Elliptic solutions of the Korteweg-de Vries equation (Q919197)

A method is proposed to construct all the solutions of the KdV equation, which are elliptic in x, associated to the Krichever curves. These curves were designed to provide solutions of the Kadomtsev-Petviashvili equations (KP) in terms of elliptic functions. The main result of this paper states that if a Krichever curve of genus g admits a holomorphic involution wit \(2g+2\) fixed points then any solution of the KP equation constructed along that curve reduces to a solution of the KdV equation. The theorem is then applied to some concrete examples and as a consequence it is offered an explanation to previous results due to Treibich and Verdier.