Two Explicit Unconnected Sets of Elliptic Solutions Parametrized by an Arbitrary Function for the Two-dimensional Toda Chain A(ii)

1. Algebraical, geometrical and analytical aspects of the theory of non-linear differential equations have been actively investigated recently. One of the problems appearing here is the problem of obtaining solutions of corresponding equations in explicit form, As known the general solution must be parametrized by the 2n arbitrary functions in the case o f n equations. Examples are well known: the Liouville equation, the Toda chain [1]. But in the last case, when the chain corresponds to the Caftan matrix of the Kac-Moody algebra, the solution is expressed in the form of infinite series of very complicated structure and its use is difficult for applications (for instance, in quasiclassics). It is reasonable to reject a number of arbitrary functions in the resulting expression and due to it to obtain a formula no more complicated than the Liouville formula. In the present letter such formulae are presented for the case of the simplest twodimensional Toda chain corresponding to the Cartan matrix of the Kac-ivloody algebra A }1) [2]. This result is obtained by means of the following principle with the use of the results obtained by one of the authors earlier [3]. The local correspondence between 0 (3 ) and O(2,1) o-models and the Toda chain under consideration is obtained which enables us to recalculate the solutions of the initial chiral model into the solutions of the reduced model [3]. In its simplest case this construction enables us to obtain the formula for the solution of the Liouville equation from the instanton sectors of 0 (3) and O(2,1) o-models. In the case considered in this letter the solutions of the initial chiral models, lying out of their instanton sector are recalculated. The second element of the construction is the rather large class of solutions of the initial chiral model. For the case of 0 (3 ) and O(2,1) a-models such solutions were obtained by one of us [4]. They are the generalization of recently obtained elliptic solutions of the 0(3) o-model [5,6], lying in the class of singular harmonic mappings [7], and they are parametrized by an arbitrary holomorphic function, and this is the most significant fact for the construction presented.