Current Preprints.

The following are, as yet, unpublished preprints with abstracts, downloadable in the specified formats (if available).

The Darboux transformation (C.Athorne.)

• Abstract:This ia an encyclopedia entry introducing the DT in an elementary fashion.

An sl(2,C) covariant theory of genus 2 hyperelliptic functions (C.Athorne, J.C.Eilbeck & V.Z.Enolskii)

• Abstract: We present an algebraic formulation of genus 2 hyperelliptic functions which exploits the underlying covariance of the {\em family} of genus 2 curves. This allows a simple interpretation of all identities in representation theoretic terms. We show how the classical theory is recovered when one branch point is moved to infinity.

Identities for hyperelliptic functions of genus 2(C.Athorne, J.C.Eilbeck & V.Z.Enolskii)

• Abstract:We present a simple method that allows one to generate and classify identities for genus two $\wp$ functions for generic algebraic curves of type (2,6). We discuss the relation of these identities to the Boussinesq equation for shallow water waves and show, in particular, that these $\wp$ functions give rise to a family of solutions to Boussinesq.

Covariant hyperelliptic functions of genus two (C.Athorne)

• Abstract:This paper reports on the author's work with Chris Eilbeck and Victor Enolskii concerning the role of representations of $SL_2(\mathbb C)$ in the theory of genus two hyperelliptic functions. We consider its role in the classical theory as well as introducing a set of new, naturally covariant $\cal P$ functions. A complete treatment is to be found in the two references \cite{AEE02a,AEE02b}.

Algebraic Hirota Maps (C.Athorne)

• Abstract:We give definitions of Hirota maps acting as intertwining operators for representations of $\SLin$. We show how these reduce to the conventional (generalised) Hirota derivatives in the limit of the dimension of the representation becoming infinite and we discuss an application to the theory of $\wp$-functions associated with hyperelliptic curves of genus 2.