Affine and Finite Lie Algebras and Integrable Toda Field Equations on Discrete Space-Time

Difference-difference systems are suggested corresponding to the Cartan matrices of any simple or affine Lie algebra. In the cases of the algebras AN, BN, CN, G₂, D₃, A₁⁽¹⁾, A₂⁽²⁾, DN⁽²⁾ these systems are proved to be integrable. For the systems corresponding to the algebras A₂, A₁⁽¹⁾, A₂⁽²⁾ genera...

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Дата:	2012
Автори:	Garifullin, R., Habibullin, I., Yangubaeva, M.
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Опубліковано:	Інститут математики НАН України 2012
Назва видання:	Symmetry, Integrability and Geometry: Methods and Applications
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Цитувати:	Affine and Finite Lie Algebras and Integrable Toda Field Equations on Discrete Space-Time / R. Garifullin, I. Habibullin, M. Yangubaeva // Symmetry, Integrability and Geometry: Methods and Applications. — 2012. — Т. 8. — Бібліогр.: 41 назв. — англ.

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spelling	nasplib_isofts_kiev_ua-123456789-1484642025-02-09T14:29:57Z Affine and Finite Lie Algebras and Integrable Toda Field Equations on Discrete Space-Time Garifullin, R. Habibullin, I. Yangubaeva, M. Difference-difference systems are suggested corresponding to the Cartan matrices of any simple or affine Lie algebra. In the cases of the algebras AN, BN, CN, G₂, D₃, A₁⁽¹⁾, A₂⁽²⁾, DN⁽²⁾ these systems are proved to be integrable. For the systems corresponding to the algebras A₂, A₁⁽¹⁾, A₂⁽²⁾ generalized symmetries are found. For the systems A₂, B₂, C₂, G₂, D₃ complete sets of independent integrals are found. The Lax representation for the difference-difference systems corresponding to AN, BN, CN, A₁⁽¹⁾,DN⁽²⁾ are presented. This paper is a contribution to the Special Issue “Geometrical Methods in Mathematical Physics”. The full collection is available at http://www.emis.de/journals/SIGMA/GMMP2012.html. The authors are grateful to the referees for their important contribution to improve the article. This work is partially supported by Russian Foundation for Basic Research (RFBR) grants 11-01-97005-r-povoljie-a, 12-01-31208-mol a and 10-01-00088-a and by Federal Task Program “Scientific and pedagogical staf f of innovative Russia for 2009–2013” contract no. 2012-1.5-12-000-1003-011. 2012 Article Affine and Finite Lie Algebras and Integrable Toda Field Equations on Discrete Space-Time / R. Garifullin, I. Habibullin, M. Yangubaeva // Symmetry, Integrability and Geometry: Methods and Applications. — 2012. — Т. 8. — Бібліогр.: 41 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 35Q53; 37K40 DOI: http://dx.doi.org/10.3842/SIGMA.2012.062 https://nasplib.isofts.kiev.ua/handle/123456789/148464 en Symmetry, Integrability and Geometry: Methods and Applications application/pdf Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	Difference-difference systems are suggested corresponding to the Cartan matrices of any simple or affine Lie algebra. In the cases of the algebras AN, BN, CN, G₂, D₃, A₁⁽¹⁾, A₂⁽²⁾, DN⁽²⁾ these systems are proved to be integrable. For the systems corresponding to the algebras A₂, A₁⁽¹⁾, A₂⁽²⁾ generalized symmetries are found. For the systems A₂, B₂, C₂, G₂, D₃ complete sets of independent integrals are found. The Lax representation for the difference-difference systems corresponding to AN, BN, CN, A₁⁽¹⁾,DN⁽²⁾ are presented.
format	Article
author	Garifullin, R. Habibullin, I. Yangubaeva, M.
spellingShingle	Garifullin, R. Habibullin, I. Yangubaeva, M. Affine and Finite Lie Algebras and Integrable Toda Field Equations on Discrete Space-Time Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Garifullin, R. Habibullin, I. Yangubaeva, M.
author_sort	Garifullin, R.
title	Affine and Finite Lie Algebras and Integrable Toda Field Equations on Discrete Space-Time
title_short	Affine and Finite Lie Algebras and Integrable Toda Field Equations on Discrete Space-Time
title_full	Affine and Finite Lie Algebras and Integrable Toda Field Equations on Discrete Space-Time
title_fullStr	Affine and Finite Lie Algebras and Integrable Toda Field Equations on Discrete Space-Time
title_full_unstemmed	Affine and Finite Lie Algebras and Integrable Toda Field Equations on Discrete Space-Time
title_sort	affine and finite lie algebras and integrable toda field equations on discrete space-time
publisher	Інститут математики НАН України
publishDate	2012
url	https://nasplib.isofts.kiev.ua/handle/123456789/148464
citation_txt	Affine and Finite Lie Algebras and Integrable Toda Field Equations on Discrete Space-Time / R. Garifullin, I. Habibullin, M. Yangubaeva // Symmetry, Integrability and Geometry: Methods and Applications. — 2012. — Т. 8. — Бібліогр.: 41 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv	AT garifullinr affineandfiniteliealgebrasandintegrabletodafieldequationsondiscretespacetime AT habibullini affineandfiniteliealgebrasandintegrabletodafieldequationsondiscretespacetime AT yangubaevam affineandfiniteliealgebrasandintegrabletodafieldequationsondiscretespacetime
first_indexed	2025-11-26T20:27:20Z
last_indexed	2025-11-26T20:27:20Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 8 (2012), 062, 33 pages Affine and Finite Lie Algebras and Integrable Toda Field Equations on Discrete Space-Time? Rustem GARIFULLIN †, Ismagil HABIBULLIN † and Marina YANGUBAEVA ‡ † Ufa Institute of Mathematics, Russian Academy of Science, 112 Chernyshevskii Str., Ufa, 450077, Russia E-mail: grustem@gmail.com, habibullinismagil@gmail.com ‡ Faculty of Physics and Mathematics, Birsk State Social Pedagogical Academy, 10 Internationalnaya Str., Birsk, 452452, Russia E-mail: marina.yangubaeva@mail.ru Received April 24, 2012, in final form September 14, 2012; Published online September 18, 2012 http://dx.doi.org/10.3842/SIGMA.2012.062 Abstract. Difference-difference systems are suggested corresponding to the Cartan matri- ces of any simple or affine Lie algebra. In the cases of the algebras AN , BN , CN , G2, D3, A (1) 1 , A (2) 2 , D (2) N these systems are proved to be integrable. For the systems corresponding to the algebras A2, A (1) 1 , A (2) 2 generalized symmetries are found. For the systems A2, B2, C2, G2, D3 complete sets of independent integrals are found. The Lax representation for the difference-difference systems corresponding to AN , BN , CN , A (1) 1 , D (2) N are presented. Key words: affine Lie algebra; difference-difference systems; S-integrability; Darboux in- tegrability; Toda field theory; integral; symmetry; Lax pair 2010 Mathematics Subject Classification: 35Q53; 37K40 1 Introduction Systems of partial differential equations of the form rix,y = e j=N∑ j=1 aijr j , i = 1, 2, . . . , N, (1.1) called generalized two-dimensional Toda lattices have very important applications in Liouville and conformal field theories, they are studied in details (see [3, 5, 6, 9, 10, 11, 13, 27, 28, 29, 30, 34, 36] and the references therein). Here the matrix A = {ai,j} is the Cartan matrix of an arbitrary finite or affine Lie algebra. It is known that in the former case system (1.1) is Darboux integrable while in the latter case – S-integrable. The widely known Drinfel’d–Sokolov formalism allows one to construct the Lax representation for the system (1.1) in terms of the Lie algebra canonically associated with the corresponding Cartan matrix A. The problem of finding discrete versions of the system is intensively studied (see, for instance, [2, 14, 15, 16, 17, 19, 20, 21, 22, 24, 25, 38, 39, 40]). Recently, in [18] integrable differential- difference analog of system (1.1) was suggested vi1,x − vix = e j=i−1∑ j=1 ai,jv j+ j=N∑ j=i+1 ai,jv j 1+ 1 2 ai,i(v i+vi1) , i = 1, 2, . . . , N. (1.2) Here the functions vj = vj(n, x), j = 1, . . . , N are the searched field variables. The subindex denotes a shift of the discrete variable n or the derivative with respect to x: rjk = rj(n + k, x) ?This paper is a contribution to the Special Issue “Geometrical Methods in Mathematical Physics”. The full collection is available at http://www.emis.de/journals/SIGMA/GMMP2012.html mailto:grustem@gmail.com mailto:habibullinismagil@gmail.com mailto:marina.yangubaeva@mail.ru http://dx.doi.org/10.3842/SIGMA.2012.062 http://www.emis.de/journals/SIGMA/GMMP2012.html 2 R. Garifullin, I. Habibullin and M. Yangubaeva and rjx = ∂ ∂xr j(n, x). A particular case of (1.2), corresponding to the algebra CN is also studied in [37]. In the present article we study the problem of further discretization of system (1.1), i.e. the problem of finding a rule allowing to assign to any Cartan matrix a system of integrable difference-difference equations approximating in the continuum limit system (1.2) and therefore system (1.1). As it was pointed out in [40] the problem of discretization is important from physical viewpoint, they might have applications in discrete field theory and in quantum physics (see also [22, 25, 38]). They can also be regarded as difference schemes in numerical computations (see [23]). Following questions were addressed in [40]: 1) whether there exists an integrable discrete version for any two dimensional Toda field equation (1.1); 2) which kind of algebraic structure (like Lie algebra or Lie group) is naturally related to discrete versions. The subject has intensively been studied during the last 10–15 years. Various discrete versions of Toda field equations were investigated in literature (see [15, 22, 24, 25, 39] and references therein). As an alternative answer to the first question we suggest a fully discrete version of system (1.1) in the following form e−u i 1,1+u i 1,0+u i 0,1−ui0,0 − 1 = e j=i−1∑ j=1 ai,ju j 0,1+ j=N∑ j=i+1 ai,ju j 1,0+ 1 2 ai,i(u i 0,1+u i 1,0) , i = 1, 2, . . . , N, (1.3) which evidently approximates (1.1) and (1.2). Here uj = uj(n,m), j = 1, 2, . . . , N , is a set of the field variables. The subindex indicates shifts of the arguments n, m as follows uji,k := uj(n+ i,m+ k). The system corresponding to the algebra AN coincides with that found years ego by Hirota (see [21]). Obviously, system (1.3) is invariant under the replacement n ↔ −m. This property is inherited from the fact that system (1.1) is invariant under the change x↔ y. The main result of the present article is in formulating of the conjecture below and proving it for numerous examples of Lie algebras. Conjecture. a) If A is the Cartan matrix of a semi-simple Lie algebra then (1.3) is Darboux integrable, in other words, it admits a complete set of integrals in both directions. Roughly speaking Darboux integrable difference-difference equations can be reduced to ordinary difference equations. b) If A is the Cartan matrix of an affine Lie algebra then (1.3) is S-integrable, i.e. it can be integrated by means of the inverse scattering transform method. It is generally accepted that existence of generalized symmetries or Lax pairs indicates S-integrability. Part a) of the conjecture is proved for the cases: A2, B2, C2, G2, D3 by finding complete sets of integrals. Systems corresponding to the algebras AN , BN , CN are studied by using Lax pair in Section 6. In the cases A2, B2 (C2), G2 the exponential system (1.3) reads as follows e−u 1 1,1+u 1 1,0+u 1 0,1−u10,0 − 1 = eu 1 0,1+u 1 1,0−u21,0 , e−u 2 1,1+u 2 1,0+u 2 0,1−u20,0 − 1 = e−cu 1 0,1+u 2 0,1+u 2 1,0 , where c takes only three values c = 1, 2, 3. Here the values c = 1, c = 2, c = 3 correspond to the algebras A2, B2 or C2, G2 respectively. Let us introduce a notation for the nonlinear second-order partial difference operator by setting ∆(u) := e−u1,1+u1,0+u0,1−u0,0 − 1, Affine and Finite Lie Algebras and Integrable Toda Field Equations 3 then the last system takes a compact form ∆ ( u1 ) = eu 1 0,1+u 1 1,0−u21,0 , ∆ ( u2 ) = e−cu 1 0,1+u 2 0,1+u 2 1,0 . (1.4) Integrals in both directions for system (1.4) are given in Section 2. Since the map defined in (1.3) converts any N×N matrix to a system of difference-difference equations one can easily specify the form of systems corresponding to any Cartan matrix canoni- cally related to a finite or affine Lie algebra both of classical series or exceptional. For instance, for the algebra D3 with the Cartan matrix A =  2 −1 −1 −1 2 0 −1 0 2  , we have an integrable system of the form ∆ ( u1 ) = eu 1 0,1+u 1 1,0−u21,0−u31,0 , ∆ ( u2 ) = e−u 1 0,1+u 2 0,1+u 2 1,0 , ∆ ( u3 ) = e−u 1 0,1+u 3 0,1+u 3 1,0 . Complete sets of integrals for this system is given in Section 2. For the Kac–Moody algebra A (2) 2 with the Cartan matrix A = ( 2 −1 −4 2 ) the corresponding system is ∆ ( u1 ) = eu 1 0,1+u 1 1,0−u21,0 , ∆ ( u2 ) = e−4u 1 0,1+u 2 0,1+u 2 1,0 . (1.5) For the difference-difference systems corresponding to the algebras A2, A (1) 1 , A (2) 2 genera- lized symmetries are found, for A2 the symmetries have usual form, for A (1) 1 , A (2) 2 they have nonlocal form (or, they are of hyperbolic type, see Definition 1.5 below). In the literature generalized symmetries are regarded as a criterion of integrability. Thus one concludes that systems (1.5), (3.1) provide new examples of 2× 2 S-integrable quad graph models. Stress that the number of the field variables in all three systems (1.1), (1.2), (1.3) coincides with the rank of the corresponding Lie algebra. It is known that any system of difference-difference equations of hyperbolic type admits a pair of characteristic Lie algebras which are effectively evaluated (see [14]). Characteristic Lie algebras of system (1.3) are closely connected with the Lie algebra canonically related to the given Cartan matrix A. In our opinion this relation could allow one to answer the second question in [40] (see the list of questions above). The problem of developing discrete versions of the Drinfel’d–Sokolov formalism is also chal- lenging since today systems of discrete equations are very popular. They have a large variety of application in theoretical physics, in discrete geometry, in the theory of tau functions in lattice Toda field equations. The present article would also provide an “experimental” background for creating a discrete theory parallel to the Drinfel’d–Sokolov formalism. The article is organized as follows. In Section 1.1 notions of the integrals and generalized symmetries of both evolutionary and hyperbolic type for quad graph equations are defined. Sufficient condition of complete set of integrals is proved. In Section 2 a method of discretization of Darboux integrable models preserving integrability is discussed. The method is explained in details by an example of the Liouville equation [19]. Then it is applied to the system (1.2) to get its difference-difference analog. The case B2 is studied in more details. These simulations allowed us to guess the formula (1.3). Complete sets of independent integrals for the systems A2, B2, C2, G2, D3 are presented. 4 R. Garifullin, I. Habibullin and M. Yangubaeva Generalized symmetries are evaluated for the systems corresponding A2, A (1) 1 , A (2) 2 in Sec- tion 3. Special attention is paid to hyperbolic type symmetries which provide a semi-discrete version of the consistency around a cube property of quad graph equations [1]. In Section 4 the concept of the characteristic Lie algebra (see [14]) of discrete models is briefly discussed. Here a complete description of such algebra is obtained for the system of difference-difference equations A2. In Section 5 cutting off conditions for the Hirota equation preserving integrability are studied by using the method suggested in [15]. It turned out that some of the systems in the class (1.3) can be obtained from the Hirota equation by imposing proper boundary conditions. Since the cutting off conditions are compatible with the Lax representation one can derive together with the reduced discrete system also its Lax pair. In Section 5 the Lax pairs for the difference- difference systems AN , BN , CN , A (1) 1 , D (2) N are found. In Section 6 an algorithm is suggested to look for integrals via the Lax pair. In Section 7 widely known periodical reduction of the Hirota equation is discussed. The Lax pair for this system found in [41] is rewritten in terms of the Cartan–Weyl basis for the algebra A (1) N . Emphasize that the difference-difference system obtained as a periodical reduction and that given by the formula (1.3) corresponding to A (1) N are not equivalent (see Remark 7.1 below). 1.1 Integrals and symmetries for quad graph systems Consider a system of quad graph equations of general form H(un,m,un+1,m,un,m+1,un+1,m+1) = 0, (1.6) where u = un,m is a vector-function depending on two integers and ranging on CN : u = (u1, u2, . . . , uN )T . As usually we request that equation (1.6) can be solved with respect to any of the arguments un,m, un+1,m, un,m+1, un+1,m+1. In other words there exists a set of functions H(±1,±1) such that un+1,m+1 = H1,1(un,m,un+1,m,un,m+1), un−1,m+1 = H(−1,1)(un−1,m,un,m,un,m+1), un+1,m−1 = H(1,−1)(un+1,m,un,m,un,m−1), un−1,m−1 = H(−1,−1)(un−1,m,un,m,un,m−1). Define the following standard set of dynamical variables which consists of the variable un,m and its shifts un+k,m and un,m+l where k, l ∈ Z S = {un+k,m,un,m+l : k, l ∈ Z}. By [u] we denote a finite set of dynamical variables, for instance notation h = h([u]) means that function h depends on a finite number of dynamical variables. Define shift operators Dm, Dn acting due to the rules Dmf(n,m) = f(n,m+ 1), Dnf(n,m) = f(n+ 1,m). Definition 1.1. i) Function F ([u], n,m) (function I([u], n,m)) is called m-integral (respectively n-integral) of the equation (1.6) if the following identity holds DmF = F (or, DnI = I) on arbitrary solutions of equation (1.6). Affine and Finite Lie Algebras and Integrable Toda Field Equations 5 ii) Integrals of the form F = F (n) (I = I(m)) are called trivial. iii) Equation admitting N non-trivial independent integrals in each direction is called Darboux integrable. iv) Set of integrals is called independent if none of them can be expressed through the other integrals and their shifts. It can be easily proved that m-integrals do not depend on the variables un,m+l with l 6= 0 and similarly n-integrals do not depend on un+k,m where k 6= 0. Now formulate a very simple and convenient sufficient condition of complete set of integrals. Theorem 1.2. Let us given a set of m-integrals of the form I(j) = I(j) ( n,m,u, Dnu, D 2 nu, . . . , D γj n u ) , γj ≥ 0, j = 1, . . . , N. (1.7) Suppose that for any j and for all n, m at least one of the derivatives ∂I(j) ∂ui differs from zero and the condition det ( ∂I(j) ∂ui ) 6= 0 (1.8) holds for all n, m. Then the integrals constitute a complete set of integrals. Proof. Suppose in contrary that the set of integrals (1.7) is not independent. Then at least one of the integrals, say for definiteness I(1) is a function of the form I(1) = Q ( I(2), I(3), . . . , I(N), DnI(2), DnI(3), . . . , DnI(N), . . . ) , (1.9) depending on the other integrals and their shifts. Differentiating (1.9) with respect to ui we obtain the following equalities ∂I(1) ∂ui − ∂I(2) ∂ui ∂Q ∂I(2) − · · · − ∂I(N) ∂ui ∂Q ∂I(N) = 0. The latter can be regarded as a linear algebraic system with the coefficient matrix {∂I(j) ∂ui } and a solution ( 1, ∂Q ∂I(2) , . . . , ∂Q ∂I(N) ) which obviously is not trivial. Therefore according to the well- known theorem the determinant of the coefficient matrix should vanish. But it contradicts (1.8). Thus our assumption that the set of integrals is dependent is not correct. Theorem is proved. � Remark 1.3. Condition (1.8) actually means that jacobian of the map u→ I = ( I(1), . . . , I(N) ) differs from zero for any value of the variables u,Dnu,D2 nu, . . . ,Dγ nu ranging in a domain. Here γ = max{γj}. Note that (1.8) is not a necessary condition for independent integrals. In the Example 6.1 we have a pair of independent integrals I(1) and I(2) for which condition (1.8) is violated. Definition 1.4. An equation of the form d dt un,m = G([u]) is called generalized symmetry of (1.6) if the following compatibility condition is satisfied d dt ( un+1,m+1 − F (1,1) )∣∣ F=0,un,m,t=G = 0. 6 R. Garifullin, I. Habibullin and M. Yangubaeva It can be shown that function G([u]) is a solution of linearized equation ∂F ∂un+1,m+1 DnDmG+ ∂F ∂un+1,m DnG+ ∂F ∂un,m+1 DmG+ ∂F ∂un,m G = 0. There are systems of the form (1.6) which do not admit evolutionary type symmetries, how- ever they admit symmetries of more complicated structure. We call them hyperbolic type symmetries. Such type symmetries for (1.6) consist of two semi-discrete equations d dt un+1,m = G ( d dt un,m, [u] ) , d dt un,m+1 = G̃ ( d dt un,m, [u] ) . (1.10) Definition 1.5. Pair of equations (1.10) define a hyperbolic type symmetry for (1.6) if equa- tions (1.6), (1.10) constitute a commuting triple, i.e. the following compatibility conditions are satisfied d dt F (1,1) = DmG = DnG̃ by means of equations (1.6), (1.10). Hyperbolic type symmetries are constructed in Section 3. They are nothing else but non-local symmetries (see, for instance, [35]) rewritten in a more convenient form. In the other hand side existence of hyperbolic type symmetries (1.10) for equation (1.6) can be regarded as a semi- discrete generalization of the well-known property of consistency around a cube [1, 4, 32] which is approved to be a criterion of integrability. 2 Systematic approach to the problem of discretization of the Darboux integrable systems The problem of finding integrable discretizations of the integrable partial differential equation is very complicated and not enough studied. The same is true for evaluating the continuum limit for discrete models. In [19] an effective algorithm of discretization (as well as evaluation of the continuum limit) of Darboux integrable equations is suggested based on the integrals. In this section we discuss the essence of the algorithm and apply it to exponential type systems. Note that sets of integrals for the systems corresponding to A2, B2, G2, D3 considered in this section as easily proved by applying sufficient condition given in Theorem 1.2 are independent. 2.1 Explanation of the method with example of Liouville equation Its well-known that the famous Liuoville equation uxy = eu admits integrals in both directions: W = uxx − 0.5u2x and W̄ = uyy − 0.5u2y. Indeed, it is very easy to check that DyW = 0 and DxW̄ = 0. Consider now the problem of finding all chains of the form tx(n+ 1, x) = f(x, t(n, x), t(n+ 1, x), tx(n, x)), (2.1) having the function I = txx − 1 2 tx 2 as their n-integral. Equality DnI = I implies fx + fttx + ft1f + ftxtxx − 1 2 f2 = txx − 1 2 tx 2. (2.2) By comparing the coefficients before txx in (2.2) we have ftx = 1. Therefore, f(x, t, t1, tx) = tx + d(x, t, t1). (2.3) Affine and Finite Lie Algebras and Integrable Toda Field Equations 7 We substitute (2.3) into (2.2) and get dx + dttx + dt1tx + dt1d − 1 2 tx 2 − dtx − 1 2d 2 = −1 2 tx 2, or equivalently, dt + dt1 − d = 0 and dx + dt1d − 1 2d 2 = 0. We solve the last two equations simultaneously and find that d = Ce 1 2 (t1+t) and C is an arbitrary constant. Therefore, chain (2.1) with n-integral I = txx− 1 2 tx 2 becomes t1x = tx+Ce(t1+t)/2. The last equation in its turn admits also an x-integral F = e(t1−t)/2 + e(t1−t2)/2. The next step consists in describing equations of the form v(n+ 1,m+ 1) = f(v(n,m), v(n+ 1,m), v(n,m+ 1)) (2.4) with m-integral F = ev1−v + ev1−v2 . Denote wij := e−vij . In the new variables F = w0,0+w2,0 w1,0 is an m-integral of equation w1,1 = g(w0,0, w1,0, w0,1). DmF = F implies w2,0 + w0,0 w1,0 = Dng + w0,1 g . (2.5) We differentiate both sides of (2.5) with respect to w2,0 and apply the shift operator D−1n , we have 1 w1,0 = ∂w2,0Dng g ⇒ D−1n ( 1 w1,0 ) = D−1n ( ∂w2,0Dng g ) ⇒ gw1,0 = w0,1 w0,0 . Therefore, g = w0,1w1,0 w0,0 + c(w0,0, w0,1). (2.6) We substitute (2.6) into (2.5) and get g w0,0 w1,0 = c(w1,0, g) + w0,1. (2.7) Substitution of (2.6) into (2.7) implies that c(w0,0, w0,1)w = c(w1,0, g)w1,0, or the same, c(w0,0, w0,1)w = Dn(c(w0,0, w0,1)w0,0). Suppose that equation w1,1 = g(w,w1,0, w0,1) does not admit an m-integral of the first order, then c(w0,0, w0,1)w0,0 = Dn(c(w0,0, w0,1)w0,0) = C = const. Thus, c(w0,0, w0,1) = C/w0,0. Finally, g(w0,0, w1,0, w0,1) = w0,1w1,0 w0,0 + C w0,0 . Therefore, the equation (2.4) searched with m-integral F = ev1,0−v0,0 + ev1,0−v2,0 becomes e−v1,1−v0,0 = C + e−v1,0−v0,1 , where C is an arbitrary constant. Note that this equation is symmetric with respect to variables v1,0 and v0,1. Therefore, n-integral for the equation can be obtained by simply changing in m-integral variables vj,0 into variables v0,j , j = 1, 2. 2.2 Application of the algorithm of discretization to the system corresponding to the algebra B2 Apply the reasonings above to the system (1.1) corresponding to the algebra B2. The first step has already been done in [18], where the differential-difference system was found u11,x − u1x = eu 1+u11−u21 , u21,x − u2x = e−2u 1+u2+u21 , (2.8) admitting two independent n-integrals having the form I(1) = 2u1xx + u2xx − 2 ( u1x )2 + 2u1xu 2 x − ( u2xx )2 , I(2) = u1xxxx + u1x ( u2xxx − 2u1xxx ) + u1xx ( 4u1xu 2 x − 2 ( u1x )2 − (u2x)2) + u1xx ( u2xx − u1xx ) + u2xxu 1 x ( u1x − 2u2x ) + ( u1x )4 + ( u1x )2( u2x )2 − 2 ( u1x )2 u2x, 8 R. Garifullin, I. Habibullin and M. Yangubaeva which are also integrals of fully continuous system (1.1) corresponding to the same algebra. It was shown in [18] that system (2.8) admits also x-integrals F(1) = e−u 1 0+u 1 1 + e−u 1 1+u 1 2+u 2 2−u23 + eu 1 1−u12−u21+u22 + eu 1 2−u13 , F(2) = e−u 2 0+u 2 1 + e−2u 1 0+2u11+u 2 1−u22 + 2e−u 1 0+2u11−u12 + e2u 1 1−2u12−u21+u22 + eu 2 2−u23 . (2.9) The goal of this section is to construct a difference-difference system having the same func- tions F(1) and F(2) as their m-integrals. To make the formulas shorter we change the variables a = e−u 1 , b = e−u 2 (2.10) where a, b are new unknowns. The given m-integrals in these variables read F(1) = a0 a1 + a1b3 a2b2 + a2b1 a1b2 + a3 a2 , F(2) = b0 b1 + a0 2b2 a12b1 + 2 a0a2 a12 + a2 2b1 a21b2 + b3 b2 . (2.11) Formulas (2.9) define integrals for a semi-discrete system (2.8) while (2.11) defines integrals for a certain difference-difference system, that is why the variables in (2.11) should be labeled by a double index, however we omitted here and below in this section the second index, because its value is zero for all considered variables. Now we will look for the equations desired in the variables a, b in the form a1,1 = f(a0,0, a1,0, a0,1, b0,0, b1,0, b0,1), b1,1 = g(a0,0, a1,0, a0,1, b0,0, b1,0, b0,1). Substitute the integrals (2.11) into the equations for m-integrals DmF(1) = F(1), DmF(2) = F(2) and bring them to the following form a0,1 f + fD2 ng DnfDng + gDnf fDng + D2 nf Dnf = a0 a1 + a1b3 a2b2 + a2b1 a1b2 + a3 a2 , (2.12a) b0,1 g + a0,1 2Dng f2g + 2 a0,1Dnf f2 + gDnf 2 f2Dng + D2 ng Dng = b0 b1 + a0 2b2 a12b1 + 2 a0a2 a12 + a2 2b1 a21b2 + b3 b2 . (2.12b) By differentiating equation (2.12b) with respect to the variables a3, b3 we get equations 1 Dng D2 n ∂g ∂a1 = 0 ⇔ ∂g ∂a1 = 0, 1 Dng D2 n ∂g ∂b1 = 1 b2 ⇔ ∂g ∂b1 = b0,1 b0 , which imply immediately g(a0,0, a1,0, a0,1, b0,0, b1,0, b0,1) = b1,0b0,1 b0,0 + g1(a0,0, a0,1, b0,0, b0,1) b0,0 . Similarly, differentiation of (2.12a) with respect to a3, b3 yields 1 Dnf D2 n ∂f ∂a1 = 1 a2 ⇔ ∂f ∂a1 = a0,1 a0 , fDng b2DnfDng + 1 Dnf D2 n ∂f ∂b1 = a1 a2b2 ⇔ ∂f ∂b1 = 1 a0b0 D−1n (a0f − a1a0,1). From these equations we get f(a0,0, a1,0, a0,1, b0,0, b1,0, b0,1) = a1,0a0,1 a0,0 + Cb1,0 a0,0 . Affine and Finite Lie Algebras and Integrable Toda Field Equations 9 By differentiating (2.12) with respect to a2 we obtain a1,1 b2b2,1 ∂ ∂a2 D2 ng1 a2,1 + b1,1 a1b2,1 = b1 a1b2 , 2 a0,1 a1,1a1 + 2 a2,1b1,1 a1a1,1b2,1 + 1 b2,1b2,0 ∂ ∂a2 D2 ng1 = 2 a0 a1 + 2 a2b1 a21b2 . The last two equations can be rewritten as follows ∂ ∂a1 Dng1 a1,1 = g1 a0a0,1 , ∂ ∂a1 Dng1 = 2 a1,1g1 a0,1a0 . By getting rid of the variable ∂a1Dng1 in these equations we find Dng1 a21,1 = g1 a20,1 ⇔ g1 = C1a 2 0,1. The last implication is due to the assumption that F(1) and F(2) are integrals of the lowest order. As a result we get equations a1,1a0,0 − a1,0a0,1 = Cb1,0, b1,1b0,0 − b1,0b0,1 = C1a 2 0,1. After some rescaling we can put C = C1 = 1. Finally, we end up with the system of discrete equations a1,1a0,0 − a1,0a0,1 = b1,0, b1,1b0,0 − b1,0b0,1 = a20,1. corresponding to the algebra B2. Turn back to the original variables u1, u2 (see (2.10)) ∆u1 = eu 1 0,1+u 1 1,0−u21,0 , ∆u2 = e−2u 1 0,1+u 2 0,1+u 2 1,0 . By construction the last system admits pair of integrals (2.9) in which the second index for the variables u1, u2 is omitted, since its value is the same for all variables. To find the integrals in the other direction one uses the discrete symmetry n↔ −m of the system. 2.3 Integrals of the systems corresponding to the algebras A2, G2, D3 2.3.1 System corresponding to the algebra A2 The Cartan matrix of A2 is A = ( 2 −1 −1 2 ) . The discrete system for A2 looks as follows ∆ ( u1 ) = eu 1 0,1+u 1 1,0−u21,0 , ∆ ( u2 ) = e−u 1 0,1+u 2 0,1+u 2 1,0 . (2.13) For system (2.13) m-integrals are F(1) = e−u 2 0,0+u 2 1,0 + e−u 1 0,0+u 1 1,0+u 2 1,0−u22,0 + eu 1 1,0−u12,0 , F(2) = e−u 1 0,0+u 1 1,0 + eu 1 1,0−u12,0−u21,0+u22,0 + eu 2 2,0−u23,0 . To find the integrals in the other direction one uses the discrete symmetry n ↔ −m of the system. 10 R. Garifullin, I. Habibullin and M. Yangubaeva 2.3.2 System corresponding to the algebra G2 The Cartan matrix of G2 is A = ( 2 −1 −3 2 ) . The discrete system for G2 is of the form ∆ ( u1 ) = eu 1 0,1+u 1 1,0−u21,0 , ∆ ( u2 ) = e−3u 1 0,1+u 2 0,1+u 2 1,0 . Its m-integrals are F(1) = eu 1 1,0−u12,0 + eu 1 −1,0−u1−2,0 + eu 2 1,0+u 1 1,0−u11,0−u22,0 + eu 1 −1,0+u 2 0,0−u10,0−u2−1,0 + eu 2 1,0+2u10,0−u20,0−2u11,0 + eu 2 0,0+2u10,0−u21,0−2u1−1,0 + 2e2u 1 0,0−u1−1,0−u11,0 , F(2) = eu 2 2,0−u23,0 + e3u 1 0,0+u 2 0,0−u21,0−3u1−1,0 + 3eu 1 0,0+u 2 1,0−u12,0−u20,0 + 3eu 1 0,0+u 1 1,0−u1−1,0−u12,0 + 3eu 2 1,0+3u11,0−2u10,0−u22,0−u12,0 + 3e3u 1 1,0+u 1 0,0−2u12,0 + 3e3u 1 0,0+u 1 1,0−u1−1,0 + 3e3u 1 0,0+u 2 1,0−u20,0−2u11,0−u1−1,0 + 3eu 2 1,0+u 2 1,0−u22,0−u1−1,0 + 2e2u 2 1,0+u 2 0,0−u22,0 + e3u 1 1,0+u 2 2,0−u21,0−3u12,0 + e3u 1 1,0+2u21,0−3u10,0−2u22,0 + e3u 1 0,0+2u21,0−2u20,0−3u11,0 + eu 2 2,0+u 2 −1,0 . To find the integrals in the other direction one uses the discrete symmetry n ↔ −m of the system. 2.3.3 System corresponding to the algebra D3 The Cartan matrix of D3 is A =  2 −1 −1 −1 2 0 −1 0 2  . The discrete system for D3 looks as ∆ ( u1 ) = eu 1 0,1+u 1 1,0−u20,1−u30,1 , ∆ ( u2 ) = eu 2 0,1+u 2 1,0−u11,0 , ∆ ( u3 ) = eu 3 0,1+u 3 1,0−u11,0 . Its m-integrals are F(1) = eu 3 0,0−u3−1,0 + eu 2 1,0−u22,0 + eu 2 1,0+u 1 1,0−u12,0−u20,0 + eu 1 1,0+u 3 0,0−u10,0−u31,0 , F(2) = eu 2 0,0−u2−1,0 + eu 3 1,0−u32,0 + eu 3 1,0+u 1 1,0−u12,0−u30,0 + eu 1 1,0+u 2 0,0−u10,0−u21,0 , F(3) = eu 1 1,0−u12,0 + eu 1 0,0−u1−1,0 + eu 2 0,0−u21,0+u30,0−u3−1,0 + eu 2 0,0−u2−1,0+u 3 0,0−u31,0+ + eu 1 1,0−u10,0+u20,0−u21,0+u30,0−u31,0 + eu 1 0,0−u11,0+u20,0−u2−1,0+u 3 0,0−u3−1,0 . By applying Theorem 1.2 one can prove that these integrals provide a complete set of inde- pendent integrals. To find the integrals in the other direction one uses the discrete symmetry n↔ −m of the system. 3 Symmetries of discrete systems In this section we demonstrate that the discrete systems admit generalized symmetries. For the case of the simple Lie algebras the systems admit local symmetries while for the case of affine algebras the symmetries are nonlocal. Affine and Finite Lie Algebras and Integrable Toda Field Equations 11 3.1 Generalized symmetries for the system A2 Higher symmetries for system corresponding to A2 are found by using the method suggested in [12] (see also [26]). The lowest order symmetry is of the form du10,0 dt = 3eu 1 0,0−u11,0 D−1m F(1) − 1, du20,0 dt = 1− 3eu 2 0,0−u2−1,0 D−1m F(1) . The next order symmetry depends on arbitrary functions f(x, y), g(x) of two and respectively one variable and has the form du10,0 dt = f(F(1), D −1 m F(2)) + g(D−1m F(1)) D−1m F(2) eu 1 0,0−u1−1,0 + f ( F(1), D −1 m F(2) ) − g(F(1)) + g ( D−1m F(1) ) − f(F(1), D −1 m F(2)) + f(D−2m F(1), D −2 m F(2))− g(F(1)) D−1m F(1) eu 1 0,0−u11,0 , du20,0 dt = f(F(1), D −1 m F(2)) + f(D−2m F(1), D −2 m F(2))− g(F(1)) D−1m F(1) eu 2 0,0−u2−1,0 − f ( D−2m F(1), D −2 m F(2) ) + f(D−1m F(1), D −2 m F(2)) + g(D−2m F(1)) D−2m F(2) eu 2 0,0−u21,0 . 3.2 Evaluation of hyperbolic type symmetries for the system A (1) 1 The method for searching nonlocal symmetries is illustrated with the following example e−u1,1−u0,0 − e−u1,0−u0,1 = e−2v1,0 , e−v1,1−v0,0 − e−v1,0−v0,1 = e−2u0,1 , (3.1) corresponding to the algebra A (1) 1 with the Cartan matrix A = ( 2 −2 −2 2 ) . Recall that the continuous version of the system is uxy = exp(2u− 2v), vxy = exp(2v − 2u). (3.2) It can be proved that system (3.2) does not have any local generalized symmetry, however it has nonlocal generalized symmetries, the simplest one can be represented in terms of s = u+ v, r = u− v as follows rt = rxxx − 2r3x, sxt = 2rxrxxx − 3r4x − r2xx + F (W,Wx). Here F is an arbitrary function and W = sxx − r2x is the y-integral of system (3.2). Note that years ago in [9] it was observed that equation rt = rxxx−2r3x is consistent with the system (3.2). For the fully discrete analogue (3.1) of the system (3.2) we have a very similar situation. The system (3.1) does not have any local generalized symmetry. One has to look for a nonlocal symmetry. The main trouble arising in this case is connected with the guessing of the structure of non-locality and the form of the symmetry. Rewrite system (3.1) in the form a1,1a0,0 − a1,0a0,1 = b21,0, b1,1b0,0 − b1,0b0,1 = a20,1, (3.3) where ai,j = exp(−ui,j), bi,j = exp(−vi,j). 12 R. Garifullin, I. Habibullin and M. Yangubaeva We will look for the hyperbolic type symmetry (see Definition 1.5) of (3.3) in such a form ∂ta1,0 = f1(a0,0, a1,0)∂ta0,0 + f(a−1,0, a0,0, a1,0, a2,0, b−1,0, b0,0, b1,0, b2,0), ∂tb1,0 = g1(b0,0, b1,0)∂tb0,0 + g(a−1,0, a0,0, a1,0, a2,0, b−1,0, b0,0, b1,0, b2,0). (3.4) Here f1, g1, f , g are unknown functions. Right hand side of (3.4) corresponds to the function G in (1.10). The function G̃ is evaluated in terms of G below by means of the compatibility conditions. From (3.4) we can find ∂ta1,1 = Dm(f1)∂ta0,1 +Dmf, ∂tb1,1 = Dm(g1)∂tb0,1 +Dmg. (3.5) After differentiation (3.1) with respect to t by means of the systems (3.4), (3.5) we get ∂ta0,0(a1,1 − a0,1f1) + ∂ta0,1(a0,0Dmf1 − a1,0)− 2∂tb0,0b1,0g1 = · · · , −2a0,1∂ta0,1 + ∂tb0,0(b1,1 − b0,1g1) + ∂tb0,1(b0,0Dmg1 − b1,0) = · · · . (3.6) Here the right hand sides do not depend on the derivatives of dynamical variables with respect to t. By applying the operator Dn to both sides of (3.6) we obtain ∂ta0,0f1Dn(a1,1 − a0,1f1) + ∂ta0,1Dmf1Dn(a0,0Dmf1 − a1,0)− 2∂tb1,0b2,0g1Dng1 = · · · , −2a1,1∂tDmf1 + ∂tb0,0g1Dn(b1,1 − b0,1g1) + ∂tb0,1Dmg1Dn(b0,0Dmg1 − b1,0) = · · · . (3.7) System of equations (3.6), (3.7) is a system of linear algebraic equations with unknowns ∂ta0,0, ∂tb0,0, ∂ta0,1, ∂tb0,1. If the determinant of this system is different from zero, then due to the Cramer’s rule the system has unique solution, and therefore the searched generalized symmetry is local. This output is in contradiction with our previous study proving the absence of local symmetries. Thus the determinant should be zero∣∣∣∣∣∣∣∣ a1,1 − f1a0,1 a0,0Dmf1 − a1,0 f1(a2,1 − a1,1Dnf1) (a1,0DnDmf1 − a2,0)Dmf1 0 −2a1,1Dmf1 0 −2a0,1 −2b1,0g1 0 −2b2,0g1Dng1 0 g1(b2,1 − b1,1Dng1) Dmg1(b1,0DnDmg1 − b2,0) b1,1 − g1b0,1 b0,0Dmg1 − b1,0 ∣∣∣∣∣∣∣∣ . Vanishing of the determinant implies f1(a0,0, a1,0) = a1,0/a0,0, g1(b0,0, b1,0) = b1,0/b0,0, (3.8) In virtue of (3.8) equations (3.6), (3.7) take the form ∂ta0,0 a0,0 + ∂ta0,1 a0,1 − 2 ∂tb0,0 b0,0 = 2 b1,0 g + a0,1 b21,0 f − a0,0 b21,0 Dmf, ∂ta0,0 a0,0 + ∂ta0,1 a0,1 − 2 ∂tb0,0 b0,0 = 2 b2,0 Dng + a1,1 b22,0 Dnf − a1,0 b22,0 DnDmf − f a1,0 − Dmf a1,1 + 2 b1,0 g, ∂tb0,0 b0,0 + ∂tb0,1 b0,1 − 2 ∂ta0,1 a0,1 = b0,1 a20,1 g + b0,0 a20,1 Dmg, ∂tb0,0 b0,0 + ∂tb0,1 b0,1 − 2 ∂ta0,1 a0,1 = b1,1 a21,1 Dng + b1,0 a21,1 DnDmg + 2 a1,1 Dmf − g b1,0 − Dmg b1,1 . (3.9) Affine and Finite Lie Algebras and Integrable Toda Field Equations 13 We can see that the left hand sides of the first and third equations as well as the left hand sides of the second and fourth equations of the last system are identical. Comparison of their right hand sides gives a system of two equation for f , g a1,0 b22,0 DnDmf − a1,1 b22,0 Dnf − a1,0a0,1 a1,1b21,0 Dmf + a1,1a0,0 a1,0b21,0 f − 2 b2,0 Dng = 0, (3.10) b1,0 a21,1 DnDmg + b1,1 a21,1 Dng − b1,0b0,1 b1,1a20,1 Dmg − b1,1b0,0 b1,0a20,1 g + 2 a1,1 Dmf = 0. (3.11) Differentiate (3.10) and (3.11) with respect to a3,0, b3,0, b−1,0 or a−1,0 and get b0,0a0,0 a21,0 ∂f ∂b−1,0 = Dm ( b0,0a0,0 a21,0 ∂f ∂b−1,0 ) , Dm ( a20,0 a21,0 ∂f ∂a−1,0 ) = a20,0 a21,0 ∂f ∂a−1,0 + 2a−1,1a 2 0,1 a21,1b0,0 Dm ∂f ∂b−1,0 , (3.12) a1,0 b0,0 ∂g ∂a2,0 = a1,1 b0,1 Dm ∂g ∂a2,0 , Dm ∂g ∂b2,0 + 2b1,0b2,0 a1,0b1,1 Dm ∂g ∂a2,0 = b1,0b0,1 b0,0b1,1 ∂g ∂b2,0 , (3.13) ∂f ∂a2,0 + 2b1,0 a0,1 ∂g ∂a2,0 = a1,1a0,0 a1,0a0,1 Dm ∂f ∂a2,0 , ∂f ∂b2,0 + 2b1,0 a0,1 ∂g ∂b2,0 = 2a0,0b2,0 a1,0a0,1 Dm ∂f ∂a2,0 + a0,0b1,1 a0,1b1,0 Dm ∂f ∂b2,0 , ∂g ∂b−1,0 + 2b1,0a 2 0,0 a1,0b1,−1b0,0 ∂f ∂b−1,0 = b21,0b 2 0,−1 b21,−1b 2 00, D−1m ∂g ∂b−1,0 , ∂g ∂a−1,0 + 2a20,0b1,0 a1,0b1,−1b0,0 ∂f ∂a−1,0 = b21,0a0,−1b0,−1 a0,0b0,0b1,−1 D−1m ∂g ∂a−1,0 + 4a20,0a−1,0b1,0 a1,0b0,−1b1,−1b0,0 ∂f ∂b−1,0 + 2a−1,0 b0,−1 ∂g ∂b−1,0 . We use the last system of eight equations in order to specify dependence of the functions f , g on the variables a−1,0, a2,0, b−1,0, b2,0 corresponding to the lowest and highest values of the first index. From the first equation in (3.12) we get ∂f ∂b−1,0 = C7a 2 1,0 b0,0a0,0 . Then the second equation takes the form Dm ( a20,0 a21,0 ∂f ∂a−1,0 ) = a20,0 a21,0 ∂f ∂a−1,0 + 2C7a−1,1a0,1 b0,1b0,0 . It can easily be proved that the expression 2a−1,1a0,1 b0,1b0,0 does not belong to the image of the operator Dm − 1, so we have to put C7 = 0 and then ∂f ∂a−1,0 = C4a 2 1,0 a20,0 . From the pair of equations (3.13) we have ∂g ∂a2,0 = 0, ∂g ∂b2,0 = C2b0,0 b1,0 . 14 R. Garifullin, I. Habibullin and M. Yangubaeva Continuing this way one can find dependence of the functions f , g on the variables a−1,0, a2,0, b−1,0, b2,0 f = a0,0b2,0 b1,0 C1 + C2 ( a0,0a2,0 a1,0 + a0,0b 2 2,0 a1,0b21,0 + 2b0,0a1,0b2,0 b21,0 ) + a21,0a−1,0 a20,0 C4 + f1(a0,0, a1,0, b0,0, b1,0), g = b0,0b2,0 b1,0 C2 + C4 ( b21,0b−1,0 b20,0 + b31,0a 2 −1,0 a20,0b 2 0,0 + 2b1,0a1,0a−1,0 a20,0 ) + b21,0a−1,0 a0,0b0,0 C5 + g1(a0,0, a1,0, b0,0, b1,0). For the functions f1, g1 from the same system (3.10), (3.11) we obtain the following non- homogeneous equations a1,0 b22,0 DnDmf1 − a1,1 b22,0 Dnf1 − a1,0a0,1 a1,1b22,0 Dmf1 + a1,1a0,0 a1,0b21,0 f1 − 2 b2,0 Dng1 = · · · , (3.14) b1,0 a21,1 DnDmg1 + b1,1 a21,1 Dng1 − b1,0b0,1 b1,1a20,1 Dmg1 − b1,1b0,0 b1,0a20,1 g1 + 2 a1,1 Dmf1 = · · · . (3.15) Differentiating (3.15) with respect to the variable a2,0 we get a0,0 b0,0 ∂g1 ∂a1,0 −Dm ( a0,0 b0,0 ∂g1 ∂a1,0 ) = 2C2(a 3 0,1a1,0 − b31,0b0,1) a0,0a0,1b1,1b1,0 + 2(C4 − C2) a30,1a−1,0 − b1,1b30,0 a0,0b0,0a0,1b0,1 . From this equation we can find C4 = C2, a0,0 b0,0 ∂g1 ∂a1,0 = 2C2a1,0b0,0 a0,0b1,0 + C6, therefore g1 = C2 a21,0b 2 0,0 a20,0b1,0 + C6 a1,0b0,0 a0,0 + g2(a0,0, b0,0, b1,0). Differentiating (3.15) with respect to b2,0 we get Dm ( ∂g2 ∂b1,0 ) − ∂g2 ∂b1,0 = 2(C1 − C6) b0,0b1,0 a0,0a0,1 + 2(C1 − C5) a20,1a0,0a−1,1 − b20,0b1,0b0,1 a0,0b0,0a0,1b0,1 , therefore C6 = C1, C5 = C1, g2 = C3b1,0 + g3(a0,0, b0,0). Now apply the operator D−1n to the equation (3.14) and then differentiate the obtained result with respect to the variable b−1,0 Dm ( a0,0b1,0 a21,0 ∂f1 ∂b0,0 ) − a0,0b1,0 a21,0 ∂f1 ∂b0,0 = 2C2 b31,0b0,1 − a1,0a30,1 a0,0a0,1b1,0b1,1 , therefore f1 = C2 a31,0b 2 0,0 a20,0b 2 1,0 + C8 a21,0b0,0 a0,0b1,0 + f2(a0,0, a1,0, b1,0). Affine and Finite Lie Algebras and Integrable Toda Field Equations 15 Substitute in (3.14) the expression found instead of f1 and differentiate with respect to a2,0 to get Dm ( ∂f2 ∂a1,0 ) − ∂f2 ∂a1,0 = 2(C1 − C8) a1,0a 3 0,1 − b31,0b0,1 a0,0a0,1b1,0b1,1 , therefore C8 = C1, f2 = C9a1,0 + f3(a0,0, b1,0). For the functions f3, g3 we have a system a1,0 b22,0 DnDmf3 − a1,1 b22,0 Dnf3 − a1,0a0,1 a1,1b22,0 Dmf3 + a1,1a0,0 a1,0b21,0 f3 − 2 b2,0 Dng3 = 2(C9 − C3), b1,0 a21,1 DnDmg3 + b1,1 a21,1 Dng3 − b1,0b0,1 b1,1a20,1 Dmg3 − b1,1b0,0 b1,0a20,1 g3 + 2 a1,1 Dmf3 = 2(C9 − C3). (3.16) We see that (3.16) is satisfied identically if one chooses C9 = C3, f3 = 0, g3 = 0. Thus we find the final form of the symmetry searched ∂ta1,0 a1,0 − ∂ta0,0 a0,0 = C1 ( a0,0b2,0 a1,0b1,0 + a1,0b0,0 a0,0b1,0 ) + C2 ( a1,0a−1,0 a20,0 + 2b0,0b2,0 b21,0 + a21,0b 2 0,0 a20,0b 2 1,0 + a0,0a2,0 a21,0 + a20,0b 2 2,0 a21,0b 2 1,0 ) + C3, ∂tb1,0 b1,0 − ∂tb0,0 b0,0 = C1 ( a−1,0b1,0 a0,0b0,0 + a1,0b0,0 a0,0b1,0 ) (3.17) + C2 ( 2a1,0a−1,0 a20,0 + b0,0b2,0 b21,0 + a21,0b 2 0,0 a20,0b 2 1,0 + b−1,0b1,0 b20,0 + a−1,0b 2 1,0 a20,0b 2 0,0 ) + C3. Here C1, C2, C3 are arbitrary constants. From (3.9) one can find the function G̃, then (1.10) looks like ∂ta0,1 a0,1 = ∂tb0,0 b0,0 − ∂ta0,0 a0,0 + C1 b1,0 a0,0 ( 2a−1,0 b0,0 − b0,0 a0,1 ) + C2 ( 2b1,0b−1,0 b20,0 + 2b21,0a 2 −1,0 a20,0b 2 0,0 + 2a1,0a−1,0 a20,0 − a−1,0b 2 1,0 a20,0a0,1 − b20,0a1,0 a20,0a0,1 ) + C3, ∂tb0,1 b0,1 = 2∂ta0,0 a0,0 − 3∂tb0,0 b0,0 + C2 ( 2b1,0a−1,0 a0,0b0,0 + b0,0a0,1 a0,0b0,1 − a−1,0a 2 0,1 a0,0b0,0b0,1 ) + C2 ( 2a1,0a−1,0 a20,0 + 2b1,0b−1,0 b20,0 + 2a0,1b1,0a−1,0 a20,0b0,1 + 2b21,0a 2 −1,0 a20,0b 2 0,0 − b20,0b1,0 a20,0b0,1 − a20,1b−1,0 b20,0b0,1 − a20,1a 2 −1,0b1,0 a20,0b 2 0,0b0,1 ) + C3. (3.18) Thus we have proved the following Theorem 3.1. Equations (3.17) and (3.18) define a hyperbolic type symmetry for equation (3.3). 16 R. Garifullin, I. Habibullin and M. Yangubaeva Remark 3.2. By applying the replacement n↔ −m to the equations (3.17) and (3.18) one can obtain the second hyperbolic type symmetry for equation (3.3). By analogy with the continuous case we have a differential constraint completely consistent with system (3.3), which is obtained from the last system by applying the operator (Dn − 1)−1 to the difference of the two equations ∂tb0,0 b0,0 − ∂ta0,0 a0,0 = C1 a−1,0b1,0 a0,0b0,0 + C2 ( a−1,0a1,0 a20,0 + b−1,0b1,0 b20,0 + a2−1,0b 2 1,0 a20,0b 2 0,0 ) . Under the Cole–Hopf type transformation â0,0 = a1,0 a0,0 , b̂0,0 = b1,0 b0,0 system (3.3) converts to the following one â1,1â0,0â0,1 − â1,0â0,1â0,0 = b̂21,0(â0,0 − â0,1), b̂1,1b̂0,0b̂0,1 − b̂1,0b̂0,1b̂0,0 = â20,1(b̂0,1 − b̂0,0). (3.19) This transformation brings our hyperbolic type symmetry to generalized symmetry of usual form for the new system (3.19) ∂tâ0,0 = C1 ( b̂1,0 + â20,0 b̂0,0 ) + C2 ( â20,0 â−1,0 + 2b̂1,0â0,0 b̂0,0 + â30,0 b̂20,0 + â1,0 + b̂21,0 â0,0 ) + C3â0,0, ∂tb̂0,0 = C1 ( b̂20,0 â−1,0 + â0,0 ) + C2 ( 2â0,0b̂0,0 â−1,0 + b̂1,0 + â20,0 b̂0,0 + b̂20,0 b̂−1,0 + b̂30,0 â2−1,0 ) + C3b̂0,0. 3.3 Hyperbolic type symmetries for the system A (2) 2 The Cartan matrix of A (2) 2 is A = ( 2 −1 −4 2 ) . The discrete system for A (2) 2 looks as ∆ ( u1 ) = eu 1 0,1+u 1 1,0−u21,0 , ∆ ( u2 ) = e−4u 1 0,1+u 2 0,1+u 2 1,0 . First part of its hyperbolic type symmetry (function G) is( u11,0 − u10,0 ) t = C1 ( e2u 1 1,0−u10,0−u12,0 + e2u 1 1,0−2u10,0+u21,0−u22,0 + e2u 1 0,0−2u11,0+u21,0−u20,0 + e2u 1 0,0−u11,0−u1−1,0 ) + C2 ( 3e4u 1 1,0−2u10,0−2u12,0 + eu 1 1,0−u10,0+u12,0−u13,0 + 3e−u 1 2,0+u 2 1,0+u 1 0,0−u20,0 + 3e4u 1 1,0−u12,0−3u10,0+u21,0−u22,0 + 2eu 1 1,0−u12,0+u10,0−u1−1,0 + e4u 1 1,0−4u10,0+2u21,0−2u22,0 + 2eu 2 2,0−2u21,0−u20,0 + 3eu 1 1,0−u22,0+u21,0−u1−1,0 + e−u 1 1,0+u 1 −1,0+u 1 0,0−u1−2,0 + e−4u 1 1,0+2u21,0−2u20,0+4u10,0 + e4u 1 1,0−u10,0+u22,0−3u12,0−u21,0 + e−u 1 1,0+u 1 −1,0+u 2 0,0−u2−1,0 + 3e−2u 1 1,0−2u1−1,0+4u10,0 + e−u 1 0,0+u 2 2,0+u 1 2,0−u23,0 Affine and Finite Lie Algebras and Integrable Toda Field Equations 17 + 3e3u 1 1,0+u 2 1,0−u1−1,0−u20,0+4u10,0 + e−u 1 1,0−u21,0−3u1−1,0+u 2 0,0+4u10,0 ) + C3,( u21,0 − u20,0 ) t = 2C1 ( e−2u 1 1,0+u 2 1,0−u20,0+2u10,0 + 2e−u 1 1,0−u1−1,0+2u10,0 + e−u 2 1,0+2u10,0+u 2 0,0−2u1−1,0 ) + 2C2 ( 2eu 1 1,0−u12,0+u10,0−u1−1,0 + 2e−u 1 2,0+u 2 1,0+u 1 0,0−u20,0 + 4e−3u 1 1,0+u 2 1,0−u1−1,0−u20,0+4u10,0 + e−4u 1 1,0+2u21,0−2u20,0+4u10,0 + 2eu 1 1,0−u22,0+u21,0−u1−1,0 + e−u 2 2,0+2u21,0−u20,0 + 2e−u 1 1,0+u 1 −1,0+u 1 0,0−u1−2,0 + 2e−u 1 1,0+u 1 −1,0+u 2 0,0−u2−1,0 + 4e−u 1 1,0−u21,0−3u1−1,0+u 2 0,0+4u10,0 + 2e−u 2 1,0+u 1 0,0+u 2 0,0−u1−2,0 + e−u 2 1,0+2u20,0−u2−1,0 + 6e−2u 1 1,0−2u1−1,0+4u10,0 + e−2u 2 1,0+4u10,0+2u20,0−4u1−1,0 ) + 2C3. Here C1, C2, C3 are arbitrary constants. Second part of hyperbolic type symmetry (function G̃) can be evaluated automatically from the compatibility conditions. And for combination we have a local constraint( 2u10,0 − u20,0 ) t = 2C1 ( e2u 1 0,0−u11,0−u1−1,0 + e2u 1 0,0−2u1−1,0+u 2 0,0−u21,0 ) + 2C2 ( e−u 1 −1,0−u12,0+u10,0+u11,0 + eu 1 1,0+u 2 1,0−u1−1,0−u22,0 + e−3u 1 1,0+u 2 1,0+4u10,0−u1−1,0−u20,0 + 3e−2u 1 1,0+4u10,0−2u1−1,0 + 3e−u 1 1,0−u21,0+4u10,0−3u1−1,0+u 2 0,0 + e−u 1 1,0+u 1 −1,0+u 2 0,0−u2−1,0 + e−u 1 1,0+u 1 0,0+u 1 −1,0−u1−2,0 + e−2u 2 1,0+4u10,0−4u1−1,0+2u20,0 + e−u 2 1,0+2u20,0−u2−1,0 + 2e−u 2 1,0+u 1 0,0+u 2 0,0−u1−2,0 ) . 4 Characteristic m-algebra for the case A2 Let us describe briefly the properties of the characteristic Lie algebras of the system a0,0a1,1 = a1,0a0,1 + b1,0, b0,0b1,1 = b1,0b0,1 + a0,1. (4.1) Recall that system (4.1) corresponds to the simple Lie algebra A2. First we concentrate on the notion of the characteristic m-algebra for the system (4.1). Lie algebra on the other destination is studied similarly. Recall that according to the definition an m-integral F (a0,0, b0,0, a1,0, b1,0, a−1,0, b−1,0, . . . ) should satisfy the equation DmF = F . In the coordinate representation this condition reads F (a0,1, b0,1, a1,1, b1,1, a−1,1, b−1,1, . . . ) = F (a0,0, b0,0, a1,0, b1,0, a−1,0, b−1,0, . . . ). (4.2) Evidently the right hand side in (4.2) does not depend on the variables a0,1 and b0,1 hence the conditions hold ∂ ∂a0,1 D−1m F = 0, ∂ ∂b0,1 D−1m F = 0 as well as Y1F = 0, Y2F = 0 where Y1 := D−1m ∂ ∂a0,1 Dm, Y2 := D−1m ∂ ∂b0,1 Dm. Denote through Lm the Lie algebra generated by the operators X1 = ∂ ∂a0,−1 , X2 = ∂ ∂b0,−1 , Y1, Y2. Algebra Lm is called characteristic m-algebra. Obviously, operators X1, X2 are the first-order linear differential operators, or vector fields. The operators Y1, Y2 can also be rewritten as vector fields of the form Y1 = ∂ ∂a0,0 + ( a1,0 a0,0 − b1,0b0,−1 a0,0b0,0a0,−1 ) ∂ ∂a1,0 + ( a−1,0 a0,0 + b0,−1 a0,0a0,−1 ) ∂ ∂a−1,0 + · · · 18 R. Garifullin, I. Habibullin and M. Yangubaeva + 1 b0,−1 ∂ ∂b1,0 − ( a−1,0 a0,0b0,−1 + 1 a0,0a0,−1 ) ∂ ∂b−1,0 + · · · , Y2 = ∂ ∂b0,0 + ( b1,0 b0,0 − a0,0 b0,0b0,−1 ) ∂ ∂b1,0 + ( b−1,0 b0,0 + a−1,0 b0,0b0,−1 ) ∂ ∂b−1,0 + · · · . We use the following lemma to show that m-algebra is of finite dimension. Lemma 4.1. Suppose that the vector field K = ∞∑ k=1 ( αk ∂ ∂ak + α−k ∂ ∂a−k ) + ∞∑ k=1 ( βk ∂ ∂bk + β−k ∂ ∂b−k ) satisfies the equality DnKD −1 n = hK, where h is a function depending on shifts of variables a and b, then K = 0. Lemma can be proved by applying both sides of the equation DnKD −1 n = hK to the dyna- mical variables ak, bk. One can easily check that DnX1D −1 n = a0,0 a1,0 X1, DnX2D −1 n = 1 a1,0 X1 + b0,0 b1,0 X2, DnY1D −1 n = a0,−1 a1,−1 Y1 − a0,−1 a1,−1b1,−1 Y2, DnY2D −1 n = b0,−1 b1,−1 Y2. Put X̃1 = a0,0X1, X̃2 = b0,0X2, Ỹ1 = a0,−1Y1, Ỹ2 = b0,−1Y2, then DnX̃1D −1 n = X̃1, DnX̃2D −1 n = b1,0 a0,0a1,0 X̃1 + X̃2, DnỸ1D −1 n = Ỹ1 − a0,−1 b0,−1b1,−1 Ỹ2, DnỸ2D −1 n = Ỹ2. Taking commutators of the vector fields X̃1, X̃2, Ỹ1, Ỹ2 we get vector fields P1 = [X1, Y1], P2 = [X2, Y1], P3 = [X2, Y2]. Lemma 4.2. DnP1D −1 n = P1 − a0,0 b0,−1b1,−1 Ỹ2, DnP2D −1 n = P2 + b1,0 a0,0a1,0 P1 − a0,−1 b0,−1b1,−1 P3 + ( a0,−1b1,0 a20,0a1,0 + a1,−1b1,0 a0,0a21,0 ) X̃1 + ( a0,−1b0,0 b20,−1b1,−1 + a1,−1a0,0b1,0 a1,0b0,−1b21,−1 ) Ỹ2, DnP3D −1 n = P3 − b1,−1 a0,0a1,0 X̃1. Lemma can be proved by direct calculations. It allows to derive the following table of com- mutators which shows that the characteristic Lie algebra Lm is of dimension seven: X̃1 X̃2 Ỹ1 Ỹ2 P1 P2 P3 X̃1 0 0 P1 0 0 R1 0 X̃2 0 0 P2 P3 R1 R2 −2X̃2 Ỹ1 −P1 −P2 0 0 2Ỹ1 R3 R4 Ỹ2 0 −P3 0 0 0 R4 0 P1 0 −R1 −2Ỹ1 0 0 Q1 Q2 P2 −R1 −R2 −R3 −R4 −Q1 0 Q3 P3 0 2X̃2 −R4 0 −Q2 −Q3 0 Affine and Finite Lie Algebras and Integrable Toda Field Equations 19 Here the following notations are used R1 = X̃2 − b1,0 b0,−1 P1 + b20,0 b20,−1 Ỹ2, R2 = 2a0,−1b 2 0,0 a0,0b20,−1 P3 − 2a0,−1b 3 0,0 a0,0b30,−1 Ỹ2 + 2a0,−1b0,0 a0,0b0,−1 X̃2, R3 = 2a20,−1b0,0 a20,0b0,−1 P1 − 2a0,−1b0,0 a0,0b0,−1 Ỹ1 + 2a30,−1b0,0 a30,0b0,−1 X̃1, R4 = −Ỹ1 − a20,−1 a20,0 X̃1 − a0,−1 a0,0 P1, Q1 = a0,−1b0,0 a0,0b0,−1 P1 − 3P2 − P3 + a20,−1b0,0 a20,0b0,−1 X̃1 + b0,0 b0,−1 Ỹ1, Q2 = 2b0,0 b0,−1 Ỹ2 − 2P3, Q3 = −3P2 + a0,−1b0,0 a0,0b0,−1 P3 − a0,−1 a0,0 X̃2 − a0,−1b 2 0,0 a0,0b20,−1 Ỹ2. Due to the reasonings above any m-integral F (. . . , a0,0, b0,0, a1,0, b1,0, a2,0, b2,0, . . . ) should satisfy equations X̃1(F ) = 0, X̃2(F ) = 0, Ỹ1(F ) = 0, Ỹ2(F ) = 0, P1(F ) = 0, P2(F ) = 0, P3(F ) = 0. (4.3) Solving system (4.3), it is enough to assume that F depends on b0,0, a0,0, b1,0, a1,0, b2,0, a2,0 or, alternatively, F depends on a−1,0, b0,0, a0,0, b1,0, a1,0, b2,0. Under such assumptions system (4.3) generates two systems of the first-order linear partial differential equations which can be solved by Jacobi method. By solving these systems we find two independent m-integrals F(1) = b0,0 b1,0 + a0,0b2,0 a1,0b1,0 + a2,0 a1,0 , F(2) = a−1,0 a0,0 + a1,0b0,0 a0,0b1,0 + b2,0 b1,0 . 5 Cutting off constraints for the Hirota equation and discrete Zakharov–Shabat systems In this section we construct Lax pairs for discrete systems corresponding to Cartan matrices of series AN , BN , CN , D (2) N and A (1) 1 . To this end we impose cutting off constrains for the Hirota equation compatible with its Lax pair. It is well known that Hirota chain tj0,0t j 1,1 − t j 1,0t j 0,1 = tj−11,0 t j+1 0,1 (5.1) admits the Lax pair consisting of two linear discrete equations [7] ψj1,0 = tj+1 1,0 t j 0,0 tj+1 0,0 t j 1,0 ψj0,0 − ψ j+1 0,0 , ψj0,1 = ψj0,0 + tj+1 0,1 t j−1 0,0 tj0,0t j 0,1 ψj−10,0 . (5.2) Here the lower indices mean as previously shifts of the arguments, and the upper index enume- rates the field variables tj and eigenfunctions ψj . Exclude from the system of equations (5.2) all 20 R. Garifullin, I. Habibullin and M. Yangubaeva the eigenfunctions with the upper index different from j. As a result one gets a linear discrete hyperbolic equation for ψj ψj1,1 − ψ j 1,0 − tj+1 1,1 t j 0,1 tj+1 0,1 t j 1,1 ψj0,1 + tj0,0t j+1 1,1 tj1,0t j+1 0,1 ψj0,0 = 0. (5.3) It is remarkable that by construction formulas (5.2) define Laplace transformations for the linear hyperbolic equation (5.3). Evidently, equation (5.1) is invariant under the transformation defined as n→ 1−m, m→ 1− n. Under this transformation the Lax pair (5.2) transforms to a Lax pair yj−1,0 = yj0,0 + tj+1 −1,0t j−1 0,0 tj−1,0t j 0,0 yj−10,0 , yj0,−1 = tj+1 0,−1t j 0,0 tj+1 0,0 t j 0,−1 yj0,0 − y j+1 0,0 , (5.4) and equation (5.3) transforms to an equation yj1,1 − tj1,0t j+1 0,1 tj+1 0,0 t j 1,1 yj1,0 − tj1,0t j 0,1 tj0,0t j 1,1 yj0,1 + tj1,0t j+1 0,1 tj+1 0,0 t j 1,1 yj0,0 = 0. (5.5) Put yj0,0 = tj+1 −1,0 tj0,0 gj0,0, then Lax pair (5.4) transforms to a Lax pair gj−1,0 = tj+1 −1,0t j −1,0 tj0,0t j+1 −2,0 ( gj0,0 + gj−10,0 ) , gj0,−1 = tj+1 0,−1t j+1 −1,0 tj+1 0,0 t j+1 −1,−1 gj0,0 − tj0,−1t j+2 −1,0 tj+1 0,0 t j+1 −1,−1 gj+1 0,0 , (5.6) and equation (5.5) transforms to an equation gj1,1 − g j 1,0 − tj1,0t j+1 −1,1 tj+1 0,1 t j 0,0 gj0,1 + tj1,0t j+1 −1,0 tj+1 0,0 t j 0,0 gj0,0 = 0. (5.7) Equations (5.6) define Laplace transformations for the linear hyperbolic equation (5.7). Thus we have two different Lax pairs for the Hirota chain and consequently two families (5.3), (5.7) of linear discrete hyperbolic equations enumerated by j. Study the question when an equation from one family can be related, by a linear transformation, to an equation from the other family. To this end we evaluate the Laplace invariants of these equations. Recall that the Laplace invariants of a discrete hyperbolic type equation of the form a0,0f1,1 + b0,0f1,0 + c0,0f0,1 + d0,0f0,0 = 0 are given by (see [2, 8, 31, 33]) K1 = b0,0c1,0 a0,0d1,0 , K2 = b0,1c0,0 a0,0d0,1 . By virtue of these formulas the invariants K1ψ, K2ψ and K1g, K2g of equations (5.3), (5.7) are, respectively, K1ψ = tj2,0t j 1,1 tj1,0t j 2,1 , K2ψ = tj+1 1,1 t j+1 0,2 tj+1 0,1 t j+1 1,2 , K1g = tj+1 0,1 t j+1 1,0 tj+1 0,0 t j+1 1,1 , K2g = tj1,0t j 0,1 tj0,0t j 1,1 . It is known that two linear hyperbolic type equations are related to one another by a linear transformation only if their corresponding Laplace invariants are equal. Evidently in generic Affine and Finite Lie Algebras and Integrable Toda Field Equations 21 case coincidence of the Laplace invariants generates two constraints on the field variables tj = tj(n,m). Only for some special cases it gives only one constraint. We are interested in such special cases. For instance pair of equations K1ψ(n,m, j) = K1g(n+ 1,m, j− 1), K2ψ(n,m, j) = K2g(n+ 1,m, j − 1) is equivalent to the constraint tj−11,0 = tj+1 0,1 which is interpreted as a cutting off boundary condition for the chain (5.1). For simplicity we put j = 0, so the boundary condition becomes t−11,0 = t10,1. (5.8) Lemma 5.1. Hirota equation (5.1) is compatible with the reduction of the type of parity t−mm+i,k = tmi,m+k and boundary condition (5.8) is a consequence of this reduction. Following [15] we can construct a Lax pair for the reduced chain. Under the boundary condition (5.8) we have coincidence of the invariants K1ψ(n,m, j) = K1g(n+ 1,m, j − 1), K2ψ(n,m, j) = K2g(n+ 1,m, j − 1), and K1g(n,m, j) = K1ψ(n,m− 1, j − 1), K2g(n,m, j) = K2ψ(n,m− 1, j − 1), and we can relate the eigenfunctions g10,1 = λψ0 0,0, g01,0 = λψ1 0,0. We study the finite reductions of the chain (5.1) on a finite interval NL ≤ j ≤ NR. The reduction is obtained by imposing the boundary conditions at the left end-point j = NL tNL−1 1,0 = tNL+1 0,1 (5.9) and respectively at the right end-point j = NR tNR+1 0,1 = tNR−1 1,0 . (5.10) First we concentrate on the left end-point. Due to the reasonings above the eigenfunctions should satisfy the following gluing conditions ψNL−1 0,0 = 1 λ gNL 0,1 , gNL−1 0,0 = λψNL −1,0. These conditions allow one to close the Lax equations at the left end-point gNL −1,0 = tNL+1 −1,0 tNL −1,0 tNL 0,0 t NL+1 −2,0 ( gNL 0,0 + λψNL −1,0 ) , ψNL 0,1 = ψNL 0,0 + 1 λ tNL+1 0,1 tNL+1 −1,1 tNL 0,0 t NL 0,1 gNL 0,1 . (5.11) From (5.2), (5.6) we have gNL 0,−1 = tNL+1 0,−1 tNL+1 −1,0 tNL+1 0,0 tNL+1 −1,−1 gNL 0,0 − tNL 0,−1t NL+2 −1,0 tNL+1 0,0 tNL+1 −1,−1 gNL+1 0,0 , ψNL 1,0 = tNL+1 1,0 tNL 0,0 tNL+1 0,0 tNL 1,0 ψNL 0,0 − ψ NL+1 0,0 . 22 R. Garifullin, I. Habibullin and M. Yangubaeva To derive similar equations at the point NR we use the right end-point constraint (5.10), for which we have gNR 0,0 = ψNR−1 0,−1 , ψNR 0,0 = gNR−1 1,0 . These conditions allow one to close the Lax equations at the right end-point ψNR−1 1,0 = tNR−1 0,0 tNR 1,0 tNR−1 1,0 tNR 0,0 ψNR−1 0,0 − gNR−1 1,0 , gNR−1 0,−1 = tNR 0,−1t NR −1,0 tNR 0,0 t NR −1,−1 gNR−1 0,0 − ( tNR−1 0,−1 )2 tNR 0,0 t NR −1,−1 ψNR−1 0,−1 . (5.12) From (5.2), (5.6) we have gNR−1 −1,0 = tNR −1,0t NR−1 −1,0 tNR−1 0,0 tNR −2,0 ( gNR−1 0,0 + gNR−2 0,0 ) , ψNR−1 0,1 = ψNR−1 0,0 + tNR 0,1 t NR−2 0,0 tNR−1 0,0 tNR−1 0,1 ψNR−2 0,0 . The Lax pair found above is not convenient to work with because the operators contain shifts in opposite directions. Below we show that it can be rewritten in a usual form. Shift of equation (5.11) forward with respect to the variable n brings it to the form gNL 1,0 = tNL+1 −1,0 tNL 1,0 tNL 0,0 t NL+1 0,0 gNL 0,0 − λψ NL 0,0 , gj1,0 = tj+1 −1,0t j 1,0 tj0,0t j+1 0,0 gj0,0 − g j−1 1,0 = j∑ k=NL (−1)j−k tk1,0t k+1 −1,0 tk0,0t k+1 0,0 gk0,0 + (−1)j+1λψNL 0,0 , NL + 1 ≤ j ≤ NR − 1. Shift of equation (5.12) forward with respect to the variable m brings it to the form gNR−1 0,1 = tNR 0,1 t NR −1,0 tNR 0,0 t NR −1,1 gNR−1 0,0 + ( tNR−1 0,0 )2 tNR 0,0 t NR −1,1 ψNR−1 0,0 , gj0,1 = tj+1 −1,0t j+1 0,1 tj+1 0,0 t j+1 −1,1 gj0,0 + tj+1 −1,0t j+1 0,1 tj+1 0,0 t j+1 −1,1 gj+1 0,1 = = tj+1 −1,0t j+1 0,1 tj+1 0,0 t j+1 −1,1 gj0,0 + NR−1∑ k=j+1 tj0,0t k+1 −1,0t k+1 0,1 tj+1 −1,1t k 0,0t k+1 0,0 gk0,0 + tj0,0t NR−1 0,0 tj+1 0,0 t NR 0,0 ψNR−1 0,0 , NL ≤ j ≤ NR − 1. So we have the following system of linear equations ψj1,0 = tj+1 1,0 t j 0,0 tj+1 0,0 t j 1,0 ψj0,0 − ψ j+1 0,0 , NL ≤ j ≤ NR − 2, (5.13) ψNR−1 1,0 = tNR−1 0,0 tNR 1,0 tNR−1 1,0 tNR 0,0 ψNR−1 0,0 + NR−1∑ k=NL (−1)NR−k tk1,0t k+1 −1,0 tk0,0t k+1 0,0 gk0,0 + (−1)NR−1λψNL 0,0 , (5.14) ψNL 0,1 = ψNL 0,0 + 1 λ tNL+1 0,1 tNR−1 0,0 tNL 0,1 t NR 0,0 ψNR−1 0,0 + NR−1∑ j=NL 1 λ tNL+1 0,1 tk+1 0,1 t k+1 −1,0 tNL 0,1 t k 0,0t k+1 0,0 gk0,0, (5.15) Affine and Finite Lie Algebras and Integrable Toda Field Equations 23 ψj0,1 = ψj0,0 + tj0,1t j−2 0,0 tj−10,0 t j−1 0,1 ψj−10,0 , NL + 1 ≤ j ≤ NR − 1, (5.16) gj1,0 = j∑ k=NL (−1)j−k tk1,0t k+1 −1,0 tk0,0t k+1 0,0 gk0,0 + (−1)j+1λψNL 0,0 , NL ≤ j ≤ NR − 1, (5.17) gj0,1 = tj+1 −1,0t j+1 0,1 tj+1 0,0 t j+1 −1,1 gj0,0 + NR−1∑ k=j+1 tj0,0t k+1 −1,0t k+1 0,1 tj+1 −1,1t k 0,0t k+1 0,0 gk0,0 + tj0,0t NR−1 0,0 tj+1 0,0 t NR 0,0 ψNR−1 0,0 , NL ≤ j ≤ NR − 1. (5.18) In the next subsection we gather these equations to a matrix form. 5.1 Lax pair for systems corresponding to the algebras D (2) N , A (1) 1 Imposing of non-degenerate boundary conditions (5.9), (5.10) at both end-points NL = 0 and NR = N leads to the system corresponding to the matrix D (2) N+1, N ≥ 2 t00,0t 0 1,1 − t01,0t00,1 = ( t10,1 )2 , tj0,0t j 1,1 − t j 1,0t j 0,1 = tj−11,0 t j+1 0,1 , 1 ≤ j ≤ N − 1, (5.19) tN0,0t N 1,1 − tN1,0tN0,1 = ( tN−11,0 )2 . Write the set of equations (5.13)–(5.18) in the form of a Lax pair for system (5.19). Denote the eigenvector as follow P = (ψ0, ψ1, . . . , ψN−1, g0, g1, . . . , gN−1)T . Introduce 2N × 2N matrices A =  t11,0t 0 0,0 t10,0t 0 1,0 −1 0 . . . 0 0 . . . 0 0 t21,0t 1 0,0 t20,0t 1 1,0 −1 . . . 0 0 . . . 0 . . . (−1)N−1λ 0 . . . tN1,0t N−1 0,0 tN0,0t N−1 1,0 (−1)N t01,0t 1 −1,0 t00,0t 1 0,0 (−1)N−1 t11,0t 2 −1,0 t10,0t 2 0,0 . . . − tN−1 1,0 tN−1,0 tN−1 0,0 tN0,0 −λ 0 . . . 0 t01,0t 1 −1,0 t00,0t 1 0,0 0 . . . 0 λ 0 . . . 0 − t01,0t 1 −1,0 t00,0t 1 0,0 t11,0t 2 −1,0 t10,0t 2 0,0 . . . 0 . . . (−1)i+1λ 0 . . . 0 . . . (−1)i−j tj1,0t j+1 −1,0 tj0,0t j+1 0,0 . . . 0 . . . (−1)Nλ 0 . . . 0 (−1)N−1 t01,0t 1 −1,0 t00,0t 1 0,0 (−1)N t11,0t 2 −1,0 t10,0t 2 0,0 . . . tN−1 1,0 tN−1,0 tN−1 0,0 tN0,0  , B =  1 0 . . . 1 λ t10,1t N−1 0,0 t00,1t N 0,0 1 λ (t10,1) 2t1−1,0 t00,0t 0 0,1t 1 0,0 1 λ t10,1t 2 −1,0t 2 0,1 t00,1t 1 0,0t 2 0,0 . . . 1 λ t10,1t N −1,0t N 0,1 t00,1t N−1 0,0 tN0,0 t00,0t 2 0,1 t10,0t 1 0,1 1 . . . 0 0 0 . . . 0 . . . 0 . . . tN−2 0,0 tN0,1 tN−1 0,0 tN−1 0,1 1 0 0 . . . 0 0 . . . 0 t00,0t N−1 0,0 t1−1,1t N 0,0 t10,1t 1 −1,0 t10,0t 1 −1,1 t00,0t 2 −1,0t 2 0,1 t1−1,1t 1 0,0t 2 0,0 . . . t00,0t N −1,0t N 0,1 t1−1,1t N−1 0,0 tN0,0 0 . . . 0 t10,0t N−1 0,0 t2−1,1t N 0,0 0 t2−1,0t 2 0,1 t20,0t 2 −1,1 . . . t10,0t N −1,0t N 0,1 t2−1,1t N−1 0,0 tN0,0 . . . 0 . . . 0 (tN−1 0,0 )2 tN−1,1t N 0,0 0 0 . . . tN−1,0t N 0,1 tN0,0t N −1,1  . 24 R. Garifullin, I. Habibullin and M. Yangubaeva It is straightforward to check that the compatibility condition of the equations P1,0 = AP, P0,1 = BP (5.20) is equivalent the system (5.19). Example 5.2. Consider a particular case when NL = 0 and NR = 1 t00,0t 0 1,1 − t01,0t00,1 = ( t10,1 )2 , t10,0t 1 1,1 − t11,0t10,1 = ( t01,0 )2 . The system corresponds to algebra A (1) 1 . Its Lax pair is given by (5.20) with the matrices A =  t11,0t 0 0,0 t10,0t 0 1,0 + λ − t01,0t 1 −1,0 t00,0t 1 0,0 −λ t01,0t 1 −1,0 t00,0t 1 0,0  , B = 1 + 1 λ t10,1t 0 0,0 t00,1t 1 0,0 1 λ (t10,1) 2t1−1,0 t00,0t 0 0,1t 1 0,0 (t00,0) 2 t10,0t 1 −1,1 t10,1t 1 −1,0 t10,0t 1 −1,1  . Example 5.3. If one imposes condition (5.9) for NL = 0 and (5.10) for NR = 2, then t00,0t 0 1,1 − t01,0t00,1 = ( t10,1 )2 , t10,0t 1 1,1 − t11,0t10,1 = t01,0t 2 0,1, t20,0t 2 1,1 − t21,0t20,1 = ( t11,0 )2 . The Lax pair is of the form (5.20) where A and B are 4× 4 matrices A =  t11,0t 0 0,0 t10,0t 0 1,0 −1 0 0 −λ t21,0t 1 0,0 t20,0t 1 1,0 t01,0t 1 −1,0 t00,0t 1 0,0 − t11,0t 2 −1,0 t10,0t 2 0,0 −λ 0 t01,0t 1 −1,0 t00,0t 1 0,0 0 λ 0 − t01,0t 1 −1,0 t00,0t 1 0,0 t11,0t 2 −1,0 t10,0t 2 0,0  , B =  1 1 λ t10,1t 1 0,0 t00,1t 2 0,0 1 λ (t10,1) 2t1−1,0 t00,0t 0 0,1t 1 0,0 1 λ t10,1t 2 −1,0t 2 0,1 t00,1t 1 0,0t 2 0,0 t00,0t 2 0,1 t10,0t 1 0,1 1 0 0 0 t00,0t 1 0,0 t1−1,1t 2 0,0 t10,1t 1 −1,0 t10,0t 1 −1,1 t00,0t 2 −1,0t 2 0,1 t1−1,1t 1 0,0t 2 0,0 0 (t10,0) 2 t2−1,1t 2 0,0 0 t2−1,0t 2 0,1 t20,0t 2 −1,1  . Remark 5.4. In Example 5.2 we give a Lax pair realized in 2 × 2 matrices, while general formula (5.20) generates 3 × 3 matrices. The matter is that in the Lax pair obtained directly from (5.20) we made in this case some additional reduction. 5.2 Lax pair for systems corresponding to the algebras AN Instead of boundary condition (5.8) we can use also the degenerate boundary conditions of the form tNL−1 = 0, tNL = 1 (5.21) at the left end-point and tNR+1 = 0, tNR = 1 (5.22) at the right end-point. The degenerate boundary conditions imply that the corresponding eigen- functions are zero: gNL−1 = 0 and ψNR = 0. Affine and Finite Lie Algebras and Integrable Toda Field Equations 25 In order to obtain the system corresponding to the Cartan matrix AN−1 we cut off the Hirota chain by imposing degenerate boundary conditions (5.21) at the point NL = 0 and (5.22) at NR = N . The resulting reduction is as follows t10,0t 1 1,1 − t11,0t10,1 = t20,1, tj0,0t j 1,1 − t j 1,0t j 0,1 = tj−11,0 t j+1 0,1 , 2 ≤ j ≤ N − 2, (5.23) tN−10,0 tN−11,1 − tN−11,0 tN−10,1 = tN−21,0 . In this case our algorithm gives the Lax pair found years ago in [21]. In order to formulate it introduce the eigenvector P = (ψ0, ψ1, . . . , ψN−1)T and N ×N matrices U =  t11,0 t10,0 −1 . . . 0 0 0 t21,0t 1 0,0 t20,0t 1 1,0 . . . 0 0 . . . 0 0 . . . tN−1 1,0 tN−2 0,0 tN−1 0,0 tN−2 1,0 −1 0 0 . . . 0 tN−1 0,0 tN−1 1,0  , V =  1 0 0 . . . 0 0 t20,1 t10,0t 1 0,1 1 0 . . . 0 0 0 t30,1t 1 0,0 t20,0t 2 0,1 1 . . . 0 0 . . . 0 0 . . . 0 tN−2 0,0 tN−1 0,0 tN−1 0,1 1  . It is easy to check that the compatibility condition of the equations P1,0 = UP, P0,1 = V P. leads to the system (5.23). 5.3 Lax pair for systems corresponding to the algebras BN We impose boundary condition (5.21) for NL = 0 and (5.10) for NR = N . The resulting reduction is as follows t10,0t 1 1,1 − t11,0t10,1 = t20,1, tj0,0t j 1,1 − t j 1,0t j 0,1 = tj−11,0 t j+1 0,1 , 2 ≤ j ≤ N − 1, (5.24) tN0,0t N 1,1 − tN1,0tN0,1 = ( tN−11,0 )2 . System (5.24) can be rewritten in form of (1.3) by changing the variables tj = e−u j . It corre- sponds to the algebra BN . The system admits a Lax pair. Let us denote P = (ψ0, ψ1, . . . , ψN−1, g0, g1, . . . , gN−1)T and introduce 2N × 2N matrices A =  t11,0 t10,0 −1 0 . . . 0 0 . . . 0 0 t21,0t 1 0,0 t20,0t 1 1,0 −1 . . . 0 0 . . . 0 . . . 0 0 . . . tN1,0t N−1 0,0 tN0,0t N−1 1,0 (−1)N t1−1,0 t10,0 (−1)N−1 t11,0t 2 −1,0 t10,0t 2 0,0 . . . − tN−1 1,0 tN−1,0 tN−1 0,0 tN0,0 0 0 . . . 0 t1−1,0 t10,0 0 . . . 0 0 0 . . . 0 − t1−1,0 t10,0 t11,0t 2 −1,0 t10,0t 2 0,0 . . . 0 . . . 0 0 . . . 0 (−1)N−1 t1−1,0 t10,0 (−1)N t11,0t 2 −1,0 t10,0t 2 0,0 . . . tN−1 1,0 tN−1,0 tN−1 0,0 tN0,0  , (5.25) 26 R. Garifullin, I. Habibullin and M. Yangubaeva B =  1 0 . . . 0 0 0 . . . 0 t20,1 t10,0t 1 0,1 1 . . . 0 0 0 . . . 0 0 t10,0t 3 0,1 t20,0t 2 0,1 . . . 0 0 0 . . . 0 . . . 0 . . . tN−2 0,0 tN0,1 tN−1 0,0 tN−1 0,1 1 0 0 . . . 0 0 . . . 0 tN−1 0,0 t1−1,1t N 0,0 t10,1t 1 −1,0 t10,0t 1 −1,1 t2−1,0t 2 0,1 t1−1,1t 1 0,0t 2 0,0 . . . tN−1,0t N 0,1 t1−1,1t N−1 0,0 tN0,0 0 . . . 0 t10,0t N−1 0,0 t2−1,1t N 0,0 0 t2−1,0t 2 0,1 t20,0t 2 −1,1 . . . t10,0t N −1,0t N 0,1 t2−1,1t N−1 0,0 tN0,0 . . . 0 . . . 0 (tN−1 0,0 )2 tN−1,1t N 0,0 0 0 . . . tN−1,0t N 0,1 tN0,0t N −1,1  . (5.26) Then according to our general scheme the compatibility condition of the equations P1,0 = AP, P0,1 = BP leads to the system (5.24). Remark 5.5. The system BN can be obtained from the system A2N−1 by imposing the cutting off constraint of the form tN+1 0,1 = tN−11,0 (see Lemma 5.1 above). 6 Method of finding integrals from Lax representation for the systems corresponding to the Cartan matrices AN , BN In this section we show that the Lax pair allows one to generate integrals for the systems corresponding to the simple Lie algebras AN , BN . Concentrate on m-integrals. Due to the definition we have two different expression for the shifted eigenvector Pk,1, k ≥ 1 Pk,1 = Vk,0Uk−1,0Uk−2,0 · · ·U1,0U0,0P, and similarly Pk,1 = Uk−1,1Uk−2,1 · · ·U1,1U0,1V0,0P. Comparison of these two formulas yields Dm(Uk−1,0Uk−2,0 · · ·U1,0U0,0) = Vk,0Uk−1,0Uk−2,0 · · ·U1,0U0,0V −1 0,0 . Due to the triangular structure of the matrices Vk,0 and V0,0 the map converting any upper triangular matrix X to a matrix X̄ = Vk,0XV −1 0,0 leaves unchanged the element of the matrix X located at the right upper corner: (X)1,N = X̄1,N . Thus the corresponding element, denote it through I(k−N) (k > N), of the upper triangular matrix Uk−1,0Uk−2,0 · · ·U1,0U0,0 is an m- integral. In such a way we get a set of integrals I(1), I(2), . . . , I(N). Examples below show that they constitute a complete set of m-integrals, however we are not able to prove this fact in general. In a similar way one can find integrals in the other direction. Let us illustrate the statement above with the following Affine and Finite Lie Algebras and Integrable Toda Field Equations 27 Example 6.1. Consider the system (5.23) for N = 3 t10,0t 1 1,1 − t11,0t10,1 = t20,1, t20,0t 2 1,1 − t21,0t20,1 = t11,0. Recall its Lax pair P1,0 = UP, P0,1 = V P, where P = (ψ0, ψ1, ψ2)T and U =  t11,0 t10,0 −1 0 0 t21,0t 1 0,0 t20,0t 1 1,0 −1 0 0 t20,0 t21,0  , V =  1 0 0 t20,1 t10,0t 1 0,1 1 0 0 t10,0 t20,0t 2 0,1 1  . Evaluate the elements at the right upper corner for the following two products U2,0U1,0U0,0 and U3,0U2,0U1,0U0,0 and find two independent m-integrals I(1) = t13 t12 + t11t 2 2 t12t 2 1 + t20 t21 , I(2) = t14 t13 I(1) + t11t 2 3 t13t 2 1 + t12t 2 0t 2 3 t13t 2 1t 2 2 + t20 t22 , where the second index for the variables t1, t2 is omitted. Since the integral I(2) is too complicated one can choose a more simple one Ĩ(2) = D−1n ( I(1)DnI(1) − I(2) ) = t10 t11 + t12t 2 0 t11t 2 1 + t22 t21 . By using Theorem 1.2 one can check that integrals I(1) and Ĩ(2) provide a complete set of integrals. In a similar way integrals for the system (5.24), corresponding to the algebra BN , are con- structed. Let us consider a matrix Φ of dimension 2N × 2N Φ = ( E11 E12 E21 E22 ) , (6.1) where E11 is the unity matrix of dimension N , E12, E21 are matrices of dimension N with zero entries, E22 is a matrix of dimension N with the unity adverse diagonal. By an automorphism X → ΦXΦ−1 the matrices (5.25) and (5.26) are transformed to the triangular matrices Ā =  t11,0 t10,0 −1 0 . . . 0 0 . . . 0 0 t21,0t 1 0,0 t20,0t 1 1,0 −1 . . . 0 0 . . . 0 . . . 0 0 . . . tN1,0t N−1 0,0 tN0,0t N−1 1,0 − tN−1 1,0 tN−1,0 tN−1 0,0 tN0,0 tN−2 1,0 tN−1 −1,0 tN−2 0,0 tN−1 0,0 . . . (−1)N t1−1,0 t10,0 0 0 . . . 0 tN−1 1,0 tN−1,0 tN−1 0,0 tN0,0 − tN−2 1,0 tN−1 −1,0 tN−2 0,0 tN−1 0,0 . . . (−1)N−1 t1−1,0 t10,0 0 0 . . . 0 0 tN−2 1,0 tN−1 −1,0 tN−2 0,0 tN−1 0,0 . . . (−1)N t1−1,0 t10,0 . . . 0 0 . . . 0 0 0 . . . t1−1,0 t10,0  , 28 R. Garifullin, I. Habibullin and M. Yangubaeva B̄ =  1 0 . . . 0 0 0 . . . 0 t20,1 t10,0t 1 0,1 1 . . . 0 0 0 . . . 0 0 t10,0t 3 0,1 t20,0t 2 0,1 . . . 0 0 0 . . . 0 . . . 0 . . . tN−2 0,0 tN0,1 tN−1 0,0 tN−1 0,1 1 0 0 . . . 0 0 . . . 0 tN−1 0,0 t1−1,1t N 0,0 tN−1,0t N 0,1 tN0,0t N −1,1 0 . . . 0 0 . . . 0 t10,0t N−1 0,0 t2−1,1t N 0,0 tN−2 0,0 tN−1,0t N 0,1 tN−1 −1,1 t N−1 0,0 tN0,0 tN−1 −1,0 t N−1 0,1 tN−1 0,0 tN−1 −1,1 . . . 0 . . . 0 . . . 0 (tN−1 0,0 )2 tN−1,1t N 0,0 tN−1,0t N 0,1 t1−1,1t N−1 0,0 tN0,0 tN−1 −1,0 t N−1 0,1 t1−1,1t N−2 0,0 tN−1 0,0 . . . t10,1t 1 −1,0 t10,0t 1 −1,1  . (6.2) By an automorphism acting as follows X → F−1XF matrix (6.2) is transformed to a lower triangular matrix for which all diagonal entries are equal to the unity. Here F is a 2N × 2N matrix of the form F = ( E11 E12 E21 F22 ) , F22 is a diagonal matrix such that F22 = diag ( tN0,0 tN−1,0 , tN−1 0,0 tN−1 −1,0 , . . . , t10,0 t1−1,0 ) and Ei,j are defined in (6.1). Reasonings similar to that of the case AN allow one to derive the integrals. Example 6.2. Consider the system (5.24) for N = 2 t10,0t 1 1,1 − t11,0t10,1 = t20,1, t20,0t 2 1,1 − t21,0t20,1 = ( t11,0 )2 . Recall its Lax pair P1,0 = AP, P0,1 = BP, where P = (ψ0, ψ1, g0, g1)T and A =  t11,0 t10,0 −1 0 0 0 t21,0t 1 0,0 t20,0t 1 1,0 t1−1,0 t10,0 − t11,0t 2 −1,0 t10,0t 2 0,0 0 0 t1−1,0 t10,0 0 0 0 − t1−1,0 t10,0 t11,0t 2 −1,0 t10,0t 2 0,0  , B =  1 0 0 0 t20,1 t10,0t 1 0,1 1 0 0 0 t10,0 t1−1,1t 2 0,0 t10,1t 1 −1,0 t10,0t 1 −1,1 t2−1,0t 2 0,1 t1−1,1t 1 0,0t 2 0,0 0 (t10,0) 2 t2−1,1t 2 0,0 0 t2−1,0t 2 0,1 t20,0t 2 −1,1  . By an automorphism X → ΦXΦ−1 transform the matrices A and B to the triangular matrices Ā =  t11,0 t10,0 −1 0 0 0 t21,0t 1 0,0 t20,0t 1 1,0 − t11,0t 2 −1,0 t10,0t 2 0,0 t1−1,0 t10,0 0 0 t11,0t 2 −1,0 t10,0t 2 0,0 − t1−1,0 t10,0 0 0 0 t1−1,0 t10,0  , B̄ =  1 0 0 0 t20,1 t10,0t 1 0,1 1 0 0 0 (t10,0) 2 t2−1,1t 2 0,0 t2−1,0t 2 0,1 t20,0t 2 −1,1 0 0 t10,0 t1−1,1t 2 0,0 t2−1,0t 2 0,1 t1−1,1t 1 0,0t 2 0,0 t1−1,0t 1 0,1 t1−1,1t 1 0,0  . (6.3) Affine and Finite Lie Algebras and Integrable Toda Field Equations 29 By automorphism X → F−1XF transform the matrices (6.3) to the triangular matrices Â =  t11,0 t10,0 −1 0 0 0 t21,0t 1 0,0 t20,0t 1 1,0 − t11,0 t10,0 1 0 0 t11,0t 2 0,0 t10,0t 2 1,0 − t20,0 t21,0 0 0 0 t10,0 t11,0  , B̂ =  1 0 0 0 t20,1 t10,0t 1 0,1 1 0 0 0 (t10,0) 2 t20,0t 2 0,1 1 0 0 t10,0 t10,1t 2 0,0 t20,1 t10,0t 1 0,1 1  . Evaluate the elements at the right upper corner for the following products Â2,0Â1,0Â0,0 and Â3,0Â2,0Â1,0Â0,0 and find two independent m-integrals I(1) = − t 1 3 t12 − t12t 2 0 t11t 2 1 − t11t 2 2 t12t 2 1 − t10 t11 , I(2) = − t 1 4 t12 − t11t 2 2t 1 4 t12t 2 1t 1 3 − t11t 2 3 t13t 2 1 − t12t 2 0t 1 4 t11t 2 1t 1 3 − (t12) 2t20t 2 3 t11t 2 1t 1 3t 2 2 − t13t 2 0 t11t 2 2 − t10t 1 4 t11t 1 3 − t10t 1 2t 2 3 t11t 2 2t 1 3 − t10t 1 3t 2 1 t11t 2 2t 1 2 − t10 t12 . Replace the integral I(2) by a more simple one Ĩ(2) = I(2) + I(1)DnI(1) = t20 t21 + (t11) 2t22 (t12) 2t21 + 2 t13t 1 1 (t12) 2 + (t13) 2t21 (t12) 2t22 + t23 t22 . Here the second index for the variables t1, t2 is omitted, since its values is zero for all considered variables. It can be proved by using Theorem 1.2 that integrals I(1) and Ĩ(2) constitute a complete set of integrals. 7 Periodic boundary conditions In this section we discuss briefly the well known periodically closed reduction of the Hirota chain (see for more details and the references [41]). Close the chain (5.1) by imposing the periodical boundary conditions t−1 = tN , tN+1 = t0. Close the Lax pair (5.2) by setting the conditions on the eigenfunctions ψ−1 = λψN , ψN+1 = 1 λ ψ0. (7.1) As a result we get a finite system of the form t00,0t 0 1,1 − t01,0t00,1 = tN1,0t 1 0,1, tj0,0t j 1,1 − t j 1,0t j 0,1 = tj−11,0 t j+1 0,1 , 1 ≤ j ≤ N − 1, (7.2) tN0,0t N 1,1 − tN1,0tN0,1 = tN−11,0 t00,1, which is closely connected with the Cartan matrix A (1) N . Boundary conditions (7.1) reduce the sequence of linear discrete equations (5.2) to a Lax pair for the reduced system (7.2). Introduce the eigenvector P = (ψ0, ψ1, . . . , ψN )T and N+1×N+1 matrices U =  t00,0t 1 1,0 t01,0t 1 0,0 −1 0 . . . 0 0 t21,0t 1 0,0 t20,0t 1 1,0 −1 . . . 0 0 0 t31,0t 2 0,0 t30,0t 2 1,0 . . . 0 . . . − 1 λ 0 . . . 0 tN0,0t 0 1,0 tN1,0t 0 0,0  , V =  1 0 . . . 0 λ t10,1t N 0,0 t00,0t 0 0,1 t20,1t 0 0,0 t10,0t 1 0,1 1 . . . 0 0 0 t30,1t 1 0,0 t20,0t 2 0,1 . . . 0 0 . . . 0 0 . . . tN−1 0,0 t00,1 tN0,0t N 0,1 1  . 30 R. Garifullin, I. Habibullin and M. Yangubaeva If the functions t0, t1, . . . , tN solve the system (7.2) then the overdetermined system of linear equations P1,0 = UP, P0,1 = V P (7.3) is compatible (see [41]). Remark 7.1. Note that system (7.2) differs from the system corresponding to the same algeb- ra A (1) N but defined by the formula (1.3). For example the system (7.2) for N = 2 is of the form t00,0t 0 1,1 − t01,0t00,1 = t21,0t 1 0,1, t10,0t 1 1,1 − t11,0t10,1 = t01,0t 2 0,1, t20,0t 2 1,1 − t21,0t20,1 = t11,0t 0 0,1. (7.4) In terms of the variables u0 = − log t0, u1 = − log t1, u2 = − log t2 it looks like ∆ ( u0 ) = eu 0 1,0+u 0 0,1−u10,1−u21,0 , ∆ ( u1 ) = e−u 0 1,0+u 1 1,0+u 1 0,1−u20,1 , ∆ ( u2 ) = e−u 0 0,1−u11,0+u21,0+u20,1 , (7.5) while formula (1.3) gives the system ∆ ( u0 ) = eu 0 1,0+u 0 0,1−u11,0−u21,0 , ∆ ( u1 ) = e−u 0 0,1+u 1 1,0+u 1 0,1−u21,0 , ∆ ( u2 ) = e−u 0 0,1−u10,1+u21,0+u20,1 . After permutations n↔ m we get ∆ ( u0 ) = eu 0 1,0+u 0 0,1−u10,1−u20,1 , ∆ ( u1 ) = e−u 0 1,0+u 1 1,0+u 1 0,1−u20,1 , ∆ ( u2 ) = e−u 0 1,0−u11,0+u21,0+u20,1 . (7.6) Obviously, systems (7.5) and (7.6) are different. Let us give also the Lax pair for the system (7.4). Let us consider vector P = (ψ0, ψ1, ψ2)T . Introduce 3× 3 matrices U =  t00,0t 1 1,0 t01,0t 1 0,0 −1 0 0 t21,0t 1 0,0 t20,0t 1 1,0 −1 − 1 λ 0 t20,0t 0 1,0 t21,0t 0 0,0  , V =  1 0 λ t10,1t 2 0,0 t00,0t 0 0,1 t20,1t 0 0,0 t10,0t 1 0,1 1 0 0 t00,1t 1 0,0 t20,0t 2 0,1 1  . If the functions t0, t1, t2 satisfy the system (7.4) then the equations P1,0 = UP, P0,1 = V P. are compatible. Remark 7.2. The Lax pair (7.3) can be rewritten in terms of the Cartan–Weyl basis of the algebra A (1) N P1,0 = ( −Λ + eU1,0−U0,0 ) P, P0,1 = ( E + eU0,1Λ̄e−U0,0 ) P, where Λ = N∑ i=0 fi, Λ̄ = N∑ i=0 ei, U = N∑ i=0 uihi, e −ui = ti, [hi, hj ] = 0, [ei, fj ] = δijhi, [hi, ej ] = Aijej , [hi, fj ] = −Aijfj , and Aij are elements of the Cartan matrix of the algebra A (1) N . Affine and Finite Lie Algebras and Integrable Toda Field Equations 31 8 Conclusions A map is suggested converting any N ×N matrix A to a finite system of difference-difference equations of exponential type (see (1.3)). A hypothesis is formulated claiming that if A coin- cides with the Cartan matrix of a finite or affine Lie algebra then the corresponding system is integrable. The hypothesis is approved by numerous examples. The systems obtained are rather simple and elegant. They essentially differ from those studied earlier (see survey [25] and references therein). For instance, the system corresponding the algebra G2 given in [25] reads as T (1) m (u− 1)T (1) m (u+ 1) = T (1) m−1(u)T (1) m+1(u) + T (2) 3m(u), T (2) 3m ( u− 1 3 ) T (2) 3m ( u+ 1 3 ) = T (2) 3m−1(u)T (2) 3m+1(u) + T (1) m ( u− 2 3 ) T (1) m (u)T (1) m ( u+ 2 3 ) , T (2) 3m+1 ( u− 1 3 ) T (2) 3m+1 ( u+ 1 3 ) = T (2) 3m(u)T (2) 3m+2(u) + T (1) m ( u− 1 3 ) T (1) m ( u+ 1 3 ) T (1) m+1(u), T (2) 3m+2 ( u− 1 3 ) T (2) 3m+2 ( u+ 1 3 ) = T (2) 3m+1(u)T (2) 3m+3(u) + T (1) m (u)T (1) m+1 ( u− 1 3 ) T (1) m+1 ( u+ 1 3 ) , while our formula (1.3) generates G2 system which can be represented as follows t1n,mt 1 n+1,m+1 − t1n+1,mt 1 n,m+1 = t2n,m+1, t2n,mt 2 n+1,m+1 − t2n+1,mt 2 n,m+1 = ( t1n+1,m )3 . In a recent article by Kimura, Yamashita and Nakamura [23] a new application of conserved quantities of discrete-time integrable systems to numerical computations is suggested. 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Phys. 113 (1997), 1347–1392, solv-int/9704001. http://dx.doi.org/10.1070/RM1997v052n05ABEH002105 http://arxiv.org/abs/math-ph/0003009 http://dx.doi.org/10.1016/0370-1573(81)90023-5 http://dx.doi.org/10.1016/0375-9601(95)00115-J http://arxiv.org/abs/1203.1764 http://dx.doi.org/10.1143/JPSJ.66.3391 http://arxiv.org/abs/solv-int/9610011 http://dx.doi.org/10.1016/0375-9601(95)00108-F http://arxiv.org/abs/solv-int/9502002 http://dx.doi.org/10.1007/BF02634165 http://arxiv.org/abs/solv-int/9704001 1 Introduction 1.1 Integrals and symmetries for quad graph systems 2 Systematic approach to the problem of discretization of the Darboux integrable systems 2.1 Explanation of the method with example of Liouville equation 2.2 Application of the algorithm of discretization to the system corresponding to the algebra B2 2.3 Integrals of the systems corresponding to the algebras A2, G2, D3 2.3.1 System corresponding to the algebra A2 2.3.2 System corresponding to the algebra G2 2.3.3 System corresponding to the algebra D3 3 Symmetries of discrete systems 3.1 Generalized symmetries for the system A2 3.2 Evaluation of hyperbolic type symmetries for the system A(1)1 3.3 Hyperbolic type symmetries for the system A(2)2 4 Characteristic m-algebra for the case A2 5 Cutting off constraints for the Hirota equation and discrete Zakharov-Shabat systems 5.1 Lax pair for systems corresponding to the algebras D(2)N, A(1)1 5.2 Lax pair for systems corresponding to the algebras AN 5.3 Lax pair for systems corresponding to the algebras BN 6 Method of finding integrals from Lax representation for the systems corresponding to the Cartan matrices AN, BN 7 Periodic boundary conditions 8 Conclusions References

Affine and Finite Lie Algebras and Integrable Toda Field Equations on Discrete Space-Time

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