Armando Treibich
Publications:
Treibich A.
Nonlinear Evolution Equations and Hyperelliptic Covers of Elliptic Curves
2011, vol. 16, nos. 34, pp. 290310
Abstract
This paper is a further contribution to the study of exact solutions to KP, KdV, sineGordon, 1D Toda and nonlinear Schrödinger equations. We will be uniquely concerned with algebrogeometric solutions, doubly periodic in one variable. According to (socalled) ItsMatveevâ€™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 $d$  1dimensional of spectral curves, of arbitrary high genus, giving rise to KdV solutions doubly periodic with respect to the dth KdV flow $(d \geqslant 1)$. Analogous results are presented, without proof, for the 1D Toda, NL Schrödinger an sineGordon equation.
