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 Funktsional. Anal. i Prilozhen., 2016, Volume 50, Issue 4, Pages 76–90 (Mi faa3251)

Tangential Polynomials and Matrix KdV Elliptic Solitons

A. Treibichab

a Université d'Artois, France
b Universidad de la República, Uruguaj

Abstract: Let $(X,q)$ be an elliptic curve marked at the origin. Starting from any cover $\pi\colon\Gamma\to X$ of an elliptic curve $X$ marked at $d$ points $\{\pi_i\}$ of the fiber $\pi^{-1}(q)$ and satisfying a particular criterion, Krichever constructed a family of $d\times d$ matrix KP solitons, that is, matrix solutions, doubly periodic in $x$, of the KP equation. Moreover, if $\Gamma$ has a meromorphic function $f\colon\Gamma\to\mathbb{P}^1$ with a double pole at each $p_i$, then these solutions are doubly periodic solutions of the matrix KdV equation $U_t=\frac14(3UU_x+3U_xU+U_{xxx})$. In this article, we restrict ourselves to the case in which there exists a meromorphic function with a unique double pole at each of the $d$ points $\{p_i\}$; i.e. $\Gamma$ is hyperelliptic and each $p_i$ is a Weierstrass point of $\Gamma$. More precisely, our purpose is threefold: (1) present simple polynomial equations defining spectral curves of matrix KP elliptic solitons; (2) construct the corresponding polynomials via the vector Baker–Akhiezer function of $X$; (3) find arbitrarily high genus spectral curves of matrix KdV elliptic solitons.

Keywords: KP equation, KdV equation, compact Riemann surface, vector Baker–Akhiezer function, ruled surface

DOI: https://doi.org/10.4213/faa3251

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English version:
Functional Analysis and Its Applications, 2016, 50:4, 308–318

Bibliographic databases:

UDC: 517.9

Citation: A. Treibich, “Tangential Polynomials and Matrix KdV Elliptic Solitons”, Funktsional. Anal. i Prilozhen., 50:4 (2016), 76–90; Funct. Anal. Appl., 50:4 (2016), 308–318

Citation in format AMSBIB
\Bibitem{Tre16} \by A.~Treibich \paper Tangential Polynomials and Matrix KdV Elliptic Solitons \jour Funktsional. Anal. i Prilozhen. \yr 2016 \vol 50 \issue 4 \pages 76--90 \mathnet{http://mi.mathnet.ru/faa3251} \crossref{https://doi.org/10.4213/faa3251} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3646711} \elib{http://elibrary.ru/item.asp?id=28119106} \transl \jour Funct. Anal. Appl. \yr 2016 \vol 50 \issue 4 \pages 308--318 \crossref{https://doi.org/10.1007/s10688-016-0161-0} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000390093200006} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85006372322} 

• http://mi.mathnet.ru/eng/faa3251
• https://doi.org/10.4213/faa3251
• http://mi.mathnet.ru/eng/faa/v50/i4/p76

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This publication is cited in the following articles:
1. Treibich A., “Tangential Covers and Polynomials Over Higher Genus Curves”, Int. Math. Res. Notices, 2019, no. 9, 2894–2918
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