Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 17 (2021), 024, 43 pages      arXiv:1912.06768

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

Yuji Kodama and Yuancheng Xie
Department of Mathematics, The Ohio State University, Columbus OH, 43210, USA

Received October 14, 2020, in final form March 02, 2021; Published online March 16, 2021

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.

Key words: space curve; soliton solution; KP hierarchy; Sato Grassmannian; numerical semigroup.

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