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.

Laplace transformations and sine-Gordon type integrable PDE

“…Note that a system of these three equations is consistent if and only if u = u(x, y) is a solution to (1.28). Due to this property the triple is very convenient basis to deriving Dubrovin type systems, describing algebro-geometric solutions of equation (1.28) (see, [12][13][14]).…”

“…By a very elementary transformation we deduce from this Lax pair the recursion operators, corresponding to the hierarchies of symmetries in the directions of x and y (see [9]). By a direct computation one reduces the order of this Lax pair and arrive at a system of three nonlinear equations from which Dubrovin type equation can be derived [12][13][14], suitable for constructing algebrogeometric solutions (about algebro-geometric solutions see [15]). By passing to new variables in a proper way one can derive from the triple of nonlinear equations the standard Lax pair (see [9][10][11]).…”