Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 15 (2019), 062, 15 pages      arXiv:1903.11893

Integrable Modifications of the Ito-Narita-Bogoyavlensky Equation

Rustem N. Garifullin and Ravil I. Yamilov
Institute of Mathematics, Ufa Federal Research Centre, Russian Academy of Sciences, 112 Chernyshevsky Street, Ufa 450008, Russia

Received April 01, 2019, in final form August 14, 2019; Published online August 23, 2019

We consider five-point differential-difference equations. Our aim is to find integrable modifications of the Ito-Narita-Bogoyavlensky equation related to it by non-invertible discrete transformations. We enumerate all modifications associated to 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 one 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.

Key words: Miura transformation; integrable differential-difference equation; Ito-Narita-Bogoyavlensky equation.

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