Abelian Toda solitons revisited

Abelian T oda solitons revisite d Kh. S. Nir ov Institute for Nuclear Resear ch of the Russian Academy of Sciences 60th October Anniversary Pro spect 7a, 117312 Moscow , Russia A. V . Razumov Institute for High Ener gy Physics 14228 1 Pro tvino, Moscow Region, Russia Abstract W e present a syste matic and det ailed r e view of the application of the metho d o f Hirota and the rational dr e ssing method to abelian T oda syst e ms associated with the untwisted loop gr oups of complex general linear groups. Emphasizing the ra- tional d ressing method, we compare the soliton s olutions const ructed within these two approaches, and s how that the solutions obtained by the Hir ota’s method ar e a subset of those obtained by t he rational dressing meth o d. 1 Introduc tion T wo-dimensional T oda equations associated with loop gr oups 1 ar e very inter esting examples of completely integrable systems, see, for example, the monog raphs [1, 2]. They possess solito n so lutions having a nice physical interpr etation as interacting ex- tended objects. Actually there is no a clear definition of a soliton solution. In the pr esent paper we call a solution of equations an n -soliton solutio n, if it depends on n linear combinations of indepe ndent variables. Soliton solutions for T oda equations can be constructed with t he help of various methods. As far as we know , first explicit solutions of T oda equations associated with loop gr oups wer e found by Mikhailov [3]. H e used the rational dr essing method be- ing a version of the inverse scattering method [4]. Note that in general the solutions obtained by Mikhailov ar e not soliton solutions. Besides, they ar e d e scribed by a r e- dundant set of parameters. Another method us ed here is the Hirota’s one. Its essence [5] is a change of the dependent variables which intro duces the so called τ -functions. Here the final goal is to come to some special bilinear partial d if fer ential equations which ar e solv ed then perturbatively . The soliton solutions arise when the perturbation series truncates at some finite or der . This method was a pplied to affi ne T oda systems, for e xample, in the papers [6, 7, 8, 9, 10]. The mai n disadvantage of t he Hir ota’s method is that ther e is no a r e gular method t o find the desir e d transformation fr om the initial dependent 1 Sometimes one d eals with T oda equations associated with affine groups being central e xtensions of loop groups. Usually it is possible to construct solutions of the e quations associated with affine groups starting from solutions of the equations a ssociated with loop groups. 1 variables to τ -functions. Therefor e, sometimes it is used in combination with other methods that helps to obtain a desir ed ansatz, see, for example, the papers [11, 12]. Ther e ar e also two additional appr oaches to the pr oblem, being a development of the Leznov–Saveliev metho d [13, 14, 15], and of the B ¨ acklund–Darboux transfo rma- tion [16, 17, 18, 19, 20]. These methods give the same so liton solutions as the Hir ota’s one a nd ar e not in the scope of the present p a per . Basic purpos e of our review is to r eproduce, in a possibly syst ematic and detailed way , the application of the Hirota’s and rational dr essing me thods to T oda systems associated with the unt wisted lo op gr oups of complex general linear gr oups, making an e mp ha sis on the rational dressing method, and compare the soliton solutions co n- stru cted along t hese appro aches. W e show that all soliton solutions obtained by the Hir ota’s me thod ar e contained among the solu tions obt ained by the rational dr e ssing method. 2 Equation s 2.1 Zero-curvature representation of T oda equations It is we ll kno wn that T oda equations can be formulated as the zero- c urvatur e co ndi- tion for a connection of a special form on t he trivial fiber bundle R 2 × G → R 2 , where G is a Lie gr oup with the Lie a lgebra G . The connection under c onsideration can be identified with a G -valued 1-form O on R 2 . One can decompose such a connection over basis 1-forms as O = O − d z − + O + d z + , wher e z − , z + ar e the standar d coor dinates on the base manifold R 2 , and the compo- nents O − , O + ar e G -valued functions on it. Let us assume that the connection O is fl at that me ans that its curvatur e is z e r o. This condition in terms of the components has the form 2 ∂ − O + − ∂ + O − + [ O − , O + ] = 0. (2.1) One can co nsider this r elation a s a system of pa rtial differ ential equations. In a sense, this system is trivial, and its general solution is well known. It is given by the relations O − = Φ − 1 ∂ − Φ , O + = Φ − 1 ∂ + Φ , wher e Φ is an arbitrary mapping of R 2 to G . A ctually the triviality of the zero-cu rvatu- r e condition is due to its gauge invariance. That means that if a connection O satis fies (2.1) then for an arbitrary mapping Ψ of R 2 to G the gauge transformed connection O Ψ = Ψ − 1 O Ψ + Ψ − 1 d Ψ , (2.2) satisfies (2.1) a s well. T o obtain a nontrivial integrable system out of the zero-c urvatur e condition one imposes on the connectio n O some rest riction destro ying the gauge invariance. Fo r the case of T oda eq uations they are the grading and gauge fixing conditions which are intr oduced as follows. 2 W e use the usual notation ∂ − = ∂ / ∂ z − and ∂ + = ∂ / ∂ z + . 2 Suppose that the Lie algebra G is e nd owed with a Z -gradation, G = M k ∈ Z G k , [ G k , G l ] ⊂ G k + l , and that a positive integer L is such that the grading subspaces G k , where 0 < | k | < L , ar e trivial. 3 The grading condition states that the components of O have the form O − = O − 0 + O − L , O + = O + 0 + O + L , (2.3) wher e O − 0 and O + 0 take values in G 0 , while O − L and O + L take values in G − L and G + L r espectively . Ther e is a residual gauge invariance. Inde ed, the gauge transforma- tion (2.2) with Ψ taking values in the Lie subgr oup G 0 corr esponding to the subalgebra G 0 does not violate the validity of the grading condition (2.3). The r efor e , one imposes an additional condition, called the gauge fixing condition, of the form O + 0 = 0. After that one can show that t he component s of the connection O can be r epr esented as O − = Ξ − 1 ∂ − Ξ + F − , O + = Ξ − 1 F + Ξ , (2.4) wher e Ξ is a mapp ing of R 2 to G 0 , F − and F + ar e some mappings of R 2 to G − L and G + L . One can easily get convinced that the zero-curv atur e condition is equivalent to the e quality 4 ∂ + ( Ξ − 1 ∂ − Ξ ) = [ F − , Ξ − 1 F + Ξ ] (2.5) and the relations ∂ + F − = 0, ∂ − F + = 0. (2.6) One supposes that the mappings F − and F + ar e fixed and considers (2.5) as an equa- tion for Ξ called the T oda equation. Whe n the gr oup G 0 is abelian the c orr esponding T oda equations are called abelian. Thus, a T oda equation associated with a Lie gr oup G is specified by a choice of a Z -gr adation of the Lie algebra G of G and mappings F − , F + satisfying the co ndi- tions (2.6). T o classify the T oda equations associated wit h a Lie gr oup G one should classify Z -gradations of the Lie algebra G of G . T wo r emarks are in order . Fir st, let Σ be an isomorphism fr om a Z -graded Lie algebra G to a Lie algebra H . One can consider H as a Z -graded Lie algebra with grading subspaces H k = Σ ( G k ) . In such situation, one says that Z -gradations of G and H ar e conjugated by Σ . Now let the Lie algebra G of the Lie gr oup G be supplied with a Z -gradation, and σ be an isomorphism fr om G to a Lie gr oup H . Denote by Σ the isomorphism fr om the Lie algebra G to the Lie algebra H of the Lie group H induced by the isomorphism σ . It is clear that if Ξ is a solution of the T oda equation (2.5), then the ma pping Ξ ′ = σ ◦ Ξ (2.7) 3 It ca n be shown that rejecting this restriction we do not come to new T oda equations , see the pa- per [21]. 4 W e assume for simplicity that G is a subgroup of the group formed by invertible elements of some unital associative algebra A . In this case G can be considered as a subalgebra of the Lie a lgebr a a ssoci- ated with A . Ac tua lly one can generalize our consideration to the case of an arbitrar y Lie group G . 3 satisfies the T oda equa tion (2.5) with the mappings F − , F + r eplaced by the mappings F ′ − = Σ ◦ F − , F ′ + = Σ ◦ F + . (2.8) In other wo r ds, the solutions to two T oda equations under consideration are con- nected via the isomorphism σ , and in this sense these eq uations ar e equivalent. Thus, to describe really differ e n t T oda equations it suffices to consider Lie gro ups and Z - gradations of their Lie algebras up to isomorphisms. Secondly , let Θ − and Θ + be some mappi ngs of R 2 to G 0 which satisfy the conditions ∂ + Θ − = 0, ∂ − Θ + = 0. If a mapping Ξ satisfies the T oda e quation (2.5), then the mapping Ξ ′ = Θ − 1 + Ξ Θ − , (2.9) satisfies the T oda eq uation (2.5) where the mappings F − , F + ar e replaced by the map- pings F ′ − = Θ − 1 − F − Θ − , F ′ + = Θ − 1 + F + Θ + . (2.10) Again, the T oda equations for Ξ and Ξ ′ ar e not actually differ ent, and it is natur al to use the transformations (2.10) to make the ma p pings F − , F + as simple as possible. If the ma p p ings Θ − and Θ + ar e such that Θ − 1 − F − Θ − = F − , Θ − 1 + F + Θ + = F + . then the mapping Ξ ′ satisfies the same T oda equation as the mapping Ξ . Hence, in this case the transformatio n described by relations (2.9) is a symmetry transformation for the T oda equation und e r consideration. 2.2 T oda equations associated with loop groups of complex simple Lie groups Let g be a finite dimensional r eal or complex Lie algebra. The loop Lie algebra of g , denoted L ( g ) , is defined a lternatively either as the linear space C ∞ ( S 1 , g ) of smoo th mappings of the circle S 1 to g , or as the linear space C ∞ 2 π ( R , g ) of smooth 2 π -periodic mappings of the real line R to g with the Lie algebra operation defined in both cases pointwise, see, for example, [22, 23, 24]. I n this paper we adopt the second definition and think of the cir cle S 1 as consisting of co mplex numbers of modulus one. There is a convenient way to supply L ( g ) with the structur e of a Fr ´ echet space, so that the Lie algebra operation becomes a continuous mapping, see , for example, [25, 23, 24]. Now , let G be a finite dimensional Lie group with the Lie algebra g . W e de fine the loop group of G , denoted L ( G ) , as the set C ∞ ( S 1 , G ) of smo oth mappings o f S 1 to G with the group law defined pointwise. W e assume that L ( G ) is supplied with the stru ctur e of a Fr ´ echet manifold modeled on L ( g ) in su ch a way that it becomes a Lie gr oup, see, for example, [25, 23, 24]. The Lie algebra of the Lie gr oup L ( G ) is naturally identified with the loop Lie algebra L ( g ) . Let A be an automorphism of a finite dimensional Lie algebra g satisfying the rela- tion A M = id g for some positiv e integer M . 5 The twisted loop Lie a lgebra L A , M ( g ) is a 5 Note that we do not assume that M is th e order of the automorphism A . It can be an arbitrary multiple of the order . 4 subalgebra of the loop Lie algebra L ( g ) formed by elements ξ that satisfy the equa lity ξ ( ǫ M ¯ p ) = A ( ξ ( ¯ p ) ) , wher e ǫ M = e 2 π i / M is the M th principal r oot of unity . Similar ly , given an automor - phism a of a Li e gro up G that satisfies the r elation a M = id G , we define the twisted loop group L a , M ( G ) as the subgr oup of the loo p group L ( G ) formed by the elements χ satisfying the equality χ ( ǫ M ¯ p ) = a ( χ ( ¯ p ) ) . The Lie al gebra o f a t wisted lo op group L a , M ( G ) is na turally identified with t he twis- ted loop Lie algebra L A , M ( g ) , wher e we denote by A the automorphism of the Lie algebra g correspo nding to the automorphism a of the Lie group G . It is clea r that loop groups and loop Lie algebras ar e partial cases of twis ted loop gr oups and twisted loop Lie algebr as respectiv ely . Therefor e, below by a loop g r oup we mean either a usual loop gr oup or a twisted loop gro up, and by a loop Lie algebra we mean either a usual loop Lie algebra or a twisted loop Lie algebra. Now let us discuss the form of the T oda e q uations assoc iated with a loop gr oup L a , M ( G ) . First of all note that the group L a , M ( G ) and its Lie algebra L A , M ( g ) are infinite dimensional ma nifolds. It appears that it is convenient to reformulate the zero curvatur e repr esentation of the T oda equations associated with L a , M ( G ) in terms of finite dimensional manifolds. T o this end we use the so-called exponential law , see, for example, [26, 27]. Let M , N , P be three finite dimensional manifolds, and N be compact. Consider a smooth mapping F o f M to C ∞ ( N , P ) . This mapping induces a mapping f of M × N to P defined by the equality f ( ¯ m , ¯ n ) = ( F ( ¯ m )) ( ¯ n ) . It can be proved that the mapping f is smooth. Conversely , if one has a smooth ma p - ping of M × N to P , rever sing the above equality one defines a mapping of M to C ∞ ( N , P ) , and this mapping is also smooth. Thus, we have the following canonical identification C ∞ ( M , C ∞ ( N , P ) ) = C ∞ ( M × N , P ) . It is this equality that is called the exponential law . In t he case under consideration the connect ion components O − and O + entering the equality (2.1 ) ar e mappings of R 2 to the lo op Lie algebra L A , M ( g ) . W e will denote the correspo nding mappings of R 2 × S 1 to g by ω − and ω + , and call them also the connection co mponents. The mapping Φ gene rating the connection is a mapping of R 2 to L a , M ( G ) . Denoting the corr esponding mapping of R 2 × S 1 by ϕ we write ϕ − 1 ∂ − ϕ = ω − , ϕ − 1 ∂ + ϕ = ω + . (2.11) Having in mind that the mapping ϕ uniquely determines the m a pping Φ , we say that the ma pping ϕ also generates the connection under consideration. The r elations (2.4) a re equivalent to the equalities ω − = γ − 1 ∂ − γ + f − , ω + = γ − 1 f + γ , 5 wher e γ is a smooth mapping of R 2 × S 1 to G corr esponding to the mapping Ξ , f − and f + ar e smooth mappings of R 2 × S 1 to the Lie algebra g of G corresponding to the mappings F − and F + . The mappings f − and f + satisfy the conditions ∂ + f − = 0, ∂ − f + = 0, (2.12) which follow fr om the conditions (2.6). The T oda equation (2.5) in the case unde r consideration is equivalent to the equation ∂ + ( γ − 1 ∂ − γ ) = [ f − , γ − 1 f + γ ] . (2.13) T o classify T oda eq uations associated with loop groups one has to classify Z -gra- dations of the corr e sponding loop Lie a lgebras. This problem was partially solved in the paper [24], see also [21]. In these papers the case of l oop L ie algebras of complex simple Lie algebras was considered and for this case a wide class o f the so-called in- tegrable Z -gradations [24] with finite dimensional grading subspaces was descr ibed. Actually it was shown that when g is a co mplex simple Lie algebra any integrable Z - gradation of a loop Lie algebra L A , M ( g ) with finite dimensional grading subspaces is conjugated by an isomorphism to the standard gradation of another loop Lie algebra L A ′ , M ′ ( g ) , where the auto morphisms A a nd A ′ diff er by an inner a utomorphism of g . In particular , if A is an inner (outer) automorphism of g , then A ′ is also an inner (outer) automorphism of g . Assume now that G is a finite dime nsional complex simple Lie gr oup, then its Lie algebra g is a complex simple Lie algebra. Consider a T oda equation associated with a loop group L a , M ( G ) . The corr esponding Z - gradation of L A , M ( g ) is conjugated by an isomorphism to the standard Z -gradation of an appropriate loop Lie algebra L A ′ , M ′ ( g ) . Since the automorphisms A and A ′ diff er by an inner aut omorphism of g , the automorphism A ′ can be lifted to an automorphism a ′ of G , and the isomorphism fr om L A , M ( g ) to L A ′ , M ′ ( g ) under consideration can be lifted to an isomorphism fr om L a , M ( G ) to L a ′ , M ′ ( G ) . Actually this me ans that the initial T oda equation associated with L a , M ( G ) is equivalent to a T oda equation associated with L a ′ , M ′ ( G ) arising when we supply L A ′ , M ′ ( g ) with the standard Z -gradation. The grading subspaces for the standard Z -gradation of a loop Lie algebra L A , M ( g ) ar e L A , M ( g ) k = { ξ ∈ L A , M ( g ) | ξ = λ k x , A ( x ) = ǫ k M x } , wher e by λ we denote the r estriction of the standar d co or d ina te on C to S 1 . It is very useful to r ea lize that every a utomorphism A of the Lie a lgebra g satisfying the relation A M = id g induces a Z M -gradation of g with the grading subspaces 6 g [ k ] M = { x ∈ g | A ( x ) = ǫ k M x } , k = 0 , . . . , M − 1. V ice versa, any Z M -gradation of g de fines in an e vident way an automorphism A of g satisfying the relation A M = id g . A Z M -gradation of g is called an inner or o uter type gradation, if the associated automorphism A of g is of inner or outer type respect ively . In terms of the corr esponding Z M -gradation t he grading subspaces for the standar d Z -gradation of a loop Lie algebra L A , M ( g ) ar e L A , M ( g ) k = { ξ ∈ L A , M ( g ) | ξ = λ k x , x ∈ g [ k ] M } . 6 W e denote by [ k ] M the element of the ring Z M corresponding to the integer k . 6 It is evident that for the standar d Z -gradation the subalgebra L A , M ( g ) 0 is isomor - phic to the subalgebra g [ 0 ] M of g , and the Lie group L a , M ( G ) 0 is isomorphic to the connected Lie subgr oup G 0 of G corr e sponding to the Lie a lgebra g [ 0 ] M . H e nce, the mapping γ is actually a mapping of R 2 to G 0 . The ma p p ings f − and f + ar e given by the relation f − ( ¯ m , ¯ p ) = ¯ p − L c − ( ¯ m ) , f + ( ¯ m , ¯ p ) = ¯ p L c + ( ¯ m ) , ¯ m ∈ R 2 , ¯ p ∈ S 1 , wher e c − and c + ar e mappings of R 2 to g − [ L ] M and g + [ L ] M r espectively . For the connec- tion components ω − and ω + we have ω − = γ − 1 ∂ − γ + λ − L c − , ω + = λ L γ − 1 c + γ , (2.14) and the T oda equation (2.13) can be written as ∂ + ( γ − 1 ∂ − γ ) = [ c − , γ − 1 c + γ ] . (2.15) The conditions (2.12) imply that ∂ + c − = 0, ∂ − c + = 0. (2.16) It is natural to call an equation of the form (2.15) also a T oda equation. Let b be an automorphism of the Lie gr oup G and B be the corr esponding automor - phism of the Lie algebra g . The mapping σ defined by the equality σ ( χ ) = b ◦ χ , χ ∈ L a , M ( G ) , is an isomorphism from L a , M ( G ) to L a ′ , M ( G ) , where a ′ is an automorphism of G de- fined as a ′ = b a b − 1 . It is clear that the mapping Σ de fine d by the equality Σ ( ξ ) = B ◦ ξ , ξ ∈ L A , M ( g ) , is an isomorphism fr om L A , M ( g ) to L A ′ , M ( g ) , where A ′ is an automorphism of g cor - r esponding to the automorphism a ′ of G . W ith such isomorphism in the c ase under consideration the transformations (2.7) and (2.8) take the form γ ′ = b ◦ γ , c ′ − = B ◦ c − , c ′ + = B ◦ c + . If a mapping γ satisfies the T oda equations (2.15), then the mapping γ ′ satisfies the T oda equation (2.15) whe r e the mappings c − , c + ar e replaced by the ma ppings c ′ − and c ′ + . Actually this means that T oda equations of the form (2.15) defined by means of conjugated Z M -gradations are equivalent. The transformat ions (2.9) and (2.10) take now the forms γ ′ = η − 1 + γ η − , (2.17) c ′ − = η − 1 − c − η − , c ′ + = η − 1 + c + η + , (2.18) 7 wher e η − and η + ar e some mapp in gs of R 2 × S 1 to G 0 that satisfy the conditions ∂ + η − = 0, ∂ − η + = 0. (2.19) Again, if a mapping γ satisfi es the T oda equations (2.15), then the mapping γ ′ satisfies the T oda equation (2.15) where the mappings c − , c + ar e replaced by the mappings c ′ − and c ′ + . I f the mappings η − and η + ar e such that η − 1 − c − η − = c − , η − 1 + c + η + = c + (2.20) then the transformat ion (2.17) is a symmetry transformation for the T oda e quation under consideration. Thus, if G is a finite dimensional complex simple Lie gro up, then the T oda equation associated with a loop gr oup L a , M ( G ) and defined with the help of an integrable Z - gradation of L A , M ( g ) with finite dime nsional grading subspaces is equivale nt to an equation of the form (2.15 ). T o describe all nonequivalent T oda equations of this type one h a s to classify finite or de r automorphisms of the Lie algebra g or , equivalently , its Z M -gradations up to conjugations. 7 This pr oblem was solved quit e a long time ago, see, for example, [28, 29]. However , it appeared that the classification described in [28, 29] is not convenient for classification of T oda equations. R estricting to the case of loop Lie a lgebras of complex clas sical Lie algebras one can use another class ification based on a convenient block matrix repr esentation of the grading subspaces [30 , 21]. Let us describe the main points of the resulting classification of T oda eq uations. Each element x of the complex classical Lie algebra g und e r consideration is consid- ered as a p × p block mat rix ( x α β ) , wher e x α β is an n α × n β matrix. Certainly , the sum of the positive integers n α is the size n of the matrices repr esenting the elements of g . For the case of T oda systems asso ciated wit h the loop gr oups L a , M ( GL n ( C ) ) , wher e a is a n inner auto morphism of GL n ( C ) , the integers n α ar e arbitrary . For the other cases they should satisfy some r estrictions dictated by the structur e of the Lie algebra g . The mapping γ has the block diagonal form γ =     Γ 1 Γ 2 Γ p     . For each α = 1 , . . . , p the mapping Γ α is a mapping of R 2 to the Lie gr oup GL n α ( C ) . For the case of T oda systems associated wit h the loop gr oups L a , M ( GL n ( C ) ) , where a is an inner automorphism of GL n ( C ) , the mappings Γ α ar e arbitrary . For the other cases they satisfy some additional r estrictio ns. The mapping c + has the following block matrix structur e: c + =         0 C + 1 0 0 C + ( p − 1 ) C + 0 0         , 7 Strictly speaking, we have to consider only conjugations by automorphisms of g which ca n be lifted to automorphisms of G . 8 wher e for each α = 1, . . . , p − 1 the mapping C + α is a ma p ping of R 2 to the space of n α × n α + 1 complex matrices, and C + 0 is a mapping of R 2 to the space of n p × n 1 complex matrices. The ma p ping c − has a similar block matrix struct ur e : c − =         0 C − 0 C − 1 0 0 C − ( p − 1 ) 0         , wher e for each α = 1, . . . , p − 1 the mapping C − α is a ma p ping of M to the space of n α + 1 × n α complex matrices, a nd C − 0 is a mapping of M to the space of n 1 × n p complex matrices. The conditions (2.16) imply ∂ + C − α = 0, ∂ − C + α = 0, α = 0, 1, . . . , p − 1. For the case of T oda systems asso ciated wit h the loop gr oups L a , M ( GL n ( C ) ) , wher e a is an inner automorphism of GL n ( C ) , the mappings C ± α ar e arbitrary . For the other cases they should satisfy some additional r estrictions . It is not difficult to show that the T oda equation (2.15) is equivalent to the following system of equations for the mappings Γ α : ∂ +  Γ − 1 1 ∂ − Γ 1  = − Γ − 1 1 C + 1 Γ 2 C − 1 + C − 0 Γ − 1 p C + 0 Γ 1 , ∂ +  Γ − 1 2 ∂ − Γ 2  = − Γ − 1 2 C + 2 Γ 3 C − 2 + C − 1 Γ − 1 1 C + 1 Γ 2 , . . . (2.21) ∂ +  Γ − 1 p − 1 ∂ − Γ p − 1  = − Γ − 1 p − 1 C + ( p − 1 ) Γ p C − ( p − 1 ) + C − ( p − 2 ) Γ − 1 p − 2 C + ( p − 2 ) Γ p − 1 , ∂ +  Γ − 1 p ∂ − Γ p  = − Γ − 1 p C + 0 Γ 1 C − 0 + C − ( p − 1 ) Γ − 1 p − 1 C + ( p − 1 ) Γ p . It appears that in the c ase under consideration without any loss of generality one can assume that the positive integer L , entering the construct ion of T oda equations, is equal to 1 . Note also that if any of the mappings C + α or C − α is a zero mapping, then the e quations (2.21) ar e actually e quivalent to a T oda equation associated with a finite dimensional gr oup o r to a set of two such eq uations. 2.3 Abelian T oda equations associated with loop groups of complex general linear Lie groups Ther e ar e thr ee types of abelian T oda equations associated with L a , M ( GL n ( C ) ) . 2.3.1 First type The abelia n T oda equations of the first type arise when the automorphism A is defined by the equality A ( x ) = h x h − 1 , x ∈ gl n ( C ) , 9 wher e h is a diagonal matrix with the diagonal matrix e lements h kk = ǫ n − k + 1 n , k = 1, . . . , n . (2.22) The corr esponding auto morphism a of GL n ( C ) is defined by the equality a ( g ) = h g h − 1 , g ∈ GL n ( C ) , (2.23) wher e again h is a diagonal matrix determined by the relation (2.22). Here the integer M is equal to n , and A is an inner automorphism which g enerates a Z n -gradation of gl n ( C ) . The block matrix structur e r elated to t his gradation is the ma trix structur e it- self. In ot her words, all blo cks are of size o ne by one. The mappings Γ α ar e mappings of R 2 to the Lie gr oup GL 1 ( C ) which is isomo rphic t o the Lie gr oup C × = C r { 0 } . The mappings C ± α ar e just complex functions on R 2 . The T oda equations under con- sideration have the form (2.21) with p = n . Let us de scribe the action o f the transformations (2.17) and (2.18) on the equations (2.21) in the case under consideration. The mapp ings η − and η + have a diagonal form and we denote ( η − ) αα = H − α , ( η + ) αα = H + α , α = 1, . . . , n . The functions H − α and H + α satisfy the relations ∂ + H − α = 0, ∂ − H + α = 0, which follow fr om the relations (2.19). In terms of the functions C − α and C + α the transformatio ns (2. 1 7) and (2.18) look a s Γ ′ α = H − 1 + α Γ α H − α , (2.24) C ′ − α = H − 1 − ( α + 1 ) C − α H − α , C ′ + α = H − 1 + α C + α H + ( α + 1 ) . (2.25) Assume that the func tions C − α and C + α have no zeros. Let us sho w that in this case the functions H − α and H + α can be chosen in such a way that C ′ − α = C − and C ′ + α = C + for some functions C − and C + which ha ve no zeros and are subject to the conditions ∂ + C − = 0, ∂ − C + = 0. (2.26) Indeed, let C − and C + be some functions which satisfy the equalities C n − = n ∏ α = 1 C − α , C n + = n ∏ α = 1 C + α . One can verify that the transformat ions (2.2 5 ) with H − α = n ∏ β = α C − C − β , H + α = n ∏ β = α C + β C + give the de sired result , C ′ − α = C − and C ′ + α = C + . The methods to find soliton solu- tions described below work for arbitrary functions C − and C + . However , to simplify 10 formulas we will only consider the case whe n C − = m , and C + = m , where m is a nonzer o constant. In other wor d s, we will assume that c − = m       0 1 1 0 0 1 0       , c + = m       0 1 0 0 1 1 0       . (2.27) The equations under consideration take in this case the form ∂ +  Γ − 1 1 ∂ − Γ 1  = − m 2 ( Γ − 1 1 Γ 2 − Γ − 1 p Γ 1 ) , ∂ +  Γ − 1 2 ∂ − Γ 2  = − m 2 ( Γ − 1 2 Γ 3 − Γ − 1 1 Γ 2 ) , . . . (2.28) ∂ +  Γ − 1 n − 1 ∂ − Γ n − 1  = − m 2 ( Γ − 1 n − 1 Γ n − Γ − 1 n − 2 Γ n − 1 ) , ∂ +  Γ − 1 n ∂ − Γ n  = − m 2 ( Γ − 1 n Γ 1 − Γ − 1 n − 1 Γ n ) . It is worth to note that when the functio ns C − and C + ar e real one can come to the T oda equations wit h C − = m and C + = m by an a p pr opriate change of the coordinates z − and z + . The symmetry transformations (2. 2 0) for the system under consideration ar e de- scribed by the relations (2.24) where H − α = H − and H + α = H + for some functions H − and H + satisfying the conditions ∂ + H − = 0, ∂ − H + = 0. In particular , multiplication of all Γ α by the same constant is a symmetry transf orma- tion. Defining Γ = Γ 1 Γ 2 . . . Γ n , one can ea sily see that ∂ + ( Γ − 1 ∂ − Γ ) = n ∑ α = 1 ∂ + ( Γ − 1 α ∂ − Γ α ) . Equations (2.28) give ∂ + ( Γ − 1 ∂ − Γ ) = 0, ther efor e , Γ = Γ + Γ − 1 − , for some functions Γ − and Γ + which satisfy t he r elations ∂ + Γ − = 0, ∂ − Γ + = 0. Thus, if we perform the symmetry transformation (2.24 ) with H − α and H + α given by H − α = Γ 1/ n − , H + α = Γ 1/ n + , 11 we will obtain functions Γ ′ i which satisfy the T oda eq uations (2.28) and obey the equal- ity Γ ′ = Γ ′ 1 Γ ′ 2 . . . Γ ′ n = 1. Actually this mean s that via app ropriate symmetry transformations we can reduce solutions of the abelia n T oda equations associated with the loop gr oup of GL n ( C ) under consideration to solutions of the corr esponding T oda equa tions associated with the loop group of SL n ( C ) . Suppose now that we have a solution of the equations (2.28) with Γ = 1. The mappings Γ α for α = 1, . . . , n − 1 are independent. Introduc e a new set of n − 1 inde- pendent mappings Φ α , α = 1, . . . , n − 1, defined as Φ α = α ∏ β = 1 Γ β . It is easy to show that the inverse transition to the mappings Γ α is described by the equalities Γ 1 = Φ 1 , Γ 2 = Φ − 1 1 Φ 2 , . . . Γ n − 1 = Φ − 1 n − 2 Φ n − 1 , Γ n = Φ − 1 n − 1 , and that the mappings Φ α satisfy the equations ∂ + ( Φ − 1 1 ∂ − Φ 1 ) = − m 2 ( Φ − 2 1 Φ 2 − Φ n − 1 Φ 1 ) , ∂ + ( Φ − 1 2 ∂ − Φ 2 ) = − m 2 ( Φ 1 Φ − 2 2 Φ 3 − Φ n − 1 Φ 1 ) , . . . ∂ + ( Φ − 1 n − 2 ∂ − Φ n − 2 ) = − m 2 ( Φ n − 3 Φ − 2 n − 2 Φ n − 1 − Φ n − 1 Φ 1 ) , ∂ + ( Φ − 1 n − 1 ∂ − Φ n − 1 ) = − m 2 ( Φ n − 2 Φ − 2 n − 1 − Φ n − 1 Φ 1 ) . This system can be written in a mor e symmetric form. T o this e nd one int r oduces an additional ma pping ∆ 0 , which satisfies the equation ∂ + ( ∆ − 1 0 ∂ − ∆ 0 ) = − m 2 Φ n − 1 Φ 1 , and denotes ∆ α = ∆ 0 Φ α , α = 1, . . . , n − 1. It is easy to see that the mappings ∆ α , α = 0, 1, . . . , n − 1, satisfy the equations ∂ + ( ∆ − 1 0 ∂ − ∆ 0 ) = − m 2 ∆ n − 1 ∆ − 2 0 ∆ 1 , ∂ + ( ∆ − 1 1 ∂ − ∆ 1 ) = − m 2 ∆ 0 ∆ − 2 1 ∆ 2 , . . . (2.29) ∂ + ( ∆ − 1 n − 2 ∂ − ∆ n − 2 ) = − m 2 ∆ n − 3 ∆ − 2 n − 2 ∆ n − 1 , ∂ + ( ∆ − 1 n − 1 ∂ − ∆ n − 1 ) = − m 2 ∆ n − 2 ∆ − 2 n − 1 ∆ 0 , which can be written as ∂ + ( ∆ − 1 α ∂ − ∆ α ) = − m 2 n − 1 ∏ β = 0 ∆ − a α β β , (2.30) 12 wher e a α β ar e the matrix e lements of the Cartan matrix of an af fine Lie algebra of type A ( 1 ) n − 1 : ( a α β ) =            2 − 1 0 · · · 0 0 − 1 − 1 2 − 1 · · · 0 0 0 0 − 1 2 · · · 0 0 0 . . . . . . . . . . . . . . . . . . . . . 0 0 0 · · · 2 − 1 0 0 0 0 · · · − 1 2 − 1 − 1 0 0 · · · 0 − 1 2            . (2.31) The equations (2.30) ar e of the standard fo rm of the T oda equations associated with the A ( 1 ) n − 1 affi ne Lie group. 2.3.2 Second type The abelian T oda equations of the second and third types arise when we use outer automorphisms of gl n ( C ) . For the equations of the second ty pe n is odd, and for the equations of the third type n is even. Consider fir st the case of an odd n = 2 s − 1, s ≥ 2. I n this case an abelian T oda equation a rises when the automorphism A is defined by the equality A ( x ) = − h ( B − 1 t x B ) h − 1 , (2.32) wher e t x mea n s the transpose of x , h is a diagonal matrix with the diagonal matrix elements h kk = ǫ n − k + 1 2 n = ǫ 2 s − k 4 s − 2 , and B is an n × n matrix of the form B =         1 1 1 − 1 − 1         . The order of the automorphism A is 2 n = 4 s − 2 and the integer p is 2 s − 1. The mapping γ is a diagonal matrix, and the mappings Γ α ar e mappings of R 2 to C × subject to the constraints Γ 1 = 1, Γ 2 s − α + 1 = Γ − 1 α , α = 2, . . . , s . The mappings C ± α ar e complex functions satisfying the equality C ± 0 = C ± 1 , (2.33) and for s > 2 the equalities C ± ( 2 s − α ) = − C ± α , α = 2, . . . , s − 1. (2.34) Let us choos e the mappings Γ α , α = 2, . . . , s , as a complete set of mappings parame- terizing the mapping γ . T aking into account the equalities (2.33) and (2.34) we come 13 to the following set of independent equations equivalent t o the T oda equation under consideration ∂ + ( Γ − 1 2 ∂ − Γ 2 ) = − C + 2 C − 2 Γ − 1 2 Γ 3 + C + 1 C − 1 Γ 2 , ∂ + ( Γ − 1 3 ∂ − Γ 3 ) = − C + 3 C − 3 Γ − 1 3 Γ 4 + C + 2 C − 2 Γ − 1 2 Γ 3 , . . . (2.35) ∂ + ( Γ − 1 s − 1 ∂ − Γ s − 1 ) = − C + ( s − 1 ) C − ( s − 1 ) Γ − 1 s − 1 Γ s + C + ( s − 2 ) C − ( s − 2 ) Γ − 1 s − 2 Γ s − 1 , ∂ + ( Γ − 1 s ∂ − Γ s ) = − C + s C − s Γ − 2 s + C + ( s − 1 ) C − ( s − 1 ) Γ − 1 s − 1 Γ s . As above, in the case whe n the functions C − α and C + α have no zeros, using the trans- formation (2.17) and (2. 1 8), one can show that the equations (2.35 ) ar e equivalent to the same equations, wher e C − α = C − and C + α = C + for some functions C − and C + which have no zeros and are subject to the conditions (2.26). If these functions are real, then with the help of an app ropriate change of the coor dinates z − and z + we can come to the eq uations ∂ + ( Γ − 1 2 ∂ − Γ 2 ) = − m 2 ( Γ − 1 2 Γ 3 − Γ 2 ) , ∂ + ( Γ − 1 3 ∂ − Γ 3 ) = − m 2 ( Γ − 1 3 Γ 4 − Γ − 1 2 Γ 3 ) , . . . ∂ + ( Γ − 1 s − 1 ∂ − Γ s − 1 ) = − m 2 ( Γ − 1 s − 1 Γ s − Γ − 1 s − 2 Γ s − 1 ) , ∂ + ( Γ − 1 s ∂ − Γ s ) = − m 2 ( Γ − 2 s − Γ − 1 s − 1 Γ s ) , wher e m is a nonzero constant, see also the pa pers [3, 31]. For s = 2 denoting Γ 2 by Γ we have the equation ∂ + ( Γ − 1 ∂ − Γ ) = − m 2 ( Γ − 2 − Γ ) . Putting Γ = exp ( F ) we obtain ∂ + ∂ − F = − m 2 [ exp ( − 2 F ) − e xp ( F ) ] . This is the Tzitz ´ eica equation [32] which is now usually called the Dodd–Bullough– Mikhailov equation [33, 3]. Let us show how the above eq uations are related to t he T oda equations asso ciated with the A ( 2 ) 2 s − 2 affi ne Lie gr oup. Assume that s > 2. I ntr oduce an additional mapping ∆ 0 which satisfies the equation ∂ + ( ∆ − 1 0 ∂ − ∆ 0 ) = − m 2 2 Γ 2 (2.36) and denote ∆ α = 2 − α ∆ 2 0 α + 1 ∏ β = 2 Γ β , α = 1, . . . , s − 2, ∆ s − 1 = 2 − s + 2 ∆ 2 0 s ∏ β = 2 Γ β . Now one can get convinced that the mappings ∆ α , α = 0, 1, . . . , s − 1 , satisfy the equa- tions of the form (2.30) where n = s and a α β ar e the matrix elements of the C a rtan 14 matrix of an affine Lie algebra of type A ( 2 ) 2 s − 2 : ( a α β ) =            2 − 1 0 · · · 0 0 0 − 2 2 − 1 · · · 0 0 0 0 − 1 2 · · · 0 0 0 . . . . . . . . . . . . . . . . . . . . . 0 0 0 · · · 2 − 1 0 0 0 0 · · · − 1 2 − 1 0 0 0 · · · 0 − 2 2            . In the case of s = 2 we again define the mapping ∆ 0 with the help of the r elation (2.36) and denote ∆ 1 = 2 − 1 / 3 ∆ 2 0 Γ 2 . After an appro priate rescaling of the co or d ina tes z − and z + we come to the equations (2.30) where n = 2 and a α β ar e the matr ix elements of the Cartan matrix of an affine Lie algebra of type A ( 2 ) 2 : ( a α β ) =  2 − 1 − 4 2  . 2.3.3 Third type In the case of an even n = 2 s , s ≥ 2, to come to an abelian T oda system one should use again the automorphism A defined by the relation (2.32) where now B =          1 1 1 1 − 1 − 1          and h is a diagonal matrix with the d i a gonal matrix elements h 11 = ǫ n − 1 2 n − 2 = ǫ 2 s − 1 4 s − 2 = − 1, h i i = ǫ n − i + 1 2 n − 2 = ǫ 2 s − i + 1 4 s − 2 , i = 2, . . . , n . The number p characterizing the block st ruct ur e is equal to n − 1 = 2 s − 1, n 1 = 2, and n α = 1 for α = 2, . . . , 2 s − 1. The mapping Γ 1 is a mapping of R 2 to the Lie gr oup SO 2 ( C ) which is isomorphic to C × . A ctually Γ 1 is a 2 × 2 complex matrix valued function satisfying the relatio n J − 1 2 t Γ 1 J 2 = Γ − 1 1 , wher e J 2 =  0 1 1 0  . It is easy to show that Γ 1 has the form Γ 1 =  ( Γ 1 ) 11 0 0 ( Γ 1 ) − 1 11  , 15 wher e ( Γ 1 ) 11 is a mapping of R 2 to C × . The mappings Γ α , α = 2, . . . , 2 s − 1, ar e map- pings of R 2 to C × satisfying the relations Γ 2 s − α + 1 = Γ − 1 α . The map pings C − 1 , C + 0 ar e complex 1 × 2 matrix valued functions, the mappings C − 0 , C + 1 ar e complex 2 × 1 matrix valued functions. Her e one has C − 0 = J − 1 2 t C − 1 , C + 0 = t C + 1 J 2 . (2.37) The ma p p ings C ± α , α = 2, . . . , p − 1 = 2 s − 2, ar e just complex functions, satisfying for s > 2 the equalities C ± ( 2 s − α ) = − C ± α , α = 2, . . . , s − 1. (2.38) The mappings ( Γ 1 ) 11 and Γ α , α = 2, . . . , s , form a complete set of mappings param- eterizing the mapping γ . T aking into account the equalities (2.37) and (2.3 8) we come to the following set of independent equations equivalent t o the T oda equation under consideration: ∂ + (( Γ 1 ) − 1 11 ∂ − ( Γ 1 ) 11 ) = − ( C + 1 ) 11 ( C − 1 ) 11 ( Γ 1 ) − 1 11 Γ 2 + ( C + 1 ) 21 ( C − 1 ) 12 Γ 2 ( Γ 1 ) 11 , ∂ + ( Γ − 1 2 ∂ − Γ 2 ) = − C + 2 C − 2 Γ − 1 2 Γ 3 + ( C + 1 ) 11 ( C − 1 ) 11 ( Γ 1 ) − 1 11 Γ 2 + ( C + 1 ) 21 ( C − 1 ) 12 Γ 2 ( Γ 1 ) 11 , ∂ + ( Γ − 1 3 ∂ − Γ 3 ) = − C + 3 C − 3 Γ − 1 3 Γ 4 + C + 2 C − 2 Γ − 1 2 Γ 3 , . . . ∂ + ( Γ − 1 s − 1 ∂ − Γ s − 1 ) = − C + ( s − 1 ) C − ( s − 1 ) Γ − 1 s − 1 Γ s + C + ( s − 2 ) C − ( s − 2 ) Γ − 1 s − 2 Γ s − 1 , ∂ + ( Γ − 1 s ∂ − Γ s ) = − C + s C − s Γ − 2 s + C + ( s − 1 ) C − ( s − 1 ) Γ − 1 s − 1 Γ s . As well as fo r the first two ty pes, under appropr iate conditions these equations can be reduced to the e quations with C − α = m , C + α = m for α = 2, . . . , s , ( C − 1 ) 11 = ( C − 1 ) 12 = m / √ 2 and ( C + 1 ) 11 = ( C + 1 ) 21 = m / √ 2, wher e m is a nonze ro constant. 8 Thus, we come to the equations ∂ + ( Γ − 1 1 ∂ − Γ 1 ) = − m 2 2 ( Γ − 1 1 − Γ 1 ) Γ 2 , ∂ + ( Γ − 1 2 ∂ − Γ 2 ) = − m 2 Γ − 1 2 Γ 3 + m 2 2 ( Γ − 1 1 + Γ 1 ) Γ 2 , ∂ + ( Γ − 1 3 ∂ − Γ 3 ) = − m 2 ( Γ − 1 3 Γ 4 − Γ − 1 2 Γ 3 ) , . . . ∂ + ( Γ − 1 s − 1 ∂ − Γ s − 1 ) = − m 2 ( Γ − 1 s − 1 Γ s − Γ − 1 s − 2 Γ s − 1 ) , ∂ + ( Γ − 1 s ∂ − Γ s ) = − m 2 ( Γ − 2 s − Γ − 1 s − 1 Γ s ) , wher e slightly abusing notation we denote ( Γ 1 ) 11 by Γ 1 . 8 This choice is convenient for applications of the ra tional dressing method. 16 Intr oduce now an additional mapping ∆ 0 which satisfies the equation ∂ + ( ∆ − 1 0 ∂ − ∆ 0 ) = − m 2 2 Γ 1 Γ 2 and denote ∆ 1 = ∆ 0 Γ 1 , ∆ α = 2 α ( α − 1 ) / ( 2 s − 1 ) − α + 1 ∆ 2 0 α ∏ β = 1 Γ β , α = 2, . . . , s . The mappings ∆ α , α = 0, 1, . . . , s , satisfy the equations which, after an appro priate r escaling of the coor d ina tes z − and z + , take the form (2.30), where now n = s + 1 and a α β ar e the matrix elements of the Cartan matrix of an affi ne Lie algebra of type A ( 2 ) 2 s − 1 : ( a α β ) =            2 0 − 1 · · · 0 0 0 0 2 − 1 · · · 0 0 0 − 1 − 1 2 · · · 0 0 0 . . . . . . . . . . . . . . . . . . . . . 0 0 0 · · · 2 − 1 0 0 0 0 · · · − 1 2 − 1 0 0 0 · · · 0 − 2 2            . 3 Soliton solution s In this section we compare two methods used to construct soliton solutions of the abelian T oda systems associated with the lo op gr oups of the complex general line ar gr oups. W e restr ict ourselves to the abelian T oda equations of the first type which have the form (2.28). 3.1 Hirota’ s method It is convenient to treat the system (2.28) a s an infinite system ∂ + ( Γ − 1 α ∂ − Γ α ) = − m 2 ( Γ − 1 α Γ α + 1 − Γ − 1 α − 1 Γ α ) , (3.1) wher e the functions Γ α ar e defined for arbitrary integer values of the index α in the periodic way , Γ α + n = Γ α . (3.2) Following the Hirota’s approach [6, 7, 8, 9, 1 0 ], one intr oduces τ -functio ns con- nected with Γ α by the r elation Γ α = τ α / τ α − 1 , ( 3.3) wher e we assume t hat t he τ -functions ar e defined for all integer value s of the index α . This change of variables is the essence of the Hiro ta’s method. The periodicity condition (3.2 ) in terms of the τ -functions takes the form τ α + n / τ α = τ α − 1 + n / τ α − 1 . It means that the ratio τ α + n / τ α does not de p e nd on α . Not ing that Γ = Γ 1 Γ 2 . . . Γ n = τ n / τ 0 , 17 we write τ α + n = Γ τ α . As was explained in the previous section, with the a ppr opriate symmetry transforma- tion one can make Γ = 1. W e will assume that the corr esponding sy mmetry transfor - mation was p e rformed, and, therefo r e, τ α + n = τ α . (3.4) The equations (3.1) in terms of the τ -functions look as ∂ + ( τ − 1 α ∂ − τ α ) − ∂ + ( τ − 1 α − 1 ∂ − τ α − 1 ) = − m 2 ( τ α − 1 τ − 2 α τ α + 1 − τ α − 2 τ − 2 α − 1 τ α ) . Consider the following decoupling of the above equatio ns ∂ + ( τ − 1 α ∂ − τ α ) = m 2 ( 1 − τ α − 1 τ − 2 α τ α + 1 ) . (3.5) It is evident that if the τ -funct ions satisfy these equations, then the functions Γ α defined by (3.3) satisfy the system (2.28). Moreover , it is easy to show that in this case the functions ∆ α = exp ( − m 2 z + z − ) τ α satisfy the system (2.29). It is convenient to rewrite the equations (3.5) in the form τ α ∂ + ∂ − τ α − ∂ + τ α ∂ − τ α = m 2 ( τ 2 α − τ α − 1 τ α + 1 ) . (3.6) These equations a r e of the Hir ota bilinear type. Their solutions, leading to multi- soliton solutions of the system (2.28), can be found perturbatively in the following way . Consider a series expansion of the functions τ α in some parameter ε which will be set t o one at the final step of the construc tion. So we repr esent the funct ions τ α in the form τ α = τ ( 0 ) α + ετ ( 1 ) α + ε 2 τ ( 2 ) α + . . . , (3.7) and assume that τ ( 0 ) α ar e constants. The periodicity condition (3.4) gives τ ( k ) α + n = τ ( k ) α , k = 0, 1, . . . . Let us try to solve equations (3.6) order by or der in ε . Actually our goal is to find solutions for which the series (3.7) truncates at some finite or der in ε . In such a case we have an exact solution. Using the expansion (3.7), one obtains τ α ∂ + ∂ − τ α = ∞ ∑ k = 0 ε k k ∑ ℓ = 0 τ ( k − ℓ ) α ∂ + ∂ − τ ( ℓ ) α . Similarly one ha s ∂ + τ α ∂ − τ α = ∞ ∑ k = 0 ε k k ∑ ℓ = 0 ∂ + τ ( k − ℓ ) α ∂ − τ ( ℓ ) α . 18 Now , using the equality τ 2 α − τ α − 1 τ α + 1 = ∞ ∑ k = 0 ε k k ∑ ℓ = 0  τ ( ℓ ) α τ ( k − ℓ ) α − τ ( ℓ ) α − 1 τ ( k − ℓ ) α + 1  , we see that the equations (3.6) a r e equivalent to the equations k ∑ ℓ = 0  τ ( k − ℓ ) α ∂ + ∂ − τ ( ℓ ) α − ∂ + τ ( k − ℓ ) α ∂ − τ ( ℓ ) α  = m 2 k ∑ ℓ = 0  τ ( ℓ ) α τ ( k − ℓ ) α − τ ( ℓ ) α − 1 τ ( k − ℓ ) α + 1  , (3. 8 ) which can be solved step by step starting from k = 0. For k = 0 one has τ ( 0 ) α − 1 τ ( 0 ) α + 1 − τ ( 0 ) α τ ( 0 ) α = 0, that can be rewrit ten as τ ( 0 ) α + 1 / τ ( 0 ) α = τ ( 0 ) α / τ ( 0 ) α − 1 . It is clear that the general solution to this relation is τ ( 0 ) α = τ ( 0 ) 0 d α , (3.9) wher e d is an arbitrar y constant. Recall that the T oda equations (3.1) are invariant with r espect to the multiplication of all Γ α by the same constant. Fr om the point of view of the τ -functions this is equivalent to the multiplication of the function τ α by the α th power of t he constant. Hence, dif fer ent values of the constant d in the r elation (3.9) corr espond to the functions Γ α connected by a r escaling. Mor eover , dividing all τ - functions by the same constant we do not change the functions Γ α . Therefor e, actually without any loss of generality , one can put τ ( 0 ) α = 1. (3.10) Using this e quality , we r ewrite (3.8) as ∂ + ∂ − τ ( k ) α − m 2 n − 1 ∑ β = 0 a α β τ ( k ) β = − k − 1 ∑ ℓ = 1  τ ( k − ℓ ) α ∂ + ∂ − τ ( ℓ ) α − ∂ + τ ( k − ℓ ) α ∂ − τ ( ℓ ) α  + m 2 k − 1 ∑ ℓ = 1  τ ( ℓ ) α τ ( k − ℓ ) α − τ ( ℓ ) α − 1 τ ( k − ℓ ) α + 1  , (3.11) wher e a α β ar e the matrix elements of the Cartan matrix (2.31) of an affi ne Lie alge- bra of type A ( 1 ) n − 1 . Thus, we see that at each st ep we should solve a system of linear diff er ential equations. In particular , for k = 1 one has to solve the system of equations ∂ + ∂ − τ ( 1 ) α − m 2 n − 1 ∑ β = 0 a α β τ ( 1 ) β = 0. (3.12) It is easy to find solut ions of these equations using the e igenvectors θ ρ of t he Ca rtan matrix ( a α β ) which ar e given by ( θ ρ ) α = ǫ ( α + 1 ) ρ n , ρ = 0, 1, . . . , n − 1. (3.13) 19 Here the corr esponding eigenvalues are κ 2 ρ = 2 − ǫ ρ n − ǫ − ρ n = 4 sin 2 ( π ρ / n ) . Let us assume that the functions τ ( 1 ) α ar e of the form τ ( 1 ) α = r ∑ i = 1 E α i , (3.14) wher e E α i = ǫ ( α + 1 ) ρ i n exp [ m κ ρ i ( ζ − 1 i z − + ζ i z + ) + δ i ] . (3.15) Here ρ i is an integer from the interval from 1 to n − 1, ζ i and δ i ar e arbitrary complex numbers. Note that the cho ice ρ i = 0 is excluded because it g ives a constant co ntri- bution to the τ -functions which can be included in to τ ( 0 ) α . Then after a corr esponding r escaling one can satisfy the normalization (3.10). For definiteness we assume that κ ρ = − i ( ǫ ρ / 2 n − ǫ − ρ / 2 n ) = 2 sin ( π ρ / n ) . (3.16) Certainly , t he ansatz (3.14) does not give a general solution to the equations (3.12) but it ensures trunc ation of the e xpansion (3.7). For k = 2 the equations (3.11) have the form ∂ + ∂ − τ ( 2 ) α − m 2 n − 1 ∑ β = 0 a α β τ ( 2 ) β = − τ ( 1 ) α ∂ + ∂ − τ ( 1 ) α + ∂ + τ ( 1 ) α ∂ − τ ( 1 ) α + m 2 ( τ ( 1 ) α τ ( 1 ) α − τ ( 1 ) α − 1 τ ( 1 ) α + 1 ) . Using the equa lities ∂ − E α i = m κ ρ i ζ − 1 i E α i , ∂ + E α i = m κ ρ i ζ i E α i , one obtains − τ ( 1 ) α ∂ + ∂ − τ ( 1 ) α + ∂ + τ ( 1 ) α ∂ − τ ( 1 ) α + m 2 ( τ ( 1 ) α τ ( 1 ) α − τ ( 1 ) α − 1 τ ( 1 ) α + 1 ) = m 2 2 r ∑ i 1 , i 2 = 1 h κ ρ i 1 κ ρ i 2  ζ i 1 ζ − 1 i 2 + ζ − 1 i 1 ζ i 2  − κ 2 ρ i 1 − κ 2 ρ i 2 + κ 2 ρ i 1 − ρ i 2 i E α i 1 E α i 2 . (3.17) Note that in the case of r = 1 we have − τ ( 1 ) α ∂ + ∂ − τ ( 1 ) α + ∂ + τ ( 1 ) α ∂ − τ ( 1 ) α + m 2 ( τ ( 1 ) α τ ( 1 ) α − τ ( 1 ) α − 1 τ ( 1 ) α + 1 ) = 0 and we can put τ ( 2 ) α = 0. This gives the one-soliton solutions Γ α = 1 + ǫ ρ ( α + 1 ) n exp [ m κ ρ ( ζ − 1 z − + ζ z + ) + δ ] 1 + ǫ ρα n exp [ m κ ρ ( ζ − 1 z − + ζ z + ) + δ ] . (3.18) In the case when r > 1 the equality (3.17) suggests to look for τ ( 2 ) α of the form τ ( 2 ) α = 1 2 r ∑ i 1 , i 2 = 1 η i 1 i 2 E α i 1 E α i 2 . 20 Here one can easily find that ∂ + ∂ − τ ( 2 ) α − m 2 n − 1 ∑ β = 0 a α β τ ( 2 ) β = m 2 2 r ∑ i 1 , i 2 = 1 h κ ρ i 1 κ ρ i 2 ( ζ i 1 ζ − 1 i 2 + ζ − 1 i 1 ζ i 2 ) + κ 2 ρ i 1 + κ 2 ρ i 2 − κ 2 ρ i 1 + ρ i 2 i η i 1 i 2 E α i 1 E α i 2 and, therefor e, η i 1 i 2 = κ ρ i 1 κ ρ i 2 ( ζ i 1 ζ − 1 i 2 + ζ − 1 i 1 ζ i 2 ) − κ 2 ρ i 1 − κ 2 ρ i 2 + κ 2 ρ i 1 − ρ i 2 κ ρ i 1 κ ρ i 2 ( ζ i 1 ζ − 1 i 2 + ζ − 1 i 1 ζ i 2 ) + κ 2 ρ i 1 + κ 2 ρ i 2 − κ 2 ρ i 1 + ρ i 2 , that can be written as η i 1 i 2 = ( ζ i 1 ζ − 1 i 2 + ζ − 1 i 1 ζ i 2 ) − 2 cos [ π ( ρ i 1 − ρ i 2 ) / n ] ( ζ i 1 ζ − 1 i 2 + ζ − 1 i 1 ζ i 2 ) − 2 cos [ π ( ρ i 1 + ρ i 2 ) / n ] . (3.19) The quantities η i 1 i 2 ar e symmetric with r espect to the indices i 1 , i 2 and they turn to zero when i 1 = i 2 . He nce one can write τ ( 2 ) α = ∑ 1 ≤ i 1 < i 2 ≤ r η i 1 i 2 E α i 1 E α i 2 . It can be shown that when r = 2 one can choose τ ( 3 ) α = 0. I n general, it can be shown that for ℓ ≤ r one can choose τ ( ℓ ) α = ∑ 1 ≤ i 1 < i 2 < . . . < i ℓ ≤ r ∏ 1 ≤ j < k ≤ ℓ η i j i k ! E α i 1 E α i 2 . . . E α i ℓ and τ ( ℓ ) i = 0 for ℓ > r . In other words, the equations (3.6) have the following solutions τ α = 1 + r ∑ i = 1 E α i + r ∑ ℓ = 2 " ∑ 1 ≤ i 1 < i 2 < . . . < i ℓ ≤ r ∏ 1 ≤ j < k ≤ ℓ η i j i k ! E α i 1 E α i 2 . . . E α i ℓ # . (3.20) 3.2 Rational dressing Since for any ¯ m ∈ R 2 the matrices c − ( ¯ m ) and c + ( ¯ m ) commut e, 9 it is obvious that γ = I n , (3.21) wher e I n is the n × n unit matrix, is a solut ion t o t he T oda equation (2.15). Denote a mapping of R 2 × S 1 to GL n ( C ) which generates the corresponding connection by ϕ . Using the equalities (2. 1 1) and (2. 1 4) and remembering that in our case L = 1, we write ϕ − 1 ∂ − ϕ = λ − 1 c − , ϕ − 1 ∂ + ϕ = λ c + , 9 Actually in the case under consideration c − and c + are constant mappings. 21 wher e the matrices c − and c + ar e defined by the relation (2.27). T o construct more interest ing solutions to the T oda equations we will look for a mapping ψ , such that the mapping ϕ ′ = ϕ ψ (3.22) would generate a connection satisfying the grading condition and the gauge-fixing constraint ω + 0 = 0. For any ¯ m ∈ R 2 the mapping ˜ ψ m defined by the equality ˜ ψ m ( ¯ p ) = ψ ( ¯ m , ¯ p ) , ¯ p ∈ S 1 , is a smooth mapping of S 1 to GL n ( C ) . R ecall that we treat S 1 as a subset of the complex plane which, in turn, will be treated as a subset of the Riemann sphere. Assume that it is possible to extend analytically each mapping ˜ ψ m to all of the Riemann sphere. As the r esult we get a mapping of the direct pro duct of R 2 and the Riemann sphere to GL n ( C ) which we also denote by ψ . Suppose that for any ¯ m ∈ R 2 the analytic extension of ˜ ψ m r esults in a rational mapping r egular at the points 0 and ∞ , hence the name rational dr essing. Below , for each p oint ¯ p of the Riemann sphere we denote by ψ p the mapping of R 2 to GL n ( C ) defined by the equality ψ p ( ¯ m ) = ψ ( ¯ m , ¯ p ) . Since we deal with the T oda equations described in sectio n 2.3.1, that is, the map- ping ψ is generated by a mapping of R 2 to the loop gro up L a , n ( GL n ( C ) ) with the automorphism a defined by the relatio ns (2.23) a n d (2.22), for any ¯ m ∈ R 2 and ¯ p ∈ S 1 we should have ψ ( ¯ m , ǫ n ¯ p ) = h ψ ( ¯ m , ¯ p ) h − 1 , (3.23) wher e h is a diagonal matrix described by the relation (2.22). The equality (3.23) means that two rational mappings coincide on S 1 , ther e for e , they must coincide on the entir e Riemann spher e. A mapp ing, satisfying the equality (3.23), can be construc ted by the following pro- cedur e. Let χ be an arbitrary mapping of the dir ect pr oduct of R 2 and the Rie mann spher e to GL n ( C ) . Let ˆ a be a linear operator acting on χ as ˆ a χ ( ¯ m , ¯ p ) = h χ ( ¯ m , ǫ − 1 n ¯ p ) h − 1 . It is easy to get convinced that the mapping ψ = n ∑ k = 1 ˆ a k χ satisfies the relation ˆ a ψ = ψ which is equivalent to the equality (3.23). N ote that ˆ a n χ = χ . T o const ruc t a rational mapping satisfying (3.23) we start with a rational mapping r egular at the points 0 and ∞ and having poles at r differ ent nonzer o points µ i , i = 1, . . . , r . Concr etely speaking, we consider a ma p p ing χ of the form χ = I n + n r ∑ i = 1 λ λ − µ i P i ! χ 0 , wher e P i ar e some smooth mappings of R 2 to the algebr a Mat n ( C ) of n × n complex matrices and χ 0 is a mapping of R 2 to the Lie subgr oup of GL n ( C ) formed by the elements g ∈ GL n ( C ) satisfy ing the equality h g h − 1 = g . (3.24) 22 Actually this subgr oup co incides with the subgr oup G 0 . The averaging procedur e leads to the mapping ψ = I n + r ∑ i = 1 n ∑ k = 1 λ λ − ǫ k n µ i h k P i h − k ! ψ 0 , (3.25) wher e ψ 0 = n χ 0 . It is convenient to assume that µ n i 6 = µ n j for all i 6 = j . Denote by ψ − 1 the ma pping of R 2 × S 1 to GL n ( C ) defined by the relation ψ − 1 ( ¯ m , ¯ p ) = ( ψ ( ¯ m , ¯ p )) − 1 . Suppose that for any fixed ¯ m ∈ R 2 the mapping ˜ ψ − 1 m of S 1 to GL n ( C ) can be extended analytically to a mapping of the R iemann sphere to GL n ( C ) and as the result we obtain a rational mapping of the same structur e as the ma p ping ψ , ψ − 1 = ψ − 1 0 I n + r ∑ i = 1 n ∑ k = 1 λ λ − ǫ k n ν i h k Q i h − k ! , (3.26) with the pole positions satisfying the conditions ν i 6 = 0, ν n i 6 = ν n j for all i 6 = j , and additionally ν n i 6 = µ n j for any i and j . W e will denote the map p in g of the direct pr oduct of R 2 and the R iemann sphere to GL n ( C ) again by ψ − 1 . By de finition, the eq ua l ity ψ − 1 ψ = I n is valid at all points of the direct product of R 2 and S 1 . Since ψ − 1 ψ is a rational map- ping, the above equality is valid at all points of the dir ect product of R 2 and the Rie- mann spher e. Hence, t he residues of ψ − 1 ψ at t he points ν i and µ i should be equal to zero . Explicitly we have Q i I n + r ∑ j = 1 n ∑ k = 1 ν i ν i − ǫ k n µ j h k P j h − k ! = 0, (3.27) I n + r ∑ j = 1 n ∑ k = 1 µ i µ i − ǫ k n ν j h k Q j h − k ! P i = 0. (3.28) In this case due to the relation (3.23) the residues of ψ − 1 ψ at the points ǫ k n µ i and ǫ k n ν i vanish for k = 1, . . . , n . W e will discuss later how to satisfy these relations , and now let us consider what connection is generated by the mapping ϕ ′ defined by (3.22) with the ma pping ψ possessing the prescr ibed propert ies. Using the repr esentation (3.2 2), we obtain for the components of the connection generated by ϕ ′ the e xpr essions ω − = ψ − 1 ∂ − ψ + λ − 1 ψ − 1 c − ψ , (3.29) ω + = ψ − 1 ∂ + ψ + λ ψ − 1 c + ψ . (3.30) W e see that the component ω − is a rational mapping which has simple poles at the points µ i , ν i and zero . 10 Similarly , the component ω + is a rational mapping which has 10 Here and be low discussing the holo morphic properties of mappings a nd functions we assume that the point of the space R 2 is arbitra ry but fixed. 23 simple poles at the points µ i , ν i and infinity . W e ar e looking for a connect ion which satisfies the g rading and gauge- fi xing conditions. The grading condition in our case is the requir ement that for each point of R 2 the component ω − is rational and has t he only simple pole at zer o, while t he component ω + is rational and has the only simple pole at infinity . Hence, we demand that t he r esidues of ω − and ω + at the points µ i and ν i should vanish. In this case, as above, due to the relation (3.23) the r esidues of ω − and ω + at the points ǫ k n µ i and ǫ k n ν i vanish for k = 1, . . . , n . The r esidues of ω − and ω + at the points ν i ar e equal to zero if and only if ( ∂ − Q i − ν − 1 i Q i c − ) I n + r ∑ j = 1 n ∑ k = 1 ν i ν i − ǫ k n µ j h k P j h − k ! = 0, (3.31) ( ∂ + Q i − ν i Q i c + ) I n + r ∑ j = 1 n ∑ k = 1 ν i ν i − ǫ k n µ j h k P j h − k ! = 0, (3.32) r espectively . Similarly , the requir eme nt of vanishing of the r esidues at the points µ i gives the relations I n + r ∑ j = 1 n ∑ k = 1 µ i µ i − ǫ k n ν j h k Q j h − k ! ( ∂ − P i + µ − 1 i c − P i ) = 0, (3.33) I n + r ∑ j = 1 n ∑ k = 1 µ i µ i − ǫ k n ν j h k Q j h − k ! ( ∂ + P i + µ i c + P i ) = 0. (3.34) T o obtain the r elations (3.31)–(3.34) we made use of the equalities (3.27), (3.28). Suppose that we have succeede d in satisfying the relations (3.27), (3.28) and (3.3 1 )– (3.34). In such a case fr om the equalities (3.29) and (3.3 0) it follows that the connection under consideration satisfies the grading condition. It is easy to see fro m (3.30) that ω + ( ¯ m , 0 ) = ψ − 1 0 ( ¯ m ) ∂ + ψ 0 ( ¯ m ) . T aking into account that ω + 0 ( ¯ m ) = ω + ( ¯ m , 0 ) , we conclude that the gauge-fixing con- straint ω + 0 = 0 is eq uivalent to the r elation ∂ + ψ 0 = 0. (3.35) Assuming that this r elation is satisfied, we come to a connectio n satisfying both the grading condition and the gauge-fixing condition. Recall that if a flat co nnection ω satisfies the grading a nd gauge-fixing conditions, then ther e exist a mapping γ from R 2 to G and mappings c − and c + of R 2 to g − 1 and g + 1 , r espectively , such that the repr esentation (2.14) for the components ω − and ω + is valid. In general, the mappings c − and c + parameterizing the connection components may be dif ferent fr om the mappings c − and c + which determine t he mapping ϕ . Let us denote the mappings corresponding to the connection under considerat ion by γ ′ , c ′ − and c ′ + . Thus, we have ψ − 1 ∂ − ψ + λ − 1 ψ − 1 c − ψ = γ ′− 1 ∂ − γ ′ + λ − 1 c ′ − , (3.36) ψ − 1 ∂ + ψ + λψ − 1 c + ψ = λ γ ′− 1 c ′ + γ ′ . (3.3 7) 24 Note that ψ ∞ is a mapping of R 2 to the Lie subgr oup of GL n ( C ) defined by the relation (3.24). Recall that this subgr oup coincides with G 0 , and denote ψ ∞ by γ . Fr om the r elation (3.36) we obtain the equality γ ′− 1 ∂ − γ ′ = γ − 1 ∂ − γ . The same relation (3.36 ) gives ψ − 1 0 c − ψ 0 = c ′ − . Impose the condition ψ 0 = I n , which is consistent with (3.35). Here we have c ′ − = c − . Finally , from (3.37) we obtain γ ′− 1 c ′ + γ ′ = γ − 1 c + γ . W e see that if we impose the condition ψ 0 = I n , then the components of the connectio n under consideration have the form given by (2.14) where γ = ψ ∞ . Thus, t o find solutions to T oda equations under consideration, we can use the fol- lowing procedur e. Fix 2 r complex numbers µ i and ν i . Find matrix-valued functions P i and Q i satisfying the relations (3.27), (3.28 ) and (3.31)–(3.34). W ith the help of (3.25), (3.26), assuming that ψ 0 = I n , constr uct the mappings ψ a nd ψ − 1 . Then, the mapping γ = ψ ∞ (3.38) satisfies the T oda equation (2.1 5). Let us ret urn to the relations (3.27), (3.28). I t can be shown that, if we suppose that the matrices P i and Q i ar e of maximum rank, then we get the trivial so lution of the T oda equa tion given by (3.21). Hence, we will assume that P i and Q i ar e not of maximum rank. The simple st case he r e is given by matrices of rank one which can be r epr esented as P i = u i t w i , Q i = x i t y i , wher e u , w , x and y a re n -dimensional column vectors. This repr esentation allows one to write the relations (3.27) and (3.28) as t y i + r ∑ j = 1 n ∑ k = 1 ν i ν i − ǫ k n µ j ( t y i h k u j ) t w j h − k = 0, (3.39) u i + r ∑ j = 1 n ∑ k = 1 µ i µ i − ǫ k n ν j h k x j ( t y j h − k u i ) = 0. (3.40) Using the identity n − 1 ∑ k = 0 z ǫ − jk n z − ǫ k n = n z n − | j | n z n − 1 , (3.41) wher e | j | n is the residue of division of j by n , we can r ewrite (3.39) in components terms, y i k + n r ∑ j = 1 ( R k ) i j w j k = 0. (3.42) 25 Here the r × r matrices R k ar e defined as ( R k ) i j = 1 ν n i − µ n j n ∑ ℓ = 1 ν n − | ℓ − k | n i µ | ℓ − k | n j y i ℓ u j ℓ . The same identity (3.41) allows one to write component form of (3.40) as u i k + n r ∑ j = 1 x j k ( S k ) ji = 0, (3.43) wher e ( S k ) ji = − 1 ν n j − µ n i n ∑ ℓ = 1 ν | k − ℓ | n j µ n − | k − ℓ | n i y j ℓ u i ℓ . W ith the help of the equality n − 1 − | i − 1 | n = | − i | n it is s traightfor war d to demonstrate that ( S k ) ji = − µ i ν j ( R k + 1 ) ji . Consequently , we can write the eq ua tion (3.4 3) as u i k − n µ i r ∑ j = 1 x j k 1 ν j ( R k + 1 ) ji = 0. (3.44) W e use the equations (3.42) and (3.44) to express the vector s w i and x i via the vectors u i and y i , w i k = − 1 n r ∑ j = 1 ( R − 1 k ) i j y j k , x i k = 1 n r ∑ j = 1 u j k 1 µ j ( R − 1 k + 1 ) ji ν i . As the result, we come to the following solution of the relations (3.27) and (3.28): ( P i ) k ℓ = − 1 n u i k r ∑ j = 1 ( R − 1 ℓ ) i j y j ℓ , ( Q i ) k ℓ = 1 n r ∑ j = 1 u j k 1 µ j ( R − 1 k + 1 ) ji ν i y i ℓ . Further , it follows fr om (3.39) and (3.40) that, to fulfill also (3.31)–(3. 3 4), it is suf fi- cient to satisfy the equations ∂ − y i = ν − 1 i t c − y i , ∂ + y i = ν i t c + y i , (3.45) ∂ − u i = − µ − 1 i c − u i , ∂ + u i = − µ i c + u i . (3.46) The n -dimensional column vectors θ ρ , defined by t he r elation (3.13), are eigenvect ors of the m a trices t c − , t c + , c − and c + , t c − θ ρ = m ǫ ρ n θ ρ , t c + θ ρ = m ǫ − ρ n θ ρ , c − θ ρ = m ǫ − ρ n θ ρ , c + θ ρ = m ǫ ρ n θ ρ , and form a basis in the space C n . H e nce, the g eneral solution of the equations (3.45) and (3.46) can be written in the form u i k = n ∑ ρ = 1 c i ρ ǫ k ρ n e − Z ρ ( µ i ) , y i k = n ∑ ρ = 1 d i ρ ǫ k ρ n e Z − ρ ( ν i ) , 26 wher e c i ρ , d i ρ ar e arbitrary constants and Z ρ ( µ ) = m ( ǫ − ρ n µ − 1 z − + ǫ ρ n µ z + ) . Thus, we see t hat it is possible to satisfy (3.2 7), (3.28) and (3.31 )–(3. 3 4). This gives us solutions of the T oda equations (2.28). Let us show that they can be written in a simple determinant form. Using (3. 3 8) and (3.25), one gets γ = ψ ∞ = I n + r ∑ i = 1 n ∑ k = 1 h k P i h − k . For the matrix elements of γ this gives the expr ession γ k ℓ = δ k ℓ 1 + n r ∑ i = 1 ( P i ) kk ! = δ k ℓ 1 − r ∑ i , j = 1 u i k ( R − 1 k ) i j y j k ! . Hence, we have Γ α = 1 − r ∑ i , j = 1 u i α ( R − 1 α ) i j y j α . T o this expressio n can also be given the form Γ α = 1 − t u α R − 1 α y α , wher e u α and y α ar e r -dimensional co lumn vectors with the components u i α and y i α , r espectively . W e assume for co nvenience that the functions u i α and y i α ar e de fined for arbit rary integral values of α and u i , α + n = u i α , y i , α + n = y i α . The periodicity of R α in the index α follows fr om the definition. I t appears that it is mor e appropr iate to use quasi- periodic quantities e u α , e y α and e R α defined as e u α = M α u α , e y α = N − α y α , e R α = N − α R α M α , wher e N and M ar e diagonal r × r matrices given by N i j = ν i δ i j , M i j = µ i δ i j . For these quantities one has quasi-periodicity conditions e u α + n = M n e u α , e y α + n = N − n e y α , e R α + n = N − n e R α M n . The expr e ssion of the matrix elements of the matrices e R α thr ough the functions e y i α and e u i α has a remarkably simple form [3] ( e R α ) i j = 1 ν n i − µ n j µ n j α − 1 ∑ β = 1 e y i β e u j β + ν n i n ∑ β = α e y i β e u j β ! . (3.47) 27 In terms of the quasi-periodic quantities, for the functions Γ α we have Γ α = 1 − t e u α e R − 1 α e y α , and it can be shown that Γ α = det ( e R α − e y α t e u α ) det e R α . Using the explicit form of e R α , one comes to the equality e R α + 1 = e R α − e y α t e u α , Ther efor e, one can write [3] Γ α = det e R α + 1 det e R α . (3.48) 3.3 Solitons through the rational dressing T o obtain a one-soliton solut ion o ne puts r = 1. In this case e R α ar e ordinary functions for which one has the expr ession e R α = 1 ν n − µ n n ∑ ρ , σ = 1 c ρ d σ e − Z ρ ( µ ) + Z − σ ( ν ) " µ n α − 1 ∑ β = 1 µ β ν − β ǫ ( ρ + σ ) β n + ν n n ∑ β = α µ β ν − β ǫ ( ρ + σ ) β n # . It is not difficult to verify that µ n α − 1 ∑ β = 1 µ β ν − β ǫ ( ρ + σ ) β n + ν n n ∑ β = α µ β ν − β ǫ ( ρ + σ ) β n = ( ν n − µ n ) µ α ν − α ǫ ( ρ + σ ) α n 1 − µ ν − 1 ǫ ρ + σ n . Thus one obtains the following expressio n for e R α : e R α = µ α ν − α n ∑ ρ , σ = 1 c ρ d σ e − Z ρ ( µ ) + Z − σ ( ν ) ǫ ( ρ + σ ) α n 1 − µ ν − 1 ǫ ρ + σ n . T o obtain a solution which de p e nds on only one combination of z − and z + we suppose t hat c ρ is differ e nt fr om zero for only one value of ρ whic h we denote by I , and that d σ is dif ferent fr om zero for o nly two values of σ which we denote by J and K . I n this case we arrive at a simplified version of e R α , that is e R α = µ α ν − α c I e − Z I ( µ ) " d J e Z − J ( ν ) ǫ ( I + J ) α n 1 − µ ν − 1 ǫ I + J n + d K e Z − K ( ν ) ǫ ( I + K ) α n 1 − µ ν − 1 ǫ I + K n # , and the corresponding solution can be written as Γ α = µ ν − 1 ǫ I + J n 1 + d ǫ ( K − J )( α + 1 ) n e Z − K ( ν ) − Z − J ( ν ) 1 + d ǫ ( K − J ) α n e Z − K ( ν ) − Z − J ( ν ) , wher e d = d K ( 1 − µ ν − 1 ǫ I + J n ) d J ( 1 − µ ν − 1 ǫ I + K n ) . 28 Making use of the fr eedom in multiplying a solution by a constant, we can wr ite the obtained solution as (3.18), where ρ = K − J , κ ρ is defined by (3.16), ζ = − i ǫ − ( K + J ) /2 n ν , and δ is a co nstant intr oduced by the r elation exp δ = d . Thus we arrive at the one- soliton solution obtained before by the Hiro ta’s method. In the case of r > 1 (multi-solito n constructio n) we suppose that for any i t he coef ficients c i ρ ar e differ ent from zero for only one value of ρ which we denote by I i , and that the coef ficients d i σ ar e differ ent from zero for only two values of σ which we denote by J i and K i . This lea d s to the following expr ession for t he matrix elements of the ma trices e R α : ( e R α ) i j = ν − α i ǫ J i α n d J i e Z − J i ( ν i ) ×   1 1 − µ j ν − 1 i ǫ I j + J i n + d K i d J i e Z − K i ( ν i ) − Z − J i ( ν i ) ǫ ( K i − J i ) α n 1 − µ j ν − 1 i ǫ I j + K i n   µ α j ǫ I j α n c I j e − Z I j ( µ j ) . Immediately we see from (3.48) that the solution in question has the form Γ α = " r ∏ i = 1 µ i ν − 1 i ǫ I i + J i n # det e R ′ α + 1 det e R ′ α , (3.49) wher e the matrices e R ′ α ar e defined by ( e R ′ α ) i j = 1 1 − µ j ν − 1 i ǫ I j + J i n + d K i d J i e Z − K i ( ν i ) − Z − J i ( ν i ) ǫ ( K i − J i ) α n 1 − µ j ν − 1 i ǫ I j + K i n . Using the matrices D d e fined in appendix A .1, we rewrite the expressio n for e R ′ α in the form ( e R ′ α ) i j = D i j ( ν ǫ − J n , µ ǫ I n ) + d K i d J i ǫ ( K i − J i ) α n e Z − K i ( ν i ) − Z − J i ( ν i ) D i j ( ν ǫ − K n , µ ǫ I n ) . It is clear that instead of e R ′ i one can use in the r elation (3.49) the matrices e R ′′ i defined as ( e R ′′ α ) i j = δ i j + d K i d J i ǫ ( K i − J i ) α n e Z − K i ( ν i ) − Z − J i ( ν i ) r ∑ k = 1 D i k ( ν ǫ − K n , µ ǫ I n ) D − 1 k j ( ν ǫ − J n , µ ǫ I n ) . Using the equa lity (A.3), one comes to the expression ( e R ′′ α ) i j = ν i r ∏ ℓ = 1 ℓ 6 = i ( ν i ǫ − J i n − ν ℓ ǫ − J ℓ n ) ǫ J i n r ∏ ℓ = 1 ( ν i ǫ − J i n − µ ℓ ǫ I ℓ n ) ( T α ) i j ǫ J j n r ∏ ℓ = 1 ( ν j ǫ − J j n − µ ℓ ǫ I ℓ n ) ν j r ∏ ℓ = 1 ℓ 6 = j ( ν j ǫ − J j n − ν ℓ ǫ − J ℓ n ) , (3.50) wher e ( T α ) i j = δ i j + d i ǫ ( K i − J i ) α n e Z − K i ( ν i ) − Z − J i ( ν i ) ν i ǫ − K i n − ν i ǫ − J i n ν i ǫ − K i n − ν j ǫ − J j n 29 and, with a slight abuse of notation, d i = d K i ǫ J i n r ∏ ℓ = 1 ℓ 6 = i ( ν i ǫ − K i n − ν ℓ ǫ − J ℓ n ) r ∏ ℓ = 1 ( ν i ǫ − J i n − µ ℓ ǫ I ℓ n ) d J i ǫ K i n r ∏ ℓ = 1 ℓ 6 = i ( ν i ǫ − J i n − ν ℓ ǫ − J ℓ n ) r ∏ ℓ = 1 ( ν i ǫ − K i n − µ ℓ ǫ I ℓ n ) . Utilizing the expression (3.50) and having in mind t he freedom in mult iplying a solu- tion by a constant, we write the solution under consideration as follows: Γ α = det T α + 1 det T α . Defining ρ i = K i − J i , ζ i = − i ǫ − ( K i + J i ) / 2 n ν i and intr oducing constants δ i satisfying the r elations exp δ i = d i , one can write ( T α ) i j = δ i j + ǫ ρ i α n exp [ m κ ρ i ( ζ − 1 i z − + ζ i z + ) + δ i ] ǫ − ρ i /2 n ζ i − ǫ ρ i /2 n ζ i ǫ − ρ i /2 n ζ i − ǫ ρ j /2 n ζ j . (3.51) It is proved in ap p endix A.2 that det T α + 1 = 1 + r ∑ i = 1 E α i + r ∑ ℓ = 2 " ∑ 1 ≤ i 1 < i 2 < . . . < i ℓ ≤ r ∏ 1 ≤ j < k ≤ ℓ η i j i k ! E α i 1 E α i 2 . . . E α i ℓ # , (3.52) wher e the functions E α i and η i j i k ar e defined by the relations (3.15) and (3.19 ) respec- tively . Thus, we come to the multi-solito n solutions which coincide with those ob- tained by the Hiro ta’s method. The Hirota’s τ -functio ns (3.20) are given by the equal- ity τ α = det T α + 1 . It is clear that the quantities η i j i k here make the same sense as do the nor mal ordering coef ficients effect ively describing the interaction between solitons in t he vertex oper- ators approach of Olive, T urok and Unde rwood [14, 15]. W e r efer the reader to the papers [10, 34, 35] for some mor e spe cific properties of such coefficients . 4 Conclus ion In this paper we have considered abelian T oda systems associated with the loop gro ups of the complex general linear gro ups. W e have reviewed two differ ent approaches to constr uct soliton solutions to these equations in the untwisted case, namely , the Hi- r ota’s and rational dr essing methods. Subsequently , basic ingr e dients repr esenting soliton solutions within the frameworks of these methods have been explicitly related. As we have seen in section 3.2, t he rational dr essing metho d allows one t o construct solutions to the loop T oda eq ua tions, presenting them as the ratio of the determinants of specific matrices (3.47), (3.48), and they actually repr esent a class of solutions be- ing wider than that formed by the soliton solutio ns of the Hirota’s method in section 30 3.1: By setting the initial-data coefficients arising in the rational dr essing method to some specific values we have shown in section 3.3 that the Hir ota’s soliton solutio ns ar e contained among the solutions construc ted by the rational dr essing appro ach. Note also that the reductio n to the systems based on the loop groups of the complex special linear gr oups can easily be performed. Our consideration can be generalized to T oda systems based on other loop groups, such as twisted loop gro ups of the complex general line a r gr oups, twisted and un- twisted loop groups of the complex orthogo nal and symplectic gr oups. However , one should take int o account that the change of field variables in the Hirot a’s method is mor e tricky ther e, and besides, when applying the rational dressing to obt ain soliton solutions, one faces that the pole positions of the dress ing m e r omorphic mappings and their inverse ones ar e to be related, just due to the group conditio ns. These cir - cumstances make part of the formulae mor e intricate than in t he general linear case consider ed in the present paper . W e will address to this problem and p resent our r esults in some future publications. This work was supported in part by the R ussian Foundation for Ba sic Research under grant #0 7 –01–00234. Appendi x A.1 Some properties of the matrices D In this appendix we investig ate r × r matrices D ( f , g ) with matr ix elements given by the e quality D i j ( f , g ) = 1 1 − f − 1 i g j = f i f i − g j . Let us show that for the matrix elements of the inverse matr ix D − 1 ( f , g ) one has the r epr esentation D − 1 i j ( f , g ) = r ∏ ℓ = 1 ℓ 6 = j ( f ℓ − g i ) r ∏ ℓ = 1 ( f j − g ℓ ) f j r ∏ ℓ = 1 ℓ 6 = i ( g ℓ − g i ) r ∏ ℓ = 1 ℓ 6 = j ( f j − f ℓ ) . (A.1) T o prov e the above equality one has to demonstrate that r ∑ k = 1 f i r ∏ ℓ = 1 ℓ 6 = j ( f ℓ − g k ) r ∏ ℓ = 1 ( f j − g ℓ ) f j ( f i − g k ) r ∏ ℓ = 1 ℓ 6 = k ( g ℓ − g k ) r ∏ ℓ = 1 ℓ 6 = j ( f j − f ℓ ) = δ i j . (A.2) 31 Consider the set of meromo rphic functions of z define d as F i j ( f , g , z ) = r ∏ ℓ = 1 ℓ 6 = j ( f ℓ − z ) ( f i − z ) r ∏ ℓ = 1 ( g ℓ − z ) . The residue of F i j ( f , g , z ) at infinity is e qual to zero, ther efor e , the su m of the r esidues at the p oint f i and at the points g ℓ , ℓ = 1, . . . , r , is also zero. Hence we have the following eq uality r ∑ k = 1 r ∏ ℓ = 1 ℓ 6 = j ( f ℓ − g k ) ( f i − g k ) r ∏ ℓ = 1 ℓ 6 = k ( g ℓ − g k ) = − r ∏ ℓ = 1 ℓ 6 = j ( f ℓ − f i ) r ∏ ℓ = 1 ( g ℓ − f i ) , and, therefor e, r ∑ k = 1 f i r ∏ ℓ = 1 ℓ 6 = j ( f ℓ − g k ) r ∏ ℓ = 1 ( f j − g ℓ ) f j ( f i − g k ) r ∏ ℓ = 1 ℓ 6 = k ( g ℓ − g k ) r ∏ ℓ = 1 ℓ 6 = j ( f j − f ℓ ) = f i r ∏ ℓ = 1 ℓ 6 = j ( f i − f ℓ ) r ∏ ℓ = 1 ( f j − g ℓ ) f j r ∏ ℓ = 1 ( f i − g ℓ ) r ∏ ℓ = 1 ℓ 6 = j ( f j − f ℓ ) . Now , taking into account the ide ntity r ∏ ℓ = 1 ℓ 6 = j ( f i − f ℓ ) / r ∏ ℓ = 1 ℓ 6 = j ( f j − f ℓ ) = δ i j , we see that the relatio n (A.2) is true. Thus the equivalent relation (A.1) is also true. In a similar way one can pro ve the validity of the equality r ∑ k = 1 D i k ( e f , g ) D − 1 k j ( f , g ) = e f i r ∏ ℓ = 1 ℓ 6 = j ( e f i − f ℓ ) r ∏ ℓ = 1 ( f j − g ℓ ) f j r ∏ ℓ = 1 ( e f i − g ℓ ) r ∏ ℓ = 1 ℓ 6 = j ( f j − f ℓ ) . (A.3) A.2 Proof of relation (3.52) Pr oceeding fr om the relation (3.51), one obtains ( T α + 1 ) i j = δ i j + E α i ˜ f i − f i ˜ f i − f j , 32 wher e e f i = ǫ − ρ i /2 n ζ i , f i = ǫ ρ i /2 n ζ i . (A. 4 ) and the functions E α i ar e defined by t he relation (3.15 ). Then it is not diffic ult to get convinced that det T α + 1 = 1 + r ∑ i = 1 E α i + r ∑ ℓ = 2 " ∑ 1 ≤ i 1 < i 2 < . . . < i ℓ ≤ r η i 1 i 2 . . . i ℓ E α i 1 E α i 2 . . . E α i ℓ # , (A.5) wher e η i 1 . . . i ℓ = ∑ π ∈ S ℓ sgn ( π ) ℓ ∏ j = 1 e f i j − f i j e f i j − f i π ( j ) . As is customary , we de note by S ℓ the symmetric group o n the set { 1, 2, . . . , ℓ } and by sgn ( π ) the signature of the permutation π . For ℓ = 2 one has η i 1 i 2 = 1 − ( e f i 1 − f i 1 )( e f i 2 − f i 2 ) ( e f i 1 − f i 2 )( e f i 2 − f i 1 ) = ( f i 1 − f i 2 )( e f i 2 − e f i 1 ) ( e f i 1 − f i 2 )( e f i 2 − f i 1 ) . Using the definition (A.4) of f i and ˜ f i , we see that the quantities η i 1 i 2 coincide with the coef ficients η i 1 i 2 defined by the relatio n (3.19). Let us pr ove by induction that η i 1 i 2 . . . i ℓ = ∏ 1 ≤ j < k ≤ ℓ η i j i k . (A.6) Certainly , for ℓ = 2 the equality (A.6) is valid. Suppose that it is valid up to some fixed value of ℓ and show that it is valid for its value incremented by one. The group S ℓ can be ide ntified with a subgr oup of S ℓ + 1 formed by the permutations π ∈ S ℓ + 1 satisfying the condition π ( ℓ + 1 ) = ℓ + 1. De note by π m , m = 1, . . . , ℓ , the transpositio n e xchanging m and ℓ + 1 and repr esent the gr oup S ℓ + 1 as the union of the right cosets S ℓ π m . This allows us to write η i 1 . . . i ℓ i ℓ + 1 = η i 1 . . . i ℓ − ℓ ∑ m = 1 ∑ π ∈ S ℓ sgn ( π ) ℓ + 1 ∏ j = 1 e f i j − f i j e f i j − f i π ( π m ( j ) ) . (A.7) It is not difficult to realize that ∑ π ∈ S ℓ sgn ( π ) ℓ + 1 ∏ j = 1 e f i j − f i j e f i j − f i π ( π m ( j ) ) = ( e f i m − f i m )( e f i ℓ + 1 − f i ℓ + 1 ) ( e f i m − f i ℓ + 1 )( e f i ℓ + 1 − f i m ) η i 1 . . . i ℓ   e f i m = e f i ℓ + 1 , and that η i 1 . . . i ℓ   e f i m = e f i ℓ + 1 = ℓ ∏ j = 1 j 6 = m ( e f i ℓ + 1 − e f i j )( e f i m − f i j ) ( e f i m − e f i j )( e f i ℓ + 1 − f i j ) η i 1 . . . i ℓ . 33 Using these equa lities in (A.7), we obtain η i 1 . . . i ℓ i ℓ + 1 = η i 1 . . . i ℓ + η i 1 . . . i ℓ ( e f i ℓ + 1 − f i ℓ + 1 ) ℓ ∏ j = 1 ( e f i ℓ + 1 − e f i j ) ℓ ∏ j = 1 ( e f i ℓ + 1 − f i j ) ℓ ∑ m = 1 ℓ ∏ j = 1 ( e f i m − f i j ) ( e f i m − f i ℓ + 1 ) ℓ + 1 ∏ j = 1 j 6 = m ( e f i m − e f i j ) . (A.8) Now consider a meromorphic function of z defined as F ( f , e f , z ) = ℓ ∏ j = 1 ( z − f i j ) ( z − f i ℓ + 1 ) ℓ + 1 ∏ j = 1 ( z − e f i j ) . 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