Singularity analysis and bilinear approach to some Bogoyavlensky equations

Abstract

We discuss singularity analysis and bilinear integrability of four Bogoyavlensky differential-difference equations. Three of them are completely integrable, and the fourth is, to our knowledge, a new one. Blending the singularity confinement with the Painlevé property reveals strictly confining and anticonfining (weakly confining) singularity patterns. Strictly confining patterns are extremely useful because they provide the nonlinear substitution needed for Hirota bilinear forms. For the new proposed equation, we also get the bilinear form and multisoliton solution, being a good candidate for a new integrable system. In addition, using the bilinear formalism, we recover integrable time-discretizations of the first three systems.