Parametric Korteweg--de Vries hierarchy and hyperelliptic sigma functions

E. Yu. Bunkova ; V. M. Bukhshtaber

Funkcionalʹnyj analiz i ego priloženiâ, Tome 56 (2022) no. 3, pp. 16-38.

Résumé

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.

Détail

Keywords: canonical Korteweg–de Vries hierarchy, parametric Korteweg–de Vries hierarchy, hyperelliptic functions, multidimensional sigma function, Buchstaber–Shorina operators, Buchstaber–\break Shorina polynomial differential operators, polynomial parametric Korteweg–de Vries hierarchy.

Bibliographie

[1] D. J. Korteweg, 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 | DOI | MR

[2] P. D. Lax, “Integrals of nonlinear equations of evolution and solitary waves”, Comm. Pure Appl. Math., 21 (1968), 467–490 | DOI | MR | Zbl

[3] V. M. Bukhshtaber, A. V. Mikhailov, “Integriruemye polinomialnye gamiltonovy sistemy i simmetricheskie stepeni ploskikh algebraicheskikh krivykh”, UMN, 76:4(460) (2021), 37–104 | DOI | MR | Zbl

[4] S. P. Novikov, “Periodicheskaya zadacha dlya uravneniya Kortevega–de Friza. I”, Funkts. analiz i ego pril., 8:3 (1974), 54–66 | MR | Zbl

[5] I. M. Gelfand, L. A. Dikii, “Asimptotika rezolventy shturm–liuvillevskikh uravnenii i algebra uravnenii Kortevega–de Friza”, UMN, 30:5(185) (1975), 67–100 | MR | Zbl

[6] B. A. Dubrovin, V. B. Matveev, S. P. Novikov, “Nelineinye uravneniya tipa Kortevega–de Friza, konechnozonnye lineinye operatory i abelevy mnogoobraziya”, UMN, 31:1(187) (1976), 55–136 | MR | Zbl

[7] V. V. Sokolov, Algebraic Structures in Integrability, World Scientific, Hackensack, NJ, 2020 | MR | Zbl

[8] V. M. Buchstaber, 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 | MR | Zbl

[9] V. M. Bukhshtaber, S. Yu. Shorina, “Kommutiruyuschie differentsialnye mnogomernye operatory tretego poryadka, zadayuschie KdF-ierarkhiyu”, UMN, 58:3(351) (2003), 187–188 | DOI | MR | Zbl

[10] V. M. Bukhshtaber, S. Yu. Shorina, “$w$-funktsiya resheniya $g$-go statsionarnogo uravneniya KdF”, UMN, 58:4(352) (2003), 145–146 | DOI | MR | Zbl

[11] H. F. Baker, “On the hyperelliptic sigma functions”, Amer. J. Math., 20:4 (1898), 301–384 | DOI | MR | Zbl

[12] V. M. Buchstaber, V. Z. Enolskii, D. V. Leikin, “Kleinian functions, hyperelliptic Jacobians and applications”, Rev. Math. and Math. Physics, 10:2 (1997), 3–120 | MR | Zbl

[13] V. M. Buchstaber, V. Z. Enolskii, D. V. Leikin, “Hyperelliptic Kleinian functions and applications”, Amer. Math. Soc. Transl. Ser. 2, 179, no. 2, Amer. Math. Soc., Providence, RI, 1997, 1–34 | MR

[14] V. M. Buchstaber, V. Z. Enolskii, D. V. Leikin, Multi-Dimensional Sigma-Functions, arXiv: 1208.0990 | MR

[15] E. T. Whittaker, G. N. Watson, A Course of Modern Analysis, Reprint of 4th (1927) ed., v. 2, Transcendental functions, Cambridge Univ. Press, Cambridge, 1996 | MR | Zbl

[16] V. M. Bukhshtaber, “Polinomialnye dinamicheskie sistemy i uravnenie Kortevega–de Friza”, Sovremennye problemy matematiki, mekhaniki i matematicheskoi fiziki. II, Sbornik statei, Tr. MIAN, 294, MAIK, M., 2016, 191–215 | DOI

[17] E. Yu. Bunkova, “Differentiation of genus 3 hyperelliptic functions”, Eur. J. Math., 4:1 (2018), 93–112 | DOI | MR | Zbl

[18] A. Nakayashiki, “On algebraic expressions of sigma functions for $(n,s)$-curves”, Asian J. Math., 14:2 (2010), 175–212 ; arXiv: 0803.2083 | DOI | MR | Zbl

[19] B. A. Dubrovin, S. P. Novikov, “Periodicheskaya zadacha dlya uravnenii Kortevega–de Friza i Shturma–Liuvillya. Ikh svyaz s algebraicheskoi geometriei”, Dokl. AN SSSR, 219:3 (1974), 531–534 | MR | Zbl

[20] V. M. Bukhshtaber, D. V. Leikin, “Reshenie zadachi differentsirovaniya abelevykh funktsii po parametram dlya semeistv $(n,s)$-krivykh”, Funkts. analiz i ego pril., 42:4 (2008), 24–36 | DOI | MR | Zbl

[21] V. M. Buchstaber, V. Z. Enolski, D. V. Leykin, “$\sigma$-functions: old and new results”, Integrable Systems and Algebraic Geometry bookvol 2, London Math. Soc. Lecture Note Series, 459, Cambridge Univ. Press, Cambridge, 2020, 175–214 | MR | Zbl