Nonlinear Evolution Equations and Hyperelliptic Covers of Elliptic Curves

This paper is a further contribution to the study of exact solutions to KP, KdV, sine-Gordon, 1D Toda and nonlinear Schrodinger equations. We will be uniquely concerned with algebro-geometric solutions, doubly periodic in one variable. According to (so-called) Its-Matveev's formulae, the Jacobians of the corresponding spectral curves must contain an elliptic curve X, satisfying suitable geometric properties. It turns out that the latter curves are in fact contained in a particular algebraic surface $S\perp$, projecting onto a rational surface $\widetilde S$. Moreover, all spectral curves project onto a rational curve inside $\widetilde S$. We are thus led to study all rational curves of $\widetilde S$, having suitable numerical equivalence classes. At last we obtain $d\,$-$\,1$-dimensional of spectral curves, of arbitrary high genus, giving rise to KdV solutions doubly periodic with respect to the $d$-th KdV flow ($d\geq 1$). Analogous results are presented, without proof, for the 1D Toda, NL Schrodinger an sine-Gordon equation.