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

Abstract

The article discusses a new method for constructing algebro-geometric solutions of nonlinear integrable lattices, based on the concept of a generalized invariant manifold (GIM). In contrast to the finite-gap integration method, instead of the eigenfunctions of the Lax operators, we use a joint solution of the linearized equation and GIM. This makes it possible to derive Dubrovin type equations not only in the time variable $t$, but also in the spatial discrete variable $n$. We illustrate the efficiency of the method using the Ruijsenaars-Toda lattice as an example, for which we have derived a real and bounded particular solution in the form of a kink, a periodic solution expressed in terms of Jacobi elliptic functions and a solution expressed through Weierstrass $\wp$-function.

26 pages, 3 figures

References in corpus (3)

• The Algebro-Geometric Solutions for the Ruijsenaars-Toda Hierarchy
• An upper order bound of the invariant manifold in Lax pairs of a nonlinear evolution partial differential equation
• Generalized invariant manifolds for integrable equations and their applications