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

Abstract

The article discusses a new method for constructing particular 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 find some new solutions: 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 -function.