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This article is cited in 2 scientific papers (total in 2 papers)
Space Curves and Solitons of the KP Hierarchy. I. The $l$-th Generalized KdV Hierarchy
Yuji Kodama, Yuancheng Xie Department of Mathematics, The Ohio State University, Columbus OH, 43210, USA
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.
Keywords:
space curve, soliton solution, KP hierarchy, Sato Grassmannian, numerical semigroup.
Received: October 14, 2020; in final form March 2, 2021; Published online March 16, 2021
Citation:
Yuji Kodama, Yuancheng Xie, “Space Curves and Solitons of the KP Hierarchy. I. The $l$-th Generalized KdV Hierarchy”, SIGMA, 17 (2021), 024, 43 pp.
Linking options:
https://www.mathnet.ru/eng/sigma1707 https://www.mathnet.ru/eng/sigma/v17/p24
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