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Funktsional. Anal. i Prilozhen., 2000, Volume 34, Issue 3, Pages 1–16 (Mi faa307)  

This article is cited in 41 scientific papers (total in 43 papers)

Uniformization of Jacobi Varieties of Trigonal Curves and Nonlinear Differential Equations

V. M. Buchstabera, D. V. Leikinb, V. Z. Ènol'skiib

a M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics
b Institute of Magnetism, National Academy of Sciences of Ukraine

Abstract: We obtain an explicit realization of the Jacobi and Kummer varieties for trigonal curves of genus $g$ ($\gcd(g,3)=1$) of the form
$$ y^3=x^{g+1}+\sum_{\alpha,\beta}\lambda_{3\alpha +(g+1)\beta}x^{\alpha}y^{\beta},\qquad 0\le3\alpha+(g+1)\beta <3g+3, $$
as algebraic subvarieties in $\mathbb{C}^{4g+\delta}$, where $\delta=2(g-3[g/3])$, and in $\mathbb{C}^{g(g+1)/2}$. We uniformize these varieties with the help of $\wp$-functions of several variables defined on the universal space of Jacobians of such curves. By way of application, we obtain a system of nonlinear partial differential equations integrable in trigonal $\wp$-functions. This system in particular contains the oussinesq equation.

DOI: https://doi.org/10.4213/faa307

Full text: PDF file (368 kB)
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English version:
Functional Analysis and Its Applications, 2000, 34:3, 159–171

Bibliographic databases:

UDC: 512.742+517.957
Received: 22.05.2000

Citation: V. M. Buchstaber, D. V. Leikin, V. Z. Ènol'skii, “Uniformization of Jacobi Varieties of Trigonal Curves and Nonlinear Differential Equations”, Funktsional. Anal. i Prilozhen., 34:3 (2000), 1–16; Funct. Anal. Appl., 34:3 (2000), 159–171

Citation in format AMSBIB
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\paper Uniformization of Jacobi Varieties of Trigonal Curves and Nonlinear Differential Equations
\jour Funktsional. Anal. i Prilozhen.
\yr 2000
\vol 34
\issue 3
\pages 1--16
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\jour Funct. Anal. Appl.
\yr 2000
\vol 34
\issue 3
\pages 159--171
\crossref{https://doi.org/10.1007/BF02482405}
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    2. Matsutani, S, “Hyperelliptic solutions of KdV and KP equations: re-evaluation of Baker's study on hyperelliptic sigma functions”, Journal of Physics A-Mathematical and General, 34:22 (2001), 4721  crossref  mathscinet  zmath  adsnasa  isi  scopus
    3. Eilbeck, JC, “Varieties of elliptic solitons”, Journal of Physics A-Mathematical and General, 34:11 (2001), 2215  crossref  mathscinet  zmath  adsnasa  isi  scopus
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    7. Eilbeck, JC, “The hyperelliptic zeta-function and the integrable massive Thirring model”, Proceedings of the Royal Society of London Series A-Mathematical Physical and Engineering Sciences, 459:2035 (2003), 1581  crossref  mathscinet  zmath  adsnasa  isi  scopus
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    9. Baldwin, S, “Genus 4 trigonal reduction of the Benney equations”, Journal of Physics A-Mathematical and General, 39:14 (2006), 3607  crossref  mathscinet  zmath  adsnasa  isi  scopus
    10. Eilbeck, JC, “Abelian Functions for Trigonal Curves of Genus Three”, International Mathematics Research Notices, 2007, rnm140  crossref  mathscinet  isi  elib  scopus
    11. Baldwin, S, “Abelian functions for cyclic trigonal curves of genus 4”, Journal of Geometry and Physics, 58:4 (2008), 450  crossref  mathscinet  zmath  adsnasa  isi  scopus
    12. Cho, K, “Differential structure of Abelian functions”, International Journal of Mathematics, 19:2 (2008), 145  crossref  mathscinet  zmath  isi  scopus
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    22. Eilbeck J.C., Enolski V.Z., Gibbons J., “Sigma, tau and Abelian functions of algebraic curves”, Journal of Physics A-Mathematical and Theoretical, 43:45 (2010), 455216  crossref  mathscinet  zmath  adsnasa  isi  scopus
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    24. England M., “Deriving Bases for Abelian Functions”, Comput. Methods Funct. Theory, 11:2 (2011), 617–654  crossref  mathscinet  zmath  isi  elib
    25. Eilbeck J.C. England M. Onishi Y., “Abelian Functions Associated with Genus Three Algebraic Curves”, LMS J. Comput. Math., 14 (2011), 291–326  crossref  mathscinet  zmath  isi
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  • Функциональный анализ и его приложения Functional Analysis and Its Applications
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