Invariant manifolds and Lax pairs for integrable nonlinear chains
type: Item
instance of: scholarly article
title: Invariant manifolds and Lax pairs for integrable nonlinear chains
zbMATH Open document ID: 1379.37121
DOI: 10.1134/S0040577917060010
author: Ismagil Habibullin
author: A. R. Khakimova
published in: Theoretical and Mathematical Physics
publication date: 2017-11-03T00:00:00Z
review text: 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 equati…
reviewed by: Eszter Gselmann
Mathematics Subject Classification ID: 37K10
Mathematics Subject Classification ID: 37K05
Mathematics Subject Classification ID: 37J35
Mathematics Subject Classification ID: 39A30
Mathematics Subject Classification ID: 39A14
Mathematics Subject Classification ID: 34K19
Mathematics Subject Classification ID: 34K31
zbMATH DE Number: 6802695
zbMATH Keywords: Lax pair
zbMATH Keywords: integrable chain
zbMATH Keywords: higher symmetry
zbMATH Keywords: invariant manifold
zbMATH Keywords: recursion operator
MaRDI profile type: Publication
cites work: A scheme for integrating the nonlinear equations of mathematical physics by the method of the inverse scattering problem. I
cites work: Integration of nonlinear equations of mathematical physics by the method of inverse scattering. II
cites work: Prolongation structures of nonlinear evolution equations
cites work: The discrete and continuous Painlevé VI hierarchy and the Garnier systems
cites work: Q4534491
cites work: Lax pair for the Adler (lattice Krichever-Novikov) system
cites work: Q3222619
cites work: Q3006825
cites work: Darboux transformations and recursion operators for differential-difference equations
cites work: On a method for constructing the Lax pairs for nonlinear integrable equations
cites work: Classification of integrable equations on quad-graphs. The consistency approach
cites work: Q3857211
cites work: Evolutionary equations with nontrivial Lie-Bäcklund group
cites work: Affine and finite Lie algebras and integrable Toda field equations on discrete space-time
cites work: Symmetries as integrability criteria for differential difference equations
cites work: On construction of recursion operators from Lax representation
cites work: Lie algebras and equations of Korteweg-de Vries type
cites work: Q3744581
cites work: T-systems andY-systems in integrable systems
cites work: Discrete Toda field equations
cites work: Discretisations of constrained KP hierarchies
full work available at URL: s0040577917060010
OpenAlex ID: W2730139977
description: scientific article; zbMATH DE number 6802695