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 Schrödinger 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 $\tilde{S}$. Moreover, all spectral curves project onto a rational curve inside $\tilde{S}$. We are thus led to study all rational curves of $\tilde{S}$, 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 $(d \geqslant 1)$. Analogous results are presented, without proof, for the 1D Toda, NL Schrödinger an sine-Gordon equation.