Acta Polytechnica


doi:10.14311/AP.2016.56.0193
Acta Polytechnica 56(3):193–201, 2016 © Czech Technical University in Prague, 2016

available online at http://ojs.cvut.cz/ojs/index.php/ap

ON PARTIAL DIFFERENTIAL AND DIFFERENCE EQUATIONS
WITH SYMMETRIES DEPENDING ON ARBITRARY FUNCTIONS

Giorgio Gubbiotti, Decio Levi∗, Christian Scimiterna

Dipartimento di Matematica e Fisica, Università degli Studi Roma Tre, e Sezione INFN di Roma Tre, Via della
Vasca Navale 84, 00146 Roma (Italy)

∗ corresponding author: decio.levi@roma3.infn.it

Abstract. In this note we present some ideas on when Lie symmetries, both point and generalized,
can depend on arbitrary functions. We show a few examples, both in partial differential and partial
difference equations where this happens. Moreover we show that the infinitesimal generators of
generalized symmetries depending on arbitrary functions, both for continuous and discrete equations,
effectively play the role of master symmetries.

Keywords: partial differential equation; partial difference equation; Lie symmetries.

1. Introduction
In a seminal work of two century ago Medolaghi [27], following Sophus Lie results on ordinary differential
equations [24, 25], proposed the following work program for Partial Differential Equations (PDE’s):

(1.) Determine all different kinds of infinite groups of point transformations in 3 variables.
(2.) For each of the obtained groups determine the invariant second order equations.

In the framework of this research Medolaghi got, among other equations, the Liouville equation

uxt(x,t) = eu(x,t). (1.1)

The symmetry algebra of the Liouville equation (1.1) is given by the vector fields

X(f(x)) = f(x)∂x −fx(x)∂u, Y (g(t)) = g(t)∂t −gt(t)∂u, (1.2)

where f(x) and g(t) are arbitrary smooth functions of their argument and by fx(x) and gt(t) we mean their
first derivatives. The commutation relations of the vector fields (1.2) are[

X(f),X(f̃)
]

= X(ff̃x − f̃fx),
[
Y (g),Y (g̃)

]
= Y (gg̃t − g̃gt),

[
X(f),Y (g)

]
= 0. (1.3)

The algebra (1.2), (1.3) is isomorphic to the direct sum of two Virasoro algebras.
We can find also S-integrable equations [5], i.e., equations integrable by the Spectral transform, which have

an infinite dimensional group of point symmetries. Classical examples are the (2 + 1)-dimensional Kadomtsev-
Petviashvili equation [7, 8, 29], the (2 + 1)-dimensional Davey-Stewartson equation [6], the (2 + 1)-dimensional
Toda [22], the three wave interaction in three dimensions [26].

In the Example 5.7 of [28] one finds the point and generalized symmetries for the nonlinear first order wave
equation

ut = uux, u = u(x,t). (1.4)

In evolutionary form they are given by the characteristic

Q = uxF
(
x + tu,u,t +

1
ux
,
uxx
u3x

)
, (1.5)

where the function F is an arbitrary function of its arguments. In particular the last term corresponds to
generalized symmetries as it depends of the second derivative of the field. For the symmetries of hydrodynamic-
type partial differential equations see [11].

Results by Shabat and Zhiber [37] show that also the Liouville equation, being Darboux integrable, has such
kind of symmetries:

Q = (Dx + ux)F(w,wx, . . . ,wk1x) + (Dt + ut)G(w̄,w̄t, . . . , w̄k2t), (1.6)

193

http://dx.doi.org/10.14311/AP.2016.56.0193
http://ojs.cvut.cz/ojs/index.php/ap


G. Gubbiotti, D. Levi, C. Scimiterna Acta Polytechnica

where the operators Dx + ux and Dt + ut are the Laplace operators, k1 and k2 are two positive integer numbers,
F and G are two arbitrary functions of their arguments, Dx and Dt stands for the total derivative with respect
to the x and t variables and w = uxx − 12u

2
x and w̄ = utt −

1
2u

2
t are the lowest order integrals of the Liouville

equation in the x and t direction. Here and in the following by wnx we mean the n times derivative of the
function w(x,t) with respect to x.

So the nonlinear wave equation (1.4) as well as the Liouville equation (1.1) admit generalized symmetries
depending on arbitrary functions. Then Medolaghi program could be extended to the case of generalized
symmetries. We here present some preliminary ideas on a possible solution of this program.

In the following Section we will present some results on the construction of PDE’s presenting generalized
symmetries depending on arbitrary functions and relate them to Darboux integrable equations. Then in Section
3 we present the counterpart of the previous results in the discrete case. At the end we present few conclusive
remarks and conjectures.

2. Factorizable differential operators, Darboux integrable equations
and symmetries depending on arbitrary functions.

The result (1.5) can be in principle generalized to any differential equation of the first order as the equation
for the symmetries, be them point or generalized ones, can be solved on the characteristics as it was in the
case of the Hopf equation (1.4). Let consider, with no loss of generality, the example of a general first order
autonomous equation in two independent variables,

ut = f(u,ux), u = u(x,t), (2.1)

where f is an arbitrary function of its arguments. The symmetries are given by their characteristics Q(x,t,u,
ux, . . . ,ukx) and their determining equations are given by

DtQ−
[∂f
∂u
Q +

∂f

∂ux
DxQ

]∣∣∣∣
ut=f

= 0. (2.2)

Equation (2.2) is a first order differential equation for Q which can be solved on the characteristics and whose
solutions provide k + 2 symmetry variables. Then, as in the case of (1.5), the symmetries of (2.1) are given
by arbitrary functions of the k + 2 symmetry variables. So any first order PDE, linear or nonlinear, will have
generically point and generalized symmetries depending on arbitrary functions of the symmetry variables.

We can easily show an interesting consequence of the existence of generalized symmetries depending on
arbitrary functions. For the sake of concreteness we consider the symmetries of (1.4). Let us consider the
subcase of (1.5) when

Qf = ux
[
f1(x + tu) + f2

(
t +

1
ux

)
+ f3

(uxx
u3x

)]
, (2.3)

i.e., the Hopf equation (1.4) has a symmetry generator of the form:

X̂f = Qf∂u (2.4)

with the functions fi, i = 1, 2, 3 analytic in their argument. When f2 and f3 are zero then (2.4) is the infinitesimal
generator of point symmetries depending on an arbitrary function f1; when f1 and f3 are zero then (2.4) is the
infinitesimal generator of contact symmetries depending on an arbitrary function f2; when f1 and f2 are zero
then (2.4) is the infinitesimal generator of generalized symmetries depending on an arbitrary function f3. If
we take two of such generators, X̂f and X̂g, where f and g are two different functions of the same argument,
what can we say of their commutator? It has been proved by Bäcklund [17] that as soon as the characteristic Q
depends on derivatives of order higher than the first one, the symmetry group is infinite. The last symmetry
group which can be finite is the one of the contact symmetries, if no arbitrary function is present. As was shown
in the case of the Liouville equation the presence of a symmetry generator of point symmetries depending on an
arbitrary function provide an infinite dimensional Lie algebra of point symmetries.

Let us carry out the the commutation of X̂f and X̂g, for f and g equal to f1 we get:[
X̂f1,X̂g1

]
= X̂f1g′1−f′1g1, (2.5)

equal to f2 we get: [
X̂f2,X̂g2

]
= 0, (2.6)

194



vol. 56 no. 3/2016 On PDE and P∆E with Symmetries Depending on Arbitrary Functions

and equal to f3 we get:

[
X̂f3,X̂g3

]
= [f′′3 g

′
3 −f

′
3g
′′
3 ]

(uxxxux − 3u2xx)2

u9x
∂u. (2.7)

The algebra (2.5) turn out to be a Virasoro algebra of Lie point symmetries. The infinitesimal generators of the
contact symmetries (2.6) commute but the infinitesimal generator depending on an arbitrary function of the
generalized symmetry take the form of a master symmetry as the commutator of two such symmetries provide a
symmetry of higher order (2.7).

The existence of arbitrary functions of generalized symmetries is not limited to the case of first order
differential equations. It can easily extended to higher order PDE’s when the differential operator which define
the equation is factorizable. Few authors [18, 34] showed through the Laplace cascade method that factorizable
linear PDE’s are Darboux integrable if the cascade terminates. We can show that a factorizable PDE admits as
a subclass of symmetries the symmetries of its first order PDE. As an example let us consider the case of second
order factorizable PDE’s. Let us consider the second order autonomous partial differential equation for one
dependent variable u in the two independent variables x and t

E1 = utt +
[
g(u,ux) −f(u,ux)ux

]
uxt −f(u,ux)uxg(u,ux)uxx −f(u,ux)u

[
ut −g(u,ux)ux

]
= 0, (2.8)

where f and g are two arbitrary functions of their arguments. Defining the first order autonomous PDE for
u = u(x,t),

E0 = ut −f(u,ux) = 0, (2.9)

(2.8) is factorizable as:

E1 =
[
∂t + g(u,ux)∂x

]
E0 = 0. (2.10)

The symmetries of E0 are given by the infinitesimal generator

X̂ = Q0(x,t,u,ux,ut,uxx, . . . )∂u (2.11)

and the determining equation is written as

DtQ0 −f(u,ux)uQ0 −f(u,ux)uxDxQ0
∣∣∣
E0=0

= 0, (2.12)

where DtQ0 and DxQ0 are the coefficient of the prolongation of X̂ with respect to ut and ux. The determining
equation for the symmetries of E1 = 0 of infinitesimal generator

Ŷ = Q1(x,t,u,ux,ut,uxx, . . . )∂u (2.13)

is given by

pr Ŷ E1
∣∣∣
E1=0

= 0. (2.14)

The explicit expression of (2.14) is long as it involves the application of the second prolongation of Ŷ . So we
do not write it down here. By direct calculation it is easy to show that (2.14) is satisfied by the solution of
(2.12). This shows that the symmetries of the second order autonomous factorizable PDE E1 = 0 contain those
of E0 = 0. So E1 = 0 can have arbitrary function dependent generalized symmetries as E0 = 0 does. This proof
can be extended to PDE’s of any order and of any number of variables. The relation of this factorization to
Laplace cascade method and Darboux integrability is still to be understood.

3. Discrete equations with generalized symmetries
depending on an arbitrary function.

Partial Difference Equations (P∆E’s) can have symmetries depending on arbitrary functions. The presence of
arbitrary functions of point symmetries has been shown to appear, as in the continuous case [3, 11, 17, 31], when
the P∆E is linearizable [21]. Here we show on some examples that one can find P∆E’s which have arbitrary
functions of generalized symmetries. When they appear the system is usually linearizable but a complete theory
in this case is absent. We have some results on Darboux integrable discrete equations [2, 9, 10, 32, 33, 35]
which correspond to P∆E’s which have two distinct conserved quantities in the two different directions of the
discrete plane. The complete classification of Darboux integrable equations on the lattice is absent and only

195



G. Gubbiotti, D. Levi, C. Scimiterna Acta Polytechnica

some simple classes are worked out in the references mentioned above. Darboux integrable equations turn out
to be linearizable but the other way around may not be true.

Here in the following we presents three linearizable P∆E’s [13, 14] which have arbitrary functions of
generalized symmetries. Then, in a subsection, we present the case of the completely discrete Liouville equations.
The first two equations are quad graph equations which do not possess the tetrahedron property. They turn out
to be linearazable and have generalized symmetries of any order [14]. Then we consider an equation of the Boll
classification, tH

(ε)
1 , which is non autonomous, linearizable and has three-point generalized symmetries [4, 13].

The three equations are:
(1.) The first equation belongs to the classification of compatible equations around the cube with no tetrahedron
property presented by Hietarinta in [14]:

vm+1,nvm,n+1 + vm,nvm+1,n+1 = 0. (3.1)

The three-point symmetries in the m-direction of (3.1) are given by the characteristic

Qm,n = vm,nF
(

(−1)n
vm+1,n
vm,n

, (−1)n
vm,n
vm−1,n

)
. (3.2)

As (3.1) is symmetric in the exchange of n and m the generalized symmetry (3.2) is valid also in the direction
n with the role of n and m interchanged.

We can easily find two conserved quantities

W1 = (−1)n
vm+1,n
vm,n

, W2 = (−1)m
vm,n+1
vm,n

(3.3)

such that

(T2 − 1)W1 = 0, T1hm,n = hm+1,n, (T1 − 1)W2 = 0, T2hm,n = hm,n+1. (3.4)

Then (3.1) is Darboux integrable equation and (3.2) is just the three-point subcase of a general characteristic
in the m direction

Qm,n = vm,nF
(

(−1)nT−N1
vm+1,n
vm,n

, . . . , (−1)nTM1
vm+1,n
vm,n

)
, (3.5)

with N and M two positive arbitrary integers such that the number of points involved in the symmetry
is given by N + M + 2. A similar characteristic exists also in the n direction with the role of n and m
interchanged.

Equation (3.1) is linearizable in three ways:
• By the point transformation v0,0

.= ex0,0 , so that x0,0 = log v0,0 (without loss of generality, log can be
always taken to stand for the principal value of the complex logarithm), is transformed into the linear
equation x0,0 −x1,0 −x0,1 + x1,1 = iπ.

• By the Hopf–Cole transformation v1,0/v0,0
.= w0,0 (v0,0 6= 0) into the ordinary difference equation

w0,1 + w0,0 = 0.
• By the Hopf–Cole transformation v0,1/v0,0

.= t0,0 (v0,0 6= 0) into the ordinary difference equation
t1,0 + t0,0 = 0.

(2.) A second equation which belongs to the classification of equations compatible around the cube with no
tetrahedron property presented by Hietarinta in [14] is:

vm,n + vm+1,n + vm,n+1 + vm+1,n+1 + vm+1,nvm,n+1vm+1,n+1
+ vm,n

[
vm+1,nvm,n+1 + vm+1,nvm+1,n+1 + vm,n+1vm+1,n+1

]
= 0, (3.6)

whose three-point symmetries in the m-direction are given by

Qm,n = (−1)n(1 −v2m,n)K
((

(1 −vm,n)(1 −vm−1,n)
(1 + vm,n)(1 + vm−1,n)

)(−1)n
,

(
(1 −vm,n)(1 −vm+1,n)
(1 + vm,n)(1 + vm+1,n)

)(−1)n)
. (3.7)

Equation (3.6) admits a fourfold discrete symmetry given by

v0,0 → v
(ε)
0,0

.=
(1 + ε)v0,0 + 1 −ε
(1 −ε)v0,0 + 1 + ε

, (3.8)

196



vol. 56 no. 3/2016 On PDE and P∆E with Symmetries Depending on Arbitrary Functions

where ε is one of the four quartic roots of unity, ε = (±i,±1). Equation (3.6) is symmetric in the exchange
of n and m and consequently the generalized symmetry (3.7) is valid also in the direction n interchanging
the role of n and m. It is not contained in the lists of Darboux integrable discrete equations [9] and [32],
however we can prove that it is Darboux integrable as it admits the following integrals which satisfy (3.4):

W1 =
(

(1 −vm,n)(1 −vm+1,n)
(1 + vm,n)(1 + vm+1,n)

)(−1)n
, W2 =

(
(1 −vm,n)(1 −vm,n+1)
(1 + vm,n)(1 + vm,n+1)

)(−1)m
. (3.9)

The symmetry characteristic (3.7) is nothing but the reduction to three-point of a general case depending on
N + M + 2 points with N and M arbitrary positive integers,

Qm,n = (−1)n(1 −v2m,n)K
(
T−N1

(
(1 −vm,n)(1 −vm−1,n)
(1 + vm,n)(1 + vm−1,n)

)(−1)n
,

. . . ,TM1

(
(1 −vm,n)(1 −vm−1,n)
(1 + vm,n)(1 + vm−1,n)

)(−1)n)
.

(3.10)

A similar characteristic exists also in the n direction with the role of n and m interchanged.
Equation (3.6) is linearizable by the point transformation:

vm,n
.=

1 + exm,n
1 −exm,n

(3.11)

into the linear equation

xm,n + xm+1,n + xm,n+1 + xm+1,n+1 = 2izπ, (3.12)

where z = −1, 0, 1, 2. The indeterminacy of the inhomogeneous term of (3.12) takes into account the fourfold
discrete symmetry (3.8) of (3.6). It is worth while to notice that inverting (3.11) we get x0,0 = log

v0,0−1
v0,0+1

,
where without loss of generality, the function log can always be taken as the principal value of the complex
logarithm [30]. Other two linearizations, similar to those showed for (3.1), are also present here.

(3.) tH
(ε)
1 is:

(xm,n −xm+1,n)(xm,n+1 −xm+1,n+1) −ε2α2
(
F (+)n xm,n+1xm+1,n+1 + F

(−)
n xm,nxm+1,n

)
−α2 = 0. (3.13)

where F (±)n = 1±(−1)
n

2 and ε and α2 are arbitrary constants. Equation (3.13) is Darboux integrable, as we
can find two integrals in the m and n directions which satisfy (3.4). The m-integral being given by

W2
.= a2F (+)n s + b2F

(−)
n (T2 + 1)u, s

.=
xm,n+1 −xm,n−1

1 + ε2xm,n+1xm,n−1
, u

.= xm,n −xm,n−1, (3.14)

where a2 and b2 are two arbitrary constants. The n direction integral is different, as tH
(ε)
1 is not symmetric

in the exchange of n and m, and is given by

W1
.= F (+)n

α2
v

+ F (−)n t, t
.=

xm,n −xm−1,n
1 + ε2xm−1,nxm,n

, v
.= xm,n −xm−1,n. (3.15)

Equations (3.14) and (3.15) are effectively four integrals as ai and bi, i = 1, 2 are independent constants.
In [12] we constructed its three-point generalized symmetries along the direction m:

Qm,n = F (+)n

(
α2(v2 + ε2α22)
(v̄ −v)(v̄ + v)

Bm
(
α2
v̄

)
−
α2(v̄2 + ε2α22)
(v̄ −v)(v̄ + v)

Bm−1
(
α2
v

)
+
(
xm,n −

(v̄ 2+ε2α22)v
(v̄−v)(v̄+v)

)
ω + γn

)

+ F (−)n

(
t̄2t2

(t̄− t)(t̄ + t)
(Bm(t̄) −Bm−1(t)) −

t̄2t

(t̄− t)(t̄ + t)
ω + δn

)
(1 + ε2x2m,n),

(3.16)

where Bm(y), γn and δn are generic functions of their arguments, ω is an arbitrary constant and v̄ = T1v
and t̄ = T1t. Let us note that any free function and free parameter may eventually depend on α2 and ε.
It is possible to demonstrate that, as long as ε 6= 0, no n−independent reduction of the above symmetry
exists. If in (3.16) the functions Bm−1 appearing in it will depend on the variables T

p
1 W1, . . . ,T

p+N
1 W1

197



G. Gubbiotti, D. Levi, C. Scimiterna Acta Polytechnica

and consequently Bm on T
p+1
1 W1, . . . ,T

p+1+N
1 W1, one obtains the generalized symmetries for (3.13) in the

direction m at all order.
The three-point generalized symmetry along the direction n is different as the equation is not symmetric.

It is:
Qm,n = F (+)n (Bn(s) + κn) + F

(−)
n (1 + ε

2x2m,n)(Cn(ū + u) + λn), (3.17)

where Bn(y) and Cn(y) are arbitrary functions of their argument and of the lattice variable n, ū = T2u
and κn and λn are arbitrary functions of the lattice variable n. Due to the complexity of the three-point
generalized symmetries we are not able, in this case, to evince the Laplace operators and thus write the
generalized symmetries depending on an arbitrary function for any number of points.

As in the case of PDE’s presented in the previous Section also here the generalized symmetry depending on
an arbitrary function play the role of a master symmetry. Let us consider , as an example, the commutation
of two generalized symmetries of (3.1) of infinitesimal generators X̂Fm = vmFm∂vm and X̂Gm = vmGm∂vm
characterized by the arbitrary functions Fm = F

(
(−1)n vm+1

vm

)
and Gm = G

(
(−1)n vm+1

vm

)
corresponding to (3.5)

with N = M = 0. We have: [
X̂Fm,X̂Gm

]
= (−1)nvm+1Hm∂vm. (3.18)

where

Hm = [∆FmG′m − ∆GmF
′
m]. (3.19)

In (3.19), F ′m means the derivative of Fm with respect to its argument and the operator ∆ is such that
∆Fm = Fm+1 − Fm. The function Hm depends on more points than the functions Fm and Gm and so the
generator X̂Fm plays the role of a master symmetry.

3.1. The discrete algebraic Liouville equations
Among the many different completely discrete Liouville equations which go in the continuous limit in the
algebraic Liouville equation

vvxy −vxvy = v3, (3.20)

obtained from (1.1) by setting v = eu, let’s mention the following four:
• The Tzitzeica-Liouville equation, given in [1]:

hm,nhm+1,n+1(hm+1,n − 1)(hm,n+1 − 1) − (hm,n − 1)(hm+1,n+1 − 1) = 0;

• The potential Hirota-Liouville equation, given in [15, 33]:

vm,nvm+1,n+1 −vm+1,nvm,n+1 + 1 = 0;

• The Hirota-Liouville equation, given in [16, 33]:

um,num+1,n+1 − (um+1,n − 1)(um,n+1 − 1) = 0;

• The Adler-Startsev Liouville equation, given in [2]:

tm,ntm+1,n+1

(
1 +

1
tm+1,n

)(
1 +

1
tm,n+1

)
− 1 = 0. (3.21)

These P∆E’s can be transformed one into the other by the following transformations:

um,n =
hm,n

hm,n − 1
, hm,n 6= 1, um,n = vm+1,nvm,n+1, um,n = −

1
tm,n

. (3.22)

The generalized symmetries of (3.21) along the direction m are given for all p ∈Z and N ∈N0 by

dtm,n
dε

= −(1 + tm,n)(T1 − 1)
(

1 +
tm,n(1 + tm−1,n)

tm−1,n

)−1
tm,nf(T

p
1 W1, . . . ,T

p+N
1 W1), (3.23)

where ε is the group parameter, f(x,y, . . . ,z) is an arbitrary function of its arguments and W1 is the Darboux
integral of the Liouville equation (3.21) along the direction n, given in [2] by

W1 =
(

1 +
tm,n(1 + tm−1,n)

tm−1,n

)(
1 +

tm,n
tm+1,n(1 + tm,n)

)
,

198



vol. 56 no. 3/2016 On PDE and P∆E with Symmetries Depending on Arbitrary Functions

satisfying (3.4). Equation (3.23) is the equivalent of formula (1.6) introduced by Zhiber and Shabat for the
continuous Liouville equation.

Choosing p = −1 and N = 1 in (3.23) we obtain the most general generalized symmetry depending at most
on the five points tm−2,n, tm−1,n, tm,n, tm+1,n and tm+2,n

dtm,n
dε

= −(1 + tm,n)(T1 − 1)
(

1 +
tm,n(1 + tm−1,n)

tm−1,n

)−1
tm,ng(T−11 W1,W1). (3.24)

If ∂xg(x,y) 6= 0 and ∂yg(x,y) 6= 0, then we get a five-point symmetry; if ∂xg(x,y) = 0 and ∂yg(x,y) 6= 0, then
we get a four-point symmetry depending on the asymmetric set tm−1,n, tm,n, tm+1,n and tm+2,n; if ∂xg(x,y) 6= 0
and ∂yg(x,y) = 0, then we get a four-point symmetry depending on the asymmetric set tm−2,n, tm−1,n, tm,n
and tm+1,n; finally if g = 1 we get the three-point symmetry

dtm,n
dε

= −(1 + tm,n)(T1 − 1)
(

1 +
tm,n(1 + tm−1,n)

tm−1,n

)−1
tm,n, (3.25)

which doesn’t depend on any arbitrary function. As it is evident from (3.23) there is no way that we can get
a Lie point symmetry in agreement with the results presented in [19, 20]. The lowest possible symmetry is a
generalized symmetry depending on three points (3.25). Equation (3.21) is symmetric in the exchange of the m
and n indices. So the symmetries in the n directions are trivially given by (3.23) with T1 substituted by T2 and
the shifts in m substituted by shifts in n.

4. Conclusive remarks
In this note we presented a set of results partially distributed in a series of articles of different authors concerning
the presence of symmetries depending on arbitrary functions both in the continuous and in the discrete setting.
Few of the words associated to these systems are Darboux integrable systems, linearizable systems, factorizable
differential operators but not all necessarily proved to be connected to the presence of symmetries depending on
arbitrary functions. Darboux integrable systems in two independent variables are systems, studied primarily by
G. Darboux, characterized by the presence of at least a couple of conserved quantities in the two independent
variables. As far as it is known in the literature Darboux integrable systems are linearizable and the primary
example is the Liouville equation (1.1) which has also arbitrary function dependent symmetries, both point and
generalized [38].

Here we showed that generically first order PDE’s have symmetries depending on arbitrary functions as
the symmetry determining equations are characterized by just first order PDE’s which can be solved on the
characteristics in term of arbitrary functions of the symmetry variables. These same symmetries can be found
in PDE’s of higher order when the differential operator is factorized in the product of lower order ones which,
at the bottom, is just a first order one. Tsarev in [34] correlate Darboux integrable systems with factorizable
differential operators.

The situation is less clear in the discrete setting. The classification of Darboux integrable systems is not
complete for P∆E’s even on the square graph. Few results [2, 35] are known on factorizable difference operators in
the framework of Darboux integrable systems and symmetries. Nothing has been done up to now to characterize
partial difference equations whose symmetries are written in term of arbitrary functions of symmetry variables.
Moreover, let us notice that the generalized symmetries we obtain for Darboux integrable equations as (3.5),
(3.10) and (3.23) do not define, in general, S-integrable differential difference equations. Indeed these symmetries
do not always satisfy the necessary condition for the S-integrability, that the highest order shift in the lattice
variable is the opposite of the lowest one [23, 36].

As a result of this note we can conjecture the following theorem:

Theorem 1. Necessary and sufficient conditions for a PDE and a P∆E to have symmetries, possibly point but
surely generalized, depending on arbitrary functions is that the system be Darboux integrable.

Some of the results presented here are new and never presented anywhere. Among them let us mention
the structure of the Lie algebra of the symmetries of the Hopf equation, the fact that both for PDE’s and
P∆E’s generalized symmetries characterized by arbitrary functions provide master symmetries. Moreover the
calculations of the generalized symmetries, the Darboux integrals and the linearizing transformation for the two
nonlinear quad graph equations introduced by Hietarinta are new here as well as the generalized symmetries for
the completely discrete Liouville equation and the Darboux integrals for tH

(ε)
1 .

Acknowledgements
CS and DL have been partly supported by the Italian Ministry of Education and Research, 2010 PRIN Continuous and
discrete nonlinear integrable evolutions: from water waves to symplectic maps.

GG and DL are supported by INFN IS-CSN4 Mathematical Methods of Nonlinear Physics.

199



G. Gubbiotti, D. Levi, C. Scimiterna Acta Polytechnica

References
[1] V. E. Adler On a discrete analog of the Tzitzeica equation, preprint arXiv:1103.5139v1, March 26th, 2011.

[2] V. E. Adler and S. Ya. Startsev, Discrete analogues of the Liouville equation, Theor. Math. Phys., 121 (1999)
1484–1495, doi:10.1007/BF02557219

[3] G. Bluman and S. Kumey, Symmetries and Differential Equations, Springer–Verlag, New York, 1989,
doi:10.1007/978-1-4757-4307-4

[4] R. Boll, Classification and Lagrangian structure od 3D consistent quad-equations, Ph. D. dissertation (2012).

[5] F. Calogero and W. Eckhaus, Nonlinear Evolution Equations, Rescalings, Model PDEs and their Integrability. I &
II. Inv. Probl. 3 (1987) 229–262, doi:10.1088/0266-5611/3/2/008 & 4 (1988) 11–33, doi:10.1088/0266-5611/4/1/005

[6] B. Champagne and P. Winternitz, On the infinite-dimensional symmetry group of the Davey-Stewartson equantions,
J. Math. Phys, 29 (1988) 1–8, doi:10.1063/1.528173

[7] D. David, N. Kamran, D. Levi and P. Winternitz, Subalgebras of loop algebras and symmetries of the
Kadomtsev-Petviashvili equation, Phys. rev. lett. 55 (1985) 2111–2113, doi:10.1103/physrevlett.55.2111

[8] D. David, N. Kamran, D. Levi and P. Winternitz, Symmetry reduction for the Kadomtsev–Petviashvili equation
using a loop algebra, J. Math. Phys. 27 (1986) 1225–1237, doi:10.1063/1.527129

[9] R. N. Garifullin and R. I. Yamilov, Generalized symmetry classification of discrete equations of a class depending
on twelve parameters, J. Phys. A: Math. Theor. 45 ( 2012) 345205 (23 pp.), doi:10.1088/1751-8113/45/34/345205

[10] R. N. Garifullin, R. I. Yamilov, Examples Of Darboux Integrable Discrete Equations Possessing First Integrals Of
An Arbitrarily High Minimal Order, Ufimsk. Mat. Zh., 4 (2012), 174–180.

[11] A. M. Grundland, M. B. Sheftel and P. Winternitz, Invariant solutions of hydrodynamic-type equations, J. Phys. A:
Math. Gen. 33 (2000) 8193-8215, doi:10.1088/0305-4470/33/46/304

[12] G. Gubbiotti, C. Scimiterna and D. Levi, Linearizability and fake Lax pair for a consistent around the cube
nonlinear non-autonomous quad-graph equation, arXiv:1510.01527, submitted to Theor. Math. Phys. .

[13] G. Gubbiotti, C. Scimiterna and D. Levi, The non autonomous YdKN equation and generalized symmetries of Boll
equations, arXiv:1510.07175, submitted to J. Phys. A.

[14] J. Hietarinta, Searching for CAC-maps, J. Nonlinear Math. Phys. 12 Suppl. 2 (2005) 223–230,
doi:10.2991/jnmp.2005.12.s2.16

[15] R. Hirota, Nonlinear partial difference equations. V. Nonlinear equations reducible to linear equations, J. Phys. Soc.
Japan 46 (1979) 312-319, doi:10.1143/jpsj.46.312

[16] R. Hirota, Discrete two-dimensional Toda molecule equation, J. Phys. Soc. Japan 56 (1987) 4285-4288,
doi:10.1143/jpsj.56.4285

[17] N. H. Ibragimov, Transformation Groups Applied to Mathematical Physics, Nauka, Moscow, 1983 (English
translation by Reidel, Dordrecht, 1985), doi:10.1007/978-94-009-5243-0

[18] M. Juráš and I. M. Anderson, Generalized Laplace Invariants and the Method of Darboux, Duke Math. J. 89 (1997)
351–375, doi:10.1215/s0012-7094-97-08916-x

[19] D. Levi, L. Martina and P. Winternitz, Lie-point symmetries of the discrete Liouville equation, J. Phys. A: Math.
Theor. 48 (2015) 025204 (18 pp.), doi:10.1088/1751-8113/48/2/025204 arXiv:1407.4043

[20] D. Levi, L. Martina and P. Winternitz, Structure Preserving Discretizations of the Liouville Equation and their
Numerical Tests, SIGMA 11 (2015) 080 (20 pp.), doi:10.3842/sigma.2015.080 arXiv:1504.01953v2

[21] D. Levi and C. Scimiterna, Linearization through symmetries for discrete equations, J. Phys. A: Math. Theor. 46
(2013) 325204 (18 pp), doi:10.1088/1751-8113/46/32/325204

[22] D. Levi and P. Winternitz, Continuous symmetries of discrete equations, Phys. Lett. A 152 (1991) 335–338,
doi:10.1016/0375-9601(91)90733-o

[23] D. Levi and R. I. Yamilov, Conditions for the existence of higher symmetries of evolutionary equations on the
lattice, J. Math. Phys. 38 (1997) 6648–6674, doi:10.1063/1.532230

[24] S. Lie, Über Gruppen von Transformationen, Gottinger Nachrichten 1874 (1874) 529–542.
[25] S. Lie, articles published mostly in the Norwegian Archive and reproduced in the essential part in: Allgemeine Theorie

der partiellen Differentialgleichungen erster Ordnung Math. Ann. 9 (1875) 245–296, doi:10.1007/bf01443377; Theorie
der Transformationsgruppen I, Math. Ann. 16 (1880), 441–528, doi:10.1007/bf01446218; Allgemeine Untersuchungen
über Differentialgleichungen, die eine continuirliche, endliche Gruppe gestatten, Math. Ann. 25 (1885) 71–151,
doi:10.1007/bf01446421; Classification und Integration von gewöhnlichen Differentialgleichungen zwischen xy, die
eine Gruppe von Transformationen gestatten, Math. Ann. 32 (1888) 213–281, doi:10.1007/bf01444068

[26] L. Martina and P. Winternitz, Analysis and applications of the symmetry group of the multidimensional three wave
resonant interaction problem, Ann. Physics 196 (1989) 231–277, doi:10.1016/0003-4916(89)90178-4

200

http://arxiv.org/abs/1103.5139v1
http://dx.doi.org/10.1007/BF02557219
http://dx.doi.org/10.1007/978-1-4757-4307-4
http://dx.doi.org/10.1088/0266-5611/3/2/008
http://dx.doi.org/10.1088/0266-5611/4/1/005
http://dx.doi.org/10.1063/1.528173
http://dx.doi.org/10.1103/physrevlett.55.2111
http://dx.doi.org/10.1063/1.527129
http://dx.doi.org/10.1088/1751-8113/45/34/345205
http://dx.doi.org/10.1088/0305-4470/33/46/304
http://arxiv.org/abs/1510.01527
http://arxiv.org/abs/1510.07175
http://dx.doi.org/10.2991/jnmp.2005.12.s2.16
http://dx.doi.org/10.1143/jpsj.46.312
http://dx.doi.org/10.1143/jpsj.56.4285
http://dx.doi.org/10.1007/978-94-009-5243-0
http://dx.doi.org/10.1215/s0012-7094-97-08916-x
http://dx.doi.org/10.1088/1751-8113/48/2/025204
http://arxiv.org/abs/1407.4043
http://dx.doi.org/10.3842/sigma.2015.080
http://arxiv.org/abs/1504.01953v2
http://dx.doi.org/10.1088/1751-8113/46/32/325204
http://dx.doi.org/10.1016/0375-9601(91)90733-o
http://dx.doi.org/10.1063/1.532230
http://dx.doi.org/10.1007/bf01443377
http://dx.doi.org/10.1007/bf01446218
http://dx.doi.org/10.1007/bf01446421
http://dx.doi.org/10.1007/bf01444068
http://dx.doi.org/10.1016/0003-4916(89)90178-4


vol. 56 no. 3/2016 On PDE and P∆E with Symmetries Depending on Arbitrary Functions

[27] P. Medolaghi, Classificazione delle equazioni alle derivate parziali del secondo ordine, che ammettono un gruppo
infinito di trasformazioni puntuali, Ann. Mat. Pura Appl. 1 (1898) 229–263, doi:10.1007/bf02419192

[28] P. J. Olver, Applications of Lie Groups to Differeial Equations, (second edition), Springer, New York, 1993,
doi:10.1007/978-1-4612-4350-2

[29] F. Schwarz, Symmetries of the Two-Dimensional Korteweg-deVries Equation, J. Phys. Soc. Jpn. 51 (1982)
2387–2389, doi:10.1143/jpsj.51.2387

[30] C. Scimiterna and D. Levi, Classification of discrete equations linearizable by point transformation on a square
lattice, Front. Math. China 8 (2013) 1067–1076, doi:10.1007/s11464-013-0280-3

[31] M. B. Sheftel, Symmetry group analysis and invariant solutions of hydrodynamic-type systems, Int. J. Math. Math.
Sci. 2004 (2004), 487–534, doi:10.1155/s0161171204206147

[32] S. Ya. Startsev, Darboux integrable discrete equations possessing an autonomous first-order integral, J. Phys. A:
Math. Theor. 47 (2014) 105204 (16pp), doi:10.1088/1751-8113/47/10/105204

[33] S. Ya. Startsev, Non-Point Invertible Transformations and Integrability of Partial Difference Equations, SIGMA 10
(2014) 066 (13 pp.), doi:10.3842/sigma.2014.066

[34] S. P. Tsarev, Factoring linear partial differential operators and the Darboux method for integrating nonlinear
partial differential equations, Teoret. Mat. Fiz. 122 (2000), 144–160; English transl., Theoret. Math. Phys. 122
(2000), 121–133, doi:10.1007/bf02551175

[35] V. L. Vereshchagin, Darboux-Integrable Discrete Systems, Theor. Math. Phys., 156 (2008) 1142–1153,
doi:10.1007/s11232-008-0084-x

[36] R. I. Yamilov, Symmetries as integrability criteria for differential difference equations, J. Phys. A: Math. Gen. 39
(2006) R541–R623, doi:10.1088/0305-4470/39/45/r01

[37] A. V. Zhiber and A. B. Shabat, Klein–Gordon equations with a nontrivial group, Dokl. Akad. Nauk SSSR 247
(1979), 1103–1107; English transl., Soviet Phys. Dokl. 24 (1979), 607–609.

[38] A. V. Zhiber and V. V. Sokolov, Exactly integrable hyperbolic equations of Liouville type, Uspekhi Mat. Nauk 56
(2001) 63–106; English transl., Russian Math. Surveys 56 (2001) 61–101, doi:10.1070/rm2001v056n01abeh000357

201

http://dx.doi.org/10.1007/bf02419192
http://dx.doi.org/10.1007/978-1-4612-4350-2
http://dx.doi.org/10.1143/jpsj.51.2387
http://dx.doi.org/10.1007/s11464-013-0280-3
http://dx.doi.org/10.1155/s0161171204206147
http://dx.doi.org/10.1088/1751-8113/47/10/105204
http://dx.doi.org/10.3842/sigma.2014.066
http://dx.doi.org/10.1007/bf02551175
http://dx.doi.org/10.1007/s11232-008-0084-x
http://dx.doi.org/10.1088/0305-4470/39/45/r01
http://dx.doi.org/10.1070/rm2001v056n01abeh000357

	Acta Polytechnica 56(3):193–201, 2016
	1 Introduction
	2 Factorizable differential operators, Darboux integrable equations and symmetries depending on arbitrary functions.
	3 Discrete equations with generalized symmetries depending on an arbitrary function.
	3.1 The discrete algebraic Liouville equations

	4 Conclusive remarks
	Acknowledgements
	References