Integrability of Nonabelian Differential-Difference Equations: the Symmetry Approach
arxiv(2024)
摘要
We propose a novel approach to tackle integrability problem for evolutionary
differential-difference equations (DΔEs) on free associative algebras,
also referred to as nonabelian DΔEs. 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
DΔEs. This work involves establishing symbolic representations for the
nonabelian difference algebra, difference operators, and formal series, as well
as introducing a novel 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), (-2,2), and (-3,3), consequently
revealing some new equations in the process.
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