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

TL;DR

This paper introduces a novel method using generalized invariant manifolds to construct explicit solutions for the Ruijsenaars-Toda lattice, expanding the toolkit for solving nonlinear integrable systems.

Contribution

The paper develops a new approach based on GIMs for deriving solutions of integrable lattices, allowing for solutions in terms of elliptic functions and Weierstrass functions.

Findings

Derived real and bounded kink solutions

Obtained periodic solutions with Jacobi elliptic functions

Expressed solutions through Weierstrass $ ext{wp}$-function

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.