Journal of Nonlinear Mathematical Physics

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
https://doi.org/10.1080/14029251.2020.1819597How 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
Β© 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=π’œ(p,px,py)   (pxyi=π’œi(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, py, pyy,...) 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 Dx 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 qk(x,n) = Dk 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,qx,qxx,...) 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

ΞΌxyi=eβˆ‘aijΞΌj,   i,j=1,2,…,N.

It was shown that such a system is Darboux integrable if and only if the matrix A = (aij) 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+vβˆ’c)uxvyvxy=vxvyu+vβˆ’c+(1u+v+c+1u+vβˆ’c)uxvy,(1.3)
where c is an arbitrary constant. This system is Darboux integrable and admits the following y-integrals
I1=2vβˆ’vx(u+v+c)ux+2clnuxu+v+c(1.4)
and
I2=uxxuxβˆ’2ux+vxu+v+c.(1.5)

The x- integrals have the same form in u, v, uy, vy,... 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) β‰  c(n + 1) for all n ∈ 𝕑, has the form
{u1x=(u1+v1+c1)uxu+v+cv1x=2(v1βˆ’v)uxu+v+c+2(c1βˆ’c)uxu+v+clnuxu+v+c+vx.(1.7)

Moreover, the system above also possesses x-integrals

F1(cβˆ’c1)(v2βˆ’v)βˆ’(cβˆ’c2)(v1βˆ’v)(cβˆ’c2)(v3βˆ’v)βˆ’(cβˆ’c3)(v2βˆ’v)(1.8)
and
F2=(c1βˆ’c2)u+(c2βˆ’c)u1+(cβˆ’c1)u2(c1βˆ’c2)v+(c2βˆ’c)v1+(cβˆ’c1)v2βˆ’(c1βˆ’c2)v+(c2βˆ’c)v1+(cβˆ’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(v1βˆ’v)uxu+v+c+vx(1.10)
with x-integrals F1=v1βˆ’vv2βˆ’v1 and F2=u2βˆ’u+vβˆ’v2v1βˆ’v, or
{u1x=(u1+v1+c)Buxu+v+cv1x=2B(v1βˆ’v+clnB)u+v+cux+Bvx,(1.11)
where B is defined by equality H(K1,K2) = 0 with
K1=v1βˆ’vB+B(1βˆ’B)u+clnB(Bβˆ’1)2+cln(Bβˆ’1)βˆ’clnB
and
K2=u1+cBβˆ’cβˆ’clnBBβˆ’1+B2vβˆ’Bv1βˆ’cBlnB(Bβˆ’1)2+cln(Bβˆ’1)βˆ’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.

  1. (I)

    System (1.11) with B=uβˆ’v+(βˆ’1)n(uβˆ’v)2+4uv12u is Darboux Integrable. (The expression for B is found from K1 = 0, with c = 0.)

  2. (II)

    System (1.11) with B=v1βˆ’u1+(βˆ’1)n(v1βˆ’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 βˆ’ 1)2K1 = 0, into a series of the form

B(u,v,v1)=βˆ‘n=0∞an(v1βˆ’v)n,(1.12)
where coefficients an depend on variables u and v, yields a0 = 1 and a1=1u+vβˆ’c. So Bcan be written as
B(u,v,v1)=1+1u+vβˆ’c(v1βˆ’v)+βˆ‘n=2∞an(v1βˆ’v)n.(1.13)

By letting u1 = u + Ξ΅uy and v = v + Ξ΅vy and taking Ξ΅ β†’ 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 Γ— 2 matrices, namely,

ΞΌxy=e2ΞΌβˆ’v,vxy=eβˆ’cΞΌβˆ’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. (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+vxxβˆ’ux2+uxvxβˆ’vx2(1.16)
    and
    I1*=uxxx+ux(vxxβˆ’2uxx)+ux2vxβˆ’uxvx2(1.17)
    has the form
    {u1x=ux+Aeu1+uβˆ’v1v1x=vx+Beβˆ’u+v+v1,(1.18)
    or
    {u1x=ux+Aeu1+uβˆ’vv1x=vx+Beβˆ’u1+v+v1,(1.19)
    where A and B are arbitrary constants.

  2. (2)

    The system (1.15) possessing n-integrals

    I2=2uxx+vxxβˆ’2ux2+2uxvxβˆ’vx2(1.20)
    and
    I2*=uxxx+ux(vxxxβˆ’2uxxx)+uxx(4uxvxβˆ’2ux2βˆ’vx2)   +uxx(vxxβˆ’uxx)+vxxux(uxβˆ’2vx)+ux4+ux2vx2βˆ’2ux3vx(1.21)
    has the form
    {u1x=ux+Aeu+u1βˆ’v1v1x=vx+Beβˆ’2u+v+v1,(1.22)
    where A and B are arbitrary constants.

  3. (3)

    The system (1.15) possessing n-integrals

    I3=uxx+13vxxβˆ’ux2+uxvxβˆ’13vx2(1.23)
    and
    I3*=u(6)βˆ’2u(5)ux+v(5)ux+u(4)(32(ux)2βˆ’30uxvx+11(vx)2βˆ’40uxxβˆ’11vxx)+v(4)(14(ux)2βˆ’15uxvx+(13/3(vx)2βˆ’10uxxβˆ’(13/3)vxx)+19(u(3))2+(13/6)(v(3))2+16u(3)v(3)+u(3)(βˆ’36uxxux+18uxxvx+80vxxuxβˆ’45vxxvx)+v(3)(βˆ’52uxxux+33uxxvxβˆ’5vxxux)+u(3)(βˆ’64(ux)3+102(ux)2vxβˆ’62ux(vx)2+13(vx)3)+v(3)(32(ux)3βˆ’58(ux)2vx+38ux(vx)2βˆ’(26/3(vx)3)+66(uxx)3+(26/3)(vxx)3βˆ’35(uxx)2(vxx)βˆ’5uxx(vxx)2+(uxx)2(30(ux)2βˆ’18uxvxβˆ’(11/2)(vx)2)+uxxvxx(βˆ’34(ux)2+32uxvxβˆ’2(vx)2)βˆ’2(vxx)2uxvx+uxx(6(ux)4βˆ’24(ux)3vx+25(ux)2(vx)2βˆ’9ux(vx)3+(vx)4+vxx(βˆ’(ux)4+8(ux)3vxβˆ’8(ux)2(vx)2+2ux(vx)3)+(βˆ’2(ux)6+6(ux)5vxβˆ’(13/2)(ux)4(vx)2+3(ux)3(vx)3βˆ’(1/2)(ux)2(vx)4)(1.24)
    has the form
    {u1x=ux+Aeu+u1βˆ’v1v1x=vx+Beβˆ’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+Beβˆ’(2+c)u+cu1+v+v1,(1.26)
and
{u1x=ux+Aeu+u12cvβˆ’(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 = ΞΌ1, u1=ΞΌ1+Ι›ΞΌy1, v = ΞΌ2 , v1=ΞΌ2+Ι›ΞΌy2 and A = Ξ΅, B = Ξ΅ in equations (1.18), (1.22), (1.25) and taking Ξ΅ β†’ 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

u1xxu1xβˆ’2u1x+v1xu1+v1+c1=uxxuxβˆ’2ux+vxu+v+c,
that is
fx+fuux+fvvx+fu1f+fv1g+fuxuxx+fvxvxxfβˆ’2f+gu1+v1+c1=uxxuxβˆ’2ux+vxu+v+c.(2.1)

Compare the coefficients by vxx and uxx, we get fvx = 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

2v1βˆ’(u1+v1+c1)gf+2c1lnfu1+v1+c1=2vβˆ’vx(u+v+c)ux+2clnuxu+v+c.(2.3)

Using (2.2) we obtain

2v1βˆ’(u1+v1+c1)gAux+2c1lnAuxu1+v1+c1=2vβˆ’vx(u+v+c)ux+2clnuxu+v+c
and find g as
g=(2(v1βˆ’v)Au1+v1+c1+2Ac1(u1+v1+c1)ln(u+v+c)A(u1+v1+c1))ux+2(c1βˆ’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 ux, vx, ux ln uxu+v+c and free term we get the following equalities

AxA=0(2.5)
2(c1βˆ’c)Av1(u1+v1+c1)βˆ’2(c1βˆ’c)A(u1+v1+c1)2=0(2.6)
AuA+Au1+(Av1Aβˆ’1(u1+v1+c1))(2(v1βˆ’v)A(u1+v1+c1)+2c1A(u1+v1+c1ln(u+v+c)A(u1+v1+c1))βˆ’2A(u1+v1+c1)+2(u+v+c)=0(2.7)
AvA+(u+v+c)Av1(u1+v1+c1)βˆ’(u+v+c)A(u1+v1+c1)2+1(u+v+c)=0.(2.8)

We have two possibilities: c1 β‰  c and c1 = c.

2.1. c depends on n

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

Av1Aβˆ’1(u1+v1+c1)=0(2.9)
AuA+Au1βˆ’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+(1βˆ’M)=0.(2.13)

Differentiating equation (2.13) with respect to v and v1 we get that Mu = 0 and Mu1 = 0 respectively. Thus, equation (2.13) implies that M = 1. So in the case c1 β‰  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(v1βˆ’v+clnB)(u+v+c)Bv1+1βˆ’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=v1βˆ’vv2βˆ’v1,   F2=u2βˆ’u+vβˆ’v2v1βˆ’v.

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

Now let us consider case when B β‰  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(v1βˆ’v+clnB)(u+v+c)Wv1+Bβˆ’1(u+v+c)WB=0(2.16)
WvB+Wv1=0.(2.17)

After the change of variables v˜=v+c, v˜1=v1+cβˆ’(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˜(BΛœβˆ’1))Wv1˜+(BΛœβˆ’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˜+(BΛœβˆ’1)Wv1˜=0
u˜B˜Wu˜+(u˜1+v˜1)Wu1˜+(2v1˜+2clnB˜)Wv1˜+(BΛœβˆ’1)WB˜=0.

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

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

The first integrals of the last equation are

K1=v1*(B*βˆ’1)2+clnB*(B*βˆ’1)2βˆ’clnB*+cln(B*βˆ’1)+cB*βˆ’1
and
K2=u1*βˆ’cβˆ’clnB*B*βˆ’1βˆ’B*v1*(B*βˆ’1)2βˆ’cB*lnB*(B*βˆ’1)2+cln(B*βˆ’1)βˆ’clnB*.

They can be rewritten in the original variables as

K1=v1βˆ’vB+B(1βˆ’B)u+clnB(Bβˆ’1)2+cln(Bβˆ’1)βˆ’clnB
and
K2=u1+cBβˆ’cβˆ’clnBBβˆ’1βˆ’B2vβˆ’Bv1βˆ’cBlnB(Bβˆ’1)2+cln(Bβˆ’1)βˆ’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 βˆ’uB2 + (u βˆ’ v)B + v1 = 0 and can be taken as B=uβˆ’v+(βˆ’1)n(uβˆ’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 βˆ’ v1)B βˆ’ u1 = 0 and can be taken as B=v1βˆ’u1+(βˆ’1)n(v1βˆ’u1)2+4u1v2v.

In both cases ((I) and (II)) let us consider the corresponding x-rings. Denote by X = Dx, Y1=βˆ‚βˆ‚ux, Y2=βˆ‚βˆ‚vx, E1=u+vB[Y1X], E2=1B[Y2X], E3 = [E1,E2]. Note that X = uxE1 + vxE2. We have,

[E1,Ej]E1E2E3E10E3Ξ±1E2+Ξ±2E3E2βˆ’E300E3βˆ’(Ξ±1E2+Ξ±2E3)00
where
Ξ±1=2v1(uβˆ’v)+2(uvβˆ’v2+2uv1)Bv1(uβˆ’v)+((uβˆ’v)2+2uv1)B,   α2=βˆ’3+2B
in case (I) and
Ξ±1=2u12+4u1vβˆ’2u1v1+2(βˆ’(u1βˆ’v1)2+vv1βˆ’3vu1)Bu1(v1βˆ’u1)+((u1βˆ’v1)2+2u1v)B,   α2=βˆ’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+v1xxβˆ’u1x2+u1xv1xβˆ’v1x2=uxx+vxxβˆ’ux2+uxvxβˆ’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˜vxvxxβˆ’f˜2+f˜gΛœβˆ’g˜2=uxx=vxxβˆ’ux2+uxvxβˆ’vx2.(3.3)

We consider the coefficients by uxx and vxx 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(v1xxβˆ’2u1xx)+u1x2v1xβˆ’u1xv1x2=uxxx+ux(vxxβˆ’2uxx)+ux2vxβˆ’uxvx2.(3.6)

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

f˜ux=1.(3.7)

Note that J=DxI1βˆ’I1*=vxxx+vx(uxxβˆ’2vxx)+vx2uxβˆ’ux2vx is an n-integral as well. Since DJ = J and DJ=v1xxx+…g˜vxvxxx+…, where the remaining terms do not depend on vxxx, 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)βˆ’2uxfβˆ’f2+uxg+vxf+fgβˆ’2vxgβˆ’g2=0.(3.10)

By considering coefficients by ux, vx and ux0vx0 in the last equality, we get

(f+g)u+(f+g)u1+(f+g)βˆ’3f=0,(3.11)
(f+g)v+(f+g)v1+(f+g)βˆ’3g=0,(3.12)
f(f+g)u1+g(f+g)v1+(f+g)xβˆ’(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)(βˆ’2fxβˆ’2fuuxβˆ’2fvvxβˆ’2fu1(ux+f)βˆ’2fv1(vx+g)βˆ’2uxx)+(ux2+2uxf+f2)(vx+g)βˆ’(vx2+2vxg+g2)(ux+f)=ux(vxxβˆ’2uxx)+ux2vxβˆ’uxvx2.(3.14)

By comparing the coefficients by uxx and vxx in the last equality, we get

fu+fu1=2ffv+fv1=βˆ’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)β€‰β€‰β€‰βˆ’2(vx+g)(gx+guux+gvvx_gu1(ux+f)+gv1(vx+g)+vxx)+(ux+f)(vx2+2vxg+g2)βˆ’(vx+g)(ux2+2uxf+f2)=vx(vxxβˆ’2uxx)+vx2vxβˆ’ux2vx.(3.16)

By comparing the coefficients by uxx and vxx in the last equality, we get

gu+gu1=βˆ’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+2fuxβˆ’fvx+fu1f+fv1g)+(ux+f)(gx+gu1f+gv1g+vxxβˆ’4fuxβˆ’2fx)   +(ux+f)(2fvxβˆ’2fu1fβˆ’2fv1gβˆ’2uxx+uxvx+fvx+fgβˆ’vx2βˆ’g2)   =ux(vxxβˆ’2uxx)+ux2vxβˆ’uxvx2.

We note that the consideration of the coefficients by uxx, vxx, ux2, vx2, uxvx in the above equality give us equations that follow immediately from (3.15) and (3.17). Considering coefficient by ux we get

fxu+fxu1+2fx+2ffu1+2fv1g+ffu1u+fu1fu+fu12+gfv1u+gfu1v1+fv1gu+fv1gu1+fu1u1f+gx+gu1f+gv1gβˆ’2fxβˆ’2fu1fβˆ’2fv1g+fgβˆ’g2βˆ’4f2=0.

Using equations (3.15) and (3.17) we get

2fx+gx+4ffu1+fv1g+gu1f+gv1g+fgβˆ’g2βˆ’4f2=0,
or using equation (3.13),
fx+3f(fu1βˆ’f)=0.(3.18)

Considering coefficient by vx we get

fxv+fxv1βˆ’fxβˆ’ffu1βˆ’fv1gβˆ’ffu1v+ffu1v1+fu1fv+fu1fv1   +gfv1v+gfv1v1+fv1gv+fv1gv1+3f2=0.

Using equations (3.15) and (3.17) we get

2fx+3f(fu1βˆ’f)=0.(3.19)

It follows from equations (3.18) and (3.19) that fx = 0 and f (fu1 βˆ’ 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+fggv1βˆ’2f2fu1βˆ’2fgfv1+f2gβˆ’fg2=0.

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

fgfv1+g2fv1v1+fv1gx+fv1gu1f+fv1gv1g+fgx+f2gu1+fggv1+f2gβˆ’fg2=0.(3.21)

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

2gx+3g(gv1βˆ’g)=0,gx+3g(gv1βˆ’g)=0.

From above equalities and (3.17) it follows that gx = 0, gv1 = g and gv = g (we assume that g β‰  0). We have

fu1βˆ’f,fu=f,fv+fv1=βˆ’fgv1=g,gv=g,gu+gu1=βˆ’gfv1g+gu1f=βˆ’fg.(3.22)

Using (3.22), the equality (3.21) takes form gu1 fv1(βˆ’g + f) = 0. This equality implies that under assumptions that f β‰  0 and g β‰  0 we have three possibilities: (I) gu1 = 0, (II) fv1 = 0 and (III) g = f. Let us consider these possibilities.

  • Case (I) From gu1 = 0, using (3.22), we get that gu = βˆ’g, gv1 = g, gv = g. Thus g = Beβˆ’u+v+v1, where B is a constant. We also get that fu = f, fu1 = f, fv = 0 and fv1 = βˆ’ f. Thus f = Aeu1+uβˆ’v1, where A is a constant. So the system (3.9) takes form (1.18).

  • Case (II) From fv1 = 0, using (3.22), we get that fu = f, fu1 = f, fv = βˆ’ f. Thus f = Aeu1+uβˆ’v, where A is a constant. We also get that gu = 0, gu1 = βˆ’g, gv = g and gv1 = g. Thus g = Beβˆ’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 uxx and vxx and get

2f˜ux+g˜ux=2,2f˜vx+g˜vx=1.(3.23)

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

D(Dx2I2βˆ’2I2*)=(Dx2I2βˆ’2I2*) by uxxxx and vxxxx 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)βˆ’2(ux+f)2+2(ux+f)(vx+g)βˆ’(vx+g)2=2uxx+vxxβˆ’2ux2+2uxvxβˆ’vx2.
By comparing the coefficients by ux, vx and ux0vx0 in the last equality we obtain the system of equations
2fu+fu1+gu+gu1βˆ’4f+2g=0,2fv+2fv1+gv+gv1+2fβˆ’2g=0,2ffu1+2gfv1+fgu1+ggv1βˆ’2f2+2fgβˆ’g2=0.
That suggests the following change of variables
u=P,u1βˆ’u=Q,v=S,v1βˆ’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 Px, Sx and Px0Sx0 gives

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

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

F+FS=0,βˆ’2F+FP=0.(3.26)

It follows from (3.25) and (3.26) that GS = 2G, GP = βˆ’2G, FS = βˆ’F and FP = 2F. Therefore, system (3.24) can be written as

{Qx=A(Q,T)eβˆ’S+2P,Tx=B(Q,T)eβˆ’2Sβˆ’2P.

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

3e4Pβˆ’2SA2βˆ’3e4Pβˆ’2SAAQ=0,
that is A = AQ. Hence, A(Q,T)=eQA˜(T). Now we compare the coefficient in DI2 = I2 by Px0Sx0 and get
A˜+A˜T=12eβˆ’4P+3Sβˆ’Q(Bβˆ’BT)βˆ’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 = BT, that is B=B˜(Q)eT. Now (3.27) becomes

βˆ’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, βˆ’2A˜+A˜TA=c and B˜QB˜=c, where c is some constant. One can see that A˜=c1eβˆ’(2c+1)T and B˜=c2ecQ and therefore system (3.24) becomes

{Qx=c1eβˆ’S+2P+Q+(2c+1)T,Tx=c2e2Sβˆ’2P+T+cQ,
where c, c1 and c2 are some constants. Equality DI2 βˆ’ I2 = 0 becomes βˆ’3cc1c2es+(c+1)Qβˆ’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 uxx and vxx and get

f˜ux+13g˜ux+1,f˜vx+13g˜vx=1.(3.28)

We also compare the coefficients in DI3*=I3* and D(Dx4I3βˆ’I3*)=(Dx4I3βˆ’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 ux, vx and ux0vx0 in DI3 = I3 we obtain the system of equations

fu+fu1+13gu+13gu1βˆ’2f+g=0,fv+fv1+13gv+13gv1+fβˆ’23g=0,ffu1+gfv1+13fgu1+13ggv1βˆ’f2+fgβˆ’13g2=0.

That suggests the following change of variables

u=P,u1βˆ’u=Q,v=S,v1βˆ’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 Px, Sx and Px0Sx0 gives

6Fβˆ’3Gβˆ’3FPβˆ’GP=0,βˆ’3F+2Gβˆ’3FSβˆ’GS=0,F2βˆ’FG+13G2βˆ’2GFTβˆ’13GGTβˆ’FFQβˆ’13FGQ=0.(3.30)

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

F+FS=0,βˆ’2F+FP=0.(3.31)

Using equations (3.30) and (3.31) we get GS = 2G, GP = βˆ’3G, FS = βˆ’F, and FP = 2F. Therefore, system (3.29) can be written as

{Qx=A(Q,T)eβˆ’S+2P,Tx=B(Q,T)e2Sβˆ’3P,
where A and B are some functions depending on Q and T only. We compare the coefficients in DI3 βˆ’ I3 = 0 by Sx0Px0 and the coefficients in DI3*=I3* by P(4), S(4) and P(3)Px 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=eβˆ’P+SB,a12=βˆ’13eβˆ’6P+4SB,a13=βˆ’e4Pβˆ’2SA,a14=βˆ’13eβˆ’P+SA,a21=βˆ’33eβˆ’P+SB,a22=βˆ’11eβˆ’6P+4SB,a23=βˆ’28e4Pβˆ’2SA,a24=βˆ’11eβˆ’P+SA,a31=βˆ’13eβˆ’P+SB,a32=βˆ’133eβˆ’6P+4SB,a33=βˆ’16e4Pβˆ’2SA,a34=βˆ’133eβˆ’P+SA,a41=18eβˆ’P+SB,a42=βˆ’79eβˆ’6P+4SB,a43=328e4Pβˆ’2SA,a44=6eβˆ’P+SA,
and
b1=e4Pβˆ’2SA2βˆ’eβˆ’P+SAB+13eβˆ’6P+4SB2,b2=28e4Pβˆ’2SA2βˆ’33eβˆ’P+SAB+11eβˆ’6P+4SB2,b3=16e4Pβˆ’2SA2βˆ’13eβˆ’P+SAB+133eβˆ’6P+4SB2,b4=βˆ’328e4Pβˆ’2SA2+18eβˆ’P+SAB+79eβˆ’6P+4SB2.

We solve the linear system of equations (3.32) with respect to AT, AQ, BT and BQ and get the following system of differential equations AT = βˆ’A, AQ = A, BT = B and BQ = 0. Thus the system (3.29) is written as

{Qx=c1e2P+Qβˆ’Sβˆ’T,Tx=c2eβˆ’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.
[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.
Journal
Journal of Nonlinear Mathematical Physics
Volume-Issue
27 - 4
Pages
616 - 632
Publication Date
2020/09
ISSN (Online)
1776-0852
ISSN (Print)
1402-9251
DOI
https://doi.org/10.1080/14029251.2020.1819597How to use a DOI?
Copyright
Β© 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/).

Cite this article

TY  - JOUR
AU  - Kostyantyn Zheltukhin
AU  - Natalya Zheltukhina
PY  - 2020
DA  - 2020/09
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  - https://doi.org/10.1080/14029251.2020.1819597
ID  - Zheltukhin2020
ER  -