Integrable Modifications of the Ito-Narita-Bogoyavlensky Equation

• Published in: Symmetry, Integrability and Geometry: Methods and Applications
• Date: 2019
• Main Authors: Garifullin, R.N., Yamilov, R.I.
• Format: Article
• Language: English
• Published: Інститут математики НАН України 2019
• Online Access: https://nasplib.isofts.kiev.ua/handle/123456789/210233
• Journal Title: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Cite this: Integrable Modifications of the Ito-Narita-Bogoyavlensky Equation / R.N. Garifullin, R.I. Yamilov // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 26 назв. — англ.

We consider five-point differential-difference equations. We aim to find integrable modifications of the Ito-Narita-Bogoyavlensky equation related to it by non-invertible discrete transformations. We enumerate all modifications associated with transformations of the first, second, and third orders. As far as we know, such a classification problem is solved for the first time in the discrete case. We analyze transformations obtained to specify their nature. A number of new integrable five-point equations and new transformations have been found. Moreover, we have derived a new, completely discrete equation. There are a few non-standard transformations which are of the Miura type or are linearizable in a non-standard way. We have also proved that the orders of possible transformations are restricted by the number five in this problem.