Heat Equations and Families of Two-Dimensional Sigma Functions

E. Yu. Bunkova ; V. M. Buchstaber

Informatics and Automation, Geometry, topology, and mathematical physics. II, Tome 266 (2009), pp. 5-32.

Résumé

In the framework of S. P. Novikov's program for boosting the effectiveness of theta-function formulas of finite-gap integration theory, a system of differential equations for the parameters of the sigma function in genus 2 is constructed. A counterpart of this system in genus 1 is equivalent to the Chazy equation. On the basis of the obtained results, a two-dimensional analog of the Frobenius–Stickelberger connection is defined and calculated.

Bibliographie

[1] Athorne C., Eilbeck J. C., Enolskii V. Z., “A $\mathrm{SL}(2)$ covariant theory of genus 2 hyperelliptic functions”, Math. Proc. Cambridge Philos. Soc., 136:2 (2004), 269–286 | DOI | MR | Zbl

[2] Athorne C., “Identities for hyperelliptic $\wp$-functions of genus one, two and three in covariant form”, J. Phys. A Math. and Theor., 41:41 (2008), Pap. 415202 | DOI | MR

[3] Baker H. F., Abelian functions: Abel's theorem and the allied theory of theta functions, Cambridge Univ. Press, Cambridge, 1995 | MR | Zbl

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

[5] Baker H. F., An introduction to the theory of multiply periodic functions, Cambridge Univ. Press, Cambridge, 1907 | Zbl

[6] Braden H. W., Enolskii V. Z., Hone A. N. W., “Bilinear recurrences and addition formulae for hyperelliptic sigma functions”, J. Nonlin. Math. Phys., 12, Suppl. 2 (2005), 46–62 ; arXiv: math.NT/0501162 | DOI | MR | Zbl

[7] Bukhshtaber V. M., Enolskii V. Z., “Abelevy blokhovskie resheniya dvumernogo uravneniya Shrëdingera”, UMN, 50:1 (1995), 191–192 | MR | Zbl

[8] Buchstaber V. M., Enolskiĭ V. Z., Leĭkin D. V., “Hyperelliptic Kleinian functions and applications”, Solitons, geometry, and topology: On the crossroad, AMS Transl. Ser. 2, 179, eds. V. M. Buchstaber, S. P. Novikov, Amer. Math. Soc., Providence (RI), 1997, 1–33 | MR | Zbl

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

[10] Bukhshtaber V. M., Leikin D. V., Enolskii V. Z., “Ratsionalnye analogi abelevykh funktsii”, Funkts. analiz i ego pril., 33:2 (1999), 1–15 | DOI | MR | Zbl

[11] Bukhshtaber V. M., Leikin D. V., Enolskii V. Z., “$\sigma$-Funktsii $(n,s)$-krivykh”, UMN, 54:3 (1999), 155–156 | DOI | MR | Zbl

[12] Bukhshtaber V. M., Leikin D. V., Enolskii V. Z., “Uniformizatsiya mnogoobrazii Yakobi trigonalnykh krivykh i nelineinye differentsialnye uravneniya”, Funkts. analiz i ego pril., 34:3 (2000), 1–16 | DOI | MR | Zbl

[13] Bukhshtaber V. M., Leikin D. V., “Mnogoobrazie reshenii uravneniya $\bigl[\frac{\partial^2}{\partial u_1\partial u_2}-6\wp_{1,2}(u_1,u_2)+E\bigr]\psi=0$”, UMN, 56:6 (2001), 141–142 | DOI | MR | Zbl

[14] Buchstaber V. M., Eilbeck J. C., Enolskii V. Z., Leykin D. V., Salerno M., “Multidimensional Schrödinger equations with Abelian potentials”, J. Math. Phys., 43:6 (2002), 2858–2881 | DOI | MR | Zbl

[15] Bukhshtaber V. M., Leikin D. V., “Polinomialnye algebry Li”, Funkts. analiz i ego pril., 36:4 (2002), 18–34 | DOI | MR | Zbl

[16] Bukhshtaber V. M., Leikin D. V., Pavlov M. V., “Egorovskie gidrodinamicheskie tsepochki, uravnenie Shazi i gruppa $SL(2,\mathbb C)$”, Funkts. analiz i ego pril., 37:4 (2003), 13–26 | DOI | MR | Zbl

[17] Buchstaber V. M., Shorina S. Yu., “$w$-Function of the KdV hierarchy”, Geometry, topology, and mathematical physics, S. P. Novikov's seminar 2002–2003, AMS Transl. Ser. 2, 212, eds. V. M. Buchstaber, I. M. Krichever, Amer. Math. Soc., Providence (RI), 2004, 41–46 | MR

[18] Bukhshtaber V. M., Leikin D. V., “Uravneniya teploprovodnosti v negolonomnom repere”, Funkts. analiz i ego pril., 38:2 (2004), 12–27 | DOI | MR | Zbl

[19] Bukhshtaber V. M., Leikin D. V., “Zakony slozheniya na yakobianakh ploskikh algebraicheskikh krivykh”, Tr. MIAN, 251, 2005, 54–126 | MR | Zbl

[20] Buchstaber V. M., “Abelian functions and singularity theory”, Analysis and singularities, Abstr. Intern. Conf. dedicated to the 70th anniversary of V. I. Arnold, Steklov Math. Inst., Moscow, 2007, 117–118; http://arnold-70.mi.ras.ru/video.html

[21] Bukhshtaber V. M., Leikin D. V., “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

[22] Clarkson P. A., Olver P. J., “Symmetry and the Chazy equation”, J. Diff. Eqns., 124 (1996), 225–246 | DOI | MR | Zbl

[23] Dubpovin B. A., Novikov S. P., “Periodicheskaya zadacha dlya uravneniya Kortevega–de Friza i Shturma–Liuvillya. Ikh svyaz s algebraicheskoi geometriei”, DAH SSSR, 219:3 (1974), 531–534

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

[25] Dubpovin B. A., “Teta-funktsii i nelineinye uravneniya”, UMN, 36:2 (1981), 11–80 | MR | Zbl

[26] Dubpovin B. A., Krichever I. M., Novikov S. P., “Integriruemye sistemy. I”, Dinamicheskie sistemy – 4, Itogi nauki i tekhniki. Sovr. probl. matematiki. Fund. napr., 4, VINITI, M., 1985, 179–285 | MR | Zbl

[27] Dubrovin B. A., “Geometry of 2D topological field theories”, Integrable systems and quantum groups, Lect. Notes Math., 1620, Springer, Berlin, 1996, 120–348 | DOI | MR | Zbl

[28] Eilbeck J. C., Enolskii V. Z., Leykin D. V., “On the Kleinian construction of Abelian functions of canonical algebraic curves”, SIDE III: Symmetries and integrability of difference equations, Proc. Conf. (Sabaudia, Italy, May 1998), CRM Proc. and Lect. Notes, 25, Amer. Math. Soc., Providence (RI), 2000, 121–138 | MR | Zbl

[29] Eilbeck J. C., Enolskii V. Z., Previato E., “On a generalized Frobenius–Stickelberger addition formula”, Lett. Math. Phys., 63 (2003), 5–17 | DOI | MR | Zbl

[30] Eilbeck J. C., Enolski V. Z., Matsutani S., Ônishi Y., Previato E., “Abelian functions for trigonal curves of genus three”, Intern. Math. Res. Not., 2008, no. 1, Article ID rnm 140 ; arXiv: math.AG/0610019 | MR | Zbl

[31] Enolskii V., Matsutani S., Ônishi Y., “The addition law attached to a stratification of a hyperelliptic Jacobian variety”, Tokyo J. Math., 31:1 (2008), 27–38 ; arXiv: math.AG/0508366 | DOI | MR | Zbl

[32] Its A. R., Matveev V. B., “Operatory Shrëdingera s konechnozonnym spektrom i $N$-solitonnye resheniya uravneniya Kortevega–de Frisa”, TMF, 23:1 (1975), 51–68 | MR

[33] Krichever I. M., “Integrirovanie nelineinykh uravnenii metodami algebraicheskoi geometrii”, Funkts. analiz i ego pril., 11:1 (1977), 15–31 | MR | Zbl

[34] Nakayashiki A., On algebraic expressions of sigma functions for $(n,s)$ curves, E-print , 2008 arXiv: 0803.2083

[35] Nakayashiki A., Sigma function as a tau function, E-print , 2009 arXiv: 0904.0846 | MR | Zbl

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

[37] Ônishi Y., “Determinant expressions for hyperelliptic functions (with an Appendix by S. Matsutani)”, Proc. Edinburgh Math. Soc., 48:3 (2005), 705–742 ; arXiv: math.NT/0105189 | DOI | MR | Zbl