Volume 72A, number 2 PHYSICS LEfl’ERS 25 June 1979 INTEGRABLE EQUATIONS AND DIFFERENTIAL GEOMETRY B.G. KONOPELCHENKO Institute of Nuclear Physics, 630090 Novosibirsk 90, USSR Received 6 February 1979 It is shown that the equations which are integrable by the inverse scattering transform method and this method itself ad- mit a natural interpretation in terms of vector and principal zero-curvature fibre bundles with certain structure groups G. 1. At present considerable attention is given to 2. Let us consider the one-parametric family {P, G, the inverse scattering transform method (1ST method) M, H; X} (pencil) of vector bundles with arbitrary struc- and to the equations integrable by this method (see re- ture group G: Mis a base space, X is a parameter nu- views [1,21). It is likely that specific properties of merating the family terms. Let M be an N-dimensional completely integrable equations are associated with manifold in which the local coordinates z = {z 1, ..., ZJ.,T} special geometric properties of these equations. In can be introduced. G is assumed to be a linear group. view of this, the technique of differential geometry Then H ‘(z) is a space Q of the representation of the seems to be quite natural and has been applied in refs. group G. Denote the basis in a bundle as F(X). As is [3—6]. The starting point of refs. [4—6] is an inter- known, its motion in the zero-curvature fibre bundle pretation of the fact that an initial non-linear equation (at a given A) is determined by the formula (see ref. [71): can be represented as the consistency condition of dF(X) = w(X) F(X), (1) some linear system in the form of zero curvature of some space. The authors of these papers point out where w(X) is the connection form of the G-bundle. that there exists a connection between the linear prob- Consider in the above base space M the n-dimen- 1cm of the 1ST method and the structure equations for sional submanifold M given by the equations z~ fibre spaces. However, only a particular case of the = p1 1x1, ... , x,~), ..., ~ = ..., x~), where structure group SL(2R) was considered. Meanwhile, ~p1(x), ..., p~,(x) are functions of the independent van- the questions concerning conservation laws, soliton ables x = {x1 —z~, I = 1, ..., n}. The remaining variables solutions, Bäckltnd transformations are left open. z,~.,,, ..., ZN are various partial derivatives ~ ..., In the present paper a general approach is proposed with respect to the variables x,. based on the theory of principal and associated (vec- Let us consider the space of all cross sections of the tor) G-bundles. The case of an arbitrary structure Lie G-bundle. After reduction to the submanifold M, eq. group G is studied. It is shown that the given corn- (1) in the local coordinates has the form pletely integrabie non-linear differential equation is = 1Z ~ Ft ~ 2 connected with some families of principal G-bundles ~ .~ W~ , j ~Z, j, and an infinite set of families of associated vector G- where ~ is the total derivative over x,. The Q-valued bundles of zero curvature. Hence, the initial differen- functions W 1(Z, A) are determined by the relationw(X) = tial equation corresponds to an infInite number of liii- w~(z, A) dx1. ear spectral problems of different dimensionality. It is Eq. (2) is just the 1ST method linear spectral prob- also shown that soliton solutions correspond to de- 1cm for the differential equation: generation of the basis in a fibre space. ( C 1.fWk(Z, A) — Dk~’i(z, A) + [w,(Z, A), Wk(Z, A)] = 0, (3) 101