More non-Abelian loop Toda solitons

arXiv:0810.1025v2 [math-ph] 19 Jun 2009 More non-Abelian loop Toda solitons Kh. S. Nirov∗and A. V. Razumov† Fachbereich C–Physik, Bergische Universit¨at Wuppertal D-42097 Wuppertal, Germany Abstract We find new solutions, including soliton-like ones, for a special case of non- Abelian loop Toda equations associated with complex general linear groups. We use the method of rational dressing based on an appropriate block-matrix repre- sentation suggested by the Z-gradation under consideration. We present solutions in a form of a direct matrix generalization of the Hirota’s soliton solution already well-known for the case of Abelian loop Toda systems. Mathematics Subject Classification (2000). 37K10, 37K15, 35Q58 Keywords. Non-Abelian loop Toda systems, rational dressing method, soliton-like solutions 1 Introduction The two-dimensional Toda equations play an essential rˆole in understanding certain structures in classical and quantum integrable systems. They are formulated as nonlin- ear partial differential equations of second order and are associated with Lie groups, see, for example, the monographs [1, 2]. The Toda equations associated with affine Kac–Moody groups are of special interest, because they possess soliton solutions hav- ing a lot of physical applications. The simplest example here is the celebrated sine- Gordon equation known for a long time. Another example of an affine Toda equation was constructed in paper [3] as a direct two-dimensional generalization of the famous mechanical Toda chain. Later on, the consideration of paper [3] was generalized in papers [4, 5] in the case of Toda chains related to various affine Kac–Moody algebras. Another approach to formulating affine Toda systems, based on folding properties of Dynkin diagrams, was implemented in [6]. Note also that papers [3, 4, 5, 7] were pi- oneering in investigating the question of integrability of affine Toda field theories by considering the corresponding zero-curvature representation. It is convenient to consider instead of the Toda systems associated with affine Kac- Moody groups the Toda systems associated with loop groups. There are two reasons to do so. First, the affine Kac-Moody groups can be considered as a loop extension of ∗On leave of absence from the Institute for Nuclear Research of the Russian Academy of Sciences, 60th October Ave 7a, 117312 Moscow, Russia †On leave of absence from the Institute for High Energy Physics, 142281 Protvino, Moscow Region, Russia 1 loop groups and the solutions of the corresponding Toda equations are connected in a simple way, see, for example, paper [8]. Second, in distinction to the loop groups, there is no a realization of affine Kac–Moody groups suitable for practical usage. It is known that a Toda equation associated with a Lie group is specified by the choice of a Z-gradation of its Lie algebra [1, 2]. Hence, to classify the Toda systems associated with some class of Lie groups one needs to describe all Z-gradations of the respective Lie algebras. Recently, in a series of papers [9, 10, 11], we classified a wide class of Toda equations associated with untwisted and twisted loop groups of complex classical Lie groups. More concretely, we introduced the notion of an integrable Z- gradation of a loop Lie 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 respective Toda equations. It appeared that despite the fact that we consider Toda equations associated with infinite-dimensional Lie groups, the resulting Toda equations are equivalent to the equations formulated only in terms of the under- lying finite-dimensional Lie groups and Lie algebras. Actually, partial cases of such type of equations appeared before, see, for example, papers [12, 13, 14] and references therein, but we demonstrated that any Toda equation of the class under consideration can be written in terms of finite-dimensional Lie groups and Lie algebras. To slightly simplify terminology and make distinction with the Toda equations associated with finite-dimensional Lie groups, we call the finite-dimensional version of a Toda equa- tion associated with a loop group of the Lie group G a loop Toda equation associated with the Lie group G. Here we consider untwisted loop Toda equations associated with complex general linear group. As was shown in papers [10, 11], any such equation has the form1 ∂+(γ−1∂−γ) = [c−, γ−1c+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 group GLn(C) having a block-diagonal form γ =     Γ1 Γ2 Γp    , so that for each α = 1, . . . , p the mapping Γα is a mapping of M to the Lie group GLnα(C) with ∑ p α=1 nα = n. Further, c+ and c−are mappings of M to the Lie algebra gln(C). The mapping c+ has the block-matrix structure c+ =         0 C+1 0 0 C+(p−1) C+0 0         , 1We denote by ∂+ and ∂−t

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