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, calledtransference, 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.     

Differential Algebras with Banach-Algebra Coefficients II: The Operator Cross-Ratio Tau-Function and the Schwarzian Derivative

Several features of an analytic (infinite-dimensional) Grassmannian of (commensurable) subspaces of a Hilbert space were developed in the context of integrable PDEs (KP hierarchy). We extended some of those features when polarized separable Hilbert spaces are generalized to a class of polarized Hilbert modules and then consider the classical Baker and τ-functions as operator-valued. Following from Part I we produce a pre-determinant structure for a class of τ-functions defined in the setting of the similarity class of projections of a certain Banach *-algebra. This structure is explicitly derived from the transition map of a corresponding principal bundle. The determinant of this map leads to an operator τ-function. We extend to this setting the operator cross-ratio which had previously been used to produce the scalar-valued τ-function, as well as the associated notion of a Schwarzian derivative along curves inside the space of similarity classes of a given projection. We link directly this cross-ratio with Fay’s trisecant identity for the τ-function. By restriction to the image of the Krichever map, we use the Schwarzian to introduce the notion of an operator-valued projective structure on a compact Riemann surface: this allows a deformation inside the Grassmannian (as it varies its complex structure). Lastly, we use our identification of the Jacobian of the Riemann surface in terms of extensions of the Burchnall–Chaundy C*-algebra (Part I) provides a link to the study of the KP hierarchy.     

Subvarieties of -divisors in the Jacobian

We explore some of the interplay between Brill-Noether subvarieties of the moduli space of rank 2 bundles with canonical determinant on a smooth projective curve and -divisors, via the inclusion of the moduli space into , singular along the Kummer variety. In particular we show that the moduli space contains all the trisecants of the Kummer and deduce that there are quadrisecant lines only if the curve is hyperelliptic; we show that for generic curves of genus , though no higher, bundles with sections are cut out by ; and that for genus 4 this locus is precisely the Donagi-Izadi nodal cubic threefold associated to the curve.

     

Isospectral hamiltonian flows in finite and infinite dimensions

A moment map is constructed from the Poisson manifold _A of rank-r perturbations of a fixedN×N matrixA to the dual of the positive part of the formal loop algebra =gl(r)[[, ^–1]]. The Adler-Kostant-Symes theorem is used to give hamiltonians which generate commutative isospectral flows on . The pull-back of these hamiltonians by the moment map gives rise to commutative isospectral hamiltonian flows in _A. The latter may be identified with flows on finite dimensional coadjoint orbits in and linearized on the Jacobi variety of an invariant spectral curveX _r which, generically, is anr-sheeted Riemann surface. Reductions of _A are derived, corresponding to subalgebras ofgl(r, ) andsl(r, ), determined as the fixed point set of automorphism groupes generated by involutions (i.e., all the classical algebras), as well as reductions to twisted subalgebras of . The theory is illustrated by a number of examples of finite dimensional isospectral flows defining integrable hamiltonian systems and their embeddings as finite gap solutions to integrable systems of PDE's.This research was partially supported by NSF grants MCS-8108814 (A03), DMS-8604189, and DMS-8601995     

The sigma function for trigonal cyclic curves

Letters in Mathematical Physics - A recent generalization of the “Kleinian sigma function” involves the choice of a point P of a Riemann surface X, namely a “pointed curve”...     

Factorization and Resultants of Partial Differential Operators

Comparatively little is known about commutative rings of partial differential operators, while in the ordinary case, concrete examples and an algebraic(-geometric) structure can be algorithmically determined for large classes. In this note, by the calculation of the partial μ-shifted differential resultant which we defined in a previous paper, we produce algebraic equations of spectral surfaces for commutative rings in two variables, and Darboux transformations of Airy-type operators that correspond to morphisms of surfaces. There are, however, many elementary differential-algebraic statements that we only observe experimentally, thus we offer open questions which seem to us quite significant in differential algebra, and access to Mathematica code to enable further experimentation.     

HPLC analysis of PCBs on porous graphitic carbon: Retention behavior and gradient elution

Summary The fractionation of PCB congeners into classes according to their planarity (i.e. amount of ortho substitution) by HPLC on porous graphitic carbon (PGC) as stationary phase has been investigated as a preliminary step before GC analysis, indispensable for a complete separation of PCB congeners. A systematic study of retention behavior on PGC eluted with different n-hexane-dichloromethane mixtures made it possible to design a linear binary gradient which separated PCB congeners in a reasonable time and with good performance. Relationships were obtained between retention behavior and the molecular structure of the PCB congeners. The beneficial effects of elevated temperature on separation efficiency were also investigated. The analysis conditions selected, i.e. continuous gradient separation at 40°C, were successfully used for fractionation of technical PCB formulations, e.g. Aroclor 1242.     

Monodromy of boussinesq elliptic operators

Verdier's program for classifying elliptic operators with a nontrivial centralizer is outlined. Examples of Boussinesq operators are developed.To J.-L. Verdier, in memoriam     

Prym varieties and the Verlinde formula

Research partially supported by the CNR Foundation     

The Riemann constant for a non-symmetric Weierstrass semigroup

The zero divisor of the theta function of a compact Riemann surface X of genus g is the canonical theta divisor of Pic\({^{(g-1)}}\) up to translation by the Riemann constant \({\Delta}\) for a base point P of X. The complement of the Weierstrass gaps at the base point P gives a numerical semigroup, called the Weierstrass semigroup. It is classically known that the Riemann constant \({\Delta}\) is a half period, namely an element of \({\frac{1}{2}\Gamma_\tau}\) , for the Jacobi variety \({\mathcal{J}(X)=\mathbb{C}^{g}/\Gamma_\tau}\) of X if and only if the Weierstrass semigroup at P is symmetric. In this article, we analyze the non-symmetric case. Using a semi-canonical divisor D₀, we express the relation between the Riemann constant \({\Delta}\) and a half period in the non-symmetric case. We point out an application to an algebraic expression for the Jacobi inversion problem. We also identify the semi-canonical divisor D₀ for trigonal pointed curves, namely with total ramification at P.