Invariant manifolds and Lax pairs for integrable nonlinear chains
- 57 Downloads
We continue the previously started study of the development of a direct method for constructing the Lax pair for a given integrable equation. This approach does not require any addition assumptions about the properties of the equation. As one equation of the Lax pair, we 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 simple but rather cumbersome calculations and, as examples show, is effectively solvable. It is remarkable that the second equation of this pair allows easily finding a recursion operator describing the hierarchy of higher symmetries of the equation. At first glance, the Lax pairs thus obtained differ from usual ones in having a higher order or a higher matrix dimensionality. We show with examples that they reduce to the usual pairs by reducing their order. As an example, we consider an integrable double discrete system of exponential type and its higher symmetry for which we give the Lax pair and construct the conservation laws.
KeywordsLax pair integrable chain higher symmetry invariant manifold recursion operator
Unable to display preview. Download preview PDF.
- 7.R. I. Yamilov, “On the classification of discrete equations [in Russian],” in: Integrable Systems (A. B. Shabat, ed.), Bashkir State Univ., Ufa (1982), pp. 95–114.Google Scholar
- 8.P. Xenitidis, “Integrability and symmetries of difference equations: The Adler–Bobenko–Suris case,” in: Proc. IV Workshop “Group Analysis of Differential Equations and Integrable Systems” (GADEIS–IV) (Protaras, Cyprus, 26–30 October 2008, N. Ivanova, C. Sophocleous, R. Popovych, P. Damianou, and A. Nikitin, eds.), Univ. of Patras, Greece (2008), pp. 226–242; arXiv:0902.3954v1 [nlin.SI] (2009).Google Scholar
- 13.B. I. Suleimanov, “The ‘quantum’ linearization of the Painlevé equations as a component of their L,A pairs [in Russian],” Ufimsk. Mat. Zh., 4, 127–135 (2012).Google Scholar