Quasi-Periodic Solutions of the Relativistic Toda Hierarchy

Abstract

On the basis of the theory of algebraic curves, the continuous flow and discrete flow related to the relativistic Toda hierarchy are straightened out using the Abel–Jacobi coordinates. The meromorphic function and the Baker–Akhiezer function are introduced on the hyperelliptic curve. Quasi-periodic solutions of the relativistic Toda hierarchy are constructed with the help of the asymptotic properties and the algebro-geometric characters of the meromorphic function and the hyperelliptic curve.

Construction of exact solutions to the Ruijsenaars–Toda lattice via generalized invariant manifolds

“…The method of invariant manifolds is one of the most common methods for constructing solutions to nonlinear equations (see, for example, [8]). As alternative approaches to the problem of constructing solutions to nonlinear equations, we should mention the method of conditional symmetries [6,7,9], the Puiseux series method [10], the method of asymptotic expansions [11] and the method of finite-gap integration [12][13][14][15][16][17][18][19][20][21].…”

Construction of exact solutions to the Ruijsenaars-Toda lattice via generalized invariant manifolds

“…The finite-gape integration method is effectively applied to nonlinear integrable chains as well. Finite-gap solutions of nonlinear Volterra and Toda lattices, as well as the relativistic Toda lattice, were investigated in a number of works [8,[10][11][12][15][16][17][18][19]. In them one can find explicit formulas for the solutions of these chains in terms of the Riemann theta functions, as well as analogues of Dubrovin's equations (1.3), which determine the dynamics in time t. As far as the authors know, analogs of the Dubrovin equations describing the dynamics in the spatial variable n ∈ Z have not been presented before.…”

Application of the trigonal curve to a hierarchy of generalized Toda lattices

“…Для (2 × 2)-матричных спектральных задач уже были получены алгебро-геометрические решения соответствующих солитонных уравнений, выражающиеся через тета-функции Римана на гиперэллиптической кривой [19]- [21]. Однако исследования алгебро-геометрических решений солитонных уравнений третьего порядка очень немногочисленны.…”