Geodesic
Application of the trigonal curve to a hierarchy of generalized Toda lattices
Teoretičeskaâ i matematičeskaâ fizika, Tome 215 (2023) no. 1, pp. 47-73 Cet article a éte moissonné depuis la source Math-Net.Ru

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Starting from the zero-curvature equation and Lenard recurrence relations, we derive a hierarchy of generalized Toda lattices. The trigonal curve is introduced through the Lax pair characteristic polynomial for the discrete hierarchy, from which a Dubrovin-type equation is established. Then the asymptotic behavior of the Baker–Akhiezer function and the meromorphic function is analyzed, and the divisors of the two functions are also discussed. Moreover, the Abel map is defined and the corresponding flows are straightened out on the Jacobian variety, such that the final algebro-geometric solutions of the hierarchy are calculated in terms of the Riemann theta function.
Keywords: discrete matrix spectral problem, generalized Toda lattices, trigonal curve, algebro-geometric solutions. 
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Qiulan Zhao; Caixue Li; Xinyue Li. Application of the trigonal curve to a hierarchy of generalized Toda lattices. Teoretičeskaâ i matematičeskaâ fizika, Tome 215 (2023) no. 1, pp. 47-73. http://geodesic.mathdoc.fr/item/TMF_2023_215_1_a2/

[1] M. Toda, “Vibration of a chain with a nonlinear interaction”, J. Phys. Soc. Japan, 22:2 (1967), 431–436 | DOI

[2] M. Toda, Nonlinear Waves and Solitons, Mathematics and its Applications (Japanese Series), 5, Kluwer, Dordrecht, 1989 | MR

[3] M. Toda, Theory of Nonlinear Lattices, Springer Series in Solid-State Sciences, 20, Springer, Berlin, Heidelberg, 2012 | DOI | MR

[4] H. Flaschka, “On the Toda lattice. II. Inverse-scattering solution”, Progr. Theor. Phys., 51:3 (1974), 703–716 | DOI | MR

[5] A. Bloch, Hamiltonian and Gradient Flows, Algorithms, and Control, AMS, Providence, RI, 1994 | DOI | MR

[6] V. Muto, A. C. Scott, P. L. Christiansen, “Thermally generated solitons in a Toda lattice model of DNA”, Phys. Lett. A, 136:1–2 (1989), 33–36 | DOI

[7] M. J. Ablowitz, J. F. Ladik, “Nonlinear differential-difference equations”, J. Math. Phys., 16:3 (1975), 598–603 | DOI | MR

[8] T. Takebe, “Toda lattice hierarchy and conservation laws”, Commun. Math. Phys., 129:2 (1990), 281–318 | DOI | MR

[9] D.-J. Zhang, “Conservation laws of the two-dimensional Toda lattice hierarchy”, J. Phys. Soc. Japan, 71:11 (2002), 2583–2586 | DOI | MR

[10] W.-X. Ma, X.-X. Xu, Y. Zhang, “Semidirect sums of Lie algebras and discrete integrable couplings”, J. Math. Phys., 47:5 (2006), 053501, 16 pp. | DOI | MR

[11] R.-G. Zhou, Q.-Y. Jiang, “A Darboux transformation and an exact solution for the relativistic Toda lattice equation”, J. Phys. A: Math. Gen., 38:35 (2005), 7735–7742 | DOI | MR

[12] C.-Z. Li, J.-S. He, “The extended multi-component Toda hierarchy”, Math. Phys. Anal. Geom., 17:3–4 (2014), 377–407 | DOI | MR

[13] A. R. Its, V. B. Matveev, “Ob operatorakh Khilla s konechnym chislom lakun”, Funkts. analiz i ego pril., 9:1 (1975), 69–70 | DOI | MR | Zbl

[14] I. M. Krichever, “Algebro-geometricheskoe postroenie uravnenii Zakharova–Shabata i ikh periodicheskikh reshenii”, Dokl. AN SSSR, 227:2 (1976), 291–294 | MR | Zbl

[15] V. E. Zakharov, S. V. Manakov, S. P. Novikov, L. P. Pitaevskii, Teoriya solitonov. Metod obratnoi zadachi, Nauka, M., 1980 | MR | MR | Zbl

[16] W. Bulla, F. Gesztesy, H. Holden, G. Teschl, “Algebro-geometric quasi-periodic finite-gap solutions of the Toda and Kac–van Moerbeke hierarchies”, Mem. Amer. Math. Soc., 135:641 (1998), 1–79 | DOI | MR

[17] C.-W. Cao, X.-G. Geng, H.-Y. Wang, “Algebro-geometric solution of the $2+1$ dimensional Burgers equation with a discrete variable”, J. Math. Phys., 43:1 (2002), 621–643 | DOI | MR

[18] X.-G. Geng, Y.-T. Wu, “Finite-band solutions of the classical Boussinesq–Burgers equations”, J. Math. Phys., 40:6 (1999), 2971–2982 | DOI | MR

[19] Q.-L. Zhao, C.-X. Li, X.-Y. Li, “Algebro-geometric constructions of a hierarchy of integrable semi-discrete equations”, J. Nonlinear Math. Phys., 2022, Publ. online | DOI

[20] L. Luo, E.-G. Fan, “Variable separation and algebraic-geometric solutions of modified Toda lattice equation”, Modern Phys. Lett. B., 24:19 (2010), 2041–2055 | DOI | MR

[21] X.-G. Geng, D. Gong, “Quasi-periodic solutions of the relativistic Toda hierarchy”, J. Nonlinear Math. Phys., 19:4 (2012), 489–523 | DOI

[22] H. Airault, H. P. McKean, J. Moser, “Rational and elliptic solutions of the Korteweg–de Vries equation and a related many-body problem”, Commun. Pure Appl. Math., 30:1 (1977), 95–148 | DOI | MR

[23] J. N. Elgin, V. Z. Enolski, A. R.Its, “Effective integration of the nonlinear vector Schrödinger equation”, Phys. D, 225:2 (2007), 127–152 | DOI | MR

[24] R. Dickson, F. Gesztesy, K. Unterkofler, “A new approach to the Boussinesq hierarchy”, Math. Nachr., 198:1 (1999), 51–108 | DOI | MR

[25] X.-G. Geng, L.-H. Wu, G.-L. He, “Algebro-geometric constructions of the modified Boussinesq flows and quasi-periodic solutions”, Phys. D, 240:16 (2011), 1262–1288 | DOI | MR

[26] X.-G. Geng, L.-H. Wu, G.-L. He, “Quasi-periodic solutions of the Kaup–Kupershmidt hierarchy”, J. Nonlinear Sci., 23:4 (2013), 527–555 | DOI | MR

[27] J. Wei, X.-G. Geng, X. Zeng, “Quasi-periodic solutions to the hierarchy of four-component Toda lattices”, J. Geom. Phys., 106 (2016), 26–41 | DOI | MR

[28] X. Zeng, X.-G. Geng, “Finite-band solutions for the hierarchy of coupled Toda lattices”, Acta Appl. Math., 154:1 (2018), 59–81 | DOI | MR

[29] L.-L. Ma, X.-X. Xu, “A family of discrete Hamiltonian equations associated with a discrete three-by-three matrix spectral problem”, Internat. J. Modern Phys. B, 23:19 (2009), 3859–3869 | DOI | MR