Degeneration of trigonal curves and solutions of the KP-hierarchy

Abstract

It is known that soliton solutions of the KP-hierarchy correspond to singular rational curves with only ordinary double points. In this paper we study the degeneration of theta function solutions corresponding to certain trigonal curves. We show that, when the curves degenerate to singular rational curves with only ordinary triple points, the solutions tend to be some intermediate solutions between solitons and rational solutions. They are considered as certain limits of solitons.The Sato Grassmannian is extensively used here to study the degeneration of solutions, since it directly connects solutions of the KP-hierarchy to the defining equations of algebraic curves. We define a class of solutions in the Wronskian form which contain soliton solutions as a subclass and prove that, using the Sato Grassmannian, the degenerate trigonal solutions are connected to those solutions by certain gauge transformations.