Journal of Nonlinear Mathematical Physics

Volume 25, Issue 1, February 2018, Pages 166 - 177

On the discretization of Laine equations

Kostyantyn Zheltukhin
Middle East Technical University, Department of Mathematics, Universiteler Mahallesi, Dumlupinar Bulvar No:1, 06800 Cankaya, Ankara, Turkey,
Natalya Zheltukhina
Department of Mathematics, Faculty of Science, Bilkent University, 06800 Bilkent, Ankara, Turkey,
Received 20 July 2017, Accepted 31 July 2017, Available Online 6 January 2021.
10.1080/14029251.2018.1440748How to use a DOI?
Semi-discrete chain; Darboux integrability; x-integral; n-integral; discretization

We consider the discretization of Darboux integrable equations. For each of the integrals of a Laine equation we constructed either a semi-discrete equation which has that integral as an n-integral, or we proved that such an equation does not exist. It is also shown that all constructed semi-discrete equations are Darboux integrable.

© 2018 The Authors. Published by Atlantis and Taylor & Francis
Open Access
This is an open access article distributed under the CC BY-NC 4.0 license (

1. Introduction

When considering hyperbolic type equations

one finds an important special subclass, so called Darboux integrable equations, that is described in terms of x-and y-integrals. Recall that a function W(x, y, u, ux, ux x, …) is called a y–integral of equation (1.1) if DyW(x, y, u, ux, …)|(1.1) = 0, where Dy represents the total derivative with respect to y( see [2] and [8]). An x-integral W¯=W¯(x,y,u,uy,uyy,) for equation (1.1) is defined in a similar way. Equation (1.1) is said to be Darboux integrable if it admits a nontrivial x-integral and a nontrivial y–integral.

The classification problem for Darboux integrable equations was considered by Goursat, Zhiber and Sokolov (see [2] and [8]). In his paper Goursat obtained a supposedly complete list of Darboux integrable equations of the form (1.1). A detailed discussion of the subject and corresponding references can be found in the survey [9].

Later Laine [7] published two Darboux integrable hyperbolic equations, which were absent in Goursat's list. The first equation found by Laine is

It has a second order y-integral
and a third order x-integral
The second equation found by Laine is
It has a second order y-integral
W2=uxx2ux(1u+X(u+X)2+ux)+u+(u+X)2+2ux(u+X)2+ux         (u+X)2+ux+(u+X)(u+X)2+uxuy(1.6)
and a third order x-integral (1.4). For the second equation Laine assumed X to be an arbitrary function of x. However Kaptsov (see [6]) has shown that X must be a constant function if equation (1.5) admits the integrals (1.6) and (1.4). Thus it can be assumed, without loss of generality, that X = 0.

One can also consider a semi-discrete analogue of Darboux integrable equations (see [1]). The notion of Darboux integrability for semi-discrete equations was developed by Habibullin (see [3]). For a function t = t(n, x) of the continuous variable x and discrete variable n we introduce notations

Then a hyperbolic type semi-discrete equation can be written as
A function F of variables x, n, and t, t1,…,tk is called an x-integral of equation (1.7) if Dx F|(1.7) = 0. A function I of variables x, n, t, t[1],…,t[m] is called an n-integral of equation (1.7) if DI|(1.7) = I, where D is a shift operator. Equation (1.7) is said to be Darboux integrable if it admits a nontrivial n-integral and a nontrivial x-integral. In what follows we consider the equalities Dx = 0 and DI = I, which define x-and n-integrals F and I, only on solutions of the corresponding equations. For more information on semi-discrete Darboux integrable equations see [3], [4] and [5].

The interest in the continuous and discrete Darboux integrable models is stimulated by exponential type systems. Such systems are connected with semi-simple and affine Lie algebras which have applications in Liouville and conformal field theories.

The discretization of equations from Goursat’s list was considered by Habibullin and Zheltukhina in [5]. In the present paper we find semi-discrete versions of Laine equations (1.2) and (1.5). In particular we find semi-discrete equations that admit functions (1.3) or (1.6) as n-integrals, and show that these equations are Darboux integrable. This is the main result of our paper given in Theorem 1.1 and Theorem 1.2 below.

Theorem 1.1.

The semi-discrete chain (1.7), which admits a minimal order n-integral

where ε(n) is an arbitrary function of n, is
where B is a function of n, t, t1, satisfying the following equation
Moreover, chain (1.9) admits an x-integral of minimal order 3.

Theorem 1.2.

The semi-discrete chain (1.7), which admits a minimal order n-integral

where ε(n) is an arbitrary function of n, is
where A is a function of n, t, t1, satisfying the following system of equations
Moreover, chain (1.12) admits an x-integral of minimal order 2.

The paper is organized as follows. In Sections 2 and 3 we give proofs of Theorems 1.1 and 1.2 respectively. In Section 4 we show that function (1.4) can not be a minimal order n-integral for any equation (1.7).

2. Proof of Theorem 1.1

Discretization by n-integral: Let us find f(x, n, t, t1, tx) such that DI1 = I1, where I1 is defined by (1.8). Equality D I1 = I1 implies

where ε = ε(n) and ε1 = ε(n + 1).

By comparing the coefficients before txx in (2.1), we get ftxf=1tx, which implies that f = A(x, n, t, t1)tx. We substitute this expression for f in (2.1) and get

The above equation is equivalent to a system of two equations
The first equation of system (2.3) can be written as x(ln|A|ln|t1x|+ln|tx|)=0 which implies that
for some function B of variables n, t, t1. Substituting expression (2.4) for A into the second equation of system (2.3), we get
We compare the coefficients before x and x0 in (2.6) and obtain
which is equivalent to
The last system is compatible, that is Bt t1=Bt1t, if and only if equality (1.10) is satisfied.

Existence of an x-integral: Let us show that equation (1.9) where function B satisfies (1.10) has a finite dimensional x-ring. We have,

t1x=t1xtxBtx,t2x=t2xtxBB1tx, and t3x=t3xtxBB1B2tx,(2.9)
where B = B(n, t, t1), B1 = B(n + 1, t1, t2) and B2 = B(n + 2, t2, t3). We are looking for a function F(x, n, t, t1, t2, t3) such that DxF = 0, that is
which is equivalent to
By comparing the coefficients of x0 and x in the last equality we get the following system
After diagonalization this system becomes
We introduce vector fields
and V = [V1, V2]. Then, we have
Direct calculation show that
[V1,V]=3ε4t+t12(εt)(tt1)V and [V2,V]=3ε1+t4t12(ε1t1)(t1t)V.(2.16)
Hence vector fields V1, V2 and V form a finite-dimensional ring. By the Jacobi Theorem the system of three equations V1(F) = 0, V2(F) = 0, V(F) = 0 has a nonzero solution F(t, t1, t2, t3). The function F(t, t1, t2, t3) is an x-integral of equation (1.9).

3. Proof of Theorem 1.2

Discretization by n-integral: Let us find a function f(x, n, t, t1, tx) such that DI2 = I2, where I2 is given by (1.11). The equality DI2 = I2 implies that

where ε = ε(n) and ε1 = ε(n + 1). Comparing the coefficients before txx in equality (3.1), we get
This can be written as
where A is some function of variables x, n, t and t1. The last equality is equivalent to
We substitute f given by (3.5) into equality (3.1), use (3.4) and equality
to get
We can solve the overdetermined system of linear equations Λi = 0, i = 1,2 … 5, with respect to Ax At, At1 and obtain
By direct calculations one can check that At t_{1} = At_{1 t}, so the above system has a solution.

Existence of an x-integral: We are looking for a function F(t, t1, t2) such that DxF = 0 that is

where t satisfies equation (1.7) with function f given by (3.5). We use
to get
By substituting these expressions for t1x and t2x into equality (3.9) and comparing the coefficients of tx+t2,tx and tx0, we obtain the following system of equations
To check for the existence of a solution we transform the above system to its row reduced form
The corresponding vector fields
commute, that is [V1, V2] = 0, provided A satisfies system (3.8). Thus by the Jacobi theorem, system (3.10) has a solution. To solve the system define a function E(t, t1, t2) by
where A = A(t, t1) and A1 = A(t1, t2).

One can check that Ett1=Et1t and Et1t2=Et2t1, so such a function E exists. Function E is a first integral of the first equation of system (3.10). We write system (3.10) using new variables

and obtain
Therefore one of the x-integrals is F(t, t1, t2) = E(t, t1, t2) /(t1ε(n + 1)) where function E defined above.

4. Nonexistence of a chain (1.7) admitting the minimal order n-integral (1.4)

Let us find a function f(x, n, t, t1, tx) such that equation (1.7) has the n-integral

We have,
Equality DI = I is equivalent to J: = L(DL)(DII) = 0, where L=2tx(tx){txx(tx)2tx(tx+1)2}. We have
where Λk, 1 ≤ k ≤ 5, are some functions of variables x, n, t, t1, tx. In particular,
Equality Λ2 = 0 implies that fftxtxftx2+2txfftxtx=0, thus
Hence, txftx2f=A2(x,n,t,t1) for some function A depending on x, n, t, t1 only. Therefore, ftxf=Atx and hence tx{fAtx=0}. We have,
where A = A(x, n, t, t1) and B = B(x, n, t, t1). We substitute f=A2tx+2ABtx+B2 into Λ1=0 and get
We solve the system of equations αk = 0,1 ≤ k ≤ 5, and obtain B = 0, that is
We substitute f=A2tx+2ABtx+B2 into Λ3 = 0 and get
We solve the system of equations βk = 0,1 ≤ k ≤ 7, and obtain B = 0, or
We equate expressions for Ax and At from (4.1) and (4.2) and find
Then, it follows from (4.1) that
Equality Att1At1t=0 becomes (t1x)2(tx)2A3(tx)2(t1x)2B=0, thus
Equality Axt1At1x=0 becomes (t1x)2+(tx)2A(1+B)2(tx)2(t1x)2B=0, thus
Equality AxtAtx = 0 becomes (t1x)2(AB)2(tx)2A3(tx)2(t1x)2B=0. It implies that
or A = B, that leads to A = B = 0 and f = 0. It follows from (4.5) and (4.7) that AB = 1 or AB = −1. It follows from (4.5) and (4.6) that 1 + B = A or 1 + B = − A. This gives rise to four possibilities:
  1. 1)

    AB = 1;

  2. 2)

    AB = 1 and A + B = − 1 which gives A = 0, B = − 1 and therefore f = 1;

  3. 3)

    AB = −1 and AB = 1 which is an inconsistent system;

  4. 4)

    AB = −1 and A + B = −1 which gives A = −1, B = 0 and therefore f = tx

We have to study case 1 ) only. In this case we get B = A − 1 and equation t1x=Atx+Bbecomes t1x+1=A(tx+1), that can be written as well as

Due to (4.5), our equation (4.8) becomes
The last equation admits an n-integral I=(tx+1)3(tx)2 of order one.

Let us consider case B = 0. We write DII = 0 for the chain t1 x = C(x, n, t, t1) tx and get

where Λk = Λk(x, n, t, t1, tx), 1 ≤ k ≤ 4. Equation Λ1 = 0 implies
where αk = αk(x, n, t, t1), 1 ≤ k ≤ 3. In particular, α2=4C((t1x)+(tx)C). Since α2 = 0 we have C = (t1x)2(tx) − 2. The chain becomes t1 x = (t1x)2(tx) − 2tx. It admits the n-integral I = (tx)−2 tx of order one.

Therefore, if equation (1.7) admits n-integral (1.4) then (1.4) is not a minimal order integral.


We are thankful to Prof. Habibullin for suggesting the Laine equations discretization problem and for his interest in our work.


[2]Goursat Goursat, Recherches sur quelques équations aux dérivés partielles du second ordre, Annales de la faculté des Sciences de l’Université de Toulouse 2e série, Vol. 1, No. 1, 1899, pp. 31-78.
[7]Laine Laine, Sur l’a application de la method de Darboux aux equations s = f (x, y, z, p, q), Comptes rendus, Vol. V.182, 1926, pp. 1127-1128.
[9]Zhiber Zhiber, Murtazina Murtazina, Habibullin Habibullin, and Shabat Shabat, Characteristic Lie rings and integrable models in mathematical physics, Ufa Math. J.,, Vol. 4, No. 3, 2012, pp. 17-85.
Journal of Nonlinear Mathematical Physics
25 - 1
166 - 177
Publication Date
ISSN (Online)
ISSN (Print)
10.1080/14029251.2018.1440748How to use a DOI?
© 2018 The Authors. Published by Atlantis and Taylor & Francis
Open Access
This is an open access article distributed under the CC BY-NC 4.0 license (

Cite this article

AU  - Kostyantyn Zheltukhin
AU  - Natalya Zheltukhina
PY  - 2021
DA  - 2021/01/06
TI  - On the discretization of Laine equations
JO  - Journal of Nonlinear Mathematical Physics
SP  - 166
EP  - 177
VL  - 25
IS  - 1
SN  - 1776-0852
UR  -
DO  - 10.1080/14029251.2018.1440748
ID  - Zheltukhin2021
ER  -