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

This article is cited in 39 scientific papers (total in 41 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:

Document Type: Article
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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\by V.~M.~Buchstaber, D.~V.~Leikin, V.~Z.~\`Enol'skii
\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. 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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    8. 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
    9. Eilbeck, JC, “Abelian Functions for Trigonal Curves of Genus Three”, International Mathematics Research Notices, 2007, rnm140  crossref  mathscinet  isi  elib  scopus
    10. 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
    11. Cho, K, “Differential structure of Abelian functions”, International Journal of Mathematics, 19:2 (2008), 145  crossref  mathscinet  zmath  isi  scopus
    12. Matsutani, S, “Jacobi inversion on strata of the Jacobian of the C-rs curve y(r) = f(x)”, Journal of the Mathematical Society of Japan, 60:4 (2008), 1009  crossref  mathscinet  zmath  isi  scopus
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    16. Feng, Y, “Hyperelliptic functions solutions of some nonlinear partial differential equations using the direct method”, Applied Mathematics and Computation, 215:11 (2010), 3868  crossref  mathscinet  zmath  isi  scopus
    17. Matthew England, “Higher Genus Abelian Functions Associated with Cyclic Trigonal Curves”, SIGMA, 6 (2010), 025, 22 pp.  mathnet  crossref  mathscinet
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    19. Yu. V. Brezhnev, “Transcendental trace formulas for finite-gap potentials”, Theoret. and Math. Phys., 164:1 (2010), 920–928  mathnet  crossref  crossref  adsnasa  isi  elib
    20. Nakayashiki A., “On algebraic expressions of sigma functions for $(n, s)$ curves”, Asian J Math, 14:2 (2010), 175–211  crossref  mathscinet  isi  scopus
    21. 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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    23. England M., “Deriving Bases for Abelian Functions”, Comput. Methods Funct. Theory, 11:2 (2011), 617–654  crossref  mathscinet  zmath  isi  elib
    24. 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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