Integrability of Nonabelian Differential–Difference Equations: The Symmetry Approach

Abstract

We extend the approach proposed in Mikhailov et al. (Commun Math Phys 393:1063–1104, 2022) to tackle the integrability problem for evolutionary differential–difference equations (D$$\Delta $$ Δ Es) on free associative algebras, also referred to as nonabelian D$$\Delta $$ Δ Es. This approach enables us to derive necessary integrability conditions, determine the integrability of a given equation, and make progress in the classification of integrable nonabelian D$$\Delta $$ Δ Es. This work involves establishing symbolic representations for the nonabelian difference algebra, difference operators, and formal series, as well as introducing a quasi-local extension for the algebra of formal series within the context of symbolic representations. Applying this formalism, we solve the classification problem of integrable skew-symmetric quasi-linear nonabelian equations of orders $$(-1,1)$$ ( - 1 , 1 ) , $$(-2,2)$$ ( - 2 , 2 ) , and $$(-3,3)$$ ( - 3 , 3 ) , consequently revealing some new equations in the process.