Lax Pair for a Novel Two-Dimensional Lattice
In the paper by I.T. Habibullin and our joint paper, the algorithm for the classification of integrable equations with three independent variables was proposed. This method is based on the requirement of the existence of an infinite set of Darboux integrable reductions and on the notion of the chara...
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Інститут математики НАН України
2021
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| Цитувати: | Lax Pair for a Novel Two-Dimensional Lattice. Maria N. Kuznetsova. SIGMA 17 (2021), 088, 13 pages |
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| description | In the paper by I.T. Habibullin and our joint paper, the algorithm for the classification of integrable equations with three independent variables was proposed. This method is based on the requirement of the existence of an infinite set of Darboux integrable reductions and on the notion of the characteristic Lie-Rinehart algebras. The method was applied for the classification of integrable cases of different subclasses of equations 𝑢ₙ‚ₓy = 𝑓(𝑢ₙ₊₁, 𝑢ₙ, 𝑢ₙ₋₁, 𝑢ₙ‚ₓ, 𝑢ₙ,y) of special forms. Under this approach, the novel integrable chain was obtained. In the present paper, we construct a Lax pair for the novel chain. To construct the Lax pair, we use the scheme suggested in papers by E.V. Ferapontov. We also study the periodic reduction of the chain.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 17 (2021), 088, 13 pages
Lax Pair for a Novel Two-Dimensional Lattice
Maria N. KUZNETSOVA
Institute of Mathematics, Ufa Federal Research Centre, Russian Academy of Sciences,
112 Chernyshevsky Street, Ufa 450008, Russia
E-mail: mariya.n.kuznetsova@gmail.com
Received February 09, 2021, in final form September 15, 2021; Published online September 26, 2021
https://doi.org/10.3842/SIGMA.2021.088
Abstract. In paper by I.T. Habibullin and our joint paper the algorithm for classifica-
tion of integrable equations with three independent variables was proposed. This method
is based on the requirement of the existence of an infinite set of Darboux integrable re-
ductions and on the notion of the characteristic Lie–Rinehart algebras. The method
was applied for the classification of integrable cases of different subclasses of equations
un,xy = f(un+1, un, un−1, un,x, un,y) of special forms. Under this approach the novel inte-
grable chain was obtained. In present paper we construct Lax pair for the novel chain. To
construct the Lax pair, we use the scheme suggested in papers by E.V. Ferapontov. We also
study the periodic reduction of the chain.
Key words: Lax pair; two-dimensional lattice; integrable reduction; characteristic algebra;
Lie–Rinehart algebra; Darboux integrable system; higher symmetry; x-integral
2020 Mathematics Subject Classification: 37K10; 37K30; 37D99
1 Introduction
In a number of recent publications [8, 13, 14, 15, 16, 18] the problem of integrable classification
of two-dimensional lattices
un,xy = f(un+1, un, un−1, un,x, un,y), −∞ < n <∞, (1.1)
was studied. Here the sought function un = un(x, y) depends on the real variables x, y and the
integer variable n. In these papers we proposed the method for seeking and classifying integrable
equations with three independent variables based on the requirement of the existence of a set
of Darboux integrable reductions and on the notion of the characteristic Lie–Rinehart algebras.
The method was applied to different subclasses of equations (1.1) of special forms.
Within this approach we use the following
Definition 1.1. A lattice of the form (1.1) is called integrable if there exist locally analytic
functions φ and ψ of two variables such that for any choice of integers N1, N2 the hyperbolic
type system
uN1,xy = φ(uN1+1, uN1),
un,xy = f(un+1, un, un−1, un,x, un,y), N1 < n < N2, (1.2)
uN2,xy = ψ(uN2 , uN2−1),
obtained from lattice (1.1) by imposing cut-off conditions at n = N1 and n = N2, is integrable
in the sense of Darboux.
Let us recall what Darboux integrability means.
mailto:mariya.n.kuznetsova@gmail.com
https://doi.org/10.3842/SIGMA.2021.088
2 M.N. Kuznetsova
Definition 1.2. A function I = I(x, ū, ūx, ūxx, . . . ) is called an y-integral if it satisfies the
equation DyI = 0 for every solution of system (1.2). A function J = J(y, ū, ūy, ūyy, . . . ) is called
a x-integral if it satisfies the equation DxJ = 0. Integrals of the form I = I(x) and J = J(y)
are called trivial.
Here ū is a vector ū = (uN1 , uN1+1, . . . , uN2), ūx is its derivative and so on. The operators Dy
and Dx are operators of the total derivative with respect to the variable y or x, correspondingly,
by virtue of system (1.2).
Definition 1.3. A system (1.2) is called Darboux integrable if it possesses N2 −N1 + 1 func-
tionally independent nontrivial integrals in both characteristic directions x and y.
Darboux integrable systems are amenable to study by the Lie–Rinehart algebras. Let I =
I(x, ū, ūx, ūxx, . . . ) be a nontrivial y-integral for the system (1.2). Then I must satisfy the
following system:
Y I = 0, XiI = 0,
where
Xi =
∂
∂ui,y
, Y =
N2∑
i=N1
(
ui,y
∂
∂ui
+ fi
∂
∂ui,x
+Dx(fi)
∂
∂ui,xx
+ · · ·
)
and fi = f(ui+1, ui, ui−1, ui,x, ui,y). The first equation follows from the fact that the operator Dy
acts on functions I = I(x, ū, ūx, ūxx, . . . ) by the rule DyI = Y I, the second one arises because I
doesn’t depend on variables ui,y.
Let us consider the Lie algebra Ly generated by the operators Y , Xi over the ring A of locally
analytic functions of the dynamical variables ūy, ū, ūx, ūxx, . . . . To the standard multiplication
operation [Z,W ] = ZW −WZ we add two conditions: [Z, aW ] = Z(a)W +a[Z,W ] and (aZ)b =
aZ(b) valid for any Z,W ∈ Ly and a, b ∈ A. These equalities means that for any Z ∈ Ly and
any a ∈ A, the element aZ ∈ Ly. In this case the algebra Ly is called the Lie–Rinehart
algebra [20, 22].
If there exists a finite basis Z1, Z2, . . . , Zk ∈ Ly such that an arbitrary element Z ∈ Ly is repre-
sented as a linear combination Z = a1Z1+a2Z2+· · ·+akZk, where coefficients a1, a2, . . . , ak ∈ A;
and if the equality Z = 0 implies that a1 = a2 = · · · = ak = 0, then algebra Ly is of a finite
dimension.
The integrability criterion of the hyperbolic type system in the sense of Darboux is formulated
as follows [28, 29]:
Theorem 1.4. System (1.2) admits a complete set of the y-integrals (a complete set of the
x-integrals) if and only if its characteristic algebra Ly (respectively, characteristic algebra Lx) is
of finite dimension.
Corollary 1.5. System (1.2) is integrable in the sense of Darboux if both characteristic alge-
bras Lx and Ly are of finite dimension.
The above statements play a key role in our classification works. Within the scope of this
paper we need one of our results: paper [16] provides a complete list of integrable two-dimensional
lattices of the form
un,xy = α(un+1, un, un−1)un,xun,y + β(un+1, un, un−1)un,x
+ γ(un+1, un, un−1)un,y + δ(un+1, un, un−1), (1.3)
with the coefficient α satisfying the conditions ∂α(un+1,un,un−1)
∂un±1
̸= 0. This list consists of two
equations:
Lax Pair for a Novel Two-Dimensional Lattice 3
Theorem 1.6. Integrable equation of the form (1.3) can be reduced by a point transformation
to one of the following forms:
un,xy = αnun,xun,y, (1.4)
un,xy = αn
(
un,x − u2n − 1
)(
un,y − u2n − 1
)
+ 2un
(
un,x + un,y − u2n − 1
)
, (1.5)
where
αn =
1
un − un−1
− 1
un+1 − un
=
un+1 − 2un + un−1
(un+1 − un)(un − un−1)
.
Equation (1.4) was found before in papers [7, 23] by Ferapontov and Shabat and Yamilov.
Equation (1.5) appeared in [16] as a result of the classification procedure.
The aim of the paper is to find Lax pair for novel chain (1.5), to explain the method of finding
Lax pairs and to prove that periodic closings of the chain possesses higher symmetries.
The Lax pair for equation (1.4)
ψn,x =
un,x
un+1 − un
(ψn+1 − ψn), ψn,y =
un,y
un − un−1
(ψn − ψn−1)
was found by E.V. Ferapontov. To construct Lax pair for chain (1.5), we use the scheme
suggested in paper [12]. Let us describe the procedure in detailed. First of all, we represent
lattice (1.5) in the equivalent following form:
uxy =
(
ux − u2 − 1
)(
uy − u2 − 1
) △zz̄u
△zu△z̄u
+ 2u
(
ux + uy − u2 − 1
)
. (1.6)
Here △z =
Tz−1
ϵ , △z̄ =
1−Tz̄
ϵ are the forward/backward discrete derivatives and △zz̄ =
Tz+Tz̄−2
ϵ2
is the symmetrised second-order discrete derivative; the operators Tz, Tz̄ are the forward and
backward ϵ-shifts operators in the variable z.
The method consists of three steps:
1) First we construct the dispersionless limit of the equation (obtained as ϵ→ 0).
2) Secondly, for the equation found at the previous step we find dispersionless Lax pair.
Usually this problem is effectively solved.
3) Finally, we reconstruct Lax pair by appropriate “quantization” of dispersionless Lax pair
as proposed in [27].
The paper is organized as follows. In Section 2 Lax pair for chain (1.5) is constructed.
Section 3 is devoted to periodic closings. Namely, we impose the periodic closure conditions
un+2 = un to infinite chains (1.4), (1.5) and obtain finite systems. Lax pairs and higher sym-
metries of the second order are constructed for obtained finite systems. Conclusion contains
a discussion of the results.
2 Construction Lax pair for equation (1.5)
The main result of this section is as follows:
Theorem 2.1. Equation (1.5) possesses the Lax pair
ψn,x =
un,x − u2n − 1
un+1 − un
(ψn+1 − ψn) + unψn,
ψn,y =
un,y − u2n − 1
un − un−1
(ψn − ψn−1) + unψn.
4 M.N. Kuznetsova
Proof. The dispersionless limit of the equation (1.6) coincides with equation:
uxy =
(
ux − u2 − 1
)(
uy − u2 − 1
)uzz
u2z
+ 2u
(
ux + uy − u2 − 1
)
. (2.1)
There exists a direct method for finding Lax pairs for equations of this form. Lax pair is sought
in the following form:
Sx = F (u, ux, uy, uz, Sz), (2.2)
Sy = G(u, ux, uy, uz, Sz). (2.3)
The compatibility condition Sxy = Syx of system (2.2), (2.3) by virtue of equation (2.1) leads
to the overdetermined equation
Fuyuyyu
2
z−Guxuxxu
2
z− (GSzFux −GuxFSz +Guz)uzxu
2
z− (GuyFSz −GSzFux − Fuz)uzyu
2
z
+ uzz
((
u2 − uy + 1
)(
u2 − ux + 1
)
(Fux −Guy)− u2z(GSzFuz −GuzFSz)
)
− u2z
(
2u
(
1 + u2 − ux − uy
)
(Fux −Guy) + uz(GSzFu −GuFSz) + uxGu − uyFu
)
= 0.
Because of the fact that variables u, ux, uy, uz, uxx, uyy, uzx, uzy, uzz are independent, this
equation splits down into the overdetermined system of equations:
Fuy = 0, Gux = 0, (2.4)
GSzFux −GuxFSz +Guz = 0, (2.5)
GuyFSz −GSzFux − Fuz = 0, (2.6)(
u2 − uy + 1
)(
u2 − ux + 1
)
(Fux −Guy)− u2z(GSzFuz −GuzFSz) = 0, (2.7)
2u
(
1 + u2 − ux − uy
)
(Fux −Guy) + uz(GSzFu −GuFSz) + uxGu − uyFu = 0. (2.8)
Equations (2.4) mean that F = F (u, ux, uz, Sz) and G = G(u, uy, uz, Sz). Substituting F and G
into (2.5), (2.6), we arrive at the equations:
Guz +GSzFux = 0, Fuz +GuyFSz = 0. (2.9)
We differentiate the first equation (2.9) by ux, the second equation (2.9) – by uy, and obtain
that GSzFuxux = 0, FSzGuyuy = 0. Obviously that the functions F and G take the following
forms:
F (u, ux, uz, Sz) = F2(u, uz, Sz)ux + F3(u, uz, Sz),
G(u, uy, uz, Sz) = F4(u, uz, Sz)uy + F5(u, uz, Sz).
Then we rewrite (2.9) and (2.7), (2.8) using the last formulas. Because of the fact that the
variables u, ux, uy, uz are independent, obtained equations split down one more time. Thus we
arrive at the system for unknown functions Fi(u, uz, Sz), i = 2, 3, 4, 5:
F2F4,Sz + F4,uz = 0, F4F2,Sz + F2,uz = 0, (2.10)
F2 − F4 + u2z(F2,SzF4,uz − F4,SzF2,uz) = 0, (2.11)
F4,u − F2,u + uz(F2,uF4,Sz − F4,uF2,Sz) = 0, (2.12)
F4F3,Sz + F3,uz = 0, (2.13)(
1 + u2
)
(F4 − F2) + u2z(F3,SzF4,uz − F4,SzF3,uz) = 0, (2.14)
2u(F4 − F2) + uz(F3,uF4,Sz − F4,uF3,Sz)− F3,u = 0, (2.15)
F2F5,Sz + F5,uz = 0, (2.16)
Lax Pair for a Novel Two-Dimensional Lattice 5
2u(F4 − F2) + uz(F2,uF5,Sz − F5,uF2,Sz) + F5,u = 0, (2.17)(
u2 + 1
)
(F4 − F2) + u2z(F2,SzF5,uz − F5,SzF2,uz) = 0, (2.18)(
u2 + 1
)2
(F2 − F4) + u2z(F3,SzF5,uz − F5,SzF3,uz) = 0, (2.19)
2u
(
u2 + 1
)
(F2 − F4) + uz(F3,SzF5,u − F5,SzF3,u) = 0. (2.20)
Now we will work with equations (2.10)–(2.12) to clarify functions F2, F4. Let us express F4,uz ,
F2,uz from (2.10) and substitute them into (2.12). This leads to the condition F4 = F2 or to the
equation(
1− u2zF2,SzF4,Sz
)
= 0. (2.21)
Let us consider case (2.21). We look for F2, F4 in the following form:
F2(u, uz, Sz) =
A(u, Sz)
uz
, F4(u, uz, Sz) =
B(u, Sz)
uz
.
Then A, B have to satisfy the system obtained using (2.21), (2.10), and (2.11),
1−ASzBSz = 0, −A+BASz = 0, −B +ABSz = 0, (2.22)
Bu −Au +BSzAu −ASzBu = 0. (2.23)
This system has the solution:
A(u, Sz) =
ea1(u)Sz+a1(u)a2(u) − 1
a1(u)
.
Here a1, a2 are arbitrary functions. Similarly, we find that
B(u, Sz) =
ea4(u)Sz+a4(u)a3(u) − 1
a4(u)
with arbitrary functions a3, a4. Under obtained A and B the first equation (2.22) becomes
1− e(a1(u)+a4(u))Sz+a2(u)a1(u)+a3(u)a4(u) = 0.
Thus one can derive that a4 = −a1, a3 = a2. Finally, equation (2.23) takes the form(
−a1(u)a′1(u)Sz − a21(u)a
′
2(u)− a1(u)a2(u)a
′
1(u) + 2a′1(u)
)
ea1(u)(Sz+a2(u))
+
(
a21(u)a
′
2(u) + a1(u)a2(u)a
′
1(u) + 2a′1(u) + a1(u)a
′
1(u)Sz
)
e−a1(u)(Sz+a2(u)) − 4a′1(u) = 0.
We assume essential dependence on Sz for functions F2, F4 and, therefore, for A, B, so the
functions ea1(u)Sz , e−a1(u)Sz , ea1(u)SzSz, e
−a1(u)SzSz are independent. Hence we have a1(u) = c1,
a2(u) = c2, where c1, c2 are arbitrary constants.
Thus, we have clarified the right hand sides of Lax pair (2.2), (2.3)
Sx = F (u, ux, uy, uz, Sz) =
(
ec1(Sz+c2) − 1
)
ux
c1uz
+ F3(u, uz, Sz),
Sy = G(u, ux, uy, uz, Sz) = −
(
e−c1(Sz+c2) − 1
)
uy
c1uz
+ F5(u, uz, Sz).
By the shift transformation S → S − c2z and by the scaling z → c1z these equations can be
reduced to
Sx = F (u, ux, uy, uz, Sz) =
(
eSz − 1
)
ux
uz
+ F3(u, uz, Sz),
6 M.N. Kuznetsova
Sy = G(u, ux, uy, uz, Sz) = −
(
e−Sz − 1
)
uy
uz
+ F5(u, uz, Sz).
To clarify F3, we substitute the above functions into (2.13), (2.14), and (2.15)(
e−Sz − 1
)
uzF3,u − 2u
(
eSz + 2
)
= 0,
uz
(
e−Sz − 1− uze
−Sz
)
F3,Sz −
(
u2 + 1
)(
eSz + e−Sz − 2
)
= 0,
−(e−Sz − 1)F3,Sz + uzF3,uz = 0.
This system has the solution
F3(u, uz, Sz) = −
(
eSz − 1
)(
u2 + 1
)
uz
.
Now we rewrite equations (2.16)–(2.20) and we obtain the system on the unknown function F5:(
eSz − 1
)
F5,Sz + uzF5,uz = 0,
uz
(
eSz − 1
)
F5,Sz + u2ze
SzF5,uz −
(
u2 + 1
)(
eSz + e−Sz − 2
)
= 0,
−uz(−e−2Sz + 3e−Sz − 3 + eSz)F5,Sz − u2z(e
Sz + e−Sz − 2)F5,uz
+
(
u2 + 1
)(
−4 + eSz − 4e−2Sz + 6e−Sz + e−3Sz
)
= 0,
uz
(
1− eSz
)
F5,u − 2u
(
eSz + e−Sz − 2
)
= 0,
2uuz
(
e−2Sz − 3e−Sz + 3− eSz
)
F5,Sz +
(
u2 + 1
)
uz
(
eSz + e−Sz − 2
)
F5,u
+ 2u
(
u2 + 1
)(
e−3Sz + 6e−Sz − 4e−2Sz + eSz − 4
)
= 0.
This system possesses the solution
F5(u, uz, Sz) = −
(
1− e−Sz
)(
u2 + 1
)
uz
.
Thus we have found the Lax pair
Sx =
ux − u2 − 1
uz
(
eSz − 1
)
+
1
uz
, (2.24)
Sy =
uy − u2 − 1
uz
(
1− e−Sz
)
− 1
uz
(2.25)
for equation (2.1).
Now we reconstruct the dispersive Lax pair by an appropriate quantization the dispersionless
Lax pair (2.24), (2.25). First, we “quantise” [27] the terms in every equation (2.24), (2.25): uz is
replaced by △zu; e
Sz − 1 by △zψ due to the formal representation e
∂
∂z ≈ 1 + ∂
∂z + · · · , and,
similarly 1− e−Sz by △z̄ψ.
In most cases, this procedure provides the necessary Lax pair. But in this case we do not
obtain the Lax pair for (1.6) if we act in the same way. It was experimentally found that we
should fit the second term in the r.h.s. of equations (2.24), (2.25) by the following way (i.e., we
guess some part):
ψx =
ux − u2 − 1
△zu
△zψ + P (u)ψ,
ψy =
uy − u2 − 1
△z̄u
△z̄ψ +Q(u)ψ.
Lax Pair for a Novel Two-Dimensional Lattice 7
The compatibility condition ψxy = ψyx is straightforward to solve. Thus we find that equa-
tion (1.6) possesses the Lax pair
ψx =
ux − u2 − 1
△zu
△zψ + uψ, ψy =
uy − u2 − 1
△z̄u
△z̄ψ + uψ.
It finally proved Theorem 2.1. ■
3 Higher symmetries of periodic closings
Let us impose the periodic closure conditions un+2 = un to infinite lattice (1.4). Then we obtain
the following finite system:
u0,xy =
2
u0 − u1
u0,xu0,y, u1,xy =
2
u1 − u0
u1,xu1,y. (3.1)
System (3.1) has the x-integral and the y-integral
w =
u0,yu1,y
(u0 − u1)2
, W =
u0,xu1,x
(u0 − u1)2
. (3.2)
Lax pair for (3.1) has the form
Ψx = (Aλ+B)Ψ, Ψy =
(
Ãλ−1 + B̃
)
Ψ, (3.3)
where Ψ = (ψ1, ψ0)
T and
A =
(
0 0
u1,x
u0 − u1
0
)
, B =
u0,x
u0 − u1
− u0,x
u0 − u1
0 − u1,x
u0 − u1
,
à =
(
0 0
0 − u0,y
u0 − u1
)
, B̃ =
− u0,y
u0 − u1
0
− u1,y
u1 − u0
− u1,y
u1 − u0
,
λ is a spectral parameter.
The classical symmetry can be found directly from the consistency condition (ui,xy)t1 =
(ui,t1)xy:
u0,t1 = u0,xF (W ) + c1u
2
0 + c2u0 + c3,
u1,t1 = u1,xF (W ) + c1u
2
1 + c2u1 + c3,
where F is an arbitrary function depending on the y-integral W defined by the second formula
of (3.2); c1, c2, c3 are arbitrary constants. The classical symmetry in the another direction is
simply found because the system is symmetric under the change of variables x↔ y:
u0,t2 = u0,yG(w) + c̃1u
2
0 + c̃2u0 + c̃3,
u1,t2 = u1,yG(w) + c̃1u
2
1 + c̃2u1 + c̃3.
Higher symmetry of the second order is sought in the following form:
ui,τ1 = ai(u0, u1, u0,x, u1,x)u0,xx + bi(u0, u1, u0,x, u1,x)u1,xx + hi(u0, u1, u0,x, u1,x),
8 M.N. Kuznetsova
i = 1, 2, where ai, bi, hi are functions to be found. To find the higher symmetry we use Lax
pair (3.3). Let us consider the linear problem
Ψτ1 =
(
αλ2 + βλ+ γ
)
Ψ, (3.4)
where α = (αi,j), β = (βi,j), γ = (γi,j), i, j = 1, 2 are matrices to be found. It is assumed that
elements of the matrices depend on the variables u0, u1, u0,x, u1,x, u0,xx, u1,xx. The compatibility
condition (Ψx)τ1 = (Ψτ1)x for the systems
Ψx = (Aλ+B)Ψ, Ψτ1 =
(
αλ2 + βλ+ γ
)
Ψ,
results in the system of relations
Aα = αA, Aβ +Bα = αx + αB + βA,
Aτ1 +Aγ +Bβ = βx + βB + γA, Bτ1 +Bγ = γx + γB.
A complete study of these equations leads to the following formulas:
u0,τ1 = H(W )u0,xx +
u20,x
(u0 − u1)2
Φ(W )u1,xx + (u0 − u1)g(u0, u1, u0,x, u1,x)
+ (u0 − u1)(c0 − c1u1 −
c2
2
)− (c1u
2
1 + c2u1 + c3),
u1,τ1 =
u1,x
u0,x
H(W )u0,xx +WΦ(W )u1,xx +
(u0 − u1)u1,x
u0,x
g(u0, u1, u0,x, u1,x)
+
(u0 − u1)u1,x
u0,x
(c0 + c1u0 +
c2
2
)− (c1u
2
1 + c2u1 + c3),
where H, Φ, g are arbitrary functions; ci are arbitrary constants. To define precisely obtained
formulas we substitute them into the compatibility condition (ui,xy)τ1 = (ui,τ1)xy. Thus, we
finally found the higher symmetry of the second order:
u0,τ1 =
(
u0,xx +
u0,x
u1,x
u1,xx −
2u0,x(u0,x − u1,x)
(u0 − u1)
)
F (W ), (3.5)
u1,τ1 =
(
u1,xx +
u1,x
u0,x
u0,xx −
2u1,x(u0,x − u1,x)
(u0 − u1)
)
F (W ), (3.6)
where F is an arbitrary function; W is the y-integral defined by the second formula of (3.2).
Also we finally found matrices α, β, γ involved in (3.4):
α =
(
α11 0
0 α11
)
, β =
(
β11 0
β21(u, ux, uxx) β11
)
,
γ =
(
γ11(u, ux, uxx) γ12(u, ux, uxx)
0 γ22(u, ux, uxx)
)
,
where
β21(u, ux, uxx) =
(
u1,x
u0,x(u0 − u1)
u0,xx +
1
(u0 − u1)
u1,xx −
2u1,x(u0,x − u1,x)
(u0 − u1)2
)
F (W ),
γ11(u, ux, uxx) =
(
1
(u0 − u1)
u0,xx +
u0,x
u1,x(u0 − u1)
u1,xx −
2u0,x(u0,x − u1,x)
(u0 − u1)2
)
F (W ),
γ12(u, ux, uxx) =
(
− 1
(u0 − u1)
u0,xx −
u0,x
u1,x(u0 − u1)
u1,xx +
u0,x(u0,x − u1,x)
(u0 − u1)2
)
F (W ),
Lax Pair for a Novel Two-Dimensional Lattice 9
α11, β11 are arbitrary constants. Thus it is seen that definitive answer is given by formu-
las (3.5), (3.6) and
Ψτ1 = (βλ+ γ)Ψ, β =
(
0 0
β21 0
)
, γ =
(
γ11 γ12
0 γ22
)
,
where β21, γij have been described just above.
Remark 3.1. The symmetry given by (3.5), (3.6) can be written as1
u0,τ1 = u0,xF (W )
Wx
W
, u1,τ1 = u1,xF (W )
Wx
W
.
Therefore this is actually the classical symmetry in disguise.
Let us consider chain (1.5). We impose the periodic closure conditions un+2 = un to infinite
chain (1.5) and obtain the following finite system:
u0,xy =
2
u0 − u1
(
u0,x − u20 − 1
)(
u0,y − u20 − 1
)
+ 2u0
(
u0,x + u0,y − u20 − 1
)
,
u1,xy =
2
u1 − u0
(
u1,x − u21 − 1
)(
u1,y − u21 − 1
)
+ 2u1
(
u1,x + u1,y − u21 − 1
)
. (3.7)
This system possesses the y-integral and the x-integral
P =
(
u0,x − u20 − 1
)(
u1,x − u21 − 1
)
(u0 − u1)2
, J =
(
u0,y − u20 − 1
)(
u1,y − u21 − 1
)(
u0 − u1
)2 . (3.8)
System (3.7) is the compatibility condition for the Lax pair
Φx = (Sλ+ T )Φ, Φy =
(
S̃λ−1 + T̃
)
Φ, (3.9)
where Φ = (ϕ0, ϕ1)
T,
S =
0 0
u1,x − u21 − 1
u0 − u1
0
, T =
−u0,x − u20 − 1
u1 − u0
+ u0
u0,x − u20 − 1
u1 − u0
0 −u1,x − u21 − 1
u0 − u1
+ u1
,
S̃ =
0 −u0,y − u20 − 1
u0 − u1
0 0
, T̃ =
u0,y − u20 − 1
u0 − u1
+ u0 0
−u1,y − u21 − 1
u1 − u0
u1,y − u21 − 1
u1 − u0
+ u1
.
To find the higher symmetry it is sufficient (as we have just seen) to consider the system
Φτ2 =
(
β̃λ+ γ̃
)
Φ, (3.10)
compatible with the first equation of (3.9). In this way we obtained the higher symmetry of
system (3.7):
u0,τ2 =
(
u0,xx +
u0,x − u20 − 1
u1,x − u21 − 1
u1,xx +
2φ(u0, u1, u0,x, u1,x)
(u1,x − u21 − 1)(u0 − u1)
)
F (P ), (3.11)
u1,τ2 =
(
u1,x − u21 − 1
u0,x − u20 − 1
u0,xx + u1,xx +
2φ(u0, u1, u0,x, u1,x)
(u0,x − u20 − 1)(u0 − u1)
)
F (P ), (3.12)
1I am grateful to the anonymous referee for this constructive comment.
10 M.N. Kuznetsova
where P is the y-integral given by the first formula of (3.8),
φ(u0, u1, u0,x, u1,x) = u0,xu1,x(u1,x − u0,x) + u20,x
(
1 + u21
)
− u21,x
(
1 + u20
)
− u0,x
(
1 + u21 + u0u1 + u0u
3
1
)
+ u1,x
(
1 + u20 + u0u1 + u30u1
)
. (3.13)
Matrices β̃, γ̃ (see (3.10)) are defined by the following formulas:
β̃ =
(
0 0
β̃21(ū, ūx, ūxx) 0
)
, γ̃ =
(
γ̃11(ū, ūx, ūxx) γ̃12(ū, ūx, ūxx)
0 γ̃22(ū, ūx, ūxx)
)
,
where
β̃21(ū, ūx, ūxx) =
(
u1,x − u21 − 1
(u0 − u1)
(
u0,x − u20 − 1
)u0,xx + u1,xx
u0 − u1
+
2φ(u0, u1, u0,x, u1,x)(
u0,x − u20 − 1
)
(u0 − u1)2
)
F (P ),
γ̃11(ū, ūx, ūxx) =
(
u0,xx
u0 − u1
+
(
u0,x − u20 − 1
)
u1,xx(
u1,x − u21 − 1
)
(u0 − u1)
+
2φ(u0, u1, u0,x, u1,x)
(u0 − u1)2
(
u1,x − u21 − 1
))F (P ),
γ̃12(ū, ūx, ūxx) =
(
− u0,xx
u0 − u1
−
(
u0,x − u20 − 1
)(
u1,x − u21 − 1
)
(u0 − u1)
u1,xx
− 2φ(u0, u1, u0,x, u1,x)
(u0 − u1)2
(
u1,x − u21 − 1
))F (P ),
γ̃22(ū, ūx, ūxx) =
(
−
(
u1,x − u21 − 1
)
(u0 − u1)
(
u0,x − u20 − 1
)u0,xx − u1,xx
u0 − u1
− 2φ(u0, u1, u0,x, u1,x)(
u0,x − u20 − 1
)
(u0 − u1)2
)
F (P ),
φ(u0, u1, u0,x, u1,x) is defined by (3.13).
Remark 3.2. The symmetry given by (3.11), (3.12) can be written as
u0,τ2 =
(
u0,x − u20 − 1
)
F (P )
Px
P
, u1,τ2 =
(
u1,x − u21 − 1
)
F (P )
Px
P
.
Therefore this is actually the classical symmetry in disguise.
Note, that periodic closing obtained by the conditions un+3 = un imposing on infinite
chain (1.5) leads to the system
u0,xy =
(
1
u0 − u2
− 1
u1 − u0
)(
u0,x − u20 − 1
)(
u0,y − u20 − 1
)
+ 2u0
(
u0,x + u0,y − u20 − 1
)
,
u1,xy =
(
1
u1 − u0
− 1
u2 − u1
)(
u1,x − u21 − 1
)(
u1,y − u21 − 1
)
+ 2u1
(
u1,x + u1,y − u21 − 1
)
,
u2,xy =
(
1
u2 − u1
− 1
u0 − u2
)(
u2,x − u22 − 1
)(
u2,y − u22 − 1
)
+ 2u2
(
u2,x + u2,y − u22 − 1
)
.
Lax Pair for a Novel Two-Dimensional Lattice 11
This system has y-integral and x-integral
W =
(
u0,x − u20 − 1
)(
u1,x − u21 − 1
)(
u2,x − u22 − 1
)
(u2 − u1)(u0 − u1)(u0 − u2)
,
w =
(
u0,y − u20 − 1
)(
u1,y − u21 − 1
)(
u2,y − u22 − 1
)
(u2 − u1)(u0 − u1)(u0 − u2)
.
Lax pair has the following form:
Ψx = (Aλ+B)Ψ, Ψy =
(
Ãλ−1 + B̃
)
Ψ,
where Ψ = (ψ0, ψ1, ψ2)
T,
A =
0 0 0
0 0 0
u2,x − u20 − 1
u1 − u0
0 0
,
B =
−u0,x − u20 − 1
u1 − u0
+ u0
u0,x − u20 − 1
u1 − u0
0
0 −u1,x − u21 − 1
u2 − u1
+ u1
u1,x − u21 − 1
u2 − u1
0 0 −u2,x − u22 − 1
u0 − u2
+ u2
,
à =
0 0 −u0,y − u20 − 1
u0 − u2
0 0 0
0 0 0
,
B̃ =
u0,y − u20 − 1
u0 − u2
+ u0 0 0
−u1,y − u21 − 1
u1 − u0
u1,y − u21 − 1
u1 − u0
+ u1 0
0 −u2,y − u22 − 1
u2 − u1
u2,y − u22 − 1
u2 − u1
+ u2
.
4 Conclusion
The problem of classification multidimensional equations is actively studied by many authors,
using different algebraic and geometry approaches [1, 2, 3, 5, 6, 9, 10, 11, 21]. We note that the
classification algorithm for integrable two-dimensional lattices proposed in our previous papers
does not provide any algorithm for constructing the Lax pair.
It is known that finite systems obtained from infinite integrable chains by degenerate bound-
ary conditions imposing at the two points of the form un+k = c1, un+s = c2 (where c1, c2 are
constants) are integrable in the sense of Darboux (they have complete set of integrals in both
characteristic directions, i.e., the number of independent integrals is equal to the order of the
system). We study finite systems obtained from infinite chains (1.4), (1.5) by periodic closure
conditions. It is interesting fact that each of these systems also has one x-integral and one
y-integral. We obtained that symmetries of these systems depend on integrals. It is known that
Darboux integrable systems possesses symmetries which depend on integrals [24, 30]. Symme-
tries of systems with incomplete sets of integrals might depend on these integrals [17, 19]. In
a discrete version, this fact is discussed in paper [26]. In papers [4, 25] an algorithm is proposed
which allows one to construct higher symmetries of arbitrary order for some special classes of
hyperbolic systems possessing the integrals.
12 M.N. Kuznetsova
Acknowledgements
The author gratefully thanks I.T. Habibulin for assignment the problem and useful discussions,
E.V. Ferapontov for explaining the method of the construction of Lax pairs and S.Ya. Startsev
for valuable comments. The author gratefully thanks anonymous referees for a contribution to
improve the paper.
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1 Introduction
2 Construction Lax pair for equation (1.5)
3 Higher symmetries of periodic closings
4 Conclusion
References
|
| id | nasplib_isofts_kiev_ua-123456789-211439 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-13T03:06:00Z |
| publishDate | 2021 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Kuznetsova, Maria N. 2026-01-02T08:33:46Z 2021 Lax Pair for a Novel Two-Dimensional Lattice. Maria N. Kuznetsova. SIGMA 17 (2021), 088, 13 pages 1815-0659 2020 Mathematics Subject Classification: 37K10; 37K30; 37D99 arXiv:2102.04207 https://nasplib.isofts.kiev.ua/handle/123456789/211439 https://doi.org/10.3842/SIGMA.2021.088 In the paper by I.T. Habibullin and our joint paper, the algorithm for the classification of integrable equations with three independent variables was proposed. This method is based on the requirement of the existence of an infinite set of Darboux integrable reductions and on the notion of the characteristic Lie-Rinehart algebras. The method was applied for the classification of integrable cases of different subclasses of equations 𝑢ₙ‚ₓy = 𝑓(𝑢ₙ₊₁, 𝑢ₙ, 𝑢ₙ₋₁, 𝑢ₙ‚ₓ, 𝑢ₙ,y) of special forms. Under this approach, the novel integrable chain was obtained. In the present paper, we construct a Lax pair for the novel chain. To construct the Lax pair, we use the scheme suggested in papers by E.V. Ferapontov. We also study the periodic reduction of the chain. The author gratefully thanks I.T. Habibulin for assigning the problem and useful discussions, E.V. Ferapontov for explaining the method of the construction of Lax pairs, and S.Ya. Startsev for valuable comments. The author gratefully thanks anonymous referees for their contribution to improving the paper. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Lax Pair for a Novel Two-Dimensional Lattice Article published earlier |
| spellingShingle | Lax Pair for a Novel Two-Dimensional Lattice Kuznetsova, Maria N. |
| title | Lax Pair for a Novel Two-Dimensional Lattice |
| title_full | Lax Pair for a Novel Two-Dimensional Lattice |
| title_fullStr | Lax Pair for a Novel Two-Dimensional Lattice |
| title_full_unstemmed | Lax Pair for a Novel Two-Dimensional Lattice |
| title_short | Lax Pair for a Novel Two-Dimensional Lattice |
| title_sort | lax pair for a novel two-dimensional lattice |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211439 |
| work_keys_str_mv | AT kuznetsovamarian laxpairforanoveltwodimensionallattice |