Integrable Modifications of the Ito-Narita-Bogoyavlensky Equation

Abstract

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 nonstandard way. We have also proved that the orders of possible transformations are restricted by the number five in this problem.

Coupled discrete solitonic equations of additive Bogoyavlensky and the periodic reduction

“…Intensively studied since the seventies, the Volterra type systems, introduced by Hirota and Satsuma [1][2][3], are still a subject of interest today from different points of view [4][5][6][7]. One of them is given by the integrable extensions of the Volterra systems, known in literature as the Bogoyavlensky lattices [8][9][10][11]. Three such integrable extensions where recently investigated from the singularity analysis and bilinear integrability point of view in [12], more precisely: the additive Bogoyavlensky equation (aB), the first multiplicative Bogoyavlensky equation (mB2) and the second multiplicative Bogoyavlensky equation (mB2).…”