Bunkova, E. Yu.; Buchstaber, V. M. Heat equations and families of two-dimensional sigma functions. (English. Russian original) Zbl 1185.37148 Proc. Steklov Inst. Math. 266, 1-28 (2009); translation from Tr. Mat. Inst. Steklova 266, 5-32 (2009). The paper is devoted to constructing a system of differential equations that describes a relationship between the parameters of two-dimensional theta and sigma functions. The authors used the classical system of heat equations, which holds for the theta function, and the system of heat equations in a nonholonomic frame which holds the sigma function. The approach is based on the fact that, under the additional requirement of quasiperiodicity of solutions, these systems characterize theta and sigma functions, respectively. Reviewer: Rakib Efendiev (Baku) Cited in 2 Documents MSC: 37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) 14H70 Relationships between algebraic curves and integrable systems Keywords:Chazy equation; elliptic sigma function; families of two-dimensional sigma functions PDFBibTeX XMLCite \textit{E. Yu. Bunkova} and \textit{V. M. Buchstaber}, Proc. Steklov Inst. Math. 266, 1--28 (2009; Zbl 1185.37148); translation from Tr. Mat. Inst. Steklova 266, 5--32 (2009) Full Text: DOI References: [1] C. Athorne, J. C. Eilbeck, and V. Z. Enolskii, ”A SL(2) Covariant Theory of Genus 2 Hyperelliptic Functions,” Math. Proc. Cambridge Philos. Soc. 136(2), 269–286 (2004). · Zbl 1063.33001 [2] C. Athorne, ”Identities for Hyperelliptic -Functions of Genus One, Two and Three in Covariant Form,” J. Phys. A: Math. Theor. 41(41), 415202 (2008). · Zbl 1149.14027 [3] H. F. Baker, Abelian Functions: Abel’s Theorem and the Allied Theory of Theta Functions (Cambridge Univ. Press, Cambridge, 1995; MTsNMO, Moscow, 2008). · Zbl 0848.14012 [4] H. F. Baker, ”On the Hyperelliptic Sigma Functions,” Am. J. Math. 20, 301–384 (1898). · JFM 29.0394.03 [5] H. F. Baker, An Introduction to the Theory of Multiply Periodic Functions (Cambridge Univ. Press, Cambridge, 1907). · JFM 38.0478.05 [6] H. W. Braden, V. Z. Enolskii, and A. N. W. Hone, ”Bilinear Recurrences and Addition Formulae for Hyperelliptic Sigma Functions,” J. Nonlinear Math. Phys. 12(Suppl. 2), 46–62 (2005); arXiv:math.NT/0501162. · Zbl 1126.11007 [7] V. M. Bukhshtaber and V. Z. Ènol’skii, ”Abelian Bloch Solutions of the Two-Dimensional Schrödinger Equation,” Usp. Mat. Nauk 50(1), 191–192 (1995) [Russ. Math. Surv. 50, 195–197 (1995)]. [8] V. M. Buchstaber, V. Z. Enolskiĭ, and D. V. Leĭkin, ”Hyperelliptic Kleinian Functions and Applications,” in Solitons, Geometry, and Topology: On the Crossroad, Ed. by V. M. Buchstaber and S. P. Novikov (Am. Math. Soc., Providence, RI, 1997), AMS Transl., Ser. 2, 179, pp. 1–33. [9] V. M. Buchstaber, V. Z. Enolskii, and D. V. Leykin, ”Kleinian Functions, Hyperelliptic Jacobians and Applications,” Rev. Math. Math. Phys. 10(2), 3–120 (1997). · Zbl 0911.14019 [10] V. M. Buchstaber, D. V. Leykin, and V. Z. Enolskii, ”Rational Analogs of Abelian Functions,” Funkts. Anal. Prilozh. 33(2), 1–15 (1999) [Funct. Anal. Appl. 33, 83–94 (1999)]. [11] V. M. Bukhshtaber, D. V. Leikin, and V. Z. Ènolskii, ”{\(\sigma\)}-Functions of (n, s)-Curves,” Usp. Mat. Nauk 54(3), 155–156 (1999) [Russ. Math. Surv. 54, 628–629 (1999)]. [12] V. M. Buchstaber, D. V. Leykin, and V. Z. Enolskii, ”Uniformization of Jacobi Varieties of Trigonal Curves and Nonlinear Differential Equations,” Funkts. Anal. Prilozh. 34(3), 1–16 (2000) [Funct. Anal. Appl. 34, 159–171 (2000)]. [13] V. M. Buchstaber and D. V. Leikin, ”The Manifold of Solutions of the Equation [ [14] V. M. Buchstaber, J. C. Eilbeck, V. Z. Enolskii, D. V. Leykin, and M. Salerno, ”Multidimensional Schrödinger Equations with Abelian Potentials,” J. Math. Phys. 43(6), 2858–2881 (2002). · Zbl 1059.81036 [15] V. M. Buchstaber and D. V. Leykin, ”Polynomial Lie Algebras,” Funkts. Anal. Prilozh. 36(4), 18–34 (2002) [Funct. Anal. Appl. 36, 267–280 (2002)]. [16] V. M. Buchstaber, D. V. Leykin, and M. V. Pavlov, ”Egorov Hydrodynamic Chains, the Chazy Equation and SL(2, \(\mathbb{C}\)),” Funkts. Anal. Prilozh. 37(4), 13–26 (2003) [Funct. Anal. Appl. 37, 251–262 (2003)]. [17] V. M. Buchstaber and S. Yu. Shorina, ”w-Function of the KdV Hierarchy,” in Geometry, Topology, and Mathematical Physics: S.P. Novikov’s Seminar 2002–2003, Ed. by V. M. Buchstaber and I. M. Krichever (Am. Math. Soc., Providence, RI, 2004), AMS Transl., Ser. 2, 212, pp. 41–46. · Zbl 1075.37024 [18] V. M. Buchstaber and D. V. Leykin, ”Heat Equations in a Nonholonomic Frame,” Funkts. Anal. Prilozh. 38(2), 12–27 (2004) [Funct. Anal. Appl. 38, 88–101 (2004)]. [19] V. M. Buchstaber and D. V. Leykin, ”Addition Laws on Jacobian Varieties of Plane Algebraic Curves,” Tr. Mat. Inst. im. V.A. Steklova, Ross. Akad. Nauk 251, 54–126 (2005) [Proc. Steklov Inst. Math. 251, 49–120 (2005)]. [20] V. M. Buchstaber, ”Abelian Functions and Singularity Theory,” in Analysis and Singularities: Abstr. Int. Conf. Dedicated to the 70th Anniversary of V.I. Arnold (Steklov Math. Inst., Moscow, 2007), pp. 117–118; see also http://arnold-70.mi.ras.ru/video_e.html [21] V. M. Buchstaber and D. V. Leykin, ”Solution of the Problem of Differentiation of Abelian Functions over Parameters for Families of (n, s)-Curves,” Funkts. Anal. Prilozh. 42(4), 24–36 (2008) [Funct. Anal. Appl. 42, 268–278 (2008)]. [22] P. A. Clarkson and P. J. Olver, ”Symmetry and the Chazy Equation,” J. Diff. Eqns. 124, 225–246 (1996). · Zbl 0842.34010 [23] B. A. Dubrovin and S. P. Novikov, ”A Periodicity Problem for the Korteweg-de Vries and Sturm-Liouville Equations. Their Connection with Algebraic Geometry,” Dokl. Akad. Nauk SSSR 219(3), 531–534 (1974) [Sov. Math., Dokl. 15, 1597–1601 (1974)]. · Zbl 0312.35015 [24] B. A. Dubrovin, V. B. Matveev, and S. P. Novikov, ”Non-linear Equations of Korteweg-de Vries Type, Finite-Zone Linear Operators, and Abelian Varieties,” Usp. Mat. Nauk 31(1), 55–136 (1976) [Russ. Math. Surv. 31 (1), 59–146 (1976)]. · Zbl 0326.35011 [25] B. A. Dubrovin, ”Theta Functions and Non-linear Equations,” Usp. Mat. Nauk 36(2), 11–80 (1981) [Russ. Math. Surv. 36 (2), 11–92 (1981)]. · Zbl 0478.58038 [26] B. A. Dubrovin, I. M. Krichever, and S. P. Novikov, ”Integrable Systems. I,” in Dynamical Systems-4 (VINITI, Moscow, 1985), Itogi Nauki Tekh., Ser.: Sovrem. Probl. Mat., Fundam. Napravl. 4, pp. 179–285; Engl. transl. in Dynamical Systems IV (Springer, Berlin, 1990), Encycl. Math. Sci. 4, pp. 173–280. [27] B. A. Dubrovin, ”Geometry of 2D Topological Field Theories,” in Integrable Systems and Quantum Groups (Springer, Berlin, 1996), Lect. Notes Math. 1620, pp. 120–348. · Zbl 0841.58065 [28] J. C. Eilbeck, V. Z. Enolskii, and D. V. Leykin, ”On the Kleinian Construction of Abelian Functions of Canonical Algebraic Curves,” in SIDE III: Symmetries and Integrability of Difference Equations: Proc. Conf., Sabaudia, Italy, 1998 (Am. Math. Soc., Providence, RI, 2000), CRM Proc. Lect. Notes 25, pp. 121–138. · Zbl 1003.14008 [29] J. C. Eilbeck, V. Z. Enolskii, and E. Previato, ”On a Generalized Frobenius-Stickelberger Addition Formula,” Lett. Math. Phys. 63, 5–17 (2003). · Zbl 1042.14015 [30] J. C. Eilbeck, V. Z. Enolski, S. Matsutani, Y. Ônishi, and E. Previato, ”Abelian Functions for Trigonal Curves of Genus Three,” Int. Math. Res. Not., article ID rnm140 (2007); arXiv:math.AG/0610019. · Zbl 1210.14032 [31] V. Enolskii, S. Matsutani, and Y. Ônishi, ”The Addition Law Attached to a Stratification of a Hyperelliptic Jacobian Variety,” Tokyo J. Math. 31(1), 27–38 (2008); arXiv:math.AG/0508366. · Zbl 1167.14016 [32] A. R. Its and V. B. Matveev, ”Schrödinger Operators with Finite-Gap Spectrum and N-Soliton Solutions of the Korteweg-de Vries Equation,” Teor. Mat. Fiz. 23(1), 51–68 (1975) [Theor. Math. Phys. 23, 343–355 (1975)]. [33] I. M. Krichever, ”Integration of Nonlinear Equations by the Methods of Algebraic Geometry,” Funkts. Anal. Prilozh. 11(1), 15–31 (1977) [Funct. Anal. Appl. 11, 12–26 (1977)]. · Zbl 0346.35028 [34] A. Nakayashiki, ”On Algebraic Expressions of Sigma Functions for (n, s) Curves,” arXiv: 0803.2083. · Zbl 1214.14028 [35] A. Nakayashiki, ”Sigma Function as a Tau Function,” arXiv: 0904.0846. · Zbl 1197.14049 [36] S. P. Novikov, ”The Periodic Problem for the Korteweg-de Vries Equation,” Funkts. Anal. Prilozh. 8(3), 54–66 (1974) [Funct. Anal. Appl. 8, 236–246 (1974)]. · Zbl 0301.54027 [37] Y. Ônishi, ”Determinant Expressions for Hyperelliptic Functions (with an appendix by S. Matsutani),” Proc. Edinb. Math. Soc. 48(3), 705–742 (2005); math.NT/0105189. · Zbl 1148.14303 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.