Rev\^etements hyperelliptiques d-osculateurs et solitons elliptiques de la hi\'erarchie KdV

TL;DR

This paper constructs large families of hyperelliptic curves with special osculation properties over elliptic curves, leading to explicit, doubly periodic solutions of the KdV hierarchy.

Contribution

It introduces a new class of hyperelliptic curves with osculating projections, generating extensive families of solutions to the KdV hierarchy with specific periodicity.

Findings

Constructed ($d-1$)-dimensional families of hyperelliptic curves with osculating properties.

Produced ($g+d-1$)-dimensional families of doubly periodic KdV solutions.

Established connections between hyperelliptic curve geometry and integrable systems.

Abstract

Let $d$ be a positive integer, $\mathbb K$ an algebraically closed field of characteristic 0 and $ X$ an elliptic curve defined over K. We study the hyperelliptic curves equipped with a projection over $ X$, such that the natural image of $ X$ in the Jacobian of the curve osculates to order $d$ to the embedding of the curve, at a Weierstrass point. We construct ($d-1$)-dimensional families of such curves, of arbitrary big genus $g$, obtaining, in particular, $(g+d-1)$-dimensional families of solutions of the $KdV$ hierarchy, doubly periodic with respect to the $d$-th $KdV$ flow.