On the discretization of Darboux Integrable Systems

Kostyantyn Zheltukhin; Natalya Zheltukhina

doi:10.1080/14029251.2020.1819597

<Previous Article In Issue

Download article (PDF)

Next Article In Issue>

Volume 27, Issue 4, September 2020, Pages 616 - 632

On the discretization of Darboux Integrable Systems

Authors

Kostyantyn Zheltukhin^*

Department of Mathematics, Middle East Technical University, Ankara, Turkey,zheltukh@metu.edu.tr

Natalya Zheltukhina

Department of Mathematics, Faculty of Science, Bilkent University, Ankara, Turkey,natalya@fen.bilkent.edu.tr

^*Corresponding author

Corresponding Author

Kostyantyn Zheltukhin

Received 2 July 2019, Accepted 6 January 2020, Available Online 4 September 2020.

DOI: 10.1080/14029251.2020.1819597 How to use a DOI?
Keywords: semi-discrete system; Darboux integrability; x-integral; discretization
Abstract: We study the discretization of Darboux integrable systems. The discretization is done using x-, y-integrals of the considered continuous systems. New examples of semi-discrete Darboux integrable systems are obtained.
Copyright: © 2020 The Authors. Published by Atlantis Press 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

The classification problem of Darboux integrable equations has attracted a considerable interest in the recent time, see the survey paper [1] and references there in. There are many classification results in the continuous case. The case of semi-discrete and discrete equations is not that well studied. Discrete models play a big role in many areas of physics and discretization of existing integrable continuous models is an important problem. There is a currently discussed conjecture saying that for each continuous Darboux integrable system it is possible to find a semi-discrete Darboux integrable system that admits the same set of x-integrals. To better understand properties of semi-discrete and discrete Darboux integrable systems it is important to have enough examples of such systems. We can test the conjecture and obtain new semi-discrete Darboux integrable systems, corresponding to given continuous ones, following an approach proposed by Habibullin et al., see [2]. In this case we take a Darboux integrable continuous equation and look for a semi-discrete equation admitting the same integrals. The method was successfully applied to many Darboux integrable continuous equations, see [2]–[4]. In almost all considered cases such semi-discrete equations exist and they are Darboux integrable.

In the present paper we apply this method of discretization to Darboux integrable systems to obtain new Darboux integrable semi-discrete systems. Let us give necessary definitions and formulate the main results of our work.

Consider a hyperbolic continuous system

pxy=𝒜(p,px,py) (pxyi=𝒜i(p1…pN,px1…pxN,py1…pyN) i=1,…N),(1.1)

where pⁱ(x,y), i = 1,...,N, are functions of continuous variables x,y ∈ ℝ. We say that a function F(x,y, p, p_y, p_yy,...) is an x-integral of the system (1.1) if

DxF(x,y,p,py,pyy,…)=0 on all the solutions of the system(1.1).

The operator D_x represents the total derivative with respect to x. The y-integral of the system (1.1) is defined in a similar way. The system (1.1) is called Darboux integrable if it admits N functionally independent non-trivial x-integrals and N functionally independent non-trivial y-integrals.

Consider a hyperbolic semi-discrete system

qx1=ℬ(q,qx,q1) (qx1i=ℬi(q1…qN,qx1…qxN,q11…q1N) i=1,…N),(1.2)

where qⁱ(x,n), i = 1,...,N, are functions of a continuous variable x ∈ ℝ and a discrete variable n ∈ N. Note that we use notation q₁(x,n) = Dq(x,n) = q(x,n+1) and q_k(x,n) = D^k q(x,n) = q(x,n+k), where D is the shift operator. To state the Darboux integrability of a semi-discrete system we need to define x- and n-integrals for such systems, see [5]. An x-integral is defined in the same way as in continuous case and a function I(x,n,q,q_x,q_xx,...) is an n-integral of system (1.2) if

DI(x,n,q,qx,qxx,…)=I(x,n,q,qx,qxx,…) on all the solutions of the system(1.2).

The system (1.2) is called Darboux integrable if it admits N functionally independent non-trivial x-integrals and N functionally independent non-trivial n-integrals.

To find new Darboux integrable semi-discrete systems we applied the discretization method proposed in [2] to one of the continuous systems derived by Zhiber, Kostrigina in [6] and continuous systems derived by Shabat, Yamilov in [7]. In [6] the authors considered the classification problem for Darboux integrable continuous systems that admit the x- and y-integrals of the first and second order. In [7] the authors considered the exponential type system

μxyi=e∑aijμj, i,j=1,2,…,N.

It was shown that such a system is Darboux integrable if and only if the matrix A = (a_ij) is a Cartan matrix of a semi-simple Lie algebra. Such systems are closely related to the classical Toda field theories, see [8]–[10] and references there in. In this case we obtain the Darboux integrable semi-discrete systems that were already described in [11].

First we consider the following system (see [6])

{uxy=uxuyu+v+c+(1u+v+c+1u+v−c)uxvyvxy=vxvyu+v−c+(1u+v+c+1u+v−c)uxvy,(1.3)

where c is an arbitrary constant. This system is Darboux integrable and admits the following y-integrals

I1=2v−vx(u+v+c)ux+2clnuxu+v+c(1.4)

and

I2=uxxux−2ux+vxu+v+c.(1.5)

The x- integrals have the same form in u, v, u_y, v_y,... variables.

Now we look for semi-discrete systems admitting these functions as n-integrals. The obtained results are given in Theorems 1.1 and 1.2 below.

Theorem 1.1.

The system

{u1x=f(x,n,u,v,u1,v1,ux,vx)v1x=g(x,n,u,v,u1,v1,ux,vx)(1.6)

possessing n-integrals (1.4) and (1.5), where c is a function of n satisfying c(n) ≠ c(n + 1) for all n ∈ 𝕑, has the form

{u1x=(u1+v1+c1)uxu+v+cv1x=2(v1−v)uxu+v+c+2(c1−c)uxu+v+clnuxu+v+c+vx.(1.7)

Moreover, the system above also possesses x-integrals

F1(c−c1)(v2−v)−(c−c2)(v1−v)(c−c2)(v3−v)−(c−c3)(v2−v)(1.8)

and

F2=(c1−c2)u+(c2−c)u1+(c−c1)u2(c1−c2)v+(c2−c)v1+(c−c1)v2−(c1−c2)v+(c2−c)v1+(c−c1)v2.(1.9)

Hence, semi-discrete system (1.7) is Darboux integrable.

Theorem 1.2.

The system (1.6) possessing n-integrals (1.4) and (1.5), where c is a constant, is either

{u1x=(u1+v1+c)uxu+v+cv1x=2(v1−v)uxu+v+c+vx(1.10)

with x-integrals F1=v1−vv2−v1 and F2=u2−u+v−v2v1−v, or

{u1x=(u1+v1+c)Buxu+v+cv1x=2B(v1−v+clnB)u+v+cux+Bvx,(1.11)

where B is defined by equality H(K₁,K₂) = 0 with

K1=v1−vB+B(1−B)u+clnB(B−1)2+cln(B−1)−clnB

and

K2=u1+cB−c−clnBB−1+B2v−Bv1−cBlnB(B−1)2+cln(B−1)−clnB,

and H being any smooth function.

Remark 1.1.

We considered some special cases of the system (1.11) and get Darboux integrable systems.

(I)
System (1.11) with B=u−v+(−1)n(u−v)2+4uv12u is Darboux Integrable. (The expression for B is found from K₁ = 0, with c = 0.)
(II)
System (1.11) with B=v1−u1+(−1)n(v1−u1)2+4u1v2v is Darboux Integrable. (The expression for B is found from K₂ = 0, with c = 0.)

Remark 1.2.

Expansion of the function B(u,v,v₁), given implicitly by (B − 1)²K₁ = 0, into a series of the form

B(u,v,v1)=∑n=0∞an(v1−v)n,(1.12)

where coefficients a_n depend on variables u and v, yields a₀ = 1 and a1=1u+v−c. So Bcan be written as

B(u,v,v1)=1+1u+v−c(v1−v)+∑n=2∞an(v1−v)n.(1.13)

By letting u₁ = u + εu_y and v = v + εv_y and taking ε → 0 one can see that the system (1.11) has a continuum limit (1.3).

Let us discuss the exponential type systems. We consider the discretization of such systems corresponding to 2 × 2 matrices, namely,

μxy=e2μ−v,vxy=e−cμ−2v,(1.14)

where c = 1, 2, 3. The obtained results are given in Theorem 1.3 below. The discretization of such systems was also considered in [11], where the form of the corresponding semi-discrete system was directly postulated and then the Darboux integrability proved. In our approach we do not make any specific assumptions about the form of the corresponding semi-discrete system. Note that the integrals corresponding to Darboux integrable exponential systems are given in the statement of Theorem 1.3.

Theorem 1.3.

(1)
The system
{u1x=f˜(u,v,u1,v1,ux,vx)v1x=g˜(u,v,u1,v1,ux,vx),(1.15)
possessing n-integrals
I1=uxx+vxx−ux2+uxvx−vx2(1.16)
and
I1*=uxxx+ux(vxx−2uxx)+ux2vx−uxvx2(1.17)
has the form
{u1x=ux+Aeu1+u−v1v1x=vx+Be−u+v+v1,(1.18)
or
{u1x=ux+Aeu1+u−vv1x=vx+Be−u1+v+v1,(1.19)
where A and B are arbitrary constants.
(2)
The system (1.15) possessing n-integrals
I2=2uxx+vxx−2ux2+2uxvx−vx2(1.20)
and
I2*=uxxx+ux(vxxx−2uxxx)+uxx(4uxvx−2ux2−vx2) +uxx(vxx−uxx)+vxxux(ux−2vx)+ux4+ux2vx2−2ux3vx(1.21)
has the form
{u1x=ux+Aeu+u1−v1v1x=vx+Be−2u+v+v1,(1.22)
where A and B are arbitrary constants.
(3)
The system (1.15) possessing n-integrals
I3=uxx+13vxx−ux2+uxvx−13vx2(1.23)
and
I3*=u(6)−2u(5)ux+v(5)ux+u(4)(32(ux)2−30uxvx+11(vx)2−40uxx−11vxx)+v(4)(14(ux)2−15uxvx+(13/3(vx)2−10uxx−(13/3)vxx)+19(u(3))2+(13/6)(v(3))2+16u(3)v(3)+u(3)(−36uxxux+18uxxvx+80vxxux−45vxxvx)+v(3)(−52uxxux+33uxxvx−5vxxux)+u(3)(−64(ux)3+102(ux)2vx−62ux(vx)2+13(vx)3)+v(3)(32(ux)3−58(ux)2vx+38ux(vx)2−(26/3(vx)3)+66(uxx)3+(26/3)(vxx)3−35(uxx)2(vxx)−5uxx(vxx)2+(uxx)2(30(ux)2−18uxvx−(11/2)(vx)2)+uxxvxx(−34(ux)2+32uxvx−2(vx)2)−2(vxx)2uxvx+uxx(6(ux)4−24(ux)3vx+25(ux)2(vx)2−9ux(vx)3+(vx)4+vxx(−(ux)4+8(ux)3vx−8(ux)2(vx)2+2ux(vx)3)+(−2(ux)6+6(ux)5vx−(13/2)(ux)4(vx)2+3(ux)3(vx)3−(1/2)(ux)2(vx)4)(1.24)
has the form
{u1x=ux+Aeu+u1−v1v1x=vx+Be−3u+v+v1,(1.25)
where A and B are arbitrary constants.

Remark 1.3.

We note that while considering systems with integrals (1.20) and (1.21) we also obtain two degenerate systems

{u1x=uxv1x=vx+Be−(2+c)u+cu1+v+v1,(1.26)

and

{u1x=ux+Aeu+u12cv−(2c+1)v1v1x=vx,(1.27)

where A, B and c are arbitrary constants, which are equivalent to a Darboux integrable equation.

Remark 1.4.

By letting u = μ¹, u1=μ1+ɛμy1, v = μ² , v1=μ2+ɛμy2 and A = ε, B = ε in equations (1.18), (1.22), (1.25) and taking ε → 0 one can see that the considered systems have corresponding continuum limit given by (1.14).

2. Proof of Theorems 1.1 and 1.2

Let us find a semi-discrete system (1.6) possessing n-integrals (1.4) and (1.5), where c is an arbitrary constant, possibly dependent on n. Let Dc = c₁. It follows from DI₂ = I₂ that

u1xxu1x−2u1x+v1xu1+v1+c1=uxxux−2ux+vxu+v+c,

that is

fx+fuux+fvvx+fu1f+fv1g+fuxuxx+fvxvxxf−2f+gu1+v1+c1=uxxux−2ux+vxu+v+c.(2.1)

Compare the coefficients by v_xx and u_xx, we get f_{v_x} = 0 and fuxf=1ux. Hence

f(x,n,u,v,u1,v1,ux,vx)=A(x,n,u,v,u1,v1)ux.(2.2)

It follows from DI₁ = I₁ that

2v1−(u1+v1+c1)gf+2c1lnfu1+v1+c1=2v−vx(u+v+c)ux+2clnuxu+v+c.(2.3)

Using (2.2) we obtain

2v1−(u1+v1+c1)gAux+2c1lnAuxu1+v1+c1=2v−vx(u+v+c)ux+2clnuxu+v+c

and find g as

g=(2(v1−v)Au1+v1+c1+2Ac1(u1+v1+c1)ln(u+v+c)A(u1+v1+c1))ux+2(c1−c)A(u1+v1+c1)uxlnuxu+v+c+(u+v+c)A(u1+v1+c1)vx.(2.4)

Substituting the expressions (2.2) and (2.4) into equality (2.1) and comparing coefficients by u_x, v_x, u_x ln uxu+v+c and free term we get the following equalities

AxA=0(2.5)

2(c1−c)Av1(u1+v1+c1)−2(c1−c)A(u1+v1+c1)2=0(2.6)

AuA+Au1+(Av1A−1(u1+v1+c1))(2(v1−v)A(u1+v1+c1)+2c1A(u1+v1+c1ln(u+v+c)A(u1+v1+c1))−2A(u1+v1+c1)+2(u+v+c)=0(2.7)

AvA+(u+v+c)Av1(u1+v1+c1)−(u+v+c)A(u1+v1+c1)2+1(u+v+c)=0.(2.8)

We have two possibilities: c₁ ≠ c and c₁ = c.

2.1. c depends on n

First we consider the case c₁ ≠ c, that is c depends on n and satisfies c(n) ≠ c(n + 1) for all n. Then equations (2.6)–(2.8) are transformed into

Av1A−1(u1+v1+c1)=0(2.9)

AuA+Au1−2A(u1+v1+c1)+2(u+v+c)=0(2.10)

AvA+1(u+v+c)=0.(2.11)

Equations (2.9) and (2.11) imply that

A=(u1+v1+c1)(u+v+c)M(n,u,u1).(2.12)

Substituting the above A into (2.10) we get that M satisfies

(u+v+c)MuM+(u1+v1+c1)Mu1+(1−M)=0.(2.13)

Differentiating equation (2.13) with respect to v and v₁ we get that M_u = 0 and M_u₁ = 0 respectively. Thus, equation (2.13) implies that M = 1. So in the case c₁ ≠ c we arrive to the system of equations (1.7). We note that the system (1.7) is Darboux integrable. It admits two n-integrals (1.4) and (1.5) and two x-integrals (1.8) and (1.9). The x-integrals can be found by considering the characteristic x-ring for system (1.7).

2.2. c does not depend on n

Now we consider the case c = c₁, that is c is a constant independent of n. Then we have equations (2.7) and (2.8). Introducing new variable B=(u+v+c)(u1+v1+c) we can rewrite the equations as

BuB+(u1+v1+c)(u+v+c)Bu1+2(v1−v+clnB)(u+v+c)Bv1+1−B(u+v+c)=0(2.14)

BvB+Bv1=0.(2.15)

The set of solutions of the above system is not empty, for example it admits a solution B = 1. Setting B = 1 we arrive to the system of equations (1.10). We note that the system (1.10) is Darboux integrable. It admits two n-integrals (1.4) and (1.5) and two x-integrals

F1=v1−vv2−v1, F2=u2−u+v−v2v1−v.

The x-integrals are calculated by considering the characteristic x-ring for system (1.10).

Now let us consider case when B ≠ 1 identically. For function W = W (u,v,u₁,v₁,B) equations (2.14) and (2.15) become

WuB+(u1+v1+c)(u+v+c)Wu1+2(v1−v+clnB)(u+v+c)Wv1+B−1(u+v+c)WB=0(2.16)

WvB+Wv1=0.(2.17)

After the change of variables v˜=v+c, v˜1=v1+c−(v+c)B, u˜=u, u˜1=u1, B˜=B equations (2.17) and (2.16) become Wv˜=0 and

u˜+v˜B˜Wu˜+(u1˜+v1˜+v˜B˜)Wu1˜+(2v1˜+2clnB˜+v˜(B˜−1))Wv1˜+(B˜−1)WB˜=0.

We differentiate the last equality with respect to v˜, use Wv˜=0, and find that W satisfies the following equations

Wu˜B˜+B˜Wu1˜+(B˜−1)Wv1˜=0

u˜B˜Wu˜+(u˜1+v˜1)Wu1˜+(2v1˜+2clnB˜)Wv1˜+(B˜−1)WB˜=0.

After doing another change of variables u1*=u˜1−B˜2u˜, v1*=v˜1+B˜(1−B˜)u˜, u*=u˜, B*=B˜, we obtain that W_u^* = 0 and

(u1*+v1*)Wu1*+(2v1*+2clnB*)Wv1*+(B*−1)WB*=0.

The first integrals of the last equation are

K1=v1*(B*−1)2+clnB*(B*−1)2−clnB*+cln(B*−1)+cB*−1

and

K2=u1*−c−clnB*B*−1−B*v1*(B*−1)2−cB*lnB*(B*−1)2+cln(B*−1)−clnB*.

They can be rewritten in the original variables as

K1=v1−vB+B(1−B)u+clnB(B−1)2+cln(B−1)−clnB

and

K2=u1+cB−c−clnBB−1−B2v−Bv1−cBlnB(B−1)2+cln(B−1)−clnB.

Therefore, system (1.6) becomes (1.11) due to (2.2) and (2.4).

2.3. Proof of Remark 1.1

Function B is any function satisfying the equality H(K₁,K₂) = 0, where H is any smooth function. (I) By taking function H as H(K₁,K₂) = K₁ we obtain one possible function B. It satisfies the equality −uB² + (u − v)B + v₁ = 0 and can be taken as B=u−v+(−1)n(u−v)2+4uv12u. (II) By taking function H as H(K₁,K₂) = K₂ we obtain another possible function B. It satisfies the equality vB² + (u₁ − v₁)B − u₁ = 0 and can be taken as B=v1−u1+(−1)n(v1−u1)2+4u1v2v.

In both cases ((I) and (II)) let us consider the corresponding x-rings. Denote by X = D_x, Y1=∂∂ux, Y2=∂∂vx, E1=u+vB[Y1X], E2=1B[Y2X], E₃ = [E₁,E₂]. Note that X = u_xE₁ + v_xE₂. We have,

[E1,Ej]E1E2E3E10E3α1E2+α2E3E2−E300E3−(α1E2+α2E3)00

where

α1=2v1(u−v)+2(uv−v2+2uv1)Bv1(u−v)+((u−v)2+2uv1)B, α2=−3+2B

in case (I) and

α1=2u12+4u1v−2u1v1+2(−(u1−v1)2+vv1−3vu1)Bu1(v1−u1)+((u1−v1)2+2u1v)B, α2=−3+2B

in case (II).

3. Proof of Theorem 1.3

3.1. Case (1)

Let us find a system

{u1x=f˜(x,n,u,v,u1,v1,ux,vx)v1x=g˜(x,n,u,v,u1,v1,ux,vx)(3.1)

possessing n-integrals (1.16) and (1.17). The equality DI = I implies

u1xx+v1xx−u1x2+u1xv1x−v1x2=uxx+vxx−ux2+uxvx−vx2,(3.2)

or the same

f˜x+f˜uux+f˜vvx+f˜u1f˜+f˜v1g˜+f˜uxuxx+f˜vxvxx+g˜x+g˜uux+g˜vvx +g˜u1f˜+g˜v1g˜+g˜uxuxx+g˜vxvxx−f˜2+f˜g˜−g˜2=uxx=vxx−ux2+uxvx−vx2.(3.3)

We consider the coefficients by u_xx and v_xx in (3.3) to get

f˜ux+g˜ux=1(3.4)

f˜vx+g˜vx=1.(3.5)

The equality DI1*=I1* implies

u1xxx+u1x(v1xx−2u1xx)+u1x2v1x−u1xv1x2=uxxx+ux(vxx−2uxx)+ux2vx−uxvx2.(3.6)

Since DI1*=u1xxx+…f˜uxuxxx+…, where the remaining terms do not depend on u_xxx, the equality (3.6) implies

f˜ux=1.(3.7)

Note that J=DxI1−I1*=vxxx+vx(uxx−2vxx)+vx2ux−ux2vx is an n-integral as well. Since DJ = J and DJ=v1xxx+…g˜vxvxxx+…, where the remaining terms do not depend on v_xxx, then

g˜vx=1.(3.8)

It follows from equalities (3.4), (3.5), (3.7) and (3.8) that f˜vs=0 and g˜ux=0. Therefore the system (3.1) and equality (3.3) become

{u1x=ux+f(x,n,u,v,u1,v1)v1x=vx+g(x,n,u,v,u1,v1)(3.9)

and

fx+fuux+fvvx+fu1(ux+f)+fv1(vx+g)+gx+guux+gvvx+gu1(ux+f) +gv1(vx+g)−2uxf−f2+uxg+vxf+fg−2vxg−g2=0.(3.10)

By considering coefficients by u_x, v_x and ux0vx0 in the last equality, we get

(f+g)u+(f+g)u1+(f+g)−3f=0,(3.11)

(f+g)v+(f+g)v1+(f+g)−3g=0,(3.12)

f(f+g)u1+g(f+g)v1+(f+g)x−(f+g)2+3fg=0.(3.13)

Now let us rewrite inequality (3.6) for the system (3.9)

Dx(fx+fuux+fvvx+fu1(ux+f)+fv1(vx+g) +(ux+f)(gx+guux+gvvx+gu1(ux+f)+gv1(vx+g)+vxx) +(ux+f)(−2fx−2fuux−2fvvx−2fu1(ux+f)−2fv1(vx+g)−2uxx)+(ux2+2uxf+f2)(vx+g)−(vx2+2vxg+g2)(ux+f)=ux(vxx−2uxx)+ux2vx−uxvx2.(3.14)

By comparing the coefficients by u_xx and v_xx in the last equality, we get

fu+fu1=2ffv+fv1=−f.(3.15)

It follows from equality DJ = J that

Dx(gx+guux+gvvx+gu1(ux+f)+gv1(vx+g) +(vx+g)(fx+fuux+fvvx+fu1(ux+f)+fv1(vx+g)+uxx) −2(vx+g)(gx+guux+gvvx_gu1(ux+f)+gv1(vx+g)+vxx)+(ux+f)(vx2+2vxg+g2)−(vx+g)(ux2+2uxf+f2)=vx(vxx−2uxx)+vx2vx−ux2vx.(3.16)

By comparing the coefficients by u_xx and v_xx in the last equality, we get

gu+gu1=−ggv+gv1=2g.(3.17)

Note that the equalities (3.11) and (3.12) follow from equalities (3.15) and (3.17). Let us use equalities (3.15) and (3.17) to rewrite equality (3.14)

Dx(fx+2fux−fvx+fu1f+fv1g)+(ux+f)(gx+gu1f+gv1g+vxx−4fux−2fx) +(ux+f)(2fvx−2fu1f−2fv1g−2uxx+uxvx+fvx+fg−vx2−g2) =ux(vxx−2uxx)+ux2vx−uxvx2.

We note that the consideration of the coefficients by u_xx, v_xx, ux2, vx2, u_xv_x in the above equality give us equations that follow immediately from (3.15) and (3.17). Considering coefficient by u_x we get

fxu+fxu1+2fx+2ffu1+2fv1g+ffu1u+fu1fu+fu12+gfv1u+gfu1v1+fv1gu+fv1gu1+fu1u1f+gx+gu1f+gv1g−2fx−2fu1f−2fv1g+fg−g2−4f2=0.

Using equations (3.15) and (3.17) we get

2fx+gx+4ffu1+fv1g+gu1f+gv1g+fg−g2−4f2=0,

or using equation (3.13),

fx+3f(fu1−f)=0.(3.18)

Considering coefficient by v_x we get

fxv+fxv1−fx−ffu1−fv1g−ffu1v+ffu1v1+fu1fv+fu1fv1 +gfv1v+gfv1v1+fv1gv+fv1gv1+3f2=0.

Using equations (3.15) and (3.17) we get

2fx+3f(fu1−f)=0.(3.19)

It follows from equations (3.18) and (3.19) that f_x = 0 and f (f_u₁ − f) = 0. Thus either f = 0 or

{f=fu1,f=fu.(3.20)

Now we consider the coefficient by ux0vx0 in (3.14) we get

f2fu1u1+fgfu1v1+ffu12+fu1fv1g+fgfu1v1+g2fv1v1+fv1gx+ffv1gu1 +gfv1gv1+fgx+f2gu1+fggv1−2f2fu1−2fgfv1+f2g−fg2=0.

First assume that f ≠ 0 then using (3.20) we can rewrite the above equality as

fgfv1+g2fv1v1+fv1gx+fv1gu1f+fv1gv1g+fgx+f2gu1+fggv1+f2g−fg2=0.(3.21)

Also we can rewrite equality (3.16), using equations (3.15), (3.17) and (3.13) then considering coefficients by u_x and v_x we obtain

2gx+3g(gv1−g)=0,gx+3g(gv1−g)=0.

From above equalities and (3.17) it follows that g_x = 0, g_v₁ = g and g_v = g (we assume that g ≠ 0). We have

fu1−f,fu=f,fv+fv1=−fgv1=g,gv=g,gu+gu1=−gfv1g+gu1f=−fg.(3.22)

Using (3.22), the equality (3.21) takes form g_u₁ f_v₁(−g + f) = 0. This equality implies that under assumptions that f ≠ 0 and g ≠ 0 we have three possibilities: (I) g_u₁ = 0, (II) f_v₁ = 0 and (III) g = f. Let us consider these possibilities.

Case (I) From g_u₁ = 0, using (3.22), we get that g_u = −g, g_v₁ = g, g_v = g. Thus g = Be^−u+v+v₁, where B is a constant. We also get that f_u = f, f_u₁ = f, f_v = 0 and f_v₁= − f. Thus f = Ae^{u₁+u−v₁}, where A is a constant. So the system (3.9) takes form (1.18).
Case (II) From f_v₁ = 0, using (3.22), we get that f_u = f, f_u₁ = f, f_v = − f. Thus f = Ae^u₁+u−v, where A is a constant. We also get that g_u = 0, g_u₁ = −g, g_v = g and g_v₁ = g. Thus g = Be^{−u₁+v₁+v}, where B is a constant. So the system (3.9) takes form (1.19).
Case (III) From g = f, using (3.22), we get that f = 0 and g = 0. So the system (3.9) takes form
{u1x=ux,v1x=vx.

3.2. Case (2)

Let us find system (1.15) possessing n-integrals (1.20) and (1.21). We compare the coefficients in DI₂ = I₂ by u_xx and v_xx and get

2f˜ux+g˜ux=2,2f˜vx+g˜vx=1.(3.23)

We also compare the coefficients in DI2*=I2* and

D(Dx2I2−2I2*)=(Dx2I2−2I2*) by u_xxxx and v_xxxx respectively and get f˜ux=1 and g˜vx=1. It follows from (3.23) that f˜vx=0 and g˜ux=0. Therefore, our system (1.15) becomes

{u1x=ux+f(u,v,u1,v1)v1x=vx+g(u,v,u1,v1).

We write equality DI₂ = I₂ and get

2uxx+2fuux+2fvvx+2fu1(ux+f)+2fv1(vx+g)+vxx+guux+gvvx+gu1(ux+f) +gv1(vx+g)−2(ux+f)2+2(ux+f)(vx+g)−(vx+g)2=2uxx+vxx−2ux2+2uxvx−vx2.

By comparing the coefficients by u_x, v_x and ux0vx0 in the last equality we obtain the system of equations

2fu+fu1+gu+gu1−4f+2g=0,2fv+2fv1+gv+gv1+2f−2g=0,2ffu1+2gfv1+fgu1+ggv1−2f2+2fg−g2=0.

That suggests the following change of variables

u=P,u1−u=Q,v=S,v1−v=T

to be made. In new variables the system (1.15) becomes

{Qx=F(P,Q,S,T),Tx=G(P,Q,S,T).(3.24)

The comparison of coefficients in DI₂ = I₂ by P_x, S_x and Px0Sx0 gives

−4F+2G+2FP+GP=0,2F−2G+2FS+GS=0,−2F2+G(−G+2FT+GT)+F(2G+2FQ+GQ)=0.(3.25)

The coefficients in DI2*=I2* by S_xxx and P_xxx are compared and we obtain the following equalities

F+FS=0,−2F+FP=0.(3.26)

It follows from (3.25) and (3.26) that G_S = 2G, G_P = −2G, F_S = −F and F_P = 2F. Therefore, system (3.24) can be written as

{Qx=A(Q,T)e−S+2P,Tx=B(Q,T)e−2S−2P.

We compare the coefficient in DI2*=I2* by S_xx and get

3e4P−2SA2−3e4P−2SAAQ=0,

that is A = A_Q. Hence, A(Q,T)=eQA˜(T). Now we compare the coefficient in DI₂ = I₂ by Px0Sx0 and get

A˜+A˜T=12e−4P+3S−Q(B−BT)−A˜2BBQ.(3.27)

Since functions A˜(T) and B(Q,T) do not depend on variable P, then it follows from (3.27) that B = B_T, that is B=B˜(Q)eT. Now (3.27) becomes

−2A˜+A˜TA˜=B˜QB˜.

Note that the right side of the last equality depends on Q only, while the left side depends on T only. Hence, −2A˜+A˜TA=c and B˜QB˜=c, where c is some constant. One can see that A˜=c1e−(2c+1)T and B˜=c2ecQ and therefore system (3.24) becomes

{Qx=c1e−S+2P+Q+(2c+1)T,Tx=c2e2S−2P+T+cQ,

where c, c₁ and c₂ are some constants. Equality DI₂ − I₂ = 0 becomes −3cc₁c₂e^{s+(c+1)Q−2cT} = 0, which implies that either c = 0, or c₁ = 0, or c₂ = 0. Note that the DI2*=I2* is also satisfied if either c = 0 or c₁ = 0 or c₂ = 0. So we have three cases:

when c = 0 the system (1.15) becomes (1.22) with c₁ = A and c₂ = B.
when c₁ = 0 the system (1.15) becomes (1.26) with c₂ = B.
when c₂ = 0 the system (1.15) becomes (1.27) with c₁ = A.

3.3. Case (3)

Let us find system (1.15) possessing n-integrals (1.23) and (1.24). We compare the coefficients in DI₃ = I₃ by u_xx and v_xx and get

f˜ux+13g˜ux+1,f˜vx+13g˜vx=1.(3.28)

We also compare the coefficients in DI3*=I3* and D(Dx4I3−I3*)=(Dx4I3−I3*) by u₍₆₎ and v₍₆₎ respectively and get f˜ux=1 and g˜vx=1. It follows from (3.28) that f˜vx=0 and g˜ux=0. Therefore, our system (1.15) becomes

{u1x=ux+f(u,v,u1,v1),v1x=vx+g(u,v,u1,v1).

By comparing the coefficients by u_x, v_x and ux0vx0 in DI₃ = I₃ we obtain the system of equations

fu+fu1+13gu+13gu1−2f+g=0,fv+fv1+13gv+13gv1+f−23g=0,ffu1+gfv1+13fgu1+13ggv1−f2+fg−13g2=0.

That suggests the following change of variables

u=P,u1−u=Q,v=S,v1−v=T

to be made. In new variables the system (1.15) becomes

{Qx=F(P,Q,S,T),Tx=G(P,Q,S,T).(3.29)

The comparison of coefficients in DI₃ = I₃ by P_x, S_x and Px0Sx0 gives

6F−3G−3FP−GP=0,−3F+2G−3FS−GS=0,F2−FG+13G2−2GFT−13GGT−FFQ−13FGQ=0.(3.30)

The comparison of coefficients in DI3*=I3* by S₍₅₎ and P₍₅₎ gives

F+FS=0,−2F+FP=0.(3.31)

Using equations (3.30) and (3.31) we get G_S = 2G, G_P = −3G, F_S = −F, and F_P = 2F. Therefore, system (3.29) can be written as

{Qx=A(Q,T)e−S+2P,Tx=B(Q,T)e2S−3P,

where A and B are some functions depending on Q and T only. We compare the coefficients in DI₃ − I₃ = 0 by Sx0Px0 and the coefficients in DI3*=I3* by P₍₄₎, S₍₄₎ and P₍₃₎P_x respectively and get

a11AT+a12BT+a13AQ+a14BQ+b1=0,a21AT+a22BT+a23AQ+a24BQ+b2=0,a31AT+a32BT+a33AQ+a34BQ+b3=0,a41AT+a42BT+a43AQ+a44BQ+b4=0,(3.32)

where

a11=e−P+SB,a12=−13e−6P+4SB,a13=−e4P−2SA,a14=−13e−P+SA,a21=−33e−P+SB,a22=−11e−6P+4SB,a23=−28e4P−2SA,a24=−11e−P+SA,a31=−13e−P+SB,a32=−133e−6P+4SB,a33=−16e4P−2SA,a34=−133e−P+SA,a41=18e−P+SB,a42=−79e−6P+4SB,a43=328e4P−2SA,a44=6e−P+SA,

and

b1=e4P−2SA2−e−P+SAB+13e−6P+4SB2,b2=28e4P−2SA2−33e−P+SAB+11e−6P+4SB2,b3=16e4P−2SA2−13e−P+SAB+133e−6P+4SB2,b4=−328e4P−2SA2+18e−P+SAB+79e−6P+4SB2.

We solve the linear system of equations (3.32) with respect to A_T, A_Q, B_T and B_Q and get the following system of differential equations A_T = −A, A_Q = A, B_T = B and B_Q = 0. Thus the system (3.29) is written as

{Qx=c1e2P+Q−S−T,Tx=c2e−3P+2S+T,

where c₁ and c₂ are arbitrary constants. It is equivalent to system (1.25) with A = c₁ and B = c₂.

References

[1]A.V. Zhiber, R.D. Murtazina, I.T. Habibullin, and A.B. Shabat, Characteristic Lie rings and integrable models in mathematical physics, Ufa Math. J, Vol. 4, No. 3, 2012, pp. 17-85.

[2]I.T. Habibullin, N. Zheltukhina, and A. Sakieva, Discretization of hyperbolic type Darboux integrable equations preserving integrability, J. Math. Phys, Vol. 52, 2011, pp. 093507-093519.

[3]I.T. Habibullin and N. Zheltukhina, Discretization of Liouville type nonautonomous equations, J. Nonlinear Math. Phys, Vol. 23, 2016, pp. 620-642.

[4]K. Zheltukhin and N. Zheltukhina, On the discretization of Laine equations, J. Nonlinear Math. Phys, Vol. 25, 2018, pp. 166-177.

[5]I.T. Habibullin and A. Pekcan, Characteristic Lie algebra and the classification of semi-discrete models, Theoret. and Math. Phys, Vol. 151, 2007, pp. 781-790.

[6]O.S. Kostrigina and A.V. Zhiber, Darboux-integrable two-component nonlinear hyperbolic systems of equations, J. Math. Phys, Vol. 52, 2011, pp. 033503-033535.

[7]A.B. Shabat and R.I. Yamilov, Exponential Systems of Type I and the Cartan Matrices (Russian), Preprint BBAS USSR Ufa, 1981.

[8]N.H. Ibragimov, A.V. Aksenov, V.A. Baikov, V.A. Chugunov, R.K. Gazizov, and A.G. Meshkov, Ibragimov (editor), CRC Handbook of Lie Group Analysis of Differential Equations, Vol. 2., Applications in Engineeringand Physical Science, CRC Press, Boca Raton, FL, 1995.

[9]E.I. Ganzha and S.P. Tsarev, Integration of Classical Series An, Bn, Cn, of Exponential Systems, Krasnoyarsk State Pedagogical University Press, Krasnoyarsk, 2001.

[10]A.N. Leznov and M.V. Savel’ev, Group Methods of Integration of Nonlinear Dynamical Systems, Progress in Physics 15, Birkhäuser Verlag, Basel, 1992.

[11]I.T. Habibullin, K. Zheltukhin, and M. Yangubaeva, Cartan matrices and integrable lattice Toda field equations, J. Phys. A, Vol. 44, 2011, pp. 465202-465222.

<Previous Article In Issue

Download article (PDF)

Next Article In Issue>

Journal: Journal of Nonlinear Mathematical Physics
Volume-Issue: 27 - 4
Pages: 616 - 632
Publication Date: 2020/09/04
ISSN (Online): 1776-0852
ISSN (Print): 1402-9251
DOI: 10.1080/14029251.2020.1819597 How to use a DOI?
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/).

Cite this article

ris enw bib

TY  - JOUR
AU  - Kostyantyn Zheltukhin
AU  - Natalya Zheltukhina
PY  - 2020
DA  - 2020/09/04
TI  - On the discretization of Darboux Integrable Systems
JO  - Journal of Nonlinear Mathematical Physics
SP  - 616
EP  - 632
VL  - 27
IS  - 4
SN  - 1776-0852
UR  - https://doi.org/10.1080/14029251.2020.1819597
DO  - 10.1080/14029251.2020.1819597
ID  - Zheltukhin2020
ER  -

download .riscopy to clipboard