Degeneration of Trigonal Curves and Solutions of the KP-Hierarchy

TL;DR

This paper investigates how solutions to the KP-hierarchy associated with trigonal curves degenerate into intermediate forms when the curves develop ordinary triple points, using the Sato Grassmannian framework.

Contribution

It introduces a new class of solutions in Wronskian form and demonstrates their connection to soliton solutions through degeneration and gauge transformations.

Findings

Degeneration leads to intermediate solutions between solitons and rational solutions.

Sato Grassmannian effectively describes the degeneration process.

New Wronskian solutions encompass solitons as a special case.

Abstract

It is known that soliton solutions of the KP-hierarchy corresponds 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 some intermediate solutions between solitons and rational solutions. They are considered as cerain 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 contains 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