Perturbative Symmetry Approach for Differential–Difference Equations

Abstract We propose a new method to tackle the integrability problem for evolutionary differential–difference equations of arbitrary order. It enables us produce necessary conditions, determine whether given equation is integrable or not, and advance in classification equations. define develop symbolic representation difference polynomial ring, operators formal series. In order formulate we introduce novel quasi-local extension ring. apply developed formalism solve anti-symmetric quasi-linear $$(-3,3)$$ ( - 3 , ) list 17 satisfying conditions. For every from present an infinite family higher relatives. Some obtained are new.