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Symmetry, Integrability and Geometry: Methods and Applications, 2021, Volume 17, 024, 43 pp.
DOI: https://doi.org/10.3842/SIGMA.2021.024
(Mi sigma1707)
 

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
Full-text PDF (610 kB) Citations (2)
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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.
Funding agency Grant number
National Science Foundation DMS-1714770
Engineering and Physical Sciences Research Council EP/P012698/1
EP/P1012655/1
He also appreciated their financial support by the EPSRC grants EP/P012698/1 and EP/P1012655/1 during his stay at University of Kent and University of Leeds. The present work is partially supported by NSF grant DMS-1714770.
Received: October 14, 2020; in final form March 2, 2021; Published online March 16, 2021
Bibliographic databases:
Document Type: Article
Language: English
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.
Citation in format AMSBIB
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\by Yuji~Kodama, Yuancheng~Xie
\paper Space Curves and Solitons of the KP Hierarchy. I.~The $l$-th Generalized KdV Hierarchy
\jour SIGMA
\yr 2021
\vol 17
\papernumber 024
\totalpages 43
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\crossref{https://doi.org/10.3842/SIGMA.2021.024}
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\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85103287508}
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  • This publication is cited in the following 2 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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    Symmetry, Integrability and Geometry: Methods and Applications
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