Classification of a subclass of quasilinear two-dimensional lattices by means of characteristic algebras Текст научной статьи по специальности «Математика»

ISSN 2074-1871 Уфимский математический журнал. Том 11. № 3 (2019). С. 110-132.

УДК 517.9

CLASSIFICATION OF A SUBCLASS OF QUASILINEAR TWO-DIMENSIONAL LATTICES BY MEANS OF CHARACTERISTIC ALGEBRAS

M.N. KUZNETSOVA

Abstract. We consider a classification problem of integrable cases of the Toda type two-dimensional lattices un,xy = f (un+i tV). The function f = f(xi,X2, ••• x5)

is assumed to be analytic in a domain D С C5. The sought function un = un(x, y) depends on real x, у and integer n. Equations with three independent variables are complicated objects for study and especially for classification. It is commonly accepted that for a given equation, the existence of a large class of integrable reductions indicates integrability. Our classification algorithm is based on this observation. We say that a constraint u0 = <p(x, y) defines a degenerate cutting off condition for the lattice if it divides this lattice into two independent semi-infinite lattices defined on the intervals —те < n < 0 and 0 < n < +то, respectively. We call a lattice integrable if there exist cutting off boundary conditions allowing us to reduce the lattice to an infinite number of hyperbolic type systems integrable in the sense of Darboux. Namely, we require that lattice is reduced to a finite system of such kind bv imposing degenerate cutting off conditions at two different points n = N1, n = N2 to arbitrary pair of integers N^ N2. Recall that a system of hyperbolic equations is called Darboux integrable if it admits a complete set of integrals in both characteristic directions. An effective criterion of the Darboux integrabilitv of the system is connected with properties of an associated algebraic structures. More precisely, the characteristic Lie-Rinehart algebras assigned to both characteristic directions have to be of a finite dimension. Since the obtained hyperbolic system is of a very specific form, the characteristic algebras are effectively studied. Here we focus on a subclass of quasilinear lattices of the form

Un,xy — P(^"n-11 Un1 ^га+1)^га,ж + 11 ^n1 ^n+1 )^n,y + Q(^"n—11 ^n1 ^ra+1).

Keywords: two-dimensional lattice, integrable reduction, characteristic Lie algebra, degenerate cutting off condition, Darboux integrable system, ж-integral.

Mathematics Subject Classification: 37K10, 37K30, 37D99

1. Introduction

The problem of classifying integrable two-dimensional chains of the form

un,xy = f {ип+1,ип,ип-1}ип,Х}ип,у )> < n < TO, (1-1)

is topical and currently remains open. The function f = f (x1,x2, ••• x5) is assumed to be analytic in a domain D с C5, and the sought function un = un(x, y) depends on real x, у and integer n.

In this paper we focus on the following subclass of quasilinear lattices (1.1):

un,xy = PÍUv^-^ un,¡ Un-1)Un,x + r{un+l, Un,¡ Un-1 )un,y + q{un+l, Un,) Un-l), то < n < TO.

(1.2)

M.N. Kuznetsova, Classification of a Subclass of Quasilinear Two-Dimensional Lattices by means of Characteristic Algebras.

© Кузнецова M.H. 2019.

Поступила 3 апреля 2019 г.

Here functions p(xi,x2,x3), r(xi,x2,x3), q(xi,x2,x3) are assumed to be analytic in a domain D c C3.

Equations with three independent variables are complicated objects for study and especially for classification. Currently, there are different approaches to studying integrable multidimensional equations [1]—[10]. The presence of a wide class of integrable reductions indicates the integrabilitv of the equation. This fact is often used in the study of multidimensional equations, see [1, 2, 3], where the existence of integrable reductions of a hydrodynamie type is taken to determine the integrabilitv. Here we use a similar idea by treating integrabilitv as the presence of an infinite sequence of Darboux integrable hyperbolic systems.

In describing Darboux integrable systems of hyperbolic equations of a special type, the concept of the characteristic Lie algebra [11, 12, 13] was used a lot. The transition to a more general characteristic Lie-Rinehart algebra opens up new possibilities [14] [18],

The characteristic Lie algebra for two-dimensional lattices was introduced in [19], Namely, the structure of this algebra was described for two-dimensional Toda lattice. It was observed in paper [16] that any integrable lattice of the form (1,1) admits a so-called degenerate cutting off boundary conditions. When such kind boundary conditions are imposed at two different points n = Ni and n = N2, the lattice reduces to a Darboux integrable system of the hyperbolic type equations. In our works [16], [17], [18], we suggested and developed a classification algorithm based on this observation. Let us briefly discuss the essence of the method.

We say that a constraint

uo = ^{x,y)

defines a degenerate cutting off condition for lattice (1,1) if it divides (1,1) into two independent semi-infinite lattices defined on the intervals < n < ^d 0 < n < respectively.

Definition 1.1. Lattice (1,1) is called integrable if there exist functions <p1 and such, that for any pair of integers Ni7 N2, where N1 < N2 — 1, the hyperbolic system

uNl = ipi (x,y),

un,xy = f {un+i,un,un-i,un,x,un,y), Ni < n < N2,

UN2 = ^2(X,y)

obtained from lattice (1,1) by imposing degenerate boundary conditions is integrable in the sense of Darboux.

Recall that a system of hyperbolic equations is called Darboux integrable if it admits a complete set of integrals in both characteristic directions, see [14], [15], An effective criterion of the Darboux integrabilitv of the system is connected with properties of an associated algebraic structures. More precisely, the characteristic Lie-Rinehart algebras [20, 21] assigned to both characteristic directions have to be of a finite dimension. Since the obtained hyperbolic system is of a very specific form, this allows us to study effectively the characteristic algebras. The method was shown to be effective in our articles [17], [18], A large class of the integrable lattices of form (1,1) was represented in [22], where they were studied in the framework of the symmetry approach. It is remarkable that all equations of this class turned out to be integrable in the sense of Definition 1,1, Another argument in favor of our definition is that the resulting hyperbolic systems admit explicit solutions, which are extended to solutions of the original nonlinear chain. So, the chains integrable in our sense have a very wide class of explicit solutions.

In this article we continue the study initiated in [17], [18], where the integrable in the sense of Definition 1,1 cases of the two-dimensional quasilinear lattices of the form

+ PnU n,x + T'nUn,y

+ In,

(1.3)

were described under the non-degeneracv condition -¡U^ = 0. Here the coefficients depend on three successive variables

an = a(un+i,un,un-i), pn = p(un+i,un,un-i), rn = r(un+i

We mention review [23], where a complete classification of lattices of the form

= f ( ) Jl__d= 0

OUn+i OUn+i

was presented. In our paper [18], we found two new equations of form (1.3), which were integrable in the sense of Definition 1.1. We note that these equations were two-dimensional generalizations of the equations from the list in paper [23].

Now we focus on a particular case (1.2) of lattice (1.3), as an vanishes identically. We suppose that the following conditions are satisfied: at least one of the following derivatives is non-zero:

= 0 or = 0, (1.4)

OUn+i OUn—i

9Pn / n 9Pn ^

= 0 or -= (1.5)

dun+i dun—i

The main result of this paper is as follows.

Theorem 1.1. If chain (1.2), (1.4) is integrable in the sense of Definition 1.1, then by point transformations it is reduced to one of the following forms:

\un—u„-i _ pun+1—un

un,xy = [eUn—Un-1 - eu"+1—u")Un,y, (1.6)

{-Un+i + 2Un - Un—i) Un,y. (1.7)

V>n,xy

Lattices (1.6), (1.7) were known before [22]. Condition (1.5) implies that lattices obtained under classification procedure coincide with these lattices up to the change x o y.

In the next section we describe briefly a theoretical base of the main research method; a detailed explanation was presented in [17, 18].

2. Preliminaries

According Definition 1.1, there exist cutting off conditions at two points that reduce (1.2) to the finite hyperbolic type system:

U—i = <fi,

^n,xy Pn^n,x + + Q_n, 0 ^ ^ ^ ^, (2.1)

uN+i = <P2.

Here pn = p(un—i,un,un+i), rn = r(un—i,Un,Un+i), qn = q(un—i,un,un+i).

We recall that a hyperbolic system of partial differential equations (2.1) is integrable in the sense of Darboux if it admits a complete set of functionally independent ^^d y-integrals (see [14]). A function I depending on finitely many dynamical variables u, ux, uxx,... is called y-integral if it solves the equation DyI = 0 (see [14]), where Dy is the operator of total derivative with respect to variable y and u is a vector with coord mates u0,ui,... ,uN. Since system (2.1) is autonomous, we consider autonomous y-integrals depending at least on one of the dynamical u, u x, u x x, . . .

We suppose that system (2.1) is Darboux integrable and denote by I(u, ux,...) its nontrivial y-integral. By definition, I solves the equation DyI = 0. Let us calculate an action of the

operator Dy on functions of the form I(u, ux,...), It is determined by the rule DyI = YI where

N

V W 9 8 t d \

Y = 1^ Ky fa + f* fa- + f*- fa~ + ■■■) ■

Here fi = PiUi- + r*uiy + q* is the right hand side of lattice (1.2). Therefore, the function I satisfies the equation YI = 0. Coefficients of the equation YI = 0 depend on the variables uiy, whereas a solution I is independent of Hence, I satisfies the system of linear equations:

YI = 0, XjI = 0, j = 0,..., N, (2.2)

where Xi = It follows from (2.2) that the commutator Y* = [X*,Y] = X*Y - YX* of

operators Y and X*, i = 0,1,..., N also annihilates I. In the case of lattice (1.2) operator Y can be represented as:

N

y = ^ u*,y Y + R, (2.3)

i=0

where Y¡ and R are defined as

d d d Yt = — + Xi(fi) — + Xl(Dxfi) --+

OUi

d d . , s ®

dm + ri + + + - (2.4)

N d d R = ^{ui,xPi + q*)--+ (Dx(uitxpi + q*) + ('u^p* + q*)n) ---I----

• „ OUixx OUi.xx

i=0 ' '

Let F be a ring of locally analytical functions of the dynamical variables u, ux, uxx,____We

consider the Lie-Rinehart algebra C(y, N) over the ring F generated by differential operators Y,Y0,Yx,... ,Yn. We call this algebra the characteristic Lie algebra of system (2.1) along the y-direetion. We shall show that we can multiply the elements in the algebra by functions depending on finitely many dynamical variables; this fact distinguishes our algebra from an ordinary Lie algebra. The characteristic Lie algebra of system (2.1) along the ^-direction is defined in the same way.

Now we shall work with the operators in the algebra C(y,N). Algebra C(y,N) is of a finite dimension if there exist a finite basis Z\,Z2,...,Zk consisting of linearly independent operators such that each element Z G C(y,N) is represented as a linear combination Z = a\Z\ + a2Z2 + ... + akZkl where the coefficients a^ a2, ..,, ak are analytic functions depending on the dynamical variables defined in an open domain. Then the identity a\Zi + a2Z2 + ... + akZk = 0 implies that a\ = a2 = ... = ak = 0. System (2.1) is integrable in the sense of Darboux if and only if the characteristic Lie algebras in both directions are of a finite dimension [14].

In our study, we shall apply the operator Dx to smooth functions depending on the dynamical

variables u, ux, uxx,____ On this class of functions, we obtain the following commutation

relations for the operators y*, R:

[Dx, R] = (uitxPi + qi)Yi The following statement holds [13, 19, 14]:

[Dx,Y] = -rlYl, (2.5)

N

'Ui

i=0

Lemma 2.1. If a vector field

d d

^ = Zl^ --+Z2,i--+'''

solves the equation [DX, Z] = 0 then Z = 0. We shall also use the standard notation adx(Z) := [X, Z],

The key method, on which the classification algorithm is based, is a test sequence method. We call a sequence of operators W0, W\, W2,... in the algebra C(y, N) a test sequence if

m

[Dx,Wm] = Wj,mWi j=o

holds true for all m. The test sequence allows us to derive integrabilitv conditions for hyperbolic type system (2,1),

see [24, 14, 15].

The first step of our study is to define the functions pn, rn. Let us note that when we search the function rn we study the subalgebra Lie generated by the operators Yi, see (2,4), It follows from (2,3), (2,4), (2,5) that this subalgebra coincides with the Lie algebra of a hyperbolic type system corresponding the lattice

Un,xy Tn(Un+\,Un,Un—i)Un,y. (^-6)

The following statement holds true for this lattice.

Lemma 2.2. If lattice (2,6) is integrable in the sense of Definition 1.1, then it is reduced by point transformations to one of the following forms:

= pun-un-1 _ pun+1-un\7 /n 7\

Ujn,xy C- J Ujn,y1 1J

^n,xy ( Un+1 + 2Un Un-1) Un,y. (2-8)

Proof of Lemma (2,2) is given in Section 3,

Remark 2.1. If the function rn depends only on the variable un, that is rn = rn(un), then

[Yk, Yj ]=0

for all k, j and system {2.Q) splits into the system of independent equations un,xy = rn(un)un>y. This system has integrals in the direction we consider. We mention that a wide class of scalar equations of the form ux,y = f (u,ux,ux) was studied in [14] within the characteristic Lie algebras approach. But the case rn = rn(un) or pn = pn(un) holds for lattice (1,2) and is to be studied, see Section 4-

3. Integrability conditions

3.1. The first test sequence. Let us define a sequence of operators in the characteristic algebra C(y,N) by the reeeurent formula:

Fc, ^, Wt = \Yo,Yl] , ^V2 =Vo,W1] , ... Wk+1 = [Yo,Wk] , ... (3.1)

The following commutation relations are valid for the first elements of the sequence (3,1), see formula (2,5):

[Dx,Yc] = -rcYc, [Dx,Yi] = -nYi. (3.2)

(3.5)

(3.6)

By using the Jaeobi identity we get the formulae

[Dx, W-] = -(n + r0)W- - Y0(r1)Y1 + Yi(rQ)Y0, (3.3)

[Dx, W2] = -(n + 2r0)W2 - Y0(2n + r0)Wi - Y02(n)Yi + Y0Yi(r0)Y0, (3.4)

[Dx, W3] = - (n + 3r0)W3 - Yq(3n + 3r0)W2 - Y02(3n + r0)Wi

- Y3(n)Yi + Y2YI(T0)Y0, [Dx, W4] = - (n + 4r0)W4 - Y0(4n + 6r0)W3 - Y02(6n + )W2

- Y03(4n + r0)Wi - Y04(n)Yi + Y^Yi(n)Y0. It can be proved by induction that (3.1) is a test sequence. Moreover, for к ^ 4

[Dx, Wk ] = ak Wk + bk Wk-i + sk Wk-2 + tkWk-3 + • • • , (3.7)

where

к — к2

ак = -(п + кг0 ), Ък = —— Yq (г0 ) - Yo (п)к, (3.8)

Sk = -Y02(3n + го) + ^(к - 3)Y0(дз + qk-i),

tk = -Y03(4n + го) + ^(к - 4)Y0(S4 + s—).

By assumption, in the algebra c(y, N) there are finitely many linearly independent elements of sequence (3.1). Therefore, there exists M such that

WM = XWM-i + • • • , (3.9)

the operators Y0, Yb W^ ..., WM-1 are linearly independent, the dots stand for a linear combination of the operators Y0, Yb W1, ..., Wm-2-Let us consider the first three elements.

Lemma 3.1. If condition (1.4) holds, then the operators Y0,Y1,W1 are linear independent. Otherwise, if ro = го(щ) depends only on the variable щ, then W1 = 0.

Доказательство. Let r0 depend on at least one of the variables u-1l u1. We are going to prove that Y0, Yb W1 are linear independent in this case. We argue by contradiction assuming that

XiWi + piYi + VoYo = 0. The operators Y0, Y1 are of the form

* = ^ + •••, я = ^ + •••,

ОЩ ou1

while W1 contains terms of the form J^ and Hence, the coefficients are equal to

zero. If \1 = 0, then W1 = 0. We apply the operator adox to both sides of the last identity, then by virtue of (3.2) we obtain the equation

^o(ri)Yi - Yi(ro)Yo = 0.

It implies that Y0(ri) = r1,U0 = 0 and Y1(ro) = r0,U1 = 0, This is equivalent to Го,u-1 = 0, ^o,U1 = 0 and we arrive at a contradiction to condition (1.4). By direct calculation of the operator

Wi = [Y0,Yi] = Y0Yi - YiY0

and using formula (2.4), we prove the second part of the lemma. The proof is complete. □

In what follows we assume that M ^ 2 and condition (1.4) holds.

Lemma 3.2. If relation (3,9) holds true for M ^ 2, then the function r0 has one of the following forms:

i) if X = 0, then

2

ro(ui,Uo,U-i] = a(u-i) - M _ iа(щ) + 6(щ); (3.10)

ii) if X = 0, then

r0 (ui,u0,u-i) = ft (и-1)е~ «(ii-!)Xu0 + ф(и0,щ), (3.11)

where functions ft and ф satisfy the equation

1 2

Хф(ио,щ) + 2M(M - 1)фи0(uo,ui) + Me-AU1ft'(щ) = 0. (3.12)

Доказательство. We apply the operator ad dx to both sides of identity (3.9). Combining the coefficients before WM-1, we get the equation:

Dx(X) = Х(ам - ам-i) + Ьм. (3.13)

We substitute formulae (3.8) into (3.13):

M(M - 1)

Dx(X) = -raX--^-^ra,U0 - Mri,Uo. (3.14)

From identity (3.14) it follows that Л is a constant and

л M(M - 1)

ra X + ^^-¿ ra,uo + Mn,U0 = 0. (3.15)

Let us apply the operator -JU^ to (3.15):

Mri,u0U2 = 0. This is equivalent to r0,U-1U1 = 0 and, hence,

r0(ui,u0, u-i) = ip(u-i,uo) + ф(и0,и\). (3.16)

We substitute function (3.15) into (3.14)

M(M - 1) ~2

We consider two different cases:

i) A = 0;

ii) A = 0.

If i) holds, then pU0U-1 = 0, so that ip(u-i,u0) = a(u-i) + ft(u0) and

ro (ui,ua,u-i) = a(u-i) + ft (uo) + ф(щ,щ). We re-denote ft + ф ^ ф and we get

r0(ui,u0,u-i) = a(u-i) + ф(и0,щ). (3.18)

We substitute (3.18) and A = 0 into (3.15):

M(M2 - ^фио(u0,ui) + Ma'(щ) = 0. (3.19)

Applying the operator J^ to identity (3.19), we obtain ф-и^ = 0 and, hence,

ф(uo,Ul) = 7 (U0) + S(ui). We substitute ф into (3.19) and we find

1 M = - M2_ a(U0) + Ci

±V± (±V± — 1) . .

X'^U-1 +--Z-^U0U-1 = 0. (3.17)

and, then

2

го(щ,щ,и-1) = a(u-i) - ——iа(щ) + 8(щ).

Let us consider case ii). Solution of equation (3,17) is the function

^(u-1,u0) = a(uo) + e Mw-1)Xu°ß(u-1).

Then function (3,16) becomes

r0(ul ,uQ,u-i) = a(u0) + e-M ^-1) Xu° ft (U-l) + ^(u0,ul). We redenote a + ^ ^ ^ and we get

r0 (ul,u0,u-l) = e- Mpf-!) Xu° ft (U-l) + ^(Uo,ul). Substituting r0 into (3,15), we obtain (3,12), The proof is complete, □

3.2. Second test sequence. We construct the test sequence containing operators Y0, Y2 and their multiple commutators:

Zq = Yo, Zl = Yl, Z2 = Y2, Z'3 = [Y^lo] , Z4 = \Y2,Y\] , Zh = [Ï2, Zs] , z6 = [Xl, Z3] , Z7 = [Yl,Z4] , Z8 = [Yl,Z5].

The elements Zmj m > 8 are defined by the recurrent formula Zm = [Yl, Zm-3], The following commutation relations hold:

[Dx ,YQ] = -roYo, [Dx,Yl] = -nYu [DX,Y2] = -^Y2, (3.20)

[Dx, Z3] = -(n + ro)Z3 + Yo(n)Yl - Yl(ro)Yo, (3.21)

[Dx, Z4] = -(T2 + n)Z4 + YI(T2)Y2 - Y2(n)Yl, (3.22) [Dx, Z5] = - (ro + n + r2)Z5 - Y2(n + ro)Zs + Yo(n)Z^

+ Y2Yo(n)Yl - Y2Yl(ro)Yo, [Dx, Z6] = -(ro + 2n)Z6 - Yl(2ro + n)Zs + YlYo(n)Yl - Y?(ro)Yo, [Dx, Z7] = -(2n + T2)Z7 - Y,(n + 2r2)Z4 + Y?(r2)Y2 - Y^2(n)YU

[Dx, Z8] = - (ro + 2n + r2)Z8 + Yo(n)Z7 - Y2(TQ + n- ^(ro + n + r^Z* + YlYo(rl)Z4 - YY2(n)Zs + Y1Y2Yq(t1 )Y - YX2Y(tq)Yq.

(3.23)

(3.24)

We recall that we assume condition (1.4), otherwise, starting with Z3, all elements of the sequence vanish.

Lemma 3.3. The operators Z0, Z\, ..., Z5 are linearly independent.

Доказательство. It is easy to show that the operators Z0, ,,,, Z4 are linearly independent; this is similar to the proof of Lemma 3,1, We prove Lemma 3,3 by arguing by contradiction. We suppose that

4

Z5 = ^ A,Zj. (3.25)

3=0

We apply the operator adox to both sides of identity (3,25), and we use formulae (3,21)-(3,23) to simplify the obtained identity:

4

j=c

4

-(rc + T'i + r2) Y, ^Z, + Yc(n)Z4 - Y2(ri)Z3 + Y2Yc(n)Yi - Y2Yi(rc)Yc

- ^a(u-i) - ^ - i a(uc) + 8(uiA4 + a'(uc) = 0. (3.28)

= J] Dx(\J)Zj + A4-(T2 + n)Z4 + Yi(r2)Y2 - Y2(n)Yi) (3.26)

j=c

+ \s{-(ri + rc)Z3 + Yc(n)Yi - Yi(rc)Yc) - \2r2Y - 2 - XiriYi - XcrcYc. Combining the coefficients at Z4 in (3.26), we get the equation:

Dx(X4) = -rcX4 + ri,uo.

This identity implies that X4 is a constant and

- rcX4 + ri,U0 = 0. (3.27)

We shall study this equation in two different cases i) and ii):

i) If rc is defined by formula (3.10), then (3.27) casts into the form

2

M- l

If X4 = 0, then (3.28) implies that functions a, S are constants since the variables u^ uc, u-i are independent. Then we get that rc is a constant that contradicts eondition (1.4). If X4 = 0, then it follows from (3.28) that a'(uc) = 0 and

rc(ui,uc,u-i) = i(wi). (3.29)

ii) If rc is defined by (3.11), then identity (3.27) becomes

- (p(u-i)e~M(M+ ^(Uc,ui)^ X4 + ft'(uc)e-^W-1)Xui = 0. (3.30)

We apply the operator to (3.30):

f3'(u-i)e-Auo A4 = 0.

If X4 = 0, then it follows from (3.30) that /3'(uc) = 0 and, hence, ft = c4, where c4 is a constant. If X4 = 0, then it follows from (3.30) that

__2_

ft(u-i)e M(M-1)A«o + ^(uc,ui) = 0.

The expression in the left hand side of the last identity coincides exactly with rc(ui,uc,u-i). Therefore, the last identity contradicts condition (1.4).

Thus, we obtain that X4 = 0 and rc is defined by the formula

rc(ui,uc ,u-i) = c4e~ rnM-1 Xuo + ^(uc,ui). (3.31)

We collect the coefficients at Z3 in (3,26), take into consideration that X4 = 0, and we obtain the equation

Dx(X3) = -T2X3 - ri,U2.

Hence, A3 is a constant, and

T2 X3 + ri,U2 = 0. Applying the shift operator, we get the equation

ri X3 + rc,ui = 0. (3.32)

(ffl \ уYi(n)+ Yi(ra + Г2)) ZSm-i + ••

i) Let us substitute the function r0 defined by formula (3,29) into (3,32):

5(щ)\3 + 5'(ui) = 0.

A simple analysis of the last equation gives the contradiction to condition (1.4).

ii) Let us substitute the function r0 defined by formula (3,31) into (3,32):

(c4e-+ ф(и1,и2)^ X3 + фи1 (ио,щ) = 0. (3.33)

We apply the operator ^ to both sides of identity (3.33) фи2 (ui,u2)X3 = 0. Studying (3.33) in this case, we arrive to contradiction to condition (1.4).

Otherwise, if A3 = 0, then the expression in the left hand side of identity (3.33), coinciding with r1, is equal to zero. Thus, we obtain the contradiction to condition (1.4). The proof is complete. □

For further purposes, it is convenient to divide sequence (3.26) into three subsequences {Z3m},

{z3m+1} {^3m+2 }

Lemma 3.4. Operator adDx acts on sequence (3.26) according the following formulae:

i ^^ — "ffl? \ [Dx, Z;im] = -(Го + mri)Z;im + ( -2-Yi(ri) - mYi(ro)j Z:im-3 +----,

/ - r^2 \

[Dx, Z;im+i] = (r2 + mri)Z;im+i + ( -2-Yi(.ri) - mYi(r2)j Z:im-2 +----,

[Dx, Z;im+2] = -(Го + mri + r2)Z3m+2 + Yo(ri)Z3m+i - Y2(ri)Z;im

m, ~2

Lemma 3.4 can be easily proved by induction. Theorem 3.1. Assume that Z3k+2 is a linear combination

Z3k+2 = Xk Z3k+i + ^k Z3k + Vk Z3k-i + ••• (3.34)

of the previous terms in sequence (3.26) and none of the operators Z3j+2 for j < к is a linear combination of operators Zs wit h s < 3j + 2. Then the coeffi cient Uk satisfies the equation

Dx(uk) = -nuk - ^tlAYi(n) - (k - 1)Yi(ro + Г2). (3.35)

Lemma 3.5. Suppose that the assumptions of Theorem 3.1 are satisfied and the operator Z3k (the opera tor Z3k+i) is linearly expressed in terms of th e operators Zi7 i < 3k. Then in this decomposition the coefficient at Z3k-i vanishes.

Доказательство. We argue by contradiction. Suppose that

Z3k = XZ3k-i +--------(3.36)

and A = 0. We apply the operator adDx to both sides of identity (3.36). Using formulae from Lemma 3.4, we get

-(го + kri)XZ3k-i +----= Dx(X)Z3k-i - Х(го + (k - 1)ri + r2)Z3k-i +----

Collecting coefficients at Z3k-i, we obtain

Dx(X) = X(r2 - r\)k.

This equation implies that A is a constant and X(r2 - ri)k = 0, Then r2 = ri = const that contradictions condition (1.4), The proof is complete. □

In order to prove Theorem 3,1, we apply the operator adox to both sides of the identity (3,34), Then we simplify a obtained identity using formulae from Lemma 3,4, Collecting coefficients at Z3k-i, we obtain equation (3,35),

The next step of our work is studying equation (3,35) as r0 is defined by formulae (3,10) or (3,11) under condition (1,4) and for M ^ 2, k ^ 2. We find exact values of coefficients in equation (3,35) and substitute them into (3,35):

Dx("k) = -riUk - k(k2 ^ r'l,ui - (k - 1)(ro,ui + r2,ui). This equation implies that vk is a constant and, hence,

Vkri + k(k2 ^ r'l'Ui + (k - 1)(ro,ui + r2,ui) = 0. (3.37)

Lemma 3.6. If relations (3,9), (3,34) hold true for some M ^ 2, k ^ 2, and condition (1,4) holds true, then

i) if X = 0 uk = 0, then

rn(un+i,Un,Un-i) = a(un-i) - M2__ 1 a(un) + ( ^ - a(u,n+i) + ci; (3.38)

ii) ifx = 0, vk = 0, k = 2, M = 3, then

rn(un+i,un, Un-i) = eu"-hu"-1 + ceu«+1-hu"; (3.39)

Hi) if X = 0, uk = 0, then rn is defined by formula (3.39).

The proof of this lemma is rather complicated and is presented in Appendix. We proceed to relations (3.9), (3.34). We need another one test sequence:

Fq, Yi, Wi = [Yi,Yo], W2 = [Yi,Wi] ,...Wk+i = [Yi,Wk] ....

The following commutation relation hold:

[Dx,Wi] = -(n + ro)Wi + YQ(n)Yi - YI(TQ)YQ, (3.40)

[Dx,W2] = -(2n + ro)W2 - Yi(n + 2ro)Wi + Y^n )Yi - Y^YQ, (3.41) [Dx,W3] = - (3n + Tq)W3 - Y1 (3n + 3ro)W2 - Y?(n + 3ro)Wi + Yi2YQ(n)Yi - Yi3(ro)YQ,

[Dx,W4] = - (4n + ro)W4 - Yi(6n + 4ro)W3 - Y?(4n + 6T-Q)W2 - Y?(n + 4ro)Wi + Y?YQ(Ti)Y, - Y4(TQ)YQ. It is easy to prove that

[Dx,Wk ] = akWk + bkWk-i + SkWk-2 + ••• , (3.44)

for k ^ 3, where

- k — k2

ak = -(kn + TQ), bk = —2—Yl(ri) - Yl(ro)k, Sk = -Yi(r\ + 3ro) + ^(k - 3)Yi(q3 + q^i).

We observe that the first terms Yq, Yi, Wi = - W\ obey Lemma 3.1.

We suppose that c(y,N) is finitely-dimensional, that is, each sequence of its elements terminates at some step. Consequently, there exists N such that:

WN = AWn-i + ••• , (3.45)

where the operators Yq, Yi, Wi, ..., WN-i are linearly independent, and the dots stand for linear combination of the operators Yq, Yi, Wi, ..., WN-2.

(3.42)

(3.43)

3.3. Case M = 2. Suppose that relation (3,9) holds true for M = 2:

W2 = \Wi + eY1 + i]Yo. (3.46)

We apply the operator adDx to both sides of identity (3,46) and we get: -(n + 2ro](XWi + eYi + VYo) - Y0(2n + rQ)Wi - Y2(n)Yi + YoYi(ro)Yo

= X(-(n + r0)Wi - Y0(n)Yi + Yi(r0)Y0) - eriYi - Vr0Y0. Collecting the coefficients at independent operators W^ Y^ Y0, we obtain the system

r0X + 2ri>uo + r0 ,uo = 0, (3.47)

2r0e + ri,UQU0 - Xri,U0 = 0, (3.48)

-(ri + r0+ rQ,UoUl - Xr0,U1 = 0. (3.49)

3.3.i) Let us consider the case when thefunetion rn is described by formula (3.38) and A = 0. We substitute function (3.38) and A = 0 into system (3,47)-(3,49), Then equation (3.47) becomes identity and we arrive to the system:

2(a(u_0 - 2a(u0) + (k - l)a(u1) + cAe + d ^^) = 0, v ' au2

(■a(u0) + 2a(u1) - (k - l)a(u2) + 2c1 + a(u-1) + (k - rq = 0.

This system yields that

e = rq = 0, a(u0 ) = C1u0 + C2,

ro(u1,uo,u-1) = (k - l)C1u1 - 2C1uo + C1u-1 + C3,

where C3 = -2C2 + kC2 + c1. We will study the lattice corresponding to this function, in Section 3,5,i, see (3.62).

3.3.ii) Let us consider the case when the function rn is described by formula (3.39) and A = 0. System (3,47)-(3,49) casts into the form:

(A + l)eu0-hu-1 + (-ch + Xc - 2h)eu1-huo = 0,

2eeu°-hu-1 + (2ec + h2 + Xh)eu1-hu° = 0,

- veu0-hu-1 + (-v - vc - ch - Xc)eU1 -hU0 - rjceU2-hU1 = 0.

A simple analysis of the last system leads us to the identities A = -1, h =l, c = -1, e = "q = 0. We get that rn has the following form:

rn(un+i,un,un-i) = eUn-Un-1 - eUn+1-Un. (3.50)

And W2 = -Wi, W2 = W1. Now let us substitute (3.50) into (3.37):

yk + 1 k2 - 2k + l] eUl-U0 + (-vk + 2k2 - 2k + l] e"1 = 0,

which implies uk = 0 k = 2.

Thus, relation (3.34) is of the form

Z8 = pZ4 + 0Z3 + tZ2 + ^Zi + (3.51)

and

Z6 = pi, [Yi,Y0]] = W2 = W1 = [Yi,Y0] = Z3,

Z7 = [Yi, [X2,Yi]] = Dn [Y0, \YI,YQ]] = -DnW2 = DnWi = -Z4.

Commutation relation (3,24) become

[Dx, Z8] = - (rc + 2n + r-2)Z8 - Yi(rc + ri + n)Z5 + (YiYc(n) - Yc(n))Z4

- (Y2(rc + ri) + YiY2(ri))Z3 + YiY2Yc(ri)Yi - YiY2Yi(rc)Yc.

We apply the operator adDx to both sides of identity (3,51) and take into consideration the formulae (3,20)-(3,23), (3,52), then we collect coefficients at independent operators Z4, Z3, Z2, Zi, Zq.

- (euo-u-1 - eu2-ui)p = 0, (-eu3-u2 + eui-uo)a = 0,

- (euo-u-1 + eui-uo - 2eu2-ui)t + peu2-ui = 0,

- (euo-u_i - ea3-u2)^ - peu2-ui + aeui-uo = 0.

- (2eui-uo - eu2-ui - eu3-u2)n - aeui-uo = 0. It is clear that p = a = t = fi = n = 0 Hence, Z8 = 0.

3.4. Case M = 3. Suppose that relation (3,9) holds true for M = 3:

W3 = XW2 + pWi + eYi + rjYc. (3.53)

We apply the operator adDx to both sides of identity (3,53) and use formulae (3,2), (3,3), (3,4), (3,5), Collecting coefficients at the independent operators, we obtain the system

rc\ + 3ri,uo + 3rc,uo = 0, (3.54)

- 2rcP + X(2ri,uo + rc,uo) - 3ri,uouo - rc,uouo = 0, (3.55)

- 3r'cs + Xri,uouo + Pri,uo - Ti,uououo = 0,

- (ri + 2rc)v - Xrc,uoui - prc,ui + rc,uououi = 0. (3.57) 3,4,i) Let us consider case (3.38), A = 0. It follows from equations (3,54)-(3,57) that

k - 2

a(un) = Ciun + C2, rn(un+i,un,un-i) = —2— Ciun+i - Ciun + Ciun-i + C3,

where C3 = 2C2k - C2 + c^ Further study of the lattice with rn defined by this formula is provided in 3,5,i, see (3,62),

3,4,ii) Let us consider case (3,39) A = 0. We substitute rn into (3,47)-(3,49), Studying this system, we obtain that A = -3 p = -2, £ = "q = 0 h = 1, c = -The function (3,39) becomes

rn(un+i,un,un-i) = eUn-Un-1 - 2eu"+i-u".

We substitute this function into equation (3.37) and we get k = 1 or k = |. These identities contradict condition k ^ 2.

3.5. Case M > 3. Let the following relation be true for M > 3

WM = XWM -i + pWM —2 + kwm-3 +■■■ (3.58)

Taking into account formula (3.7), we apply the operator adDx to both sides of the above identity:

aM (xwm-i + pwm-2 + kwm-3 +----) + bM wm-i + sm wm-2 + tM wm-3 +----

=X(aM-iwm-i + bM-Iwm-2 + sm-i wm-3 +----)

+ p(a,M-2wm-2 + bM-2wm-3 +----) + K(aM-3^m-3 +----)

We collect coefficients at the independent operators WMWM-2, WM-3:

X(aM - aM-i) + bM = 0, (3.59)

p(aM - aM-2) + sm - XbM-i = 0, (3.60)

n(aM - aM-3) + tM - xsm-i - pbM-2 = 0. (3.61)

3,5,i) By system (3.59)-(3.61) we obtain that a(un) = Ciun + C2 and

, k - (M - l) 2Ci

rn(un+i,un,un-i) = —M _ l— Ciun+i - M _ lun + CiUn-i + C3,

where

_ciM - ci - 2C2 + kC2

= W-l .

Now we consider the function

rn(un+i,un, un-i) = CiUn+i + C2Un + c3un-i + C4. (3.62)

Commutation relations (3,7), (3,44) become

[Dx, Wk ] = ak Wk + bk Wk-i, [Dx,Wk ] = akWk + bkWk-i, (3.63)

where

k — k2

ak = -(ri + kr0), bk = —2— ^2 - c3k, (3.64)

- k — k2

ak = -(kri + r0), bk = —2— ^2 - cik. (3.65)

Assume that sequence {Wn} is terminated at the step M\

m -i

WM = ^ AM-kWM-k + faiYi + faY0. (3.66)

k=i

We apply the operator adDx to both sides of identity (3.66)

m-1 \

aM I -kWm-k + + ) + bMWm-1

k=i J

m-2

Am-k(aM-kWm-k + bM-kWm-k-i)

k=i

+ ki(-(n + rQ)Wi - c3Yi + ciY0) - far-iYi - forQYQ. We collect the coefficients at WM-i in this identity:

AM-i(a>M - -i) + bM =

We substitute formulae (3.64),(3.65) into the last equation:

f \ M - M 2

-AM-i [ciUi + C2U0 + c3u-i + d) +--2-c2 - = 0.

A simple analysis of this equation shows that AM-i = 0 and c3 = c2. Then, collecting coefficients before Wm-k, k = 2,... ,M - 2, we arrive at the equations

AM-k (aM - aM-k ) = 0, k = 2,..., M - 2,

which implies AM-k = 0 k = 2,... ,M - 2. The coefficient at Wi is Ai(aM + t1 + r0 ) = 0. Then Ai = 0. The coefficient s at Yi and Y0 read as (aM + ri = 0, (aM + Tç, = 0 and hence, = fa = 0. Thus, WM = 0-

Similarly, if sequence {Wk} is terminated at step N, then ci = c2 and Wn = 0- As a result, we obtain:

, l - N l - M

rn(un+i,un, Un-i) = -2-C2Un+i + C2Un +--2-C2Un-i + C4.

Bv rescaling Ui ^ Vi, the original lattice is reduced to a lattice of the same form with function rn defined by the formula

(Un+i ) = (l - N)un+i + 2un + (l - M)un-i + c,

where c is an arbitrary const ant. If 4 — M — N = 0, then we exclude constant c by the shift transformation u ^ u — 4_^_N- If M + N = 4, then M = N = ^^d c is excluded by the transformation un ^ un + |n2. Thus, the function rn becomes:

(Un+1 , ^ n , ^ n— 1 ) = (1 — N )un+i + 2un + (1 — M )un—i (3.67)

and, in particular,

Tn (Un+1,Un,Un—l) = —Un+1 + 2Un — Un—1. (3.68)

We substitute (3.67) into (3.37), and we get that k = M + N — 2. We substitute (3.68) into (3.37), and we get that k = 2.

Let us consider lattice (2.6) when rn is defined by (3.67). We impose cut-off conditions u0 = 0, uL+1 = 0 and we reduce this lattice to the following hyperbolic system:

ui,xy = (2u1 + pu2)u1,y,

Uk,xy = (quk—1 + 2uk + puk+1)uk,y, 2 ^ k ^ L — 1, (3.69)

UL,xy = (quL—1 + 2uL)uL}y,

where p = 1 — N, q = 1 — M; we recall that N > 1, M > 1. This system is reduced by differential substitution Vi = ln Ui,y to the exponential system:

vhxy = 2evi + pev2,

vk,xy = qeVk-1 + 2eVk + peVk+1, 2 ^ k ^ L — 1, (3.70)

VL,xy = qeVL-1 + 2e"L.

vxy = Ae

We denote by A the matrix of coefficients before exponents in the right hand side of the system and we denote by v = (v1,v2,..., vK)T, ev = (eV1 ,e"2,..., e"K)T the column vectors. System (3.70) is related with the system

wl>xy = e2wi+pw2,

wk>xy = e«wk-i+2wk+Pwk+i, 2 ^ k ^ L — 1, (3.71)

H)T = p^wL-1+2wL UJL,xy — c

wxy 6

by the following point change of variables

v1 = 2w1 + pw2,

vk = qwk—1 + 2wk + pwk+1, 2 ^ k ^ L — 1, (3.72)

vl = qwL—1 + 2Wl. v = Aw ).

System (3.71) is reduced to system (3.69) by differential substitution

Ui = Wi,x. (3.73)

It is shown in [11, 25] (see also [14]) that if A is the Cartan matrix of a simple Lie algebra, then the system (3.70) ((3.71)) is integrated in quadratures. Comparing the Cartan matrix and matrix A, one can see that p = q = — 1 Thus, we have that M = N = 2. In this case we find that W2 = 0 W2 = 0 Zg = 0.

Let us show that if systems (3.70), (3.71) is integrable in the sense of Darboux then system (3.69) is integrable in the sense of Darboux, too. Suppose that I(wx, wxx, • • •) is an y-integral of system (3.71). We change variables by the rule wi,x = u^ wi,xx = uix and so on, due to

(3,73), then we obtain an y-integral I(u, ux, ■ ■ ■) of system (3.69). Assume that I( is an ^-integral of system (3,70), Using (3,73) and (3,72), we derive:

u = wx = A-lvx.

Hence, by virtue (3,70)

y, "yyj

Uy — A vxy — A Ae — g .

We change variables in the function I(vy, vyy,...) by the rule vi = ln Ui,y, Vi,y = (ln Ui,y)y and so on. Thus, we get an ^-integral I ((ln uiy )y, (ln ui,y )yy,...) of system (3,69),

3,5,ii) Let us consider case (3,39) and A = 0, We substitute rn into system (3,59)-(3,61) and into equation (3,37), we get the following system:

AieU0 - -hu- 1 + Bieui-huo = 0,

A2eu0 - -hu- 1 + B2eui-huo = 0,

A3euo - -hu- 1 + B3eui-huo = 0,

AieUl - -huo + B4eu2-hui = 0.

Obviously, the coefficients Au Bi at independent exponent functions have to be equal to zero. Thus, we obtain a system of 8 algebraic equations in 8 unknowns c, h, M, X, p, k, k, uk. Studying this system, we get the following possible variants:

M = 4, k = 10; M = 5, k = —; M = 2, k = 2. 3 ' 4 '

All of these variants contradict our assumptions about values of k, M.

Thus, we have proved the following statement.

Lemma 3.7. If relations (3.9), (3.34), (3.45) hold true for some M ^ 2, k ^ 2, N ^ 2, then the function rn casts into one of the forms (3.50) or (3.68) up to point transformations.

Lemma 2.2 is implied immediately by Lemma 3.7.

Summarizing the rezults of this section, we observe that we have lattice (1.2) for further study, where ther function rn is defined by one of the formulae rn = rn(un), (3.50), (3.68). Similarly, function pn is defined by one of the following formulae:

Pn Pn(U'n) ,

Pn(un+!,un,un-i) = eUn-Un-1 - eUn+1-Un,

Pn(Un+l,Un,Un-\) = Un+1 + 2Un — Un-\.

4. Function qn

We recall that the operator Y can be represented as follows, see formula (2.3):

y = ^ ui y Yi + R,

where

d d d Yi = to- + ^6kT + + £kT~ + • •

Ct^i Ct>i * ^ Ct¡i *X

d d

R = ^^(ui'xlpi + qi) A--+ (Dx(ui,xPi + qi) + (Ui,*pi + qi)ri) ---1----

OUi,* UUi,*x

We shall determine the function qn by using the operator R. We define a sequence of operators in the characteristic algebra C(y, N) by the following recurrent formula:

Y-i, Yo, Y, Yo,-i = [Yo,Y-i] , Yo =[Y,Yo] , (4.1)

Ro =[Yo,R], Ri = [Yo,Ro], R2 =[Yo,Ri], ... Rk+i = [Yo,Rk].

)

For elements of the sequence the following commutation relations hold:

[DX,Y-1] = -Г-iY-i, [DX,Y0 ] = -r0Y0, [DX,Y1] = -rlYl, [DX,Y0-1] = -(r-i + rQ)Y0-1 - Y0(r-i)Y-i + Y-i(r0)Y0, [Dx,Ylfl] = -(r0 + r1)Yh0 - Y1(r0)Y0 + Y0(n)Yu

[DX,R] = hiYi, hi = piUitX + qi,

г

[Dx, Rq] = - r0R0 + h1Ylfl - h-{Y0—

- Y0(h1)Y1 + (R(r0) - Y0(h0))Y0 - Y0(h-1)Y-1, [DX,R1] = -2rQR1 - Y0 (r0 )R0 + ••• , [Dx, Щ = -3rQR2 - 3Y0 (r0 )R1 - Y02(r0 )R0 + ••• , [Dx, R3] = -4r0R3 - 6Y0 (r0 )R2 - 4Y02(r0 )R1 - Y03(r0 )R0 + ••• ,

where the dots stand for a linear combinations of operators Y1}0, Y0 —1, Уь y0, Y-1. By induction we prove that the following formula holds for all n ^ 2:

Rn] 1 + • • • ,

where

n2 + n

an = - in + 1)r0, bn =--2— ^o ),

and the dots stand for a linear combinations of the operators Rk, к < n-1, Y1i0,Yo—1 ,Y1,Y0, Y—1.

Lemma 4.1. If the operator R0 is linearly expressed in terms of operators (4,1)

До = pYifl + fiY0-1 + vYx + I]YQ + eY-1, (4.2)

then chain (1,2) is reduced to one of forms (2,7), (2,8) by point transformations.

Доказательство. We apply the operator adox to both sides of identity (4,2), Collecting the coefficients at independent operators Yip, Y0—1, Уь Y0, Y—1, we get the system of equations

Dx(y) = ny + h1, (4,3)

Dx(p) = г-ф - h-1, (4.4)

Dx(u) = (n - rQ)u - Y0(h1) - ^Y0(n), (4.5)

DX(V) = R(r0) - Y0(h0) + №(г0) - p,Y-1(r0), (4.6)

Dx(e) = (r-1 - r0)e - Y0(h-1) + jlY0(r-1). (4.7) We consider equation (4.3):

Г1(и2,Щ,и0)rf + P1(U2 ,щ,щ )uhx + q1(u2,U1,u0) = Dx(p).

A simple analysis of this equation shows that ц = ц(и1) and, hence, this equation splits into two equations

rf (щ) = P1(U2,U1,U0), Г1 (U2,U1,UQ )y(u1) + q1(u2,u1 ,щ) = 0.

Hence,

Pn(un+1,un,un-1) = rf'(un), qn(un+1,un,un-1) = -rn(un+1,un,un-1)rf(un). (4.8)

Using equation (4.4), we obtain that Д = jj,(u-1), fi(v) = We simplify identity (4.5) using (4.8) and we get

Dx(P) = (Г1 - r0)v.

It easy to see that v = 0. Similarly, it follows from (4.7) that e = 0. We simplify identity (4.6) as follows:

Dx(v) = -Po,UqЩ,x - ®,ua - ropo + ^rotUl - jlrotU_1.

A simple analysis of this equation shows that ^ = r](u0) and, hence, this equation splits into two equations

г/'(щ) = Po,uo, -Qo,uo - Г0Р0 + pro,Ul - pr0,u_1 = 0. (4,9)

We substitute formulae (4,8) into identities (4,9) and we obtain r/'(u0) = —p"(u0),

ro,ua p(uo) + )ro,ui + p(u-i)ro,u_i = 0. (4.10)

We substitute the function rn defined by formula (3.50) into (4.10), and we get that p = с is an arbitrary constant. Therefore, pn = 0 Qn = —crn, and lattice (1.2) becomes

„. _ ( „un-u„_1 _ pUn+1-Un )„. _ „(pun-un_1 _ pUn+1-Un )

u>n,xy (с с ) Un,y ь(с с ).

The transformation un — су ^ un reduces this lattice to (2.7).

If rn is defined by (3.68), then (4.10) implies p = c, where с is an arbitrary constant. Hence, pn = 0 Qn = —crn, and lattice (1.2) takes the following form:

Un,xy = ( Un+1 + 2Un Un-l)Un,y + C( — Un+1 + 2Un — Un-l).

The transformation un — су ^ un reduces this lattice to (2.8).

If rn = rn(un), then it follows from (4.10) that p = 0 or r0,UQ = 0. In the first case formulae (4.8) imply pn = 0 Qn = 0. Then chain (1.2) becomes un,xy = rn(un)uny. In the second case r0 = Ci, where C\ is ад arbitrary constant, hence, by (4.8)), pn = p'(un), qn = —Cip(un), and chain (1.2) casts into the form un,xy = p'(un)un,x + ciuny — c\p(un). The proof is complete. □

Suppose that ^depends linearly on Rk, к < n, YY0-l, Y, Y0, Y_l for some n:

Rn = XRn-i + ••• , n> 0. (4.11)

Lemma 4.2. If function rn has one of forms (3.50), (3.68), then case (4.11) is not realized.

Доказательство. We apply the operator ad Dx to both sides of identity (4.11). Collecting coefficients at Rn-l in obtained relation, we get the equation:

Dx(\) = —Г0 A — П 2+ П Г0,uo.

A simple analysis of this equation shows that A is a constant, hence

ri2 + n

Г0Х + Г0,ио = 0. (4.12)

Substituting formulae (3.50), (3.68) into (4.12), we get that A = 0 and n2 + n = ^Лсe, n = 0 orn = — 1. Both solutions contradict the assumption n > 0. The proof is complete. □

heorem 1.1 is implied Lemma 4.1, 4.2.

5. Appendix. Proof of Lemma 3.6 The proof is a study of equation (3.37):

k(k — 1)

Vkrl 2--2-rl,u1 + (k — 1)(r0m + r2,u1) = 0 (5.1)

in different cases (3.10) and (3.11) under conditions (1.4), M ^ 2, k ^ 2. We denote uk = u. i) We substitute function r0 defined by formula (3.10) into (5.1)

(2 \ k(k — 1) v i а(щ) — M _ 1 a(ul) + S(u2)j — M — 1 а'(щ) + (k — 5'(щ) + a'(wi)) = 0. (5.2)

We apply the operator ^ to this identity, and we get u5'(u2) = 0. It is easy to show that the case v = 0 leads us to a contradiction to (1.4). Assume that v = 0, then from (5.2) we obtain that the function r0 becomes

ro(ui,uo,U-i) = a(u-i) - M — 1 a(uo) + (- ^ a(ui) + ci-

ii) We substitute the function r0 defined by formula (3,11) into equation (5,1),

p (u0)(vM2— vM + kX — k2X)__,, .__

^^----e M(M-r} + (k — i)p>(Ul)e MiM

M (M — 1) (5.3)

+ u^(u1}u2) + (k — 1) + '-(k — 1)k= 0.

OUi 2 OUi

We apply the operator qQ^ to both sides of identity (5,3)

p'(u0)(uM2 — vM + k\ — k2X)_ d2^(u0,Ui)

' K --e m(M-i) + (k- 1)—u' 1 = 0.

M (M — 1) e + (f^ 1) Qu0dui 0.

This equation has the following solution::

, p (u0 )(vM2 — vM + kX — k2X)__. . . „

^(u0,ui) = -— ')X-^e M(M-i) + fi(uq) + f2(ui). (5.4)

We substitute function (5.4) into equation (5.3), then we differentiate an obtained identity with

2\u2

respect to u2, and we multiple both sides of the obtained identity by eM(M-i): u( vM2 — vM — k2X + kX)/3 (ui)

(5.5)

+ F2(u2)eM(M-i) v = 0.

M (k — 1)(M — 1)

1 ( Xk3 + k2X + kM2v — kMv + 4kX — 4X)3' (ui)

2 ( M — 1) M

Let us consider two different cases v = 0 and v = 0. ii. 1) If v = 0, then (5.5) becomes

1 X(k — 1)(k — 2)(k + 2)3' (ui)

2 M (M — 1)

It follows from this identity that k = 2 or 3'(ui) = 0.

ii. 1.1) If k = 2, then equation (5.3) casts into the form Fi(ui) + F2(ui) = 0. It is clear that F2(ui) = —Fi(ui) + cx. Equation (3.12) becomes

0

1 3 \

—Xp U) — -p (UQ )M 2 + ^Mp' U )j

Q M(M-i)

(5.7)

+ 1m (M — 1)F[(u0 ) + X(Fi(uo) — Fi (Ui) + a) = 0. We apply the operator QuQ2Quo to both sides of identity (5.7)

MM-) 2xp'(up) + M(M — 3)P'(Up) = 0 M (M — 1)

By the condition X = 0 we see that

2 Xp' (u0) + M (M — 3)p''(u0) = 0. (5.8)

ii.1.1.1) If M = 3, then p(uo) = Co, where c0 is an arbitrary constant. The function r0 defined by formula (3.11) becomes

r0(ui,U0,u_i) = c0e~ 3Xu0 — c0e~iXui + Fi(u0) — Fi(ui) + cu (5.9)

and equation (3.12) reads as

— Xc0 e~ 3Xui — XFi(ui) + XFi(u0) + Xc^ + 3F[(u0 ) = 0. (5.10)

We apply the operator qQ- to identity (5.10) :

Qui

1

3X2Co e~ iXui — XFi(ui) = 0,

i

hence,

F1(u1) = -cae-3XU1 + Cv

We substitute F1 into (5,9) and we get r0 = C1. This contradicts condition (1.4), ii.1.1.2) If M = 3, then equation (5.8) has the solution

2 Auq

ft(uo) = C1 + c2e-M(M-3). We differentiate equation (3.12) with respect to u1 and, since A = 0, this equation gives

2 AU1

F1(u1) = -C1e- M(M-1) + C2. Equation (3.12) becomes Xc1 = 0, henee, c1 = 0, and, finally,

__2 A__2A __2A__2A

r0(u1,u0,u-1) = C2e M(M-1)U0 M(M-3)U-1 - C2e M(M-3)U0 m(m-dU1 . (5,11)

We return back to equation (5.6) and consider the following case.

ii. 1.2) If P'(u1) = 0, then P(u]) = c3, where c3 is an arbitrary constant. By equation (5.1) we find

F2(U1) = - 2 kF1(u1) + cA.

Equation (3.12) is transformed as

1 1 / 2 AU1 \

XF1(u0) + -M(M - 1)F[(uo) --A [c3ke-+ kF1(u1) - 2c4) = 0. (5.12)

We apply the operator -J^ to both sides of identity (5.12) 1 k\ ( „ , 2A-i

(2Xui \

-2c3Xe~+ M(M - 1)F'l(u1)) = 0.

2 M(M - 1) This equation has the solution

2 AU1

F1(u1) = -c3e- M(M-1) + c5.

We substitute ^ into (5.12), and we find c4\ c4 = 1 c5(k - 2). We substitute the found functions and constants into (3.11) and we get r0(u1,u0,u-1) = 0, which contradicts condition (1.4). We return back to equation (5.5).

2 AU2

ii.2) If v = 0, then F2(u2)eM(M-1) = c1 and, hence,

2 AU2

, , 1 M(M - 1)c1e- M(M-1) F2(U2) = - 2~-J- + ^2.

Equation (5.5) reads as

v(vM2 - vM - k2X + kX)p(m)

M(k - 1)(M - 1)

1 (Xk3 + k2X + kM2v - kMv + 4kX - 4X)p'(m)

2 (M - 1)M

(5.13)

+ ciu = 0.

We denote:

A = vM2 - vM - k2X + kX, (5.14)

B = Xk3 + k2X + kM2u - kMv + 4kX - 4X. (5.15)

We shall consider the following different cases:

11.2.1) A = 0 B = 0;

11.2.2) A = 0 B = 0

11.2.3) A = 0 B = 0

11.2.4) A = 0 B = 0

In case ii.2.1), that is, as

uM2 -vM - k2X + kX = 0, Xk3 + k2X + kM2v - kMv + 4kX - 4X = 0.

equation, and we get 4(k - 1)X = 0, which contradicts to k ^ 2, X = 0, ii.2.2) Assume that

A = vM2 -vM - k2X + kX = 0.

P(ui) = 2kciui + c3. Equation (5,3) becomes

1,,, ^dFi(ui) k( k - 1)X ^ . . . k(k - 1) c2X

2 -1 ~kr + MM-I) + c'(k -1)e+ MM-iy = 0

This equation has the solution

^ , . (2c\Ui - c4k)__

Fi(ui) = - ,-—e "("-1) - C2.

k

Equation (3,12) casts into the form

dM (M - 1) _ 2Xu0 CiM (M -k- 1) _ 2Xui

1 y ' g "("-i) — —_-_-Q "("-1) = 0.

2

i = 0

M ^ 2), Hence, we have

2 A UQ

r0(ui,u0,u-i) = (C3 + c4)e~"M-), which contradicts condition (1.4), ii.2.3) Suppose that

B = Xk3 + k2X + kM2v - kMv + 4kX - 4X = 0. X(k - 1)(k - 2)(k + 2)

v

kM(M - l)

Since v = 0, then k = 2. Equation (5,13) becomes X(k - l)(k - 2)(k + 2)

k2M2(M - l)2

We find the function 3-

(cikM2 - c\kM + 4X3(щ)) = 0.

p ^=- i . (5,6)

Taking into consideration the obtained function, we simplify equation (5,3):

1 k(k 1) dFi(ui) i X(k - 1)(k - 2)(k + 2) F (u )

2 k- + m{M-1) Fi(ui)

n ^ - 2A^i C2X(k - 2)(k - 1)(k + 2) n

+ ( k - 1 ae "2"-1) + ( Jlu-l]-= 0

This equation has the solution:

. CikM(M - 1)__2A^i_ _ - 2A(k-2)(k+2)

Fi(ui) = —-j---e "("-1) + Cie k'2"("-1) 1 - c2.

4 X

Let us transform equation (3,12)

4 X C

-e k^M (M-l) U1 = 0.

2

Ci = 0

we obtain that r0 = 0, that contradicts condition (1.4),

ii.2.4) If A = 0 and B = 0, then equation (5,13) has the solution

nt ,__2v(vM2rM-k2x+kx)u1_ McAk — 1)(M — 1)

p(ui) = c3e (k-l)—k3+k2A+4kA+kM2v-kMv-4\) + v— -f-. (5.17)

uM2 — uM — k2X + kX

We substitute (5,17) into (5,3) and we obtain

1 dF (u) 2Xui

-k( k — 1)—^^ + uFi(ui) + cAk — 1) e-M (M-1) + uc2. (5.18)

2 dui

Equation (5,18) has the solution

^ . . CXM (M — 1)(k — 1) 2\u-\

Function (3.11) becomes

2vuQ 2XUQ 2vAu-i AC3 2Xu- 2VAUQ

r0(ui ,u0,u-i) = c4e - k(k=1 + c3e - M (M-1) - 'B(k=T> + ——— e-M (M-1)-B(k=T).

2( k — 1) X

A B

Ac4 2uuo Ac3(XB + uM2A — uMA — XkB — 4MvX + 4MukX)_ 2Xu- 2„Au0

4 g k(k-1)__^_ig M (M-1) B(k-1) = 0

k( k — 1) 2B k X( k — 1)2

Since A = 0, v = 0, it follows from the last identity that c4 = 0 and

XB + uM2 A — vMA — XkB — 4MvX + 4MvkX = 0.

Thus, we have specified the function r0:

2\u0 2vAu-1 A C3 2Xu1 2vAuo

r0(ul, u0, u_l) = c3e-M(M-1)-'B0k=1) + —-— e-M(M-1)-Bok=°.

2( k — 1)X

We can rewrite r0 in the following form:

ro(ui,uo,u-i) = Cieh1U0-h2U-1 + C2eh1U1-h2U0,

where ClC2 = 0, hlh2 = 0 are some constants.

Lattice (1,2) is reduced to one with rn of the following form

rn(un+i,un ,un-i) = eu"-hu"-1 + ceu"+1-hu"

by resealing hiun ^ un, cihix ^ x. Similarly transformations one can apply to the lattice in

case ii.1.1.2 (see (5,11)), The proof of Lemma 3,6 is complete,

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Mariva Nikolaevna Kuznetsova, Institute of Mathematics, Ufa Federal Research Center, Russian Academy of Sciences, Chernvshevsky str. 112, 450008, Ufa, Russia

E-mail: mariya .11 .kuznetsovaSgmail. com