Abstract
In this paper, a parametric Korteweg–de Vries hierarchy is defined that depends on an infinite set of graded parameters \(a = (a_4,a_6,\dots)\). It is shown that, for any genus \(g\), the Klein hyperelliptic function \(\wp_{1,1}(t,\lambda)\) defined on the basis of the multidimensional sigma function \(\sigma(t, \lambda)\), where \(t = (t_1, t_3,\dots, t_{2g-1})\) and \(\lambda = (\lambda_4, \lambda_6,\dots, \lambda_{4 g + 2})\), specifies a solution to this hierarchy in which the parameters \(a\) are given as polynomials in the parameters \(\lambda\) of the sigma function. The proof uses results concerning the family of operators introduced by V. M. Buchstaber and S. Yu. Shorina. This family consists of \(g\) third-order differential operators in \(g\) variables. Such families are defined for all \(g \geqslant 1\), the operators in each of them pairwise commute with each other and also commute with the Schrödinger operator. In this paper a relationship between these families and the Korteweg–de Vries parametric hierarchy is described. A similar infinite family of third-order operators on an infinite set of variables is constructed. The results obtained are extended to the case of such a family.
Similar content being viewed by others
References
D. J. Korteweg and G. de Vries, “On the change of form of long waves advancing in a rectangular canal and on a new type of long stationary waves”, Philos. Mag., 39:5 (1895), 422–443.
P. D. Lax, “Integrals of nonlinear equations of evolution and solitary waves”, Comm. Pure Appl. Math., 21 (1968), 467–490.
V. M. Buchstaber and A. V. Mikhailov, “Integrable polynomial Hamiltonian systems and symmetric powers of plane algebraic curves”, Russian Math. Surveys, 76:4 (2021), 587–652.
S. P. Novikov, “The periodic problem for the Korteweg–de Vries equation.”, Funct. Anal. Appl., 8:3 (1974), 54–66.
I. M. Gelfand and L. A. Dikii, “Asymptotic behaviour of the resolvent of Sturm–Liouville equations and the algebra of Korteweg–de Vries equations”, Russian Math. Surveys, 30:5 (1975).
B. A. Dubrovin, V. B. Matveev, and S. P. Novikov, “Nonlinear equations of Korteweg–de Vries type, finite-zone linear operators, and Abelian varieties”, Russian Math. Surveys, 31:1 (1976).
V. V. Sokolov, Algebraic Structures in Integrability, World Scientific, Hackensack, NJ, 2020.
V. M. Buchstaber and S. Yu. Shorina, “The \(w\)-function of the KdV hierarchy. Geometry, topology, and mathematical physics”, Amer. Math. Soc. Transl. Ser. 2, no. 212, Amer. Math. Soc., Providence, RI, 2004, 41–66.
V. M. Buchstaber and S. Yu. Shorina, “Commuting multidimensional third-order differential operators defining a KdV hierarchy”, Russian Math. Surveys, 58:3 (2003), 610–612.
V. M. Buchstaber and S. Yu. Shorina, “The \(w\)-function of a solution of the \(g\)-th stationary KdV equation”, Russian Math. Surveys, 58:4 (2003), 780–781.
H. F. Baker, “On the hyperelliptic sigma functions”, Amer. J. Math., 20:4 (1898), 301–384.
V. M. Buchstaber, V. Z. Enolskii, and D. V. Leikin, “Kleinian functions, hyperelliptic Jacobians and applications”, Rev. Math. and Math. Physics, 10:2 (1997), 3–120.
V. M. Buchstaber, V. Z. Enolskii, and D. V. Leikin, “Hyperelliptic Kleinian functions and applications”, Amer. Math. Soc. Transl. Ser. 2, no. 179, Amer. Math. Soc., Providence, RI, 1997, 1–34.
V. M. Buchstaber, V. Z. Enolskii, and D. V. Leikin, Multi-Dimensional Sigma-Functions, arXiv: 1208.0990.
E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, Cambridge Univ. Press, Cambridge, 1996.
V. M. Buchstaber, “Polynomial dynamical systems and the Korteweg–de Vries equation”, Modern problems of mathematics, mechanics, and mathematical physics. II, Collected papers, no. 294, Proc. Steklov Inst. Math, 2016, 176–200.
E. Yu. Bunkova, “Differentiation of genus 3 hyperelliptic functions”, Eur. J. Math., 4:1 (2018), 93–112.
A. Nakayashiki, “On algebraic expressions of sigma functions for \((n,s)\)-curves”, Asian J. Math., 14:2 (2010), 175–212; arXiv: 0803.2083.
B. A. Dubrovin and S. P. Novikov, “The periodic problem for the Korteweg–de Vries and Sturm–Liouville equations. Their connection with algebraic geometry.”, Dokl. Akad. Nauk SSSR, 219:3 (1974), 531–534.
V. M. Buchstaber and D. V. Leikin, “Solution of the Problem of Differentiation of Abelian Functions over Parameters for Families of \((n,s)\)-curves”, Funct. Anal. Appl., 42:4 (2008), 262–278.
V. M. Buchstaber, V. Z. Enolski, and D. V. Leykin, “\(\sigma\)-Functions: old and new results”, Integrable Systems and Algebraic Geometry, Vol. 2, London Math. Soc. Lecture Note Series, 459 Cambridge Univ. Press, Cambridge, 2020, 175–214.
Funding
This research was supported by the Russian Science Foundation (grant no. 20-11-19998), https://rscf.ru/project/20-11-19998/.
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Funktsional'nyi Analiz i ego Prilozheniya, 2022, Vol. 56, pp. 16–38 https://doi.org/10.4213/faa4020.
The article is dedicated to the memory of the remarkable mathematician Emma Previato (1952–2022)
Translated by S. Konstantinou-Rizos
Rights and permissions
About this article
Cite this article
Bunkova, E.Y., Bukhshtaber, V.M. Parametric Korteweg–de Vries Hierarchy and Hyperelliptic Sigma Functions. Funct Anal Its Appl 56, 169–187 (2022). https://doi.org/10.1134/S0016266322030029
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0016266322030029