Multi-Wroński Determinant Solutions of the Darboux System

Sequences of Levy transformations for the Darboux system of conjugates nets in multidimensions are studied. We show that after a suitable number of Levy transformations, with at least a Levy transformation in each direction, we get closed formulae in terms of multiWroński determinants. These formulae are for the tangent vectors, Lamè coefficients, rotation coefficients and points of the surface. On leave of absence from Beijing Graduate School, CUMT, Beijing 100083, China Supported by Beca para estancias temporales de doctores y tecnólogos extranjeros en España: SB95-A01722297 Partially supported by CICYT: proyecto PB95–0401 1 1. The interaction between Soliton Theory and Geometry is a growing subject. In fact, many systems that appear by geometrical considerations have been studied independently in Soliton Theory, well-known examples include the Liouville and sine-Gordon equations which characterize minimal and pseudo-spherical surfaces, respectively. Another relevant case is given by the the Darboux equations that were solved 12 years ago in its matrix generalization, using the ∂̄–dressing, by Zakharov and Manakov [14]. In this note we want to iterate a transformation that preserves the Darboux equations which is known as Levy transformation [10]. 2. The Darboux equations ∂βij ∂uk = βikβkj, i, j, k = 1, . . . , N, i 6= j 6= k 6= i, (1) for the N(N−1) functions {βij}i,j=1,...,N i6=j of u1, . . . , uN , characterize N -dimensional submanifolds of R , N ≤ P , parametrized by conjugate coordinate systems [3, 4], and are the compatibility conditions of the following linear system ∂X j ∂ui = βjiX i, i, j = 1, . . . , N, i 6= j, (2) involving suitable P -dimensional vectors X i, tangent to the coordinate lines. The so called Lamé coefficients satisfy ∂Hj ∂ui = βijHi, i, j = 1, . . . , N, i 6= j, (3) and the points of the surface x can be found by means of ∂x ∂ui = X iHi, i = 1, . . . , N, (4) which is equivalent to the more standard Laplace equation ∂x ∂ui∂uj = ∂ lnHi ∂uj ∂x ∂ui + ∂ lnHj ∂ui ∂x ∂uj , i, j = 1, . . . , N, i 6= j. A Darboux type transformation for this system was found by Levy [10, 5, 9]. In fact, in [10] the transformation is constructed only for two-dimensional surfaces, N = 2, being the Darboux equations in this case trivial and Levy 2 only presents the transformation for the points of the surface. However, in [9] the Levy transformation is extended to the first non trivial case of Darboux equations, namely N = 3. The extension to arbritary N is straightforward and reads as follows. Given a solution ξj of ∂ξj ∂uk = βjkξk, for each of the N possible directions in the coordinate space there is a corresponding Levy transformation that reads for the i-th case: x[1] = x− Ω(ξ,H) ξi X i,