The rational dressing for abelian twisted loop Toda systems

The rational dressing for abelian twisted loop T oda syste ms 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 consider abelian twiste d loop T od a equations associated with t he complex general linear groups. The Dodd–Bu lloug h –Mikhailov equ ation is a simplest par - ticular case of the equations unde r consideration. W e construct ne w s oliton solu- tions of these T oda equations using the method of ra tional dressing. 1 Introduc tion The compr ehen sive investigation of completely integrable systems has at least two r easons. First, such systems serve as a testing ar ea for de veloping various me thods to solve nonlinear partial differ ential equations. And second, they possess an inter e sting class of solutions, calle d solitons, which have pr operties attractive fr om the point of view of possible physical applications. The two-dimensional loop T oda equations provide an illustrative and very rich example of co mpletely integrable nonlinea r equations, see, for example, the mono- graphs [1, 2]. Dif fer ent methods applicable to loop T oda equations for constr ucting their soliton-like solut ions wer e analysed in the paper [3]. Namely , multi- soliton s o- lutions of abelian untwisted loop T oda equations associated with the general linear gr oups were explicitly constr ucted by means of the Hirota’s [4, 5, 6, 7, 8, 9] and the rational dr e ssing [10, 11] methods, and a dir ect relationship between these approaches was e stablished. 1 . In this paper we continue the investigation of abelian loop T oda equations asso ci- ated with the complex general linear gr oups started in the paper [3]. Here we consider abelian twis ted loop T oda equations. It is inter e sting that the famous Dodd–Bullough– Mikhailov equation is a simplest particular case of suc h twis ted loop T oda equa tions. W e develop the rational dressing me thod in application to t hese classes o f nonlinear equations and construct for them new soliton solutions. 1 It is worth to note that sometim es it is helpful to employ a com bination of such complementary methods, see, for example, [1 2, 13] 1 2 Loop T oda equation s In this section, mainly following the monographs [1, 2] and our pape rs [14, 15], we r e- call basic notions and intr oduce notations to be used below . W e start our consideration with a Lie group G whose Lie algebra G is endowed with a Z -gradation, G = M k ∈ Z G k , [ G k , G l ] ⊂ G k + l , and denote by L such a positive integer that the grading subspaces G k , where 0 < | k | < L , are trivial. W e de note by G 0 the clo sed Lie subgr oup of G corr esponding to the zero-grade Lie subalgebra G 0 . Then, the T oda equation associated with G is an equation for a mapping Ξ of the Euclidean plane R 2 to G 0 , explicitly of the form ∂ + ( Ξ − 1 ∂ − Ξ ) = [ F − , Ξ − 1 F + Ξ ] , (2.1) wher e F − and F + ar e some fixed mappin gs of R 2 to G − L and G + L , respectively , satis- fying the relations ∂ + F − = 0, ∂ − F + = 0. (2.2) Here we use the customary not ation ∂ − = ∂ / ∂ z − , ∂ + = ∂ / ∂ z + for the partial deriva- tives over the standard coor dinates on R 2 . C e rtainly , to obtain a nontrivial T oda equa- tion we have to assume that the subspaces G − L and G + L ar e nontrivial. When the Lie gr oup G 0 is abelian, the corr esponding T oda equation is said to be abelian , otherwise we dea l with a non-abelian T oda equation . W e see that a T oda equation is specified by a choice of a Z -gradation of the Lie alge- bra G of G and mappings F − , F + satisfying the conditions (2.2). Therefor e , to classify the T oda equations associated with a Lie group G we should classify Z -gradations of the L ie al gebra G of G up to isomorphisms. It is essential for our purposes that the T oda equation (2.1) together with the r ela - tions (2.2) are equivalent to the ze r o-curvatur e condition fo r a flat connection on the trivial fiber bundle R 2 × G → R 2 . Indeed, writing the zer o-curvatur e condition as the equation ∂ − O + − ∂ + O − + [ O − , O + ] = 0 (2.3) for the G -valued components of the flat connection under consideration, imposing the grading conditions O − = O − 0 + O − L , O + = O + 0 + O + L , and destr oying the residual gauge invariance by the condition O + 0 = 0, we bring the connection components to the form O − = Ξ − 1 ∂ − Ξ + F − , O + = Ξ − 1 F + Ξ , (2.4) and then derive the equation (2.1) and the r elations (2.2) d irectly fr om the zero -cur- vatur e condit ion (2.3), as we ll as vice versa [16, 2, 17]. It follows from the equality (2.3) that there is a mapping Φ of R 2 to G such that Φ − 1 ∂ − Φ = O − , Φ − 1 ∂ + Φ = O + . (2.5) 2 W e say in this situat ion that the connect ion with the components O − and O + is gen- erated by the mapping Φ . W e consider the case where G is a twisted loop gro up of a complex classical Lie gr oup G which is defined as f ollows. Let a be an automo rphism of G satisfying the r elation a M = id G for some positive integer M . 2 The twisted loop group L a , M ( G ) is formed by the mappings χ o f the unit circle S 1 to G satisfying the equality χ ( ǫ M p ) = a ( χ ( p )) , wher e ǫ M = e 2 π i / M is the M th princ ipal roo t o f unity . W e think the cir cle S 1 as con- sisting of complex numbers of modulus one. The group law in L a , M ( G ) is defined pointwise. The Lie algebra of L a , M ( G ) is the twisted loop Lie algebra L A , M ( g ) , wher e g is the Lie alg ebra of G and A is the automor phism o f g corr esponding to t he auto- morphism a of G . The Lie algebra L A , M ( g ) is formed by the mappings ξ of S 1 to g satisfying the e q uality ξ ( ǫ M p ) = A ( ξ ( p ) ) with the Lie algebra operation defined pointwis e. Note that the r ela tion A M = id g is satisfied automatically . In the paper [18] we classified a wide class of the so-called integrable Z -gradations with finit e-dimensional grading subspaces of the twisted loop Lie algebras of the finite-dimensional complex simple Lie algebras. Namely , we showed that any such Z -gradation of a loop L ie algebra L A , M ( g ) is conjugated by an isomorphism to the standar d Z -gradation of another loop Lie algebra L A ′ , M ′ ( g ) , where the automorphisms A and A ′ diff er by an inner automorphism of g . Recall that for the standard Z -gradation of the Lie algebra L A , M ( g ) the grading sub- spaces a r e L A , M ( g ) k = { ξ = λ k x ∈ L A , M ( g ) | x ∈ g , A ( x ) = ǫ k M x } , wher e by λ we denote the rest riction of the standard coordinate on C to S 1 . It is well known that twisted loop Lie algebras defined by automo rphisms which diff er by an inner automorphism ar e isomorphic, and r eally dif fer ent twisted loop Lie algebras can be labele d by the eleme nts of the corr e sponding outer automorphism gr oup. In particular , if A is a n inner automor phism, the loop Lie algebra L A , M ( g ) is isomorphic to an untwisted loop Lie algebra L ( g ) = L id g ,1 ( g ) . Ther efore, in this case a T oda eq uation associated with L a , M ( G ) and spe cified by some choice of a Z -gradation of L A , M ( g ) is eq uivalent to a T oda equation associated with L ( G ) = L id G ,1 ( G ) and specified by the corr esponding choice of a Z -gradation of L ( g ) . Thus, t o descr ibe T oda equations associated with the loop gr oups L a , M ( G ) , wher e a is an inner automorphism of G , it suffic es to d e scribe the T oda equa tions associated with the untwisted loop gr oups L ( G ) . However , due to simplicity of the standard Z -gradation, to study T oda equations it is more convenient, instead of using one un- twisted loop gr oup L ( G ) and dif fer ent Z -gradations of L ( g ) , to use dif ferent twisted loop gro ups L a , M ( G ) and the standar d Z -gradation of L A , M ( g ) . Similarly , to describe the T oda equations assoc iated with the loop gr oups L a , M ( G ) , wher e a is a n out er au- tomorphism of G satisfying the relation a M = id G , it suf fices to use only the standar d Z -gradation of the loop Lie algebras L A , M ( g ) . Having all this in mind and slightly abusing terminology , we say that when a is an out er automorphism of G t hen a T oda 2 Here M is not necessar ily the order of the automorphism a , b ut ca n be its arbitrary multiple. 3 equation associated with L a , M ( G ) is a twisted loop T oda equation associated with G , and when a is an inner automorphism of G then a T oda equation associated with L a , M ( G ) is a n untwisted loop T oda equation ass ociated with G . 3 The gro up L a , M ( G ) and its Lie algebra L A , M ( g ) are infinite-dimensional manifolds. Nevertheless, using the so-called exponential law [19, 20], it is possible to write the zero -curvatur e repr esentation of the T oda e q uations associated with L a , M ( G ) in terms of finite-dimensional manifolds. The essence of this useful law can be expressed by the canonic al identification C ∞ ( M , C ∞ ( N , P ) ) = C ∞ ( M × N , P ) , wher e M , N and P are finite-dimensional manifolds, among which N is compact. The connection components O − and O + entering the equality (2.3) ar e mappings of R 2 to the loop Lie algebra L A , M ( g ) . W e denote the mappings of R 2 × S 1 to g , corr e- sponding to O − and O + in accordance with the e xponential la w , by ω − and ω + , and call them also the connection components. The mapping Φ generating the connection under consideration is a mapp ing of R 2 to L a , M ( G ) . Denoting the r espective mapp ing of R 2 × S 1 to G by ϕ we can write ϕ − 1 ∂ − ϕ = ω − , ϕ − 1 ∂ + ϕ = ω + , (2.6) which is equivalent to the expr essions (2.5). Having in mind that the mapping ϕ uniquely determines the connect ion generating mapp ing Φ , we say that the mapping ϕ a lso g enerates the connection under consideration. W e intr oduce, accor ding to the exponential law , the smooth mapping γ of R 2 × S 1 to G respect ive to Ξ , and smooth mappings of R 2 × S 1 to g respective to F − and F + . Now , explicitly describing the grading subspaces of the standard Z -gradation of the loop Lie algebra L A , M ( g ) , we find that the subalgebra L A , M ( g ) 0 is isomorphic to the subalgebra g [ 0 ] M of g , and the Lie gr oup L a , M ( G ) 0 is isomorphic to the connect ed Lie subgr oup G 0 of G corresponding to the Lie algebra g [ 0 ] M . Here g [ k ] M mean the grading subspaces of the Z M -gradation of g induced by the automorphism A , and [ k ] M denotes the element of the ring Z M corr esponding to the integer k . For the connection components ω − and ω + we can write the expr essions ω − = γ − 1 ∂ − γ + λ − L c − , ω + = λ L γ − 1 c + γ , (2.7) which are equivalent to the equalities (2.4). Here c − and c + ar e map p ings of R 2 to g − [ L ] M and g + [ L ] M r espectively . Hence, the T oda equation (2.1) can be written as ∂ + ( γ − 1 ∂ − γ ) = [ c − , γ − 1 c + γ ] , (2.8) and the conditions (2.2) imply that ∂ + c − = 0, ∂ − c + = 0. (2.9) W e call an equation of the form (2.8) also a loop T oda equation. Our classification of loop T oda equations is based on a convenient block-matrix r epr esentation of the grading subspaces [14, 15] we have implemented. Each ele ment x of the complex classical Lie algebra g under consideration is treat ed as a p × p block matrix ( x α β ) , whe r e x α β is an n α × n β matrix. The sum of the positive integers n α is the size n of the matrices r epresenting the elements of g . For t he case of T oda e q uations 3 It is common to omit the word ‘untwisted’. 4 associated with the loop gr oups L a , M ( GL n ( C ) ) , where a is an inner automorphism of GL n ( C ) , the integers n α ar e arbitrary . For the other cases they should satisfy some r estrictions dictated by the struct ur e of the L ie a lgebra g . The mapping γ has the block-diagonal form γ =     Γ 1 Γ 2 Γ p     . (2.10) 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 equations asso ciated with the loop gro ups L a , M ( GL n ( C ) ) , where a is an inner automorphism o f GL n ( C ) , the mappings Γ α ar e a rbitrary . For t he 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         , (2.11) wher e for each α = 1, . . . , p − 1 the mapping C + α is a map p ing 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 map p ing c − has a similar block-matrix struct ur e: c − =         0 C − 0 C − 1 0 0 C − ( p − 1 ) 0         , (2.12) wher e for each α = 1, . . . , p − 1 the mapping C − α is a map p ing of R 2 to the space of n α + 1 × n α complex matrices, and C − 0 is a mapping of R 2 to the space of n 1 × n p complex matrices. The conditions (2.9) imply ∂ + C − α = 0, ∂ − C + α = 0, α = 0, 1, . . . , p − 1. (2.13) For the case of T oda equations asso ciated with the loop gro ups L a , M ( GL n ( C ) ) , where a is an inner automorphism o f GL n ( C ) , the mappings C ± α ar e arbitrary . For the o ther cases they should satisfy some additional r estrictio ns. The T oda equation (2.8) is equivalent to t he following syst em of equations for the 5 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 , . . . (2.14) ∂ +  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 case under consideration without any loss of generality we can assume that the positive integer L , entering the construction of T oda equations, is equal to 1. Note also that if any of the mappings C + α or C − α is a zer o mapping, then the equations (2.14) are equivale nt to a T oda equation associated with a finite-dimensional gr oup or to a set of two such equations. 3 Abelian T o da equations ass ociated with loop groups of complex general linear groups It can be shown that there are thr ee types of abelian loop T oda equations associated with the groups GL n ( C ) , see , for example the paper [3]. 3.1 First type: untw isted loop T oda equat ions The abelian 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 ) , wher e h is a diagonal matrix with the diagonal matrix e le ments h kk = ǫ n − k + 1 n , k = 1, . . . , n . (3.1) 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 ) , wher e again h is a diagonal matrix determined by the relation (3.1). Here the integer M is equal to n , and A is an inner aut omorphism which generates a Z n -gradation of gl n ( C ) . The block-matrix str uctur e r ela ted to t his gradation is the matrix str uctur e it- self. In ot her words, all blocks ar e of size one by o ne. The mappings Γ α ar e mappings of R 2 to the Lie gr oup GL 1 ( C ) which is isomorphic to the Lie group 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 (2.14) with p = n . W e have shown in the p a per [3] that if the functions C − α and C + α have no ze- r os then the T oda equations (2.14) are equivalent to the same equations, but where C − α = C − and C + α = C + for some functions C − and C + which ha ve no zer os. If these functions are r eal, then with the help of an appropriat e change of the coordinates z − 6 and z + we can come to the T oda equations with C ± α equal to a nonzero constant m . This system of equations gives the T oda equations associated with untwisted loop gr oups of general linear gr oups. I n the pape r [3] we investigated the soliton solutions of the above T oda e quations obtained by two dif ferent appr oaches, the Hirot a’s and rational dr essing methods, and established explicit r e lationships between these meth- ods. 3.2 Second type: twist ed loop T oda equa tions, odd-dimensional case The abe li a n T oda equations of the other two types arise when we use outer a utomor - phisms of gl n ( C ) . For the equations of the second type n is odd, and for the equations of the third type n is e ven. Consider fir st the c ase 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 e quality A ( x ) = − h ( B − 1 t x B ) h − 1 , (3.2) wher e t x mean s the transpose of x , h is a diagonal matrix with the dia gonal 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 corr esponding gr oup automorphism a is defined as a ( g ) = h ( B − 1 t g − 1 B ) h − 1 . (3.3) The or d e r M of the automo rphism A is 2 N = 4 s − 2 and the integer p is 2 s − 1. The mapping γ is a diagonal matrix of t he for m (2.10), wher e the mappings Γ α ar e map- pings of R 2 to C × , subject to the constraint s Γ 1 = 1, Γ 2 s − α + 1 = Γ − 1 α , α = 2, . . . , s . The ma p p ings C ± α in the r elations (2.1 1 ) and (2.12) are complex funct ions satisfy ing the e quality C ± 0 = C ± 1 , (3.4) and for s > 2 the equalities C ± ( 2 s − α ) = − C ± α , α = 2, . . . , s − 1. (3.5) Let us choose the mappings Γ α , α = 2, . . . , s , as a complete set of mappings parameter- izing the mapping γ . T aking into account the equalities (3.4) and (3.5) we come to a set of s − 1 independ ent equations equivalent to the T oda eq ua tion under co nsideration. 7 As well as in the untwisted case, under appropriate conditions the T oda equations under consideration are eq uivale nt to the same equations, but where C ± 0 = C ± 1 = C ± s = m , (3.6) and C ± α = − C ± ( 2 s − α ) = m , α = 2, . . . , s − 1. (3.7) Explicitly , we have the equations ∂ + ( 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 ) , . . . (3.8) ∂ + ( 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 again a nonzero constant, see also the papers [11, 21]. 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 ) − exp ( F ) ] . This is the well-known Dodd–Bulloug h–Mikhailov equation [22 , 11], formulated for the first time by Tzitz ´ eica [23] in geometry of hyperbolic surfaces. 3.3 Third type: twi sted loop T oda equations, even-dimensional case In the case of an even n = 2 s , s ≥ 2, to come to an abelian T oda equation we should use again the Lie algebra automorphism A a nd the corresponding grou p automorphism a defined by the relations (3.2) and (3.3), r espectively , where now B =          1 1 1 1 − 1 − 1          and h is a diagonal matrix with the diagonal 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 or der M of the automo rphism A is again 2 N = 4 s − 2 , and t he number p char- acterizing the block structur e is e q ual to n − 1 = 2 s − 1, n 1 = 2, and n α = 1 for α = 2, . . . , 2 s − 1. 8 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 relation 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  , wher e ( Γ 1 ) 11 is a mapping of R 2 to C × . The mappings Γ α , α = 2, . . . , 2 s − 1, are map- pings of R 2 to C × satisfying the relations Γ 2 s − α + 1 = Γ − 1 α . The mappings 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 we have C − 0 = J − 1 2 t C − 1 , C + 0 = t C + 1 J 2 . (3.9) 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. (3.10) 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 (3.9) and (3.10) we come to a set of s independe nt e quations equivalent to the T oda equation under consideration. As well as for the first two types, unde r appro priate conditions these equations can be r educed to equations with C − α = m , C + α = m , α = 2, . . . , s , (3.11) and ( C − 1 ) 11 = ( C − 1 ) 12 = m / √ 2, ( C + 1 ) 11 = ( C + 1 ) 21 = m / √ 2, (3.12) wher e m is a nonzero constant. 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 ) , . . . (3.13) ∂ + ( 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, with a slight abuse of notation, we have denoted ( Γ 1 ) 11 by Γ 1 . W e also note t hat all thr ee systems of T oda equations descr ibed above can be r ep- r esented in standar d forms with e xplicit indication of the Cartan matrices of the corr e- sponding affine Lie algebras of the types A ( 1 ) n − 1 , A ( 2 ) 2 s − 2 and A ( 2 ) 2 s − 1 , respectiv ely , see, for example, the paper [3]. 9 4 Rational dressin g In this section we apply the method of rational dressing to co nstru ct solutions of the abelian T oda systems associated with t he loop gr oups of the complex general linear gr oups. Here we solve the abelian T oda equations of the second and thir d types which have the forms (3.8) and (3.13)respectively . In fact, s ome preliminary r elations o f the rational dressing formalism can be intr oduced on a common basis in application to the both types of abelian T oda systems. Because in the cases under consideration the matrices c − and c + ar e commuting, it is obvious that γ = I n , (4.1) wher e I n is the n × n unit matrix, is a solution to the T oda equation (2.8). Denote a mapping of R × S 1 to GL n ( C ) , which generates t he corr esponding connect ion, by ϕ . Using the equalities (2.6) and (2.7) and remembering that in our case L = 1, we write ϕ − 1 ∂ − ϕ = λ − 1 c − , ϕ − 1 ∂ + ϕ = λ c + , (4.2) wher e the matrices c + and c − having generally the forms (2.11) and (2.12), ar e speci- fied by the relatio ns (3.6), ( 3.7) for the T oda equatio ns of the second type, and by the r elations (3.11), (3.12) for the T oda equations of the thir d ty pe. T o constr uct some other solutions to the T oda equations we will look for a mapp ing ψ , such that the map p ing ϕ ′ = ϕ ψ (4.3) would generate a connection satisfying the grading condition ω − = ω − 0 + ω − 1 , ω + = ω + 0 + ω + 1 (4.4) and the gauge-fixing constraint ω + 0 = 0. (4.5) 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 ) . W e tr eat S 1 as a subset of the complex plane which, in turn, will be tr eated as a subset o f the Riemann spher e. Assume that it is possible to extend analytically each mapping ˜ ψ m to all of the R iemann sphere. As the r esult we obtain a m a pping o f the direct pr oduct 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 mapp ing r egular at the points 0 and ∞ , he nce the name rational dressing . Below , for e ach point p of the Riemann spher e we denote by ψ p the ma pping 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 Sections 3.2 and 3.3, for any m ∈ R 2 and p ∈ S 1 we should have ψ ( m , ǫ 2 N p ) = h B − 1 t ψ − 1 ( m , p ) B h − 1 , (4.6) wher e h is a block-diagonal m a trix described by the relation h α β = ǫ N − α + 1 2 N I n α δ α β , α , β = 1, . . . , p , 10 with n 1 = 1 for the T oda equations of the second type, and n 1 = 2 for the T oda equations of the third type, while for all other indices α = 2, . . . , p we a lwa ys ha ve n α = 1. Note that h 11 = − I n 1 . He re we also use the notation B =          J n 1 1 1 − 1 − 1          common for the both cases. The equality (4.6) means that for any m ∈ R 2 two rational mappings coincide on S 1 , therefor e, they must coincide on the entire Riemann sphere. W e define a linear mapping ˆ a acting on a mapping χ of the dir ect pr oduct of R 2 and the R iemann sphere to the algebra Mat n ( C ) of n × n complex matrices as 4 ˆ a χ ( m , p ) = h B − 1 t χ − 1 ( m , ǫ − 1 2 N p ) B h − 1 . The relation (4.6) is equivalent to the e quality ˆ a ψ = ψ . T o construct rational mappings satisfying this relation we will use the following procedur e. First, we constr uct a fam- ily of mappings ψ satis fying the relation ˆ a 2 ψ = ψ , and then select from it the mappings satisfying the e q uality ˆ a ψ = ψ . It is easy to see that the mapping ψ = N ∑ k = 1 ˆ a 2 k χ (4.7) satisfies the r elation ˆ a 2 ψ = ψ . It is worth to note that ˆ a 2 N χ = χ . W e start with a rational mapping χ r egular at the p oints 0 and ∞ and having poles at r differ ent nonzero points µ i , i = 1, . . . , r . More specifically , we consider a mappi ng χ of the form χ = χ 0 I n + r ∑ i = 1 λ λ − µ i P i ! , wher e P i ar e some smooth mappings of R 2 to the algebra Mat n ( C ) and χ 0 is a mapping of R 2 to the Lie subgr oup of GL n ( C ) formed b y the elements g ∈ GL n ( C ) satisfying the e quality h 2 g h − 2 = g . (4.8) W ith a ccount of the equality ˆ a 2 χ ( m , p ) = h 2 χ ( m , ǫ − 1 N p ) h − 2 the a veraging pr ocedur e (4.7 ) le ads to the mapping ψ = ψ 0 I n + r ∑ i = 1 N ∑ k = 1 λ λ − ǫ 2 k 2 N µ i h 2 k P i h − 2 k ! , (4.9) 4 Note that below χ is a mapping to the Lie group GL n ( C ) , although to justify the relation (4.7) it is convenient to think GL n ( C ) as a subset of Mat n ( C ) . 11 wher e ψ 0 = N χ 0 . W e assume that µ 2 N i 6 = µ 2 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 ) , defined by the equality ˜ ψ − 1 m ( p ) = ψ − 1 ( m , p ) , can be extended a nalytically to a mapp ing of the Riemann spher e to GL n ( C ) , which we also denote by ψ − 1 , and as the r esult we obtain a rational mapping of the same structur e as the mapp ing ψ , ψ − 1 = I n + r ∑ i = 1 N ∑ k = 1 λ λ − ǫ 2 k 2 N ν i h 2 k Q i h − 2 k ! ψ − 1 0 , (4.10) with the pole po sitions satisfying the conditions ν i 6 = 0, ν 2 N i 6 = ν 2 N j for all i 6 = j , and additionally ν N i 6 = µ N j for any i and j . 5 The map p in gs ψ and ψ − 1 given by the equalities (4.9) and (4.1 0), r espectively , sat- isfy the relations ˆ a 2 ψ = ψ and ˆ a 2 ψ − 1 = ψ − 1 . T o satisfy the relations ˆ a ψ = ψ and ˆ a ψ − 1 = ψ − 1 we have to assume that the pole positions of the mappings ψ and ψ − 1 ar e necessarily connected as ν i = µ i / ǫ 2 N , i = 1, . . . , r , and the ma trices P i and Q i ar e r elated as Q i = h − 1 B − 1 t P i B h , i = 1, . . . , r , (4.11) By de finition, the eq ua lity 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 sphere. Hence, the r esidues of ψ − 1 ψ at the points ν i = µ i / ǫ 2 N and µ i should be equal to zero. Explicitly we have h − 1 B − 1 t P i B h I n + r ∑ j = 1 N ∑ k = 1 µ i / ǫ 2 N µ i / ǫ 2 N − ǫ 2 k 2 N µ j h 2 k P j h − 2 k ! = 0, (4.12) I n + r ∑ j = 1 N ∑ k = 1 µ i µ i − ǫ 2 k − 1 2 N µ j h 2 k − 1 B − 1 t P j B h − 2 k + 1 ! P i = 0. (4.13) W e will d iscuss later how to satisfy these r elations, and now let us consider what con- nection is generated by the mapping ϕ ′ defined by (4.3) with the mapping ψ possessing the p roperties d escribed above. Using the equality (4.3) and the relations (4.2), 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 , (4.14) ω + = ψ − 1 ∂ + ψ + λψ − 1 c + ψ . (4.15) 5 Actually , as it will be clear , for the extended mappings ψ a nd ψ − 1 we hav e ψ − 1 ψ = I n . This justi fies the notation used. 12 W e see that the component ω − is a rational mapping which has simple poles at t he points µ i , ν i = µ i / ǫ 2 N and zero . 6 Similarly , t he component ω + is a r ational mapping which has simple poles at the points µ i , ν i = µ i / ǫ 2 N and infinity . W e are looking for a connection which satisfies the grading co ndition (4.4) and the gauge-fixing co ndition (4.5). The grading condition in our case is the requir ement that for each point of R 2 the component ω − is rat ional and has the only simple pole at zer o, while the compo nent ω + is rational and has the only simple p ole at infinity . Hence, we demand that the r esidues of ω − and ω + at the points µ i and ν i = µ i / ǫ 2 N should vanish. The residues o f ω − and ω + at t he points ν i = µ i / ǫ 2 N ar e e q ual to zer o if and only if ( ∂ − Q i − ǫ 2 N µ − 1 i Q i c − ) I n + r ∑ j = 1 N ∑ k = 1 µ i / ǫ 2 N µ i / ǫ 2 N − ǫ 2 k 2 N µ j h 2 k P j h − 2 k ! = 0, (4.16) ( ∂ + Q i − ǫ − 1 2 N µ i Q i c + ) I n + r ∑ j = 1 N ∑ k = 1 µ i / ǫ 2 N µ i / ǫ 2 N − ǫ 2 k 2 N µ j h 2 k P j h − 2 k ! = 0, (4.17) r espectively , with the equality (4.11) to be taken into account . Similarly , the r equire- ment of vanishing of the residues at the points µ i gives the relations I n + r ∑ j = 1 N ∑ k = 1 µ i µ i − ǫ 2 k − 1 2 N µ j h 2 k − 1 B − 1 t P j B h − 2 k + 1 ! ( ∂ − P i + µ − 1 i c − P i ) = 0, (4.18) I n + r ∑ j = 1 N ∑ k = 1 µ i µ i − ǫ 2 k − 1 2 N µ j h 2 k − 1 B − 1 t P j B h − 2 k + 1 ! ( ∂ + P i + µ i c + P i ) = 0. (4.19) T o obtain the r elations (4.16)–(4.19) we made use of the equalities (4.12), (4.13). Suppose that we have succeeded in satisfying the relations (4.12), (4.1 3) and (4.16)– (4.19). In such a case fr om the equalities (4.14) and (4.1 5) it follows that the connection under consideration satisfies the grading condition. It follows from the equality (4.15) 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. (4.20) Assuming that this relation is satisfi ed, we come to a connectio n satisfying both the grading condition and the gauge-fixing condition. Recall that if a flat connection ω satisfies the grading and 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 r epresentation (2.7) for the compo nents 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 6 Here and below discussing the holomorphic properties of ma ppings and functions we assume that the point of the space R 2 is arbitrary but fixed. 13 us denote the mappings corresponding to t he connection under consideration by γ ′ , c ′ − and c ′ + . Thus, we have ψ − 1 ∂ − ψ + λ − 1 ψ − 1 c − ψ = γ ′− 1 ∂ − γ ′ + λ − 1 c ′ − , (4.21) ψ − 1 ∂ + ψ + λ ψ − 1 c + ψ = λ γ ′− 1 c ′ + γ ′ . (4.22) Note that ψ ∞ is a ma pping of R 2 to the Lie subgroup of GL n ( C ) defined by the r elation (4.8). W e recall that this subgr oup coincides with G 0 and d e note ψ ∞ by γ . Fr om the r elation (4.21) we obtain the equality γ ′− 1 ∂ − γ ′ = γ − 1 ∂ − γ . The same relation (4.21 ) gives ψ − 1 0 c − ψ 0 = c ′ − . Impose the condition ψ 0 = I n , which is consistent with the condition (4.20). Here we have c ′ − = c − . Finally , from the equality (4.22) 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 connection under consideration have the form (2.7) wher e γ = ψ ∞ . Thus, to find solutions to the T oda equations under consideration, we can use the following pr ocedur e. W e fix 2 r complex numbers µ i and ν i and find matrix-valued functions P i and Q i satisfying the relations (4.12), (4.13) an d (4.16)–(4.19). W ith the help of the relations (4.9), (4.10), assuming that ψ 0 = I n , we constr uct the mappings ψ and ψ − 1 . Then, the mapping γ = ψ ∞ (4.23) satisfies the T oda equation (2.8 ). Let us r eturn to the r elations (4. 1 2 ), (4. 1 3). It is easy to see that they ar e equivalent, and so, we will use the relatio n (4.13) for further calculations. W e can show that, if we suppose that the matrix P i has the ma ximum rank, then we get the trivial solut ion of the T oda equation given by (4.1). Hence, we will assume that P i is no t o f maximum rank. The simplest case here is given by matrices of rank one which can be repr esented as P i = u i t w i , wher e u and w are n -dime nsional co lumn vectors. This repr e sentation allows writ ing the relations (4.13) as u i + r ∑ j = 1 N ∑ k = 1 µ i µ i − ǫ 2 k − 1 2 N µ j h 2 k − 1 B − 1 w j ( t u j B h − 2 k + 1 u i ) = 0. (4.24) 14 Using the identity N ∑ k = 1 z ǫ − 2 k j 2 N z − ǫ 2 k 2 N = N z N − | j | N z N − 1 , wher e | j | N is the residue of division of j by N , we can rewrit e the e quality (4.24) in terms of the components of u i as follows: t u i ,1 J n 1 + N r ∑ j = 1 ( R 1 ) i j t w j ,1 = 0, wher e u i ,1 and w i ,1 gather first n 1 components of the corr esponding n -dimensional column vectors, so these are in fact n 1 -dimensional column vectors, 7 u i , N + 2 − k − N r ∑ j = 1 ( R k ) i j w j , k = 0, k = 2, . . . , s , and u i , N + 2 − k + N r ∑ j = 1 ( R k ) i j w j , k = 0, k = s + 1, . . . , p = N . Here the r × r ma trices R 1 and R k ar e defined as ( R 1 ) i j = 1 µ N i + µ N j µ N i ( t u i ,1 J n 1 u j ,1 ) − s ∑ ℓ = 2 µ N − | ℓ − 1 | N i µ | ℓ − 1 | N j ( u i , N + 2 − ℓ u j , ℓ ) + N ∑ ℓ = s + 1 µ N − | ℓ − 1 | N i µ | ℓ − 1 | N j ( u i , N + 2 − ℓ u j , ℓ ) ! , (4.25) ( R k ) i j = 1 µ N i + µ N j − µ N − | 1 − k | N i µ | 1 − k | N j ( t u i ,1 J n 1 u j ,1 ) + k − 1 ∑ ℓ = 2 µ N − | ℓ − k | N i µ | ℓ − k | N j ( u i , N + 2 − ℓ u j , ℓ ) − s ∑ ℓ = k µ N − | ℓ − k | N i µ | ℓ − k | N j ( u i , N + 2 − ℓ u j , ℓ ) + N ∑ ℓ = s + 1 µ N − | ℓ − k | N i µ | ℓ − k | N j ( u i , N + 2 − ℓ u j , ℓ ) ! (4.26) for k = 2 , . . . , s , and ( R k ) i j = 1 µ N i + µ N j − µ N − | 1 − k | N i µ | 1 − k | N j ( t u i ,1 J n 1 u j ,1 ) + s ∑ ℓ = 2 µ N − | ℓ − k | N i µ | ℓ − k | N j ( u i , N + 2 − ℓ u j , ℓ ) − k − 1 ∑ ℓ = s + 1 µ N − | ℓ − k | N i µ | ℓ − k | N j ( u i , N + 2 − ℓ u j , ℓ ) + N ∑ ℓ = k µ N − | ℓ − k | N i µ | ℓ − k | N j ( u i , N + 2 − ℓ u j , ℓ ) ! (4.27) for k = s + 1, . . . , N . Recall that for all cases consider e d her e N = p = 2 s − 1. 7 W e remember that either n 1 = 1 or n 1 = 2. 15 W e use the equations (4.25), (4.26) and (4.27) to expr ess the vectors w i via the vec- tors u i , t w i ,1 = − 1 N r ∑ j = 1 ( R − 1 1 ) i j t u j ,1 J n 1 , w i , k = 1 N r ∑ j = 1 ( R − 1 k ) i j u j , N + 2 − k for k = 2 , . . . , s , and w i , k = − 1 N r ∑ j = 1 ( R − 1 k ) i j u j , N + 2 − k for k = s + 1, . . . , N = p . As the r esult, having expr essed the matric es P i and Q i in terms of the components of the vectors u i , we find a solution of the relations (4.12) a nd (4.13). Further , it follows from t he equality (4.24) that, to fulfi ll also the r elations (4.16)– (4.19), it is sufficient to satisfy the e quations ∂ − u i = − µ − 1 i c − u i , ∂ + u i = − µ i c + u i . The general solution to these equations is given formally by the e xpr ession u i ( z − , z + ) = exp ( − µ − 1 i c − z − − µ i c + z + ) u 0 i , (4.28) wher e u 0 i = u i ( 0, 0 ) . W e will make explicit this formal solution when later construct ing soliton solutions. Thus, we see that it is possible to satisfy the relatio ns (4.12), (4.13) and (4.16)–(4.19). This g ives us so lutions of the T oda equatio n (2.8), and so, to the equations ( 3.8) and (3.13) by specifying the above formal expr ession of u i for the two corr esponding cases. Let us show that they can be written in a simple determinant form. Using the equalities (4.23) and (4.9), we get γ = ψ ∞ = I n + r ∑ i = 1 N ∑ k = 1 h 2 k P i h − 2 k . For the matrix elements of γ this gives the e xpr ession γ k ℓ = δ k ℓ 1 + N r ∑ i = 1 ( P i ) kk ! = δ k ℓ Γ k . Hence, we have Γ 1 = I n 1 − r ∑ i , j = 1 u i ,1 ( R − 1 1 ) i j t u j ,1 J n 1 , Γ α = 1 + r ∑ i , j = 1 u i , α ( R − 1 α ) i j u j ,2 s + 1 − α , wher e α = 2 , . . . , s , and Γ α = 1 − r ∑ i , j = 1 u i , α ( R − 1 α ) i j u j ,2 s + 1 − α , α = s + 1, . . . , 2 s − 1. W e assume for convenience that the functions u i , α ar e defined for arbitrary integral values of α so that u i ,2 s − 1 + α = u i , α . 16 By definition the matrices R α ar e pe riodic in the index α . It ap p e ars that it is more appro priate to use quasi-periodic quantities e u α and e R α defined as e u α = M α u α , e R 1 = M R 1 M 2 s , e R α = M 2 s + 1 − α R α M α , wher e α = 2 , . . . , 2 s − 1; her e M is a diagonal r × r matr ix given by M i j = µ i δ i j . For these quantities we have quasi-periodicity conditions e u 2 s − 1 + α = M 2 s − 1 e u α , e R 2 s = M 2 s − 1 e R 1 M 2 s − 1 , e R 2 s − 1 + α = M − 2 s + 1 e R α M 2 s − 1 . The e xpression of the matrix elements of the matric es e R α thr ough the quasi- periodic quantities e u i α has a remarkably simple form. W e have for α = 1 ( e R 1 ) i j = 1 µ 2 s − 1 i + µ 2 s − 1 j µ 2 s − 1 i ( t e u i ,1 J n 1 e u j ,1 ) µ 2 s − 1 j − µ 2 s − 1 j s ∑ β = 2 e u i ,2 s + 1 − β e u j , β + µ 2 s − 1 j 2 s − 1 ∑ β = s + 1 e u i ,2 s + 1 − β e u j , β ! . Further , we have for α = 2, . . . , s ( e R α ) i j = 1 µ 2 s − 1 i + µ 2 s − 1 j − µ 2 s − 1 i ( t e u i ,1 J n 1 e u j ,1 ) µ 2 s − 1 j + µ 2 s − 1 j α − 1 ∑ β = 2 e u i ,2 s + 1 − β e u j , β − µ 2 s − 1 i s ∑ β = α e u i ,2 s + 1 − β e u j , β + µ 2 s − 1 i 2 s − 1 ∑ β = s + 1 e u i ,2 s + 1 − β e u j , β ! , and for α = s + 1, . . . , 2 s − 1 ( e R α ) i j = 1 µ 2 s − 1 i + µ 2 s − 1 j − µ 2 s − 1 i ( t e u i ,1 J n 1 e u j ,1 ) µ 2 s − 1 j + µ 2 s − 1 j s ∑ β = 2 e u i ,2 s + 1 − β e u j , β − µ 2 s − 1 j α − 1 ∑ β = s + 1 e u i ,2 s + 1 − β e u j , β + µ 2 s − 1 i 2 s − 1 ∑ β = α e u i ,2 s + 1 − β e u j , β ! . Here we used the ide ntity | − k | N = N − 1 − | k − 1 | N . The q uasi-periodic funct ions have the following useful pro perties: ( e R α + 1 ) i j = ( e R α ) i j + e u i ,2 s + 1 − α e u j , α , α = 2, . . . , s , (4.29) ( e R α + 1 ) i j = ( e R α ) i j − e u i ,2 s + 1 − α e u j , α , α = s + 1, . . . , 2 s − 1, (4.30) and ( e R 1 ) i j = − ( e R 2 ) ji , ( e R α ) i j = ( e R 2 s + 2 − α ) ji , α = 2, . . . , 2 s − 1. (4.31) 17 In terms of the quasi-periodic quantities, for the n 1 × n 1 matrix-valued function Γ 1 and for the functions Γ α we have Γ 1 = I n 1 − r ∑ i , j = 1 µ 2 s − 1 i e u i ,1 ( e R − 1 1 ) i j t e u j ,1 J n 1 , Γ α = 1 + r ∑ i , j = 1 e u i , α ( e R − 1 α ) i j e u j ,2 s + 1 − α , for α = 2, . . . , s , and Γ α = 1 − r ∑ i , j = 1 e u i , α ( e R − 1 α ) i j e u j ,2 s + 1 − α , for α = s + 1, . . . , 2 s − 1. The expressions for the functions Γ α for α > 1 can be writ ten as Γ α = 1 + t e u α e R − 1 α e u 2 s + 1 − α , α = 2, . . . , s , and Γ α = 1 − t e u α e R − 1 α e u 2 s + 1 − α , α = s + 1, . . . , 2 s − 1. Here e R α is an r × r matrix and e u α is an r -dimensional column vector . W e r emember that in the cases under consideration we should have J − 1 n 1 t Γ 1 J n 1 = Γ − 1 1 , Γ 2 s + 1 − α = Γ − 1 α , α = 2, . . . , 2 s − 1. (4.32) T o verify these r elations we use that γ − 1 = ψ − 1 ∞ = I n + r ∑ i = 1 2 s − 1 ∑ k = 1 h 2 k ( h − 1 B − 1 t P i B h ) h − 2 k , ther efor e we find the following expr ession of Γ − 1 1 in terms of quasi-periodic quantities, Γ − 1 1 = I n 1 − r ∑ i , j = 1 e u i ,1 µ 2 s − 1 j ( e R − 1 1 ) ji t e u j ,1 J n 1 . Comparing now this express ion with what we have for Γ 1 above, we conclude that the first relation of equations (4.32) is satisfied. The expressio ns just given above allow writing a r emarkable determinant repr e - sentation for the functions Γ α . I t can be shown that Γ α = det ( e R α + e u 2 s + 1 − α t e u α ) det e R α , α = 2, . . . , s , and Γ α = det ( e R α − e u 2 s + 1 − α t e u α ) det e R α , α = s + 1, . . . , 2 s − 1. Using the properties (4.29 ) and (4.30) we can see Γ α = det e R α + 1 det e R α , α = 2, . . . , s , s + 1, . . . , 2 s − 1. For these functions we can also easily de monstrate that Γ 2 s + 1 − α = det e R 2 s + 2 − α det e R 2 s + 1 − α = det ( t e R α ) det ( t e R α + 1 ) = det e R α det e R α + 1 = Γ − 1 α , 18 using for this purpose the relatio ns (4.30). Hence all equations (4.32) are fulfilled. W e remember also that for the case n 1 = 1 corr esponding to the second type of abelian twisted loop T oda equations considered here ( n = p = 2 s − 1), we have I 1 = J 1 = 1, and so, we can write for the function Γ 1 the e xpr ession Γ 1 = 1 − t e u 1 M 2 s − 1 e R − 1 1 e u 1 , wher e e u 1 is a lso a n r -dimensional column vector . It can be shown that Γ 1 = det ( e R 1 − e u 1 t u 1 M 2 s − 1 ) det e R 1 , W e obtain fro m the expr essions of e R 1 and e R 2 dir ectly that ( e R 1 ) i j µ − N j = t e u i ,1 J n 1 e u j ,1 + µ − N i ( e R 2 ) i j , and so, for n 1 = 1 we can write M − 2 s + 1 e R 2 + e u 1 t e u 1 = e R 1 M − 2 s + 1 . Using this r ela tion in the above expr ession for Γ 1 as the ratio of de terminants, we easily derive Γ 1 = det e R 2 det e R 1 . (4.33) But we have from the equality (4.31) that e R 1 = − t e R 2 , and so, for n 1 = 1 the expression (4. 3 3) gives Γ 1 = ( − 1 ) r . 5 Soliton solution s 5.1 Odd-dimensional case Here we co nsider the case of n = p = N = 2 s − 1. It means also that we ha ve n 1 = 1 . The eigenvectors of the matrices t c − , t c + , c − and c + ar e n -dimensional column vectors Ψ ρ , ρ = 1, . . . , 2 s − 1, satisfying the relations t c − Ψ ρ = m ǫ s + 2 ρ 2 N Ψ ρ , t c + Ψ ρ = m ǫ − s − 2 ρ 2 N Ψ ρ , c − Ψ ρ = m ǫ − s − 2 ρ 2 N Ψ ρ , c + Ψ ρ = m ǫ s + 2 ρ 2 N Ψ ρ , wher e the 2 s − 1 components of Ψ ρ ar e defined as ( Ψ ρ ) α = ǫ α ( s + 2 ρ ) 2 N , α = 1, . . . , s , ( Ψ ρ ) α = ( − 1 ) α − s − 1 ǫ α ( s + 2 ρ ) 2 N , α = s + 1, . . . , 2 s − 1. 19 Consequently , we can give a concr ete e xpr ession to the formal solution (4.28) as u i , α = 2 s − 1 ∑ ρ = 1 c i ρ ǫ α ( s + 2 ρ ) 2 N e − Z ρ ( µ i ) , α = 1, . . . , s u i , α = 2 s − 1 ∑ ρ = 1 c i ρ ( − 1 ) α − s − 1 ǫ α ( s + 2 ρ ) 2 N e − Z ρ ( µ i ) , α = s + 1, . . . , 2 s − 1, wher e c i ρ ar e arbitrary constants and we have intr oduced the notation Z ρ ( µ i ) = m ( ǫ − s − 2 ρ 2 N µ − 1 i z − + ǫ s + 2 ρ 2 N µ i z + ) . Then, after some calculation using, in particular , properties of ǫ 2 N , we write for the matrix elements of e R α for α ≥ 2: ( e R α ) i j = ( − 1 ) α µ 2 s + 1 − α i µ α j 2 s − 1 ∑ ρ , σ = 1 c i ρ c j σ ǫ 4 ρ + 1 − 2 ( ρ − σ ) α 2 N 1 + µ j µ − 1 i ǫ − 2 ( ρ − σ ) 2 N e − Z ρ ( µ i ) − Z σ ( µ j ) . (5.1) It is clear that to obtain no ntrivial solutions to the T oda equations we should requir e that at least two coefficients c i ρ for any i = 1, . . . , r are diff er ent fr om z e r o. In this, we construct solutions depending on only r combinations of independ e nt variables z − and z + . W e denote such nonzer o constants by C J i and C K i . The expr ession for the matrix elements (5.1) takes then the form ( e R α ) i j = ( − 1 ) α µ 2 s + 1 − α i ǫ 4 J i + 1 − 2 α J i 2 N C J i e − Z J i ( µ i ) ( e R ′ α ) i j µ α j C J j ǫ 2 α J j 2 N e − Z J j ( µ j ) , wher e ( e R ′ α ) i j = 1 1 + µ j µ − 1 i ǫ 2 ( J j − J i ) 2 N + C K j C J j ǫ 2 ( K j − J j ) α 2 N 1 + µ j µ − 1 i ǫ 2 ( K j − J i ) 2 N e Z J j ( µ j ) − Z K j ( µ j ) + C K i C J i ǫ 4 ( K i − J i ) − 2 ( K i − J i ) α 2 N 1 + µ j µ − 1 i ǫ 2 ( J j − K i ) 2 N e Z J i ( µ i ) − Z K i ( µ i ) + C K i C K j C J i C J j ǫ 4 ( K i − J i ) − 2 ( K i − J i + K j − J j ) α 2 N 1 + µ j µ − 1 i ǫ 2 ( K j − K i ) 2 N e Z J i ( µ i ) − Z K i ( µ i )+ Z J j ( µ j ) − Z K j ( µ j ) . It is easy to show that Γ α = det e R α + 1 det e R α = ( − 1 ) r det e R ′ α + 1 det e R ′ α . Recalling also that Γ 1 = ( − 1 ) r , we see that we can take e R ′ α instead of e R α to constr uct solutions of the T oda eq uations using for that the above de terminant repr esentation. Defining a new set of parameters ρ i = J i − K i , θ ρ i = π ρ i 2 s − 1 , κ ρ i = − i ( ǫ ρ i 2 N − ǫ − ρ i 2 N ) = 2 sin θ ρ i , exp δ i = C K i C J i , ζ i = i ǫ s + J i + K i 2 N µ i , f i = ǫ ρ i 2 N ζ i , e f i = ǫ − ρ i 2 N ζ i , 20 and intr oducing the notation D i j ( f , g ) = f i f i + g j , we can rewrite the expression for e R ′ α as ( e R ′ α ) i j = D i j ( f , f ) + ǫ 2 ρ i ( α − 1 ) 2 N e Z i ( ζ ) + δ i − 2i θ ρ i D i j ( e f , f ) + D i j ( f , e f ) e Z j ( ζ ) + δ j − 2i θ ρ j ǫ − 2 ρ j ( α − 1 ) 2 N + ǫ 2 ρ i ( α − 1 ) 2 N e Z i ( ζ ) + δ i − 2i θ ρ i D i j ( e f , e f ) e Z j ( ζ ) + δ j − 2i θ ρ j ǫ − 2 ρ j ( α − 1 ) 2 N , (5.2) wher e now the depende nce on independent variables is given thro ugh Z i ( ζ ) = m κ ρ i ( ζ − 1 i z − + ζ i z + ) . In fact, it appears that it is appropriate to use the matrices T α = D − 1 ( f , f ) e R ′ α and write the solutions unde r construct ion as Γ α = det T α + 1 det T α . The pr oblem of construct ing multi-soliton solutions for the T oda eq ua tions (3.8) is thus r educed to calculating the determinant of the r × r matrix T α . T o obtain a one-soliton solution we set r = 1 . In this case T α ar e ordinary functions, and we easily find T α + 1 = 1 + 2 cos ( 2 α − 1 ) θ ρ cos θ ρ e Z ( ζ ) + δ − 2i θ ρ + e 2 ( Z ( ζ ) + δ − 2i θ ρ ) . (5.3) Setting r = 2 we work out the determinant of the respect ive 2 × 2 matrix explicitly given above and thus obtain for the two-soliton solution the expr ession det T α + 1 = 1 + 2 cos ( 2 α − 1 ) θ ρ 1 cos θ ρ 1 e e Z 1 + 2 cos ( 2 α − 1 ) θ ρ 2 cos θ ρ 2 e e Z 2 + e 2 e Z 1 + e 2 e Z 2 +  2 η + 12 cos ( 2 α − 1 ) ( θ ρ 1 − θ ρ 2 ) cos θ ρ 1 cos θ ρ 2 + 2 η − 12 cos ( 2 α − 1 ) ( θ ρ 1 + θ ρ 2 ) cos θ ρ 1 cos θ ρ 2  e e Z 1 + e Z 2 + 2 η + 12 η − 12  cos ( 2 α − 1 ) θ ρ 1 cos θ ρ 1 e e Z 1 + 2 e Z 2 + cos ( 2 α − 1 ) θ ρ 2 cos θ ρ 2 e 2 e Z 1 + e Z 2  +  η + 12 η − 12  2 e 2 ( e Z 1 + e Z 2 ) , (5.4) with the ‘ soliton interaction factors’ η + 12 = ( ζ 1 ζ − 1 2 + ζ 2 ζ − 1 1 ) + 2 cos ( θ ρ 1 + θ ρ 2 ) ( ζ 1 ζ − 1 2 + ζ 2 ζ − 1 1 ) + 2 cos ( θ ρ 1 − θ ρ 2 ) , η − 12 = ( ζ 1 ζ − 1 2 + ζ 2 ζ − 1 1 ) − 2 cos ( θ ρ 1 − θ ρ 2 ) ( ζ 1 ζ − 1 2 + ζ 2 ζ − 1 1 ) − 2 cos ( θ ρ 1 + θ ρ 2 ) , and the constant parameters e δ ′ 1 = ( f 1 + f 2 )( e f 1 − f 2 ) ( f 1 − f 2 )( e f 1 + f 2 ) , e δ ′ 2 = ( f 1 + f 2 )( f 1 − e f 2 ) ( f 1 − f 2 )( f 1 + e f 2 ) 21 giving rise to a shift in the exponents as e Z i = Z i ( ζ ) + δ i + δ ′ i − 2i θ ρ i . It can also be shown that performing the corr esponding change of variables as sug- gested in the paper [ 3], one can reach the same r esult along the lines of the Hiro ta’s appro ach. Here, the quantities det T α + 1 constr ucted by mea ns of the rational dr essing formalism, will coincide with the Hiro ta’s τ -functions τ α , see the pa p e r [3] wher e such corr espondence was established for the untwisted case. Now , considering s = 2 , so that N = 3, we describe the Dodd–Bullough–Mikhailov equation fr om Section 3.2. Here we have Γ 1 = ( − 1 ) r , Γ 3 = Γ − 1 2 , and so, the mapping γ is parameterized by the only nontrivial function Γ 2 , denoted he re by Γ . The cor - r esponding soliton solutions can ea sily be derived from the relations (5.3) and (5.4) putting α = 2 and α = 3 in or der and taking into account that θ ρ = π ρ / 3. R emember here that ρ = J − K , wher e J and K take values 1, 2 or 3 only . In particular , it is easy to see that the one-soliton solution can be written as Γ = 1 − 4 e Z ( ζ ) + δ − 2i θ ρ + e 2 ( Z ( ζ ) + δ − 2i θ ρ ) ( 1 + e Z ( ζ ) + δ − 2i θ ρ ) 2 . For the two-soliton solution, we should respectively simplify the expression (5.4). The corr esponding expr essions r epr oduce the one- a nd two-soliton solutions of the Dodd– Bullough–Mikhailov equation obtained in the paper [13] by means of the Hirot a’s method. 5.2 Even-dimensional case Here we consider the case of n = 2 s , with p = N = 2 s − 1 . It mea ns also that now we have n 1 = 2. The e igenvectors of the matrices t c − , t c + , c − and c + ar e 2 s -dimensional column vectors Ψ ρ , ρ = 1, . . . , 2 s − 1, satisfying the relations t c − Ψ ρ = m ǫ s + 2 ρ 2 N Ψ ρ , t c + Ψ ρ = m ǫ − s − 2 ρ 2 N Ψ ρ , c − Ψ ρ = m ǫ − s − 2 ρ 2 N Ψ ρ , c + Ψ ρ = m ǫ s + 2 ρ 2 N Ψ ρ , wher e we define the 2 s compo nents of Ψ ρ as ( Ψ ρ ) 0 = ( Ψ ρ ) 1 = ǫ α ( s + 2 ρ ) 2 N , ( Ψ ρ ) α = √ 2 ǫ α ( s + 2 ρ ) 2 N , α = 2, . . . , s and ( Ψ ρ ) α = ( − 1 ) α − s − 1 √ 2 ǫ α ( s + 2 ρ ) 2 N , α = s + 1, . . . , 2 s − 1. Besides, r espective to the only zer o eigenvalue, c − , c + and their transposed matrices have one and the same null-vector that can be defined as t Ψ 0 = ( 1, − 1, 0, . . . , 0 ) . Consequently , the solution (4.28) takes the form ( u i ,1 ) 0 = c i 0 + 2 s − 1 ∑ ρ = 1 c i ρ ǫ s + 2 ρ 2 N e − Z ρ ( µ i ) , ( u i ,1 ) 1 = − c i 0 + 2 s − 1 ∑ ρ = 1 c i ρ ǫ s + 2 ρ 2 N e − Z ρ ( µ i ) , and u i , α = 2 s − 1 ∑ ρ = 1 c i ρ √ 2 ǫ α ( s + 2 ρ ) 2 N e − Z ρ ( µ i ) , α = 2, . . . , s u i , α = 2 s − 1 ∑ ρ = 1 c i ρ ( − 1 ) α − s − 1 √ 2 ǫ α ( s + 2 ρ ) 2 N e − Z ρ ( µ i ) , α = s + 1, . . . , 2 s − 1, 22 wher e c i 0 and c i ρ ar e arbitrary constants and , as usual, we have intr oduced the notation Z ρ ( µ i ) = m ( ǫ − s − 2 ρ 2 N µ − 1 i z − + ǫ s + 2 ρ 2 N µ i z + ) . Note t hat u i ,1 is now a 2-dimensional column vecto r with the components ( u i ,1 ) 0 and ( u i ,1 ) 1 given in order . For the quasi-periodic quantities e R α intr oduced in Section 4 we obtain the expr es- sions ( e R 1 ) i j = − 2 µ 2 s i µ 2 s j µ 2 s − 1 i + µ 2 s − 1 j c i 0 c j 0 + 2 µ i µ 2 s j 2 s − 1 ∑ ρ , σ = 1 c i ρ c j σ ǫ 2 ( s + ρ + σ ) 2 N 1 + µ j µ − 1 i ǫ − 2 ( ρ − σ ) 2 N e − Z ρ ( µ i ) − Z σ ( µ j ) , ( e R α ) i j = 2 µ 2 s i µ 2 s j µ 2 s − 1 i + µ 2 s − 1 j c i 0 c j 0 + 2 ( − 1 ) α µ 2 s + 1 − α i µ α j 2 s − 1 ∑ ρ , σ = 1 c i ρ c j σ ǫ 4 ρ + 1 − 2 ( ρ − σ ) α 2 N 1 + µ j µ − 1 i ǫ − 2 ( ρ − σ ) 2 N e − Z ρ ( µ i ) − Z σ ( µ j ) , that ar e to be used in the determinant r epresentation derived earlier for constr ucting the soliton solutions. It is easy to see that if we set he r e c i 0 = 0, then we come to the same solutions given for n = 2 s − 1 by the relations (5.2)–(5.4), with the only une ssential dif ference that for n = 2 s the rational dressing gives Γ 1 = ( − 1 ) r J r 2 . Therefor e, to obtain new solutions we consider that in what follows c i 0 does not vanish. T o construct such new simplest soliton solutions we thus assume that for each value of the index i only one arbitrary constant c i ρ , apart fr om the c i 0 , is differ ent fr om zero. T o keep up with the notations used in the pr eceding section, we den ote such nonvanishing coeffic ients by C I i and C 0 i . Then we can write for the above r × r matrices e R 1 and e R α ( e R 1 ) i j = − 2 µ i C 0 i ( e R ′ 1 ) i j C 0 j µ 2 s j , ( e R α ) i j = 2 µ i C 0 i ( e R ′ α ) i j C 0 j µ 2 s j , wher e e R ′ 1 and e R ′ α can be repr esented as ( e R ′ 1 ) i j = D i j ( ζ 2 s − 1 , ζ 2 s − 1 ) − e − Z ′ i D i j ( ζ , ζ ) e − Z ′ j , ( e R ′ α ) i j = D i j ( ζ 2 s − 1 , ζ 2 s − 1 ) − ( − 1 ) α ζ 2 s − α i e − Z ′ i D i j ( ζ , ζ ) e − Z ′ j ζ α − 2 s j . Here we use the same notation for the matrices D ( f , g ) intr oduced in the preceding Section 5 .1, and besides, Z ′ i = Z i ( ζ ) − δ i − i θ s + 2 I i , Z i ( ζ ) = m ( ζ − 1 i z − + ζ i z + ) , with the set of parameters ζ i = ǫ s + 2 I i 2 N µ i , e δ i = C I i C 0 i , θ s + 2 I i = π ( s + 2 I i ) 2 s − 1 . 23 W e also r e write the explicit forms of the components of the 2-dimensional column vector e u i ,1 in terms of the notations intr oduced above. W e have ( e u i ,1 ) 0 = µ i C 0 i ( 1 + exp ( − Z ′ i )) , ( e u i ,1 ) 1 = − µ i C 0 i ( 1 − exp ( − Z ′ i )) . Hence, a ccording to the general relatio ns d e rived in Section 4, we can take the ma- trices T α = D − 1 ( ζ 2 s − 1 , ζ 2 s − 1 ) e R ′ α instead of e R α and write for the solutions of the T oda equations (3.13 ) the following expr essions: Γ 1 = I 2 + r ∑ i , j = 1 v i ( e R ′− 1 1 ) i j t v j J 2 , wher e v i ar e 2-dimensional column vectors with the components v i ,0 = 1 √ 2 ( 1 + exp ( − Z ′ i ) ) , v i ,1 = − 1 √ 2 ( 1 − exp ( − Z ′ i ) ) , and Γ α = det T α + 1 det T α , α = 2, . . . , s . T o obt ain a one-solito n solut ion of the type under consideration, we put r = 1, for which T α ar e or din a ry functions. It is easy to show that in this case we have Γ 1 =  0 Γ Γ − 1 0  , Γ = 1 + exp ( − Z ′ ) 1 − exp ( − Z ′ ) , and Γ α = 1 + ( − 1 ) α exp ( − 2 Z ′ ) 1 − ( − 1 ) α exp ( − 2 Z ′ ) , α = 2, . . . , s . Note that ap a rt fr om the r elation Γ 2 s + 1 − α = Γ − 1 α here we also have Γ α + 1 = Γ − 1 α . I t is clear that to have a mapping γ belonging to G 0 we should take Γ 1 J 2 instead of the above Γ 1 . Setting r = 2 we work out the corr esponding 2 × 2 matrices and thus obtain new two-solit on so lutions to (3.13). The calculations lead to the expressions Γ 1 =  Γ 0 0 Γ − 1  , Γ = 1 + e − e Z 1 − e − e Z 2 − η 12 e − ( e Z 1 + e Z 2 ) 1 − e − e Z 1 + e − e Z 2 − η 12 e − ( e Z 1 + e Z 2 ) , wher e the ‘ soliton interaction factor ’ is now η 12 = ζ 1 − ζ 2 ζ 1 + ζ 2 · ζ 2 s − 1 1 − ζ 2 s − 1 2 ζ 2 s − 1 1 + ζ 2 s − 1 2 , and we have intro duced a new pa rameter δ ′ defined by e δ ′ = ζ 2 s − 1 1 + ζ 2 s − 1 2 ζ 2 s − 1 1 − ζ 2 s − 1 2 and pr oducing a shift in the exponents, e Z i = Z ′ i − δ ′ = Z i ( ζ ) − δ i − δ ′ − i θ s + 2 I i . 24 W e also have det T α + 1 = 1 + ( − 1 ) α ( e − 2 e Z 1 + e − 2 e Z 2 ) − 4 ( − 1 ) α ζ α 1 ζ 2 s − α 2 + ζ α 2 ζ 2 s − α 1 ( ζ 1 + ζ 2 )( ζ 2 s − 1 1 + ζ 2 s − 1 2 ) e − ( e Z 1 + e Z 2 ) + η 2 12 e − 2 ( e Z 1 + e Z 2 ) . Note finally that under the permutation of the p a rameters ζ 1 and ζ 2 the function Γ transforms into Γ − 1 , thus Γ 1 goes to Γ − 1 1 , while Γ α for the other values of α all stay invariant. 6 Conclus ion W e have considered the abelian T oda systems associated with the loop gr oups o f the complex general linear gr oups. Using the metho d of rational dress ing, along the lines of [3], we have constructed soliton solutio ns to these equations in the twisted cases, that is, when the gradations ar e generated by outer automorphisms of the str ucture Lie algebras. Our consideration can be generalized to T oda syst ems connected with other loop groups , such as twisted and untwisted loop gr oups of the complex orthog- onal and symplectic gro ups. It is worth noting that, as we have alr eady observed her e, the p ole positions of the dressing meromorphic mappings and their inverse ones turn out to be bound up with each other because of the specific structur e of the outer auto- morphism leading to the twisted cases. This cir cumstance made pa rt of the formulae mor e intricat e than in the untwisted general linear case considered in t he pr eceding paper [3]. Actually , similar prob lems of coinciding pole posit ions arise also due to the specific gr oup conditions. W e will a d d ress to this problem and pr esent our respective r esults in some future publications. This work was supported in part by the Russian Foundation for Basic Research under grant #07–01– 0 0234 and by the joint DFG–RFBR grant #08 –01–91953. One of the authors (A.V .R .) wishes to ackno wledge the warm hospitality of the Erwin Schr ¨ o- dinger International Institute for Mathematical Physics where a part of this work was carried out. Referenc es [1] A . 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