Emma Previato

Hyperelliptic quasi-periodic and soliton solutions of the nonlinear Schrödinger equation

On considere la construction de solutions par des methodes de geometrie algebriques et en particulier par les fonctions theta hyperelliptiques

Abelian Functions for Trigonal Curves of Genus Three

We develop the theory of generalized Weierstrass sigma- and \wp-functions defined on a trigonal curve of genus three. In particular we give a list of the associated partial differential equations satisfied by the \wp-functions, a proof that the coefficients of the power series expansion of the sigma-function are polynomials of moduli parameters, and the derivation of two addition formulae.

Monodromy of boussinesq elliptic operators

Verdiers program for classifying elliptic operators with a nontrivial centralizer is outlined. Examples of Boussinesq operators are developed.

On the Hitchin system

1.1 What is known as the Hitchin system is a completely integrable hamiltonian system (CIHS) involving vector bundles over algebraic curves, identified by Hitchin in ([H1], [H2]). It was recently generalized by Faltings [F]. In this paper we only consider the case of ranktwo vector bundles with trivial determinant. In that case the Hitchin system corresponding to a curve C of genus g is obtained as follows. Let

Darboux Transformations for Higher-Rank Kadomtsev—Petviashvili and Krichever—Novikov Equations

It is shown that the action of a special ‘rank 2’ or ‘rank 3’ Darboux transformation, called transference, upon a pair of commuting ordinary differential operators of orders 4 and 6 implements the Abelian sum on their elliptic joint spectrum. A consequence of this is that, under the deformation of these commuting operators by the KP flow, every ‘rank 2’ KP solution corresponds to a solution of the Krichever—Novikov (KN) equation, and vice versa, with the transference process providing the correspondence between (2 + 1) and (1 + 1) dimensions. For a singular joint spectrum, it is further shown that transference at the singular point produces a correspondence between solutions of the singular KN equation and those of the KdV equation. These correspondences are illustrated by considering examples of a nondecaying rational KdV or Boussinesq solutions and the corresponding rational, singular-KN and rational KP solutions which the transference process generates.

On a generalized Frobenius-Stickelberger addition formula

In this Letter we obtain a generalization of the Frobenius–Stickelberger addition formula for the (hyperelliptic) σ-function of a genus 2 curve in the case of three vector-valued variables. The result is given explicitly in the form of a polynomial in Kleinian ℘-functions.

Higher Rank Darboux Transformations

The Darboux transformation has been discovered several times in history (cf. references in [EK] and in [G3]): the reason for this is its deep geometric significance. In this note we explore some geometric aspects of the transformation and their relevance to the difficult problems of describing explicitly higher-rank commutative algebras of ordinary differential operators and KP flows.

Commutative partial differential operators

In one variable, it is possible to describe explicitly the differential operators that commute with a given one, at least when the centralizer of the given operator has rank 1. So far, a generalization of the theory to several variables has been developed (inexplicitly) only for matrices, whose size increases with the number of variables. We propose to develop an algebraic theory of commuting partial differential operators (PDOs) by formulating a generalization of the one-variable techniques, in particular Darboux transformations and differential resultants. In this paper, we present a counter example to a one-variable feature of maximal-commutative rings and some facts, examples and questions on differential resultants.

Differential Algebras with Banach-Algebra Coefficients I: from C*-Algebras to the K-Theory of the Spectral Curve

We present an operator-coefficient version of Sato’s infinite-dimensional Grassmann manifold, and τ-function. In this setting the classical Burchnall–Chaundy ring of commuting differential operators can be shown to determine a C*-algebra. For this C*-algebra topological invariants of the spectral ring become readily available, and further, the Brown–Douglas–Fillmore theory of extensions can be applied. We construct KK classes of the spectral curve of the ring and, motivated by the fact that all isospectral Burchnall–Chaundy rings make up the Jacobian of the curve, we compare the (degree-1) K-homology of the curve with that of its Jacobian. We show how the Burchnall–Chaundy C*-algebra extension by the compact operators provides a family of operator τ-functions.

Curvature of Universal Bundles of Banach Algebras

Given a Banach algebra we construct a principal bundle with connection over the similarity class of projections in the algebra and compute the curvature of the connection. The associated vector bundle and the connection are a universal bundle with attendant connection. When the algebra is the linear operators over a Hilbert module, we establish an analytic diffeomorphism between the similarity class and the space of polarizations of the Hilbert module. Likewise, the geometry of the universal bundle over the latter is studied. Instrumental is an explicit description of the transition maps in each case which leads to the construction of certain functions. These functions are in a sense pre-determinants for the universal bundles in question.