Invariant manifolds and Lax pairs for integrable nonlinear chains
The authors continue their former work [J. Phys. A, Math. Theor. 49, No. 3, Article ID 035202, 35 p. (2016; Zbl 1359.37133)] for the development of a direct method for constructing the Lax pair for a given integrable equation \[ \dfrac{d u_{n}}{dt}= f\left(u_{n+k}, u_{n+k-1}, \ldots, u_{n-k}\right), \] where the function \(u = u_{n}(t)\) depends on \(t\in \mathbb{R}\) and \(n\in \mathbb{Z}\). The approach presented in the paper does not require any additional assumption on the properties of the equation. As one equation of the Lax pair, one can take the linearization of the considered nonlinear equation, and the second equation of the pair is related to its generalized invariant manifold. The problem of seeking the second equation reduces to a simple but rather cumbersome calculations and, as examples show, is effectively solvable. It is pointed out that the second equation of this pair allows easily to find a recursion operator describing the hierarchy of higher symmetries of the equation. At first glance, the Lax pairs obtained in this way differ from the usual ones in having a higher order or a higher matrix dimensionality. The authors show with examples that they reduce to the usual pairs by reducing their order. As an example, they consider an integrable double discrete system of exponential-type equations. In summary, the authors propose the following scheme for constructing the Lax pair for an integrable equation: {\parindent00.7cm\begin{itemize}\item[1.] Linearize the given integrable equation. \item[2.] Construct a linear manifold invariant with respect to the linearized equation. The pair consisting of the linearized equation and the equation of its invariant manifold defines a certain Lax pair for the considered equation. This is called its preliminary Lax pair because it allows reducing its order. \item[3.] Construct the integral of the preliminary Lax pair and reduce its order. As a result, one obtains a pair of nonlinear equations (or systems of equations). \item[4.] Transform the nonlinear system to a linear one via an appropriate change of variables. This system is the desired Lax pair. \end{itemize}}
scientific article; zbMATH DE number 6802695
scientific article