More non-Abelian loop Toda solitons

More non-Abelian loop T oda solitons Kh. S. Nir ov ∗ and A. V . Ra z umov † Fachbereic h C–Physik, Ber gische Universit ¨ at W uppertal D-4209 7 W uppertal, Germany Abstract W e find new solutions, including soliton-like ones, for a special case of non- Abelian loop T oda equations associated with complex general linear groups. W e use the method o f rational dressing based on an appropriate block-matrix repr e- sentation sugges ted b y the Z -gradation under consideration. W e present solutions in a fo r m of a direct matrix gene raliza tion of t he Hirota’s soliton solution already well-known for the case of Abe lian loop T oda systems. Mathematics Su bject Classification (200 0). 37K10 , 37K15, 35Q58 Keywords. Non-Abelian loop T oda s ystems , rational d ressing method, soliton-like solutions 1 Introduc t ion The two-dimensional T oda equations play an essential r ˆ ole in understanding certain stru ctur es in classical and quantum integrable systems. They are formulated as nonlin- ear pa rtial dif ferential equations of second or der and ar e associated with Lie gr oups, see, for example, the monogr aphs [1, 2]. The T oda equations associated with affi ne Kac–Moody gr oups are of special interest , because they possess solito n solut ions hav- ing a lot of physical applications. The simplest example here is the celebrated sine- Gor don equation known for a long time. A nother example of an affine T oda equation was constr ucted in paper [3] as a direct two-dimensional generalization of the famous mechanical T oda chain. Later on, the consideration of paper [3] was generalized in papers [4, 5] in the case of T oda chains r ela ted to various affine Kac–Moody algebras. Another approach to formulating affine T oda systems, based on folding properties of Dynkin d ia grams, was impleme nted in [6]. Note also that papers [3, 4, 5, 7] were pi- oneering in investigating the question of integrability of affine T oda field theories by considering the corresponding zero-cu rvatur e repr esentation. It is convenient to consider instead of the T oda systems associated with af fine Kac- Moody gr oups the T oda systems associated with loop gr oups. There ar e two reasons to do so. First , the affine Kac-Moody gr oups can be considered as a loop extension of ∗ On leave of a bsence from the Institute for Nuclear Resear ch of the Ru ssian Academy of Sciences, 60th October A ve 7a, 117312 Moscow , Russia † On leave of absence from the Institute for High Energy Physics, 142281 Protv ino, Moscow R egion, Russia 1 loop gr oups and the solutions of the corr esponding T oda equations are connected in a simple way , see, for example , pap er [8]. Second, in distinction to the loop groups , ther e is no a realization of a ffine Kac–Moody gr oups suitable for practical usage. It is known that a T oda equation associated with a Lie gr oup is spe cified by the choice of a Z -gradation of its Lie algebra [1, 2]. Hence, to classify the T oda syst ems associated with some class of Lie gro ups one needs to describe all Z -gradations of the r espective Lie algebras. R e cently , in a series of pape rs [9, 10, 11], we classified a wide class of T oda eq uations associated with untwisted and twisted loop grou ps of complex classical Lie gr oups. Mor e concr etely , we introduced the notion of an integrable Z - gradation of a loop L ie algebra, and found all such gradations with finite-dimensional grading subspaces for the loop Lie algebras of complex classical Lie algebras. Then we described the r espective T oda equations. I t appeared that despite the fact that we consider T oda equations associated with infinite-dimensional Lie gro ups, the resulting T oda equations are equivalent to the equations formulated only in terms of the under- lying finite-dimensional Lie grou ps and Lie al gebras. Actually , pa rtial cases of such type of equations appea red befor e, see, for example, pa p e rs [12, 13, 14] a nd r eferences ther e in, but we demonstrated that any T oda eq ua tion of the class under consideration can be written in terms of finite-dimensional Lie gr oups and Lie algebras. T o slightly simplify terminology and make distinction with the T oda equations associated with finite-dimensional Lie groups, we call the finite-dimensional version of a T oda equa- tion associated with a loop gr oup of the Lie gro up G a loop T oda equation associated with the Lie group G . Here we consider untwist ed loop T oda equations associated with complex general linear grou p. A s was shown in papers [10, 11], any such equation ha s the form 1 ∂ + ( γ − 1 ∂ − γ ) = [ c − , γ − 1 c + γ ] , (1) supplied with the conditions ∂ + c − = 0, ∂ − c + = 0. (2) Here, γ is a mapping of the two-dimensional manifold M to the complex general linear grou p GL n ( C ) having a block-diagonal form γ =     Γ 1 Γ 2 Γ p     , so that for each α = 1, . . . , p the ma pping Γ α is a mapping of M to the Lie group GL n α ( C ) with ∑ p α = 1 n α = n . Further , c + and c − ar e mappings of M to the Lie algebra gl n ( C ) . The mapping c + has the block-matrix str ucture c + =         0 C + 1 0 0 C + ( p − 1 ) C + 0 0         , 1 W e denote by ∂ + and ∂ − the partial derivatives over the standa rd coordinates z + and z − of a sm ooth two-dimensional manifold M , where M is either the E uclidean plane R 2 or the complex line C ; in the latter case z − denotes the standard complex coordinate on C , and z + – its complex c onjugate. 2 wher e for each α = 1, . . . , p − 1 the ma p p ing C + α is a mapping of M to the space of n α × n α + 1 complex matrices, a nd C + 0 is a mapping of M to the space of n p × n 1 complex matrices. The mapping c − has a similar block-mat rix str ucture, c − =         0 C − 0 C − 1 0 0 C − ( p − 1 ) 0         , wher e for each α = 1, . . . , p − 1 the ma p p ing C − α is a mapping 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. It is assumed that the mappings c + and c − ar e fixed, and the e qua- tion (1) is considered as an equation for the mapping γ , which can be written explicitly as a system of equations for the mappings Γ α . It is worth noting that for arbitrary complex classical Lie gro ups loop T oda e qua- tions belonging to the class under consideration have the same form (1) with the same block-matrix structur e of the mappings γ , c + and c − , but with some additional restric - tions imposed on the blocks. Note also that the T oda equation unde r consideration is Abelian if the mapping γ is effectively a mapping to an Abelian Lie group, otherwise we have a non-Abelian T oda equation. In the pr esent pa per we consider a particular case of non-Abelian loop T oda equa- tions associated with the complex general linear gr oup GL n ( C ) , where n α = n / p = n ∗ for a l l α = 1, . . . , p . Moreover , we assume for simplicity that all nonzero entries of the block-matrix repr esentation of c + and c − ar e unit n ∗ × n ∗ matrices. In this case the T oda equation (1) can be written as an infinite periodic system, ∂ + ( Γ − 1 α ∂ − Γ α ) + Γ − 1 α Γ α + 1 − Γ − 1 α − 1 Γ α = 0, (3) with Γ α subject to the condition Γ α + p = Γ α . This particular case of T oda systems was intr oduced in the remarkable paper by Mikhailov [4]. Here we are interest ed in explicit solutions of the system (3), in particular , in the soliton-like ones in the non-Abelian case when n ∗ > 1. Soliton solutions of the A be lian loop T oda equations can be found by various methods. The most known and elabo- rated among them are the Hir ota’s me thod [15], successfully applied to many par- ticular cases of Abelian af fine T oda systems [16, 8, 17, 18, 19, 20]; the vertex opera- tors appro ach of [21, 22, 23] ba sed on a pr oper specialisation of the Leznov–Saveliev method [24], see also [25, 26] for mor e details and the relation to the dressing symme- try; and the formalism of rational dr essing developed by Mikhailov [4] on the basis of a gene ral dressing procedur e proposed by Zakhar ov and Shabat [27]. Actually , in all known cases of Abelian T oda systems the vertex operators constructio ns repr oduce the same soliton solutions found by the Hirota’s approach. In the paper [28] we consid- ered Abelian untwisted loop T oda equations associated with complex general linear gr oups within the frameworks of the Hir ota’s and rational dressing methods and e s- tablished the explicit r elationships between solutions given by these two appro aches. Further , in the paper [29], using the rational dressing method, we have constr ucted multi-solito n solut ions for Abe li a n twisted loop T oda systems associated with general linear grou ps. 3 Ther e are n ot many papers dealing with soliton solutions of non-Abelian loop T oda equations. W e would like to me ntion here paper [30] where a combination of the no- tion of a q uasi-determinant and the Marchenko lemmas were used to construct soliton- like solutions of equations (3). In the pa p e r [31] we have developed the rational dress- ing method in application to non-Abelian untwisted loop T oda equations associated with complex general linear groups a nd found certain multi-solito n solutions. Her e we restr ict our attention to equations (3) and show that the rational dressing method in this most symmetric case allows one to construct new soliton-like solutio ns which can be pr esented in a form of a direct matrix generalization of the Hirota’s soliton solutions well-known for the case of Abelian loop T oda systems. The main ide a of this paper , a s of the paper [3 1 ], is to demonstrate the power of the rational dr essing formalism appro priately de veloped for explicitly constru cting solutions to the non-Abelian loop T oda equations. I t is a distinct ive feature of this method that it allows for such remarkable generalizations to the non-Abelian case, wher e other well-known me thods fail to work. Among the solutions to be presented in what follows, we single out a class of soliton-like ones thus justifying the title of our paper . Her e, by an n -soliton, or soliton-like, solution we mea n a solution depending on n line a r combinations of inde pendent variables and having an appropriate number of characteristic parameters. 2 Rational dressing W e see that the constant matrices c − and c + commute. Hence, it is obvious that γ = I n , (4) wher e I n is the n × n unit m a trix, is a solution to the T oda eq ua tion (1). For the formal- ism of rational dr e ssing it is crucial that the T oda equation (1), (2) can be r epresent ed as the zero-curvat ur e condition for a fl at connection in a trivial principal fiber bundle, satisfying the grading and gauge-fixing conditio ns, see, for example, the books [1, 2]. Actually , having such a connection we have a solution to the T oda equations (1). In our case, the corr esponding base manifold M is either the Euclidea n plane R 2 or the com- plex manifold C , and the fiber coincides with the untwisted loop gr oup L a , p ( GL n ( C ) ) , wher e a denotes the inner automorphism of GL n ( C ) of or der p acting on an element g ∈ GL n ( C ) in accor dance with the equality a ( g ) = h g h − 1 . Here h is a block-diagonal matrix defined by the r e lation h α β = ǫ p − α + 1 p I n ∗ δ α β , α , β = 1, . . . , p , (5) wher e ǫ p = e 2 π i / p is the p th principal r oot of unity . Using the exponential law [32, 33], it is convenient to identify the mapping gen- erating a flat connection under consideration with a smooth mapping of M × S 1 to GL n ( C ) and the connectio n components with smooth mappings of M × S 1 to gl n ( C ) . Below we think of the circle S 1 as consisting of complex numbers of modulus one. Denote the mapping generating the connection corr esponding to the solution (4) by ϕ . For the case under consideration, the rational dr e ssing method consists in finding a 4 mapping ψ of M × S 1 to GL n ( C ) , such that the grading and gauge-fixing conditions for the components of the flat connection generated by the mapping ϕ ψ are satisfied. W e assume that the analytic extension of the mapping ψ fr om S 1 to the whole Riemann spher e is a rational mapping given by the expressio n ψ = I n + r ∑ i = 1 p ∑ k = 1 λ λ − ǫ k p µ i h k P i h − k ! ψ 0 . (6) Here λ is the standard coor dinate in C , ψ 0 is a mapping of M to the Lie subgroup of GL n ( C ) formed by the elements g ∈ GL n ( C ) subject to the equality h g h − 1 = g , and P i ar e some smooth mappings of M to the algebra Mat n ( C ) of n × n complex matrices. It is also assumed her e that µ i 6 = 0, µ p i 6 = µ p j for all i 6 = j . Express ion (6) is obtained from an initial rational mapping by averaging over the action of the inner automorphism a , wher e it was used that a p = id GL ( C ) . Furt her , we suppose that the analytic extension of the corr esponding inverse mapping to the whole R iemann sphere has a similar str ucture, ψ − 1 = ψ − 1 0 I n + r ∑ i = 1 p ∑ k = 1 λ λ − ǫ k p ν i h k Q i h − k ! , with the pole positions satisfying the conditions ν i 6 = 0, ν p i 6 = ν p j for all i 6 = j , and additionally ν p i 6 = µ p j for any i and j . Note that the mappings ψ ( λ ) and ψ − 1 ( λ ) are r egular at the points λ = 0 and λ = ∞ . By definition, the equality ψ − 1 ψ = I n is valid on S 1 . Since ψ and ψ − 1 ar e ratio nal ma p pings, this equality is valid on the whole Riemann sphere. Ther efor e, the r esidues of ψ − 1 ψ at the points µ i and ν i must vanish. This leads to certain relatio ns to be satisfied by the mappings P i and Q i , Q i I n + r ∑ j = 1 p ∑ k = 1 ν i ν i − ǫ k p µ j h k P j h − k ! = 0, (7) I n + r ∑ j = 1 p ∑ k = 1 µ i µ i − ǫ k p ν j h k Q j h − k ! P i = 0. (8) Further , for the components of the flat connection generated by the mapping ϕ ψ we find the expressions ω − = ψ − 1 ∂ − ψ + λ − 1 ψ − 1 c − ψ , ω + = ψ − 1 ∂ + ψ + λψ − 1 c + ψ . W e see that ω − is a rational mapping having simple poles at µ i , ν i and zero. Similarly , ω + is a rational ma p p ing having simple poles at µ i , ν i and infinity . W e need a connec- tion satisfying the grading a nd gauge-fixing conditio ns. The grading condition in our case means that for each point of M the component ω − ( λ ) is rational and has the only simple pole at ze r o, and the component ω + ( λ ) is rational and has the only simple pole at infinity . Therefor e, we r e quir e that the r esidue s of ω − and ω + at the points µ i and 5 ν i must vanish. And this requir eme nt imposes additional conditions on the mappings P i and Q i , that ar e ( ∂ − Q i − ν − 1 i Q i c − ) I n + r ∑ j = 1 p ∑ k = 1 ν i ν i − ǫ k p µ j h k P j h − k ! = 0, (9) ( ∂ + Q i − ν i Q i c + ) I n + r ∑ j = 1 p ∑ k = 1 ν i ν i − ǫ k p µ j h k P j h − k ! = 0 (10) for the residues at the points ν i , and also I n + r ∑ j = 1 p ∑ k = 1 µ i µ i − ǫ k p ν j h k Q j h − k ! ( ∂ − P i + µ − 1 i c − P i ) = 0, (11) I n + r ∑ j = 1 p ∑ k = 1 µ i µ i − ǫ k p ν j h k Q j h − k ! ( ∂ + P i + µ i c + P i ) = 0 (12) for the r esidues at the points µ i . Having these and the above me ntioned relations fulfilled by P i and Q i , and besides assuming that ψ 0 = I n that reso lves the gauge- fixing constraint ω + 0 = 0 , where we put ω + 0 = ω + ( 0 ) , we can see that the mapping γ = ψ ( ∞ ) , with fixed 2 r complex numbers µ i , ν i , satisfies the T oda e quation (1). In the block-matrix form we have the expressions γ α β = δ α β I n ∗ + p r ∑ i = 1 ( P i ) αα ! , γ − 1 α β = δ α β I n ∗ + p r ∑ i = 1 ( Q i ) αα ! , (13) wher e ( P i ) α β and ( Q i ) α β ar e n ∗ × n ∗ complex matrices satisfying certain conditions (7)–(12). These conditions ensuring the vanishing of the r esidues of ψ − 1 ψ , ω − and ω + at the points ν i , µ i can be non-trivially fulfilled, see also the papers [28, 31]. T o make the solutio n (13) explicit we should specify the ma trix-valued functions P i and Q i . W e first note that, if we suppose that the functions P i and Q i take values in the space of the matrices of maximum rank, then we come to the tr ivial solution (4). H e nce, we assume that the values of the functions P i and Q i ar e not matrices of maximum rank. The case given by matrices of rank one was elaborated in p a per [31 ]. Now we consider another inter esting case, where the functions P i and Q i take values in the space of n × n ma trices of rank n ∗ . Such functions can be repr esented as 2 P i = u i t w i , Q i = x i t y i , (14) wher e u , w , x and y ar e functions on M taking values in the space of n × n ∗ complex matrices of rank n ∗ . The used Z -gradation suggests the most convenient block-matrix r epresent ation for the matrices of the form (14) as ( P i ) α β = u i , α t w i , β , ( Q i ) α β = x i , α t y i , β , wher e the standard matrix multiplication of the n ∗ × n ∗ matrix-valued functions u i , α , x i , α by the n ∗ × n ∗ matrix-valued functions t w i , β , t y i , β is implie d . Considering the rank 2 Hereafter , the super script t stands for the usual matrix transposition. 6 of the matrices P i and Q i not to be eq ua l to 1, we ar e looking for a novel generalization of the A belian loop T oda soliton construc tions [28] that would be dif fe rent from those given in pape r [31]. Then, within the formalism of rational dr essing we see that the functions w a nd x can be e xpr e ssed via the functio ns y and u , and we find the express ions ( P i ) α β = − 1 p u i , α r ∑ j = 1 ( R − 1 β ) i j t y j , β , ( Q i ) α β = 1 p r ∑ j = 1 u j , α 1 µ j ( R − 1 α + 1 ) ji ν i t y i , β , with n ∗ r × n ∗ r matrix-valued functions R α defined thr ough its n ∗ × n ∗ blocks as ( R α ) i j = 1 ν p i − µ p j p ∑ β = 1 ν p − | β − α | p i µ | β − α | p j t y i , β u j , β , wher e | x | p denotes the residue of division of x by p . It is important to note here that, unlike the Abe li a n a nd non-Abelian cases considered in papers [28, 29, 3 1], for any i , j the block ( R α ) i j is now an n ∗ × n ∗ matrix-valued function, that adds to the explicitly indicated summations over the pole indices respective matrix multiplications. It is convenient to use quantities defined as e u i , α = u i , α µ α i , e y i , α = y i , α ν − α i and ( e R α ) i j = ν − α i ( R α ) i j µ α j . For the matrices e R α we have explicitly the relation ( e R α ) i j = 1 ν p i − µ p j µ p j α − 1 ∑ β = 1 t e y i , β e u j , β + ν p i p ∑ β = α t e y i , β e u j , β ! . Then, in terms of these quantities, the n ∗ × n ∗ matrix-valued functions Γ α can be writ- ten as Γ α = I n α − r ∑ i , j = 1 e u i , α ( e R − 1 α ) i j t e y j , α . Similarly , for the corr esponding inverse mappin gs we obtain the expressio n Γ − 1 α = I n α + r ∑ i , j = 1 e u i , α ( e R − 1 α + 1 ) i j t e y j , α , which can be useful for verifying the T oda equations. T o finally satisfy the conditions imp osed earlier on the matrix-valued functio ns P i and Q i , we also demand the validity of the equations ∂ − u i = − µ − 1 i c − u i , ∂ + u i = − µ i c + u i , (15) ∂ − y i = ν − 1 i t c − y i , ∂ + y i = ν i t c + y i , (16) that ar e suf ficient to fulfill r e la tions (9), (10) and (11), (12). Using the explicit forms of the matrices c ± , we write d own the general solutions to (15), (16) as 3 u i , β = p ∑ α = 1 ǫ βα p exp  − µ − 1 i ǫ − α p z − − µ i ǫ α p z +  c i , α , (17) y i , β = p ∑ α = 1 ǫ βα p exp  ν − 1 i ǫ α p z − + ν i ǫ − α p z +  d i , α , (18) 3 It should be instructive to confer these expressions with those ones derived in paper [31] for the case of general Z -grada tions of inner type. 7 wher e c i , α and d i , α ar e n ∗ × n ∗ complex matrices meaning the initial-value data for equations (15), (16). W ith these solutio ns we immedia tely obtain for the blocks of the matrix-valued functions e R α the following expression: ( e R α ) i j = p ∑ β , δ = 1 e Z − β ( ν i ) − Z δ ( µ j ) ǫ α ( β + δ ) p 1 − µ j ν − 1 i ǫ β + δ p ν − α i ( t d i , β c j , δ ) µ α j , wher e we have intr oduced the notation Z α ( µ i ) = µ − 1 i ǫ − α p z − + µ i ǫ α p z + . 3 Soliton-l ike soluti o n s T o constr uct solutions making sense a s r -solitons, that is, by the definition we use he r e, solutions de pending on r linear combinations of inde pendent variables z + and z − , we assume that for each value of the inde x i = 1, . . . , r the initial-value data of the T oda system under consideratio n ar e such that matrix-valued coeffic ients c i , α ar e dif ferent fr om zero for only one value of α , which we d e note by I i , a nd that the matrix-valued coef ficients d i , α ar e differ ent fr om ze r o for only two values of α , which we de note by J i and K i . W e a lso use for such non-vanishing initial-data n ∗ × n ∗ matrices the notation d J i = d i , J i , d K i = d i , K i and c I i = c i , I i . For the n ∗ × n ∗ blocks of the matrix- valued functions e u i and e y i this assumption gives e u i , α = µ α i ǫ α I i p e − Z I i ( µ i ) c I i , (19) t e y i , α = ν − α i ǫ α J i p e Z − J i ( ν i ) t d J i + ν − α i ǫ α K i p e Z − K i ( ν i ) t d K i . (20) W ith these relations, we can write the expr ession for the mappings Γ α in the form Γ α = I n α − r ∑ i , j = 1 c I i ( e R ′− 1 α ) i j ( t d J j + E α , j t d K j ) , (21) wher e we have used the notation ( e R ′ α ) i j = e D i j ( J ) + E α , i e D i j ( K ) , E α , i = ǫ αρ i p e Z i ( ζ ) , (22) with the dependence on the variables z + and z − given thr ough the functions Z i ( ζ ) = κ ρ i ( ζ − 1 i z − + ζ i z + ) , and the convenient parameters ρ i = K i − J i , ζ i = − i ν i ǫ − ( K i + J i ) / 2 p , κ ρ i = 2 sin ( π ρ / p ) , and besides e D i j ( A ) = t d A i c I j 1 − ν − 1 i µ j ǫ A i + I j p , i , j = 1, . . . , r , (23) for the n ∗ × n ∗ blocks of the n ∗ r × n ∗ r matrices e D ( A ) , A = J , K . Similar r × r matrices wer e introduced already in our pr evious paper [31] for the rank-1 case, however now , in the case of rank- n ∗ , we have a differ ent situation, such that e D i j itself is a complex n ∗ × n ∗ matrix for e a ch i and j . 8 W e assume that the n ∗ × n ∗ matrices c I i ar e non-degenerate. Let us multiply Γ α in (21) by c I ℓ fr om the right ha nd side and write Γ α c I ℓ = r ∑ i , j = 1 c I i ( e R ′− 1 α ) i j [ ( e R ′ α ) j ℓ − ( t d J j + E α , j t d K j ) c I ℓ ] . (24) It is not dif ficult to see that ( e R ′ α ) j ℓ − ( t d J j + E α , j t d K j ) c I ℓ = ν − 1 j ǫ J j p ( e R ′ α + 1 ) j ℓ µ ℓ ǫ I ℓ p . Now , multiplying (24) from the right ha nd side by the inverse matrix c − 1 I ℓ and summing up over ℓ = 1, . . . , r , we obtain the following expressio n: Γ α = 1 r r ∑ i , j , k = 1 c I i ( e R ′− 1 α ) i j ν − 1 j ǫ J j p ( e R ′ α + 1 ) j k µ k ǫ I k p c − 1 I k . T o give the soliton solutions a final form, it is a lso convenient to make use of simplest symmetries of the T oda equation. It is clear from r ela tions (3) that the transformations Γ α → ξ Γ α , Γ α → x − 1 Γ α x (25) for a nonzero constant ξ and a non-singular constant n ∗ × n ∗ matrix x are symmetry transformatio ns of the T oda system under consideration. In particular , using the symmetry transformations (25) with ξ = µ − 1 ν ǫ − ( I + J ) p and x = c I , we can write the one-soliton solut ion as Γ α = e R ′− 1 α e R ′ α + 1 , (26) wher e the matrices e R ′ α ar e explicitly given by the relations (22), (23). I t r eproduces exactly the one-solit on solution constr ucted in the pa per [30] by means of a differ e nt appro ach. The solution (26) can also be written in a convenient form Γ α = T − 1 α T α + 1 , wher e T α = I n α + E α H and H = e D ( K ) e D − 1 ( J ) . Also the multi-soliton solutions, r ≥ 2, can be written in a compact form, Γ α = T c I ( e R ′− 1 α ) N − 1 J ( e R ′ α + 1 ) M I c − 1 I . (27) Here we use the notation T c I and c − 1 I for n ∗ × n ∗ r a nd n ∗ r × n ∗ matrices, r espectively , defined as T c I = ( c I 1 . . . c I r ) and t c − 1 I = ( t c − 1 I 1 . . . t c − 1 I r ) , and the notation N − 1 J and M I for block-diagonal n ∗ r × n ∗ r matrices defined as N − 1 J =    ν − 1 1 ǫ J 1 p I n ∗ . . . ν − 1 r ǫ J r p I n ∗    , M I =    µ 1 ǫ I 1 p I n ∗ . . . µ r ǫ I r p I n ∗    . Putting n ∗ = 1 we can r e cover the Abe lian case analyzed in paper [28]. The solutions (26) a nd (27) may be regar ded as a novel non-Abelian generalization, compleme ntary 9 to that of p a per [31], of the Hirota’s soliton constru ctions [15] successfully used for the Abelian loop T oda systems. Recall that the τ -functions of the Abelian constr uction of Hir ota, the τ α , were generalized by the set of matrix-valued functions e T X α and ordinary functions e T α in the non-Abelian case [31], while now , in a higher-rank case, we have the set of matrix-valued functions e R ′ α + 1 M I c − 1 I and T c I e R ′− 1 α N − 1 J , taken always in a r espective combination, instead of the functions τ α . T o obtain mor e solutions to these equations, not necessarily soliton-like, one should keep nonzero mor e initial-value data c i , α and d i , α entering (17), (18), that leads to more general expressions for e u i , α and e y i , α than (19) and (20). 4 Discuss ion It should be rather illuminative to repr oduce our r esults along the lines of any other appro aches. One such a possibilit y would be , probably , to try the general dressing pr ocedur e [34, 25 , 26]. Here, after the dr essing transformations ar e completed, the pr oblem is reduced to certain spectral pr oblems for the matrices c + and c − . Thus, using specific vertex operators V i r elated to an appr opriate basis, one could bring the generalized τ -functio ns to the form r ∏ i = 1 ( I n α + E α , i V i + . . . ) , wher e the vertex operators ar e supposed to obey certain nilpotency conditions [25, 26]. As we mentioned at the beginning, the resulting expr essions can be then dir ectly compar e d to what one finds in the case of Abelian T oda systems. However , it is not clear yet how to pr ovide the corresponding statement for the non-Abelian case, while pr eserving the block -matrix structur e suggest ed by the Z -gradation. The ne xt step, most naturally following our constr uctions, would be a thor ough investigation of the physical content of the solutions. In this way , one should describe standar d p roperties defining the solitons, for e xample, in the spirit of paper [16]. Note that such a pr ogram, in general, must be based on the spe cification of real forms of the loop T oda e quations under consideration, so that the solutions making sense as ‘physical solitons’ could be found in the present ones by certain reductio ns. Here, such reductions impose certain conditions on the characteristic parameters entering the explicit forms of the soliton-like solutions . In particular , it can be shown that to compact real forms specified by the conditio ns Γ † α = Γ − 1 α , wher e dagger means the Hermitian conjugation, actually corr espond such ‘physical solitons’, and there are no such solutions in the case of non-compact real forms corr esponding to the condition Γ ∗ α = Γ α , where star denotes the usual complex conjugation. A few comments are in or der . T o have a solution m a king sense as a soliton, we must r equire that µ ∗ i = ν i . Putt ing r = 1 we see that it should be valid H † ≡ ( H ′ exp δ ) † = ǫ ρ p H ′ exp δ , wher e we have used the notatio n e xp δ = ( 1 − µ ν − 1 ǫ I + J p ) / ( 1 − µ ν − 1 ǫ I + K p ) . Note that the function Z ( ζ ) should be real, and so, if z − = x − i t , z + = x + i t , this function can be written in a familiar form, as 2 κ ρ ( x − v t ) / √ 1 + v 2 , whe r e v is the ratio of the imaginary and r eal parts of the parame ter ζ , | ζ | 2 = 1, and if z − = x − t , z + = x + t , as 2 κ ρ ( x + vt ) / √ 1 − v 2 , where v = ( ζ − 1 / ζ ) / ( ζ + 1/ ζ ) for a r eal ζ . In the Abelian limit, whe n n ∗ = 1 , the matrix H ′ is just a complex number , a nd it can be 10 lifted up to exp δ , and then the above r elation, that is a r estriction on the initial-value data, fixes the ima ginary part of total δ to be − π ρ / p . W e would like to r efer to the paper [11], where the above real forms of loop T oda equations were obtained for the case of p = 2, and to the p a per [12], wher e basic phys- ical properties of one-soliton and two-solito n (soliton – anti-soliton and breather) solu- tions were investigated for a matrix generalization of the sine-Gor don equation based on the coset SU 2 × SU 2 /SU 2 . In the A belian case the physical properties of loop T oda solitons, including masses, topolog ical charges, scattering processes, were described in pap e r [16]. W e will addr ess these issues about our non-Abelian constr uctions in forthcoming publications. Acknowledgments W e express our gratitude to Pr ofs. H. Boos, F . G ¨ ohmann and A . Kl ¨ umper for hos- pitality at the University of W uppe rtal and interesting discussio ns. This wor k was supported in p a rt by the by the joint DFG–RFBR grant #08–01–91953. Referenc es [1] A. N. Leznov a nd M. V . Saveliev , Gro up-theoretical m e thods for the integration of non- linear dynamical syste m s , (Birkhauser , Basel, 19 9 2). [2] A. V . Razumov and M. V . Saveliev , Lie algebras, geometry and T oda-type systems , (Cambridge University Press, Cambridge, 1997). [3] A. V . Mikhailov , Integrabil i ty of a two-dimensi onal generalization of the T oda c h ain , Soviet Physics JETP L e tt. 30 (1979) 414 –418. [4] A. V . Mikhailov , The reduction problem and the inver s e scatteri ng method , Physica 3D (1981) 73–117. [5] A. V . Mikhailov , M. A. Olshanetsky and A. M. Per elomov , T wo-dimensional gener- alized T oda l attic e , Commun. Math. Phys. 79 (1981) 473– 4 88. [6] S. P . Khastgir , and R. Sasaki, Non-canonical folding of Dynkin diagram s and reduction of affine T oda theories , Pro g. Theor . Phys. 95 (1996) 503–518. [7] D. I. Olive, N. T ur ok, The T oda lattice fie ld theory hierarchies and zero-curvatu re condi- tions in Kac–Moody algebras , Nucl. Phys. B265 (19 8 6) 469–484 . [8] C. P . Constantinidis, L. A. Ferr e i ra, J. F . Gomes and A. H. Zimerman, Connection between affine and conformal affine T oda models and the ir Hirota’ s solution , Phys. L e tt. B298 (1993) 88–94 [arXiv :hep-th/920 7061 ] . [9] Kh. S. Niro v and A . V . Razumov , On Z -gradations of twisted loop Lie alge- bras of c omplex s imple Lie algebras , Commun. Math. Phys. 267 (2006) 587–610, [arXiv:math -ph/0504038 ] . [10] Kh. S. Nir ov and A. V . Razumov , T oda equations associated with loop grou ps of complex classical Lie groups , Nucl. Phys. B782 (2007) 2 4 1–275, [arXiv:math -ph/0612054 ] . 11 [11] Kh. S. Nir ov and A. V . Razumov , Z -graded loop Lie algebras, loop groups, and T oda equations , Theor . Math. Phys. 154 (2008) 3 85–404, [arXiv:0705 .2681 ] . [12] Q-Han Park and H. J. Shin, Classical matrix si ne-Gordon theory , Nucl. Phys. B458 (1996) 327–354, [arXiv:hep-th/9505 017 ] . [13] C. R. Fern ´ andez-Pousa, M. V . Gallas, T . J. Hollowood and J. L. Miramontes, The symmetric space and homogeneous sine-Gordon theorie s , Nucl. Phys. B484 (1997) 609 – 630, [arXiv:hep-th /9606032 ] . [14] L. Miramontes, T au-functions generating the conservation laws for g e neralized in- tegrable h i erarchies of KdV and affi ne-T oda type , Nucl. Phys. B547 (1999) 623–663, [arXiv:hep- th/9809052 ] . [15] R. Hirot a , The Direct Method i n Soliton Theory , (Cambridge U niversity Press, Cam- bridge, 2004). [16] T . Hollowood, Solitons in affine T oda field theories , Nucl. Phys. B384 (1992) 523–540. [17] N. J. MacKay and W . A. McGhee, Affine T oda solutions and automorphis ms of Dynkin diagrams , I nt. J. Mod. Phys. A8 (1993) 279 1–2807, er ratum ibid. A8 (1993) 3830 [arXiv:hep- th/9208057 ] . [18] H. Aratyn, C. P . Constantinidis, L. A. Ferr eira, J. F . Gomes and A . H. Zimerman, Hirota’ s solitons in the affine and the c onformal af fi ne T oda models , Nucl. Phys. B406 (1993) 727–770 [arXiv:hep-th/9212 086 ] . [19] Z. Zhu and D. G. Caldi, Multi-soli ton s ol utions of affine T oda models , Nucl. Phys. B436 (1995) 659–6 80 [arXiv:hep-t h/9307175 ] . [20] S. P . Khastgir , and R. Sasaki, Instability of solitons in imaginary c oupling affi ne T oda field theory , Prog. Theor . Phys. 95 (1996) 485–5 02. [21] D. I. Olive, N. T ur ok and W . R. Underwood, Solitons and the energy-momentum tensor for affine T oda theory , Nucl. Phys. B401 (1993) 663– 6 97. [22] D. I . Olive, N. T ur ok and W . R. Underwood, Affine T oda soli tons and vertex operators , Nucl. Phys. B409 (1993) 509–546. [23] M. A. C. Kneipp, and D. I. Olive, Solitons and vertex operators in twisted affine T oda field theories , Commun. Math. Phys. 177 (1996) 561–582. [24] D. I. Olive, M. V . Saveliev and J. W . R. Underwood, On a solitonic spe cialisation for the ge neral s olutions of some two-dime nsional comp l etely integrable system s , Phys. Lett. B311 (1993) 117–1 22 [arXiv:hep-t h/9212123 ] . [25] L. A. Ferreir a, J. L. Miramontes and J. S. Guill ´ en, Solitons, τ -functions and Hamil to- nian reduction for non-Abelian conformal affi ne T oda theories , Nucl. Phys. B449 (1995) 631–679, [arXiv:hep-t h/9412127 ] . [26] L. A. Ferr eira, J. L. Miramontes and J. S. Guill ´ en, T au-functions and dressing transfor- mations for zero-curvatur e affine integrable equations , J. Math. Phys. 38 (1997) 882–9 01, [arXiv:hep- th/9606066 ] . 12 [27] V . E. Zakharov an d A . B. Shabat, Integration of nonlinear e quations of mathem atical physics by the m ethod of inverse scattering. II , Func. Anal. 13 (1979) 166–175. [28] Kh. S. Nirov and A. V . Razumov , Abelian T oda solitons revisited , Rev . Math. Phys. 20 (2008) 120 9 –1248, [arXiv:080 2.0593 ] . [29] Kh. S. Nir ov and A. V . R azumov , The rational d ressing for Abelian twi sted loop T oda systems , JHEP 0812 (20 0 8) 048, [arXiv:0806.2597 ] . [30] P . Etingof, I. Gelfand, and V . Retakh, Non-Abelian integrable systems, quasideterminants, and Marchenko Lem ma , Math. R es. Lett. 5 (1998) 1–12 , [arXiv:q-al g/9707017 ] . [31] Kh. S. Nirov and A. V . Razumov , Solving non-Abelian loop T oda equations , Nucl. Phys. B (2009 ), doi:10.1016/j.nuclphysb.2009.01.010, [arXiv:0809. 3944 ] . [32] A. Kriegl and P . Michor , Aspects of the theory of infinite dimensional m anifolds , Diff. Geom. Appl. 1 (199 1 ) 159–176, [arXiv:math.DG/ 9202206 ] . [33] A. Kriegl and P . Michor , The Convenient Setting of Global Analysis , Mathematical Surveys and Monog raphs, vol. 53, (American Mathematical Society , Prov idence, RI, 1997). [34] O. Babelon and D. Bernard, D ressing symm etries , Commun. Math. Phys. 1 4 9 (1992) 279–306. 13