Triangular Reductions of the $2D$ Toda Hierarchy
Funkcionalʹnyj analiz i ego priloženiâ, Tome 51 (2017) no. 1, pp. 60-81.

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New reductions of the 2D Toda equations associated with lower-triangular difference operators are proposed. Their explicit Hamiltonian description is obtained.
Keywords: integrable systems, bi-Hamiltonian theory, Baker–Akhiezer function.
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A. V. Il'ina; I. M. Krichever. Triangular Reductions of the $2D$ Toda Hierarchy. Funkcionalʹnyj analiz i ego priloženiâ, Tome 51 (2017) no. 1, pp. 60-81. http://geodesic.mathdoc.fr/item/FAA_2017_51_1_a4/

[1] O. Babelon, D. Bernard, M. Talon, Introduction to classical integrable systems, Cambridge University Press, Cambridge, 2003 | MR | Zbl

[2] I. M. Krichever, “Algebraicheskie krivye i nelineinye raznostnye uravneniya”, UMN, 33:4 (1978), 215–216 | MR | Zbl

[3] I. M. Krichever, “Periodicheskaya neabeleva tsepochka Toda i ee dvumernoe obobschenie”, Prilozhenie k state B. A. Dubrovina «Teta-funktsii i nelineinye uravneniya», UMN, 36:2 (1981), 72–77

[4] I. M. Krichever, “Metod usredneniya dlya dvumernykh «integriruemykh» uravnenii”, Funkts. analiz i ego pril., 22:3 (1988), 37–52 | MR | Zbl

[5] I. M. Krichever, “Kommutiruyuschie raznostnye operatory i kombinatornye preobrazovaniya Geila”, Funkts. analiz i ego pril., 49:3 (2015), 175–188 | DOI | MR | Zbl

[6] I. M. Krichever, “Elliptic solutions to difference nonlinear equations and nested Bethe ansatz equations”, Calogero–Moser–Sutherland models (Monréal, QC, 1997), CRM Ser. Math. Phys, Springer-Verlag, New-York, 2000, 249–271 ; arXiv: solv-int/9804016 | MR

[7] I. Krichever, D. H. Phong, “On the integrable geometry of $N=2$ supersymmetric gauge theories and soliton equations”, J. Differential Geometry, 45 (1997), 445–485 ; arXiv: hep-th/9604199 | DOI | MR

[8] I. Krichever, D. Phong, “Symplectic forms in the theory of solitons”, Survey in Differential Geometry, 4 (1998), 239–313 | DOI | MR | Zbl

[9] I. Krichever, D. H. Phong, “Spin chain models with spectral curves from M Theory”, Commun. Math. Phys., 213:3 (2000), 539–574 | DOI | MR | Zbl

[10] I. Krichever, T. Shiota, “Soliton equations and the Riemann–Schottky problem”, Handbook of Moduli, Advanced Lectures Math., 25, International Press, Boston, 2013 | MR | Zbl

[11] S. Morier-Genoud, V. Ovsienko, R. E. Schwartz, S. Tabachnikov, “Linear difference equations, frieze patterns and combinatorial Gale transform”, Forum Math. Sigma, 2 (2014) ; arXiv: 1309.3880 | DOI | MR | Zbl

[12] V. Ovsienko, R. Schwartz, S. Tabachnikov, “The pentagram map: A discrete integrable system”, Commun. Math. Phys., 299:2 (2010), 409–446 | DOI | MR | Zbl

[13] F. Soloviev, “Integrability of the pentagram map”, Duke. Math. J., 162:15 (2013), 2815–2853 | DOI | MR | Zbl