# On the discretization of Laine equations

- DOI
- 10.1080/14029251.2018.1440748How to use a DOI?
- Keywords
- Semi-discrete chain; Darboux integrability; x-integral; n-integral; discretization
- Abstract
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.- Copyright
- © 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 (http://creativecommons.org/licenses/by-nc/4.0/).

## 1. Introduction

When considering hyperbolic type equations

*x*-and

*y*-integrals. Recall that a function

*W*(

*x*,

*y*,

*u*,

*u*,

_{x}*u*, …) is called a

_{x}_{x}*y*–integral of equation (1.1) if

*D*(

_{y}W*x*,

*y*,

*u*,

*u*, …)|

_{x}_{(1.1)}= 0, where

*D*represents the total derivative with respect to

_{y}*y*( see [2] and [8]). An

*x*-integral

*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

*y*-integral

*x*-integral

*y*-integral

*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

*F*of variables

*x*,

*n*, and

*t*,

*t*

_{1},…,

*t*is called an

_{k}*x*-integral of equation (1.7) if

*D*|

_{x}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

*D*= 0 and

_{x}*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*,

*t*

_{1},

*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*,

*t*

_{1},

*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*,

*t*

_{1},

*t*) such that

_{x}*DI*

_{1}=

*I*

_{1}, where

*I*

_{1}is defined by (1.8). Equality

*D I*

_{1}=

*I*

_{1}implies

*ε*=

*ε*(

*n*) and

*ε*

_{1}=

*ε*(

*n*+ 1).

By comparing the coefficients before *t _{xx}* in (2.1), we get

*f*=

*A*(

*x*,

*n*,

*t*,

*t*

_{1})

*t*. We substitute this expression for

_{x}*f*in (2.1) and get

*B*of variables

*n*,

*t*,

*t*

_{1}. Substituting expression (2.4) for

*A*into the second equation of system (2.3), we get

*x*and

*x*

^{0}in (2.6) and obtain

** 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,

*B*=

*B*(

*n*,

*t*,

*t*

_{1}),

*B*

_{1}=

*B*(

*n*+ 1,

*t*

_{1},

*t*

_{2}) and

*B*

_{2}=

*B*(

*n*+ 2,

*t*

_{2},

*t*

_{3}). We are looking for a function

*F*(

*x*,

*n*,

*t*,

*t*

_{1},

*t*

_{2},

*t*

_{3}) such that

*D*= 0, that is

_{x}F*x*

^{0}and

*x*in the last equality we get the following system

*V*= [

*V*

_{1},

*V*

_{2}]. Then, we have

*V*

_{1},

*V*

_{2}and

*V*form a finite-dimensional ring. By the Jacobi Theorem the system of three equations

*V*

_{1}(

*F*) = 0,

*V*

_{2}(

*F*) = 0,

*V*(

*F*) = 0 has a nonzero solution

*F*(

*t*,

*t*

_{1},

*t*

_{2},

*t*

_{3}). The function

*F*(

*t*,

*t*

_{1},

*t*

_{2},

*t*

_{3}) 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*,

*t*

_{1},

*t*) such that

_{x}*DI*

_{2}=

*I*

_{2}, where

*I*

_{2}is given by (1.11). The equality

*DI*

_{2}=

*I*

_{2}implies that

*ε*=

*ε*(

*n*) and

*ε*

_{1}=

*ε*(

*n*+ 1). Comparing the coefficients before

*t*in equality (3.1), we get

_{xx}*A*is some function of variables

*x*,

*n*,

*t*and

*t*

_{1}. The last equality is equivalent to

*f*given by (3.5) into equality (3.1), use (3.4) and equality

*= 0,*

_{i}*i*= 1,2 … 5, with respect to

*A*,

_{x}A_{t}*A*

_{t}_{t}_{1} =

*A*

_{t}_{1

*t*}, so the above system has a solution.

** Existence of an x-integral:** We are looking for a function

*F*(

*t*,

*t*

_{1},

*t*

_{2}) such that

*D*= 0 that is

_{x}F*t*satisfies equation (1.7) with function

*f*given by (3.5). We use

*t*

_{1}

*and*

_{x}*t*

_{2}

*into equality (3.9) and comparing the coefficients of*

_{x}*V*

_{1},

*V*

_{2}] = 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*,

*t*

_{1},

*t*

_{2}) by

*A*=

*A*(

*t*,

*t*

_{1}) and

*A*

_{1}=

*A*(

*t*

_{1},

*t*

_{2}).

One can check that *E* exists. Function *E* is a first integral of the first equation of system (3.10). We write system (3.10) using new variables

*x*-integrals is

*F*(

*t*,

*t*

_{1},

*t*

_{2}) =

*E*(

*t*,

*t*

_{1},

*t*

_{2}) /(

*t*

_{1}−

*ε*(

*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*, *t*_{1}, *t _{x}*) such that equation (1.7) has the

*n*-integral

*DI*=

*I*is equivalent to

*J*: =

*L*(

*DL*)(

*DI*−

*I*) = 0, where

*, 1 ≤*

_{k}*k*≤ 5, are some functions of variables

*x*,

*n*,

*t*,

*t*

_{1},

*t*. In particular,

_{x}_{2}= 0 implies that

*A*depending on

*x*,

*n*,

*t*,

*t*

_{1}only. Therefore,

*A*=

*A*(

*x*,

*n*,

*t*,

*t*

_{1}) and

*B*=

*B*(

*x*,

*n*,

*t*,

*t*

_{1}). We substitute

*α*= 0,1 ≤

_{k}*k*≤ 5, and obtain

*B*= 0, that is

_{3}= 0 and get

*β*= 0,1 ≤

_{k}*k*≤ 7, and obtain

*B*= 0, or

*A*and

_{x}*A*from (4.1) and (4.2) and find

_{t}*A*−

_{xt}*A*= 0 becomes

_{tx}*A*=

*B*, that leads to

*A*=

*B*= 0 and

*f*= 0. It follows from (4.5) and (4.7) that

*A*−

*B*= 1 or

*A*−

*B*= −1. It follows from (4.5) and (4.6) that 1 +

*B*=

*A*or 1 +

*B*= −

*A*. This gives rise to four possibilities:

- 1)
*A*−*B*= 1; - 2)
*A*−*B*= 1 and*A*+*B*= − 1 which gives*A*= 0,*B*= − 1 and therefore*f*= 1; - 3)
*A*−*B*= −1 and*A*−*B*= 1 which is an inconsistent system; - 4)
*A*−*B*= −1 and*A*+*B*= −1 which gives*A*= −1,*B*= 0 and therefore*f*=*t*_{x}

We have to study case 1 ) only. In this case we get *B* = *A* − 1 and equation

*n*-integral

Let us consider case *B* = 0. We write *DI* − *I* = 0 for the chain *t*_{1} * _{x}* =

*C*(

*x*,

*n*,

*t*,

*t*

_{1})

*t*and get

_{x}*= Λ*

_{k}*(*

_{k}*x*,

*n*,

*t*,

*t*

_{1},

*t*), 1 ≤

_{x}*k*≤ 4. Equation Λ

_{1}= 0 implies

*α*=

_{k}*α*(

_{k}*x*,

*n*,

*t*,

*t*

_{1}), 1 ≤

*k*≤ 3. In particular,

*α*

_{2}= 0 we have

*C*= (

*t*

_{1}−

*x*)

^{2}(

*t*−

*x*)

^{− 2}. The chain becomes

*t*

_{1}

*= (*

_{x}*t*

_{1}−

*x*)

^{2}(

*t*−

*x*)

^{− 2}

*t*. It admits the

_{x}*n*-integral

*I*= (

*t*−

*x*)

^{−2}

*t*of order one.

_{x}Therefore, if equation (1.7) admits *n*-integral (1.4) then (1.4) is not a minimal order integral.

## Acknowledgment

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

## References

### Cite this article

TY - JOUR 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 - https://doi.org/10.1080/14029251.2018.1440748 DO - 10.1080/14029251.2018.1440748 ID - Zheltukhin2021 ER -