Invariant manifolds and Lax pairs for integrable nonlinear chains

Abstract

A notion of the generalized invariant manifold for a nonlinear integrable lattice is considered. Earlier it has been observed that this kind 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 ordinary difference equation defining the generalized invariant manifold has a solution polynomially depending on one of the spectral parameters and derive ordinary difference and differential equations, for the roots of the polynomials. In the simplest nontrivial case we constructed exact common solutions to these equations. As a result we derived for the Volterra chain a new explicit solution, displaying unusual behavior.

A Direct Algorithm for Constructing Recursion Operators and Lax Pairs for Integrable Models

“…for the equation (1) (see [5,6]). Moreover, as it is demonstrated in [3,4] by numerous examples the Lax pair (7) is effectively converted into the classical one. A clear expression of this circumstance is that a linear invariant manifold is transformed into a nonlinear one by decreasing order in the corresponding ordinary differential equation (3).…”

“…By the construction equations (2) and (3) are compatible if u = u(x, t) is a solution of the equation (1). It is remarkable that for an appropriate choice of the equation (3) the converse is also true: the compatibility of the equations (2) and (3) implies (1). As it is commonly known just this property is principal for the Lax pairs.…”

“…It is supposed that in (4) all of the derivatives are expressed due to the equations (1) and (2) and the variables U s+m with s ≥ 0 are expressed due to the equation (3).…”

On description of generalized invariant manifolds for nonlinear equations

“…. are changed by virtue of equality (5). To emphasize that ( , ) is an arbitrary solution, we treat the variables , , , .…”

“…Let an invariant manifold be defined by equation(5).A pair of numbers ( , ) is called the order of the manifold . The manifold is said to be trivial if an arbitrary solution of equation (5) reads as = ( , , , , .…”