4 Conclusions
References
|
| id | nasplib_isofts_kiev_ua-123456789-212047 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-17T03:55:02Z |
| publishDate | 2023 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Levi, Decio Rodríguez, Miguel A. 2026-01-23T10:12:14Z 2023 Non-Existence of S-Integrable Three-Point Partial Difference Equations in the Lattice Plane. Decio Levi and Miguel A. Rodríguez. SIGMA 19 (2023), 084, 7 pages 1815-0659 2020 Mathematics Subject Classification: 39A14; 39A36 arXiv:2304.06956 https://nasplib.isofts.kiev.ua/handle/123456789/212047 https://doi.org/10.3842/SIGMA.2023.084 Determining if an (1+1)-differential-difference equation is integrable or not (in the sense of possessing an infinite number of symmetries) can be reduced to the study of the dependence of the equation on the lattice points, according to Yamilov's theorem. We shall apply this result to a class of differential-difference equations obtained as partial continuous limits of 3-point difference equations in the plane and conclude that they cannot be integrable. We thank the anonymous referees for corrections, useful suggestions, and constructive criticism that helped improve this article. MAR acknowledges the support of Universidad Complutense de Madrid (Spain), under grant G/6400100/3000. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Non-Existence of S-Integrable Three-Point Partial Difference Equations in the Lattice Plane Article published earlier |
| spellingShingle | Non-Existence of S-Integrable Three-Point Partial Difference Equations in the Lattice Plane Levi, Decio Rodríguez, Miguel A. |
| title | Non-Existence of S-Integrable Three-Point Partial Difference Equations in the Lattice Plane |
| title_full | Non-Existence of S-Integrable Three-Point Partial Difference Equations in the Lattice Plane |
| title_fullStr | Non-Existence of S-Integrable Three-Point Partial Difference Equations in the Lattice Plane |
| title_full_unstemmed | Non-Existence of S-Integrable Three-Point Partial Difference Equations in the Lattice Plane |
| title_short | Non-Existence of S-Integrable Three-Point Partial Difference Equations in the Lattice Plane |
| title_sort | non-existence of s-integrable three-point partial difference equations in the lattice plane |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/212047 |
| work_keys_str_mv | AT levidecio nonexistenceofsintegrablethreepointpartialdifferenceequationsinthelatticeplane AT rodriguezmiguela nonexistenceofsintegrablethreepointpartialdifferenceequationsinthelatticeplane |