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 TMF, 2017, Volume 191, Number 3, Pages 369–388 (Mi tmf9216)

Invariant manifolds and Lax pairs for integrable nonlinear chains

I. T. Habibullinab, A. R. Khakimovaab

a Institute of Mathematics with Computing Centre, Ufa Science Centre, RAS, Ufa, Russia
b Bashkir State University, Ufa, Russia

Abstract: 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.

Keywords: Lax pair, integrable chain, higher symmetry, invariant manifold, recursion operator.

 Funding Agency Grant Number Russian Science Foundation 15-11-20007 This research is supported by a grant from the Russian Science Foundation (Project No. 15-11-20007).

DOI: https://doi.org/10.4213/tmf9216

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English version:
Theoretical and Mathematical Physics, 2017, 191:3, 793–810

Bibliographic databases:

MSC: 35Q53
Revised: 06.02.2017

Citation: I. T. Habibullin, A. R. Khakimova, “Invariant manifolds and Lax pairs for integrable nonlinear chains”, TMF, 191:3 (2017), 369–388; Theoret. and Math. Phys., 191:3 (2017), 793–810

Citation in format AMSBIB
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• https://doi.org/10.4213/tmf9216
• http://mi.mathnet.ru/eng/tmf/v191/i3/p369

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Habibullin I.T. Khakimova A.R., “On a Method For Constructing the Lax pairs For Integrable Models Via a Quadratic Ansatz”, J. Phys. A-Math. Theor., 50:30 (2017), 305206
2. I. T. Habibullin, A. R. Khakimova, “A direct algorithm for constructing recursion operators and Lax pairs for integrable models”, Theoret. and Math. Phys., 196:2 (2018), 1200–1216
3. I. T. Habibullin, A. R. Khakimova, “On the recursion operators for integrable equations”, J. Phys. A-Math. Theor., 51:42 (2018), 425202
4. A. R. Khakimova, “On description of generalized invariant manifolds for nonlinear equations”, Ufa Math. J., 10:3 (2018), 106–116
5. I. T. Khabibullin, A. R. Khakimova, “Invariantnye mnogoobraziya integriruemykh uravnenii giperbolicheskogo tipa i ikh prilozheniya”, Kompleksnyi analiz. Matematicheskaya fizika, Itogi nauki i tekhn. Ser. Sovrem. mat. i ee pril. Temat. obz., 162, VINITI RAN, M., 2019, 136–150
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