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 ⊥, projecting onto a rational surface \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\tilde S$\end{document}. Moreover, all spectral curves project onto a rational curve inside \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\tilde S$\end{document}. We are thus led to study all rational curves of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\tilde S$\end{document}, 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 ≥ 1). Analogous results are presented, without proof, for the 1D Toda, NL Schrodinger an sine-Gordon equation.