Space Curves and Solitons of the KP Hierarchy. I. The $l$-th Generalized KdV Hierarchy

TL;DR

This paper explores the relationship between KP solitons, space curves, and numerical semigroups, providing explicit constructions, Schur polynomial expansions, and deformations of singular curves into smooth ones.

Contribution

It introduces a novel connection between KP solitons of the generalized KdV hierarchy and rational space curves linked to specific numerical semigroups, including explicit tau-function expansions.

Findings

KP solitons are related to rational space curves from numerical semigroups.

Explicit Schur polynomial expansions of tau-functions are provided.

Smooth curves are constructed by deforming singular curves associated with solitons.

Abstract

It is well known that algebro-geometric solutions of the KdV hierarchy are constructed from the Riemann theta functions associated with hyperelliptic curves, and that soliton solutions can be obtained by rational (singular) limits of the corresponding curves. In this paper, we discuss a class of KP solitons in connections with space curves, which are labeled by certain types of numerical semigroups. In particular, we show that some class of the (singular and complex) KP solitons of the $l$-th generalized KdV hierarchy with $l\ge 2$ is related to the rational space curves associated with the numerical semigroup $\langle l,lm+1,\dots, lm+k\rangle$, where $m\ge 1$ and $1\le k\le l-1$. We also calculate the Schur polynomial expansions of the $\tau$-functions for those KP solitons. Moreover, we construct smooth curves by deforming the singular curves associated with the soliton solutions. For these KP solitons, we also construct the space curve from a commutative ring of differential operators in the sense of the well-known Burchnall-Chaundy theory.