Two-dimensionalized Toda lattice, commuting difference operators, and holomorphic vector bundles.

*(English. Russian original)*Zbl 1060.37068
Russ. Math. Surv. 58, No. 3, 473-510 (2003); translation from Usp. Mat. Nauk 58, No. 3, 51-88 (2003).

The present paper is devoted to a circle of problems related to the construction of higher-rank algebro-geometric solutions of the two-dimensional Toda lattice
\[
\partial _{\xi\eta}^2\varphi_n =e^{\varphi_n-\varphi_{n-1}}-e^{\varphi_{n+1}-\varphi_n},
\]
which constitutes a counterpart theory of the algebro-geometric solution theory of the Kadomtsev-Petviashvili equation [see the authors, Funct. Anal. Appl. 12, 276–286 (1979); translation from Funkts. Anal. Prilozh. 12, No. 4, 41–52 (1978; Zbl 0393.35061)]. The primary part of the paper is a classification theory of commuting difference operators
\[
L=\sum_{i=-N_-}^{N_+} u_i(n)T^i,\quad A=\sum_{i=-M_-}^{M_+}v_i(n)T^i,
\]
where \(Ty_n=y_{n+1}\), \(N_\pm\) and \(M_\pm\) are nonnegative integers, and the coefficients \(u_i(n)\) and \(v_i(n)\) are scalar. It is shown that the problem of recovering the coefficients of commuting operators can be effectively solved by means of the equations of the discrete dynamics of the Tyurin parameters characterizing the stable holomorphic vector bundles over an algebraic curve. The algebro-geometric solutions of the two-dimensional Toda lattice then come from constructing the multiparameter Baker-Akhiezer vector functions by means of deformations of eigenfunctions of the corresponding commuting difference operators. These deformations are completely determined by the behavior of the Baker-Akhiezer functions in a neighborhood of the marked points determined by certain grafting matrix functions. The principal difference between the one-point and two-point constructions are highlighted.

In two previous papers, the authors also briefly discussed the construction of higher-rank algebro-geometric solutions of the two-dimensional Toda lattice [Russ. Math. Surv. 55, 180–181 (2000); translation from Usp. Mat. Nauk 55, No. 1, 187–188 (2000; Zbl 1101.14315) and Russ. Math. Surv. 55, No. 3, 586–588 (2000); translation from Usp. Mat. Nauk 55, No. 3, 181–182 (2000; Zbl 0978.35066)]. The obtained results are a beautiful contribution to the integrable theory of the two-dimensional Toda lattice, among which web-structural interaction and Pfaffianization are two other recently explored integrable characteristics.

In two previous papers, the authors also briefly discussed the construction of higher-rank algebro-geometric solutions of the two-dimensional Toda lattice [Russ. Math. Surv. 55, 180–181 (2000); translation from Usp. Mat. Nauk 55, No. 1, 187–188 (2000; Zbl 1101.14315) and Russ. Math. Surv. 55, No. 3, 586–588 (2000); translation from Usp. Mat. Nauk 55, No. 3, 181–182 (2000; Zbl 0978.35066)]. The obtained results are a beautiful contribution to the integrable theory of the two-dimensional Toda lattice, among which web-structural interaction and Pfaffianization are two other recently explored integrable characteristics.

Reviewer: Ma Wen-Xiu (Tampa)

##### MSC:

37K20 | Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions |

14H70 | Relationships between algebraic curves and integrable systems |

39A70 | Difference operators |

47B39 | Linear difference operators |