Invariant manifolds and separation of the variables for integrable chains

Abstract

A notion of the generalized invariant manifold for a nonlinear integrable lattice is considered. Earlier, it has been observed that this kind of objects provide an effective tool for evaluating the recursion operators and Lax pairs. In this article we show with an example of the Volterra chain that the generalized invariant manifold can be used for constructing exact particular solutions as well. To this end we first find an invariant manifold depending on two constant parameters. Then we assume that an ordinary difference equation (ODE) defining the generalized invariant manifold has a solution polynomially depending on one of the spectral parameters and derive an ODE, for the roots of the polynomials. The efficiency of the method is approved by some illustrative examples.

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

“…In the work [6], we introduced the concept of a generalized invariant manifold (GIM) of a nonlinear differential (difference, differential-difference) equation, which turned out to be useful in the study of nonlinear equations. It was shown in [7], [9] that the GIM can be effectively used to construct particular solutions of integrable equations. In [7], using the GIM, analogs of equations (1.2) and (1.3) for the Volterra chain were derived.…”

“…It was shown in [7], [9] that the GIM can be effectively used to construct particular solutions of integrable equations. In [7], using the GIM, analogs of equations (1.2) and (1.3) for the Volterra chain were derived. However, they are of the second order, and not the first, as should be in the Dubrovin equations.…”

“…However, they are of the second order, and not the first, as should be in the Dubrovin equations. In this paper, we improve the approach proposed in [7] and select the GIM in such a way that the equations of Dubrovin type that follow from it have the correct structure. As a result, we derived discrete analogs of the Dubrovin equations for the relativistic Toda lattice (see §5 formulas (5.12), (5.13)).…”

Generalized invariant manifolds for integrable equations and their applications

“…In the article a notion of the generalized invariant manifold for nonlinear integrable equation is discussed. Recently in our works [1]- [7] it has been observed that this kind objects provide an effective tool for evaluating the Lax pairs and recursion operators.…”

“…The approach developed in [1]- [7] explains the essence of the Lax pair phenomenon. In fact, the Lax pair in 1 + 1 dimension is naturally (internally) derived from the nonlinear equation under consideration.…”