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Algebraic Curves and Flag Varieties in Solutions of the KP Hierarchy and the Full Kostant-Toda Hierarchy

Xie, Yuancheng
http://rave.ohiolink.edu/etdc/view?acc_num=osu1626565722059012

Abstract Details

2021, Doctor of Philosophy, Ohio State University, Mathematics.
This thesis contains two parts. In the first part, we discuss certain class of KP solitons in connections with singular projective 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 $\langle l,lm+1,\ldots, lm+k\rangle$ 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, then we check that quasi-periodic solutions of $l$-th generalized KdV hierarchy indeed degenerate to soliton solutions we begin with when we degenerate the underlying algebraic curve and the line bundle over it properly. For these KP solitons, we also construct the space curves from commutative rings of differential operators in the sense of the well-known Burchnall-Chaundy theory. This part is mainly based on a published paper \cite{Kodama-Xie2021KP}. In the second part, we discuss solutions of the full Kostant-Toda (f-KT) lattice and their connections with the flag varieties. Firstly, we carry out Kowalevski-Painlev\'e analysis for f-KT equation. In particular, we associate each solution of the indicial equations with a Weyl group element, provide explicit formulas for eigenvalues of Kowalevski matrix and at last parameterize all the Laurent series solutions by $\mathcal{G} \slash \mathcal{B} \times \mathbb{C}^n$ where $\mathcal{G} \slash \mathcal{B}$ is the flag variety and $\mathbb{C}^n$ represents the spectral parameters. Secondly, we use iso-spectral deformation theory to study f-KT in the Hessenberg form, and give explicit form of the wave functions and entries in the Lax matrix expressed by $\tau$-functions with which we study $\ell$-banded Kostant-Toda hierarchy. We also explicit construct some semi-invariants for f-KT equation. Thirdly, we use representation theory to construct rational solutions of f-KT in type $A$ and type $B$, and we also give explicit form of soliton solutions of type $B$ in rank $2$ and $3$. The Kowalevski-Painlev\'e analysis is new, and part of the other parts is based on a joint work with Yuji Kodama which is under preparation \cite{Kodama-Xie2021f-KT}.
Yuji Kodama (Advisor)
David Anderson (Committee Member)
Herb Clemens (Committee Member)
James Cogdell (Committee Member)
168 p.
Mathematics
space curve, soliton solution, KP hierarchy, Sato Grassmannian, numerical semigroup, flag variety, Painleve analysis, full Kostant-Toda, isospectral deformation, tau-function, polynomial solutions, B-Toda

Recommended Citations

Citations

  • Xie, Y. (2021). Algebraic Curves and Flag Varieties in Solutions of the KP Hierarchy and the Full Kostant-Toda Hierarchy [Doctoral dissertation, Ohio State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=osu1626565722059012

    APA Style (7th edition)

  • Xie, Yuancheng. Algebraic Curves and Flag Varieties in Solutions of the KP Hierarchy and the Full Kostant-Toda Hierarchy. 2021. Ohio State University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=osu1626565722059012.

    MLA Style (8th edition)

  • Xie, Yuancheng. "Algebraic Curves and Flag Varieties in Solutions of the KP Hierarchy and the Full Kostant-Toda Hierarchy." Doctoral dissertation, Ohio State University, 2021. http://rave.ohiolink.edu/etdc/view?acc_num=osu1626565722059012

    Chicago Manual of Style (17th edition)

osu1626565722059012
195
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