Finite-Band Solutions for the Hierarchy of Coupled Toda Lattices
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Finite-Band Solutions for the Hierarchy of Coupled Toda Lattices Xin Zeng1 · Xianguo Geng1
Received: 6 August 2015 / Accepted: 1 November 2017 © Springer Science+Business Media B.V., part of Springer Nature 2017
Abstract Based on the characteristic polynomial of Lax matrix for the hierarchy of coupled Toda lattices associated with a 3 × 3 discrete matrix spectral problem, we introduce a trigonal curve with two infinite points, from which we establish the associated Dubrovin-type equations. The asymptotic properties of the meromorphic function and the Baker-Akhiezer function are studied near two infinite points on the trigonal curve. Finite-band solutions of the entire hierarchy of coupled Toda lattices are obtained in terms of the Riemann theta function. Keywords Coupled Toda lattices · Finite-band solutions · Trigonal curve
1 Introduction The study of finite-band solutions (quasi-periodic solutions, algebro-geometric solutions) for soliton equations is very important if these equations under consideration are of interest for physics or mathematics. However, it is very difficult to obtain finite-band solutions for a soliton equation because of concerning the theory of hyperelliptic curves. There have been several systematic methods to obtain finite-band solutions for soliton equations associated with the 2 × 2 matrix spectral problem, such as the algebro-geometric method [1–5], the inverse scattering transformation for periodic problem [6–8], and others [9–18]. It has been applied extensively to 2 × 2 matrix spectral problem of continuous and discrete cases, from which finite-band solutions of many soliton equations are obtained such as the KdV, nonlinear Schrödinger, sine-Gordon, discrete KdV, Toda lattice, Ablowitz-Ladik equations, and so on. The situation is not so good for the soliton equations associated with the 3 × 3 matrix spectral problem, which are more complicated and more difficult because of concerning
B X. Geng
[email protected]
1
School of Mathematics and Statistics, Zhengzhou University, Zhengzhou, Henan 450001, People’s Republic of China
X. Zeng, X. Geng
the theory of trigonal curves [19–23] rather than hyperelliptic ones as for the 2 × 2 matrix spectral problems. Only a few of literature [24–31] studied finite-band solutions of the Boussinesq equation related to a third-order differential operator by the reduction theory of Riemann theta functions. In Refs. [32, 33], a unified framework is proposed which yields all quasi-periodic solutions of the entire Boussinesq hierarchy associated with the third-order differential operator. On the basis of that, we give a systematical method to define the trigonal curve, from which the finite-band solutions of the entire soliton hierarchies associated with the 3 × 3 matrix spectral problems are obtained with the help of the algebro-geometric characteristic of the trigonal curve and asymptotic properties of the Baker-Akhiezer function and the meromorphic function [34–37]. The present paper is devoted to study the hierarchy of coupled Toda lattices and
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