On weakly commutative triples of partial differential operators

摘要

We investigate algebraic properties of weakly commutativetriples, appearing in the theory of integrable nonlinear partial differentialequations. Algebraic technique of skew fields of formal pseudodifferentialoperators as well as skew Ore fields of fractions are applied to thisproblem, relating weakly commutative triples to commuting elements ofskew Ore fields of formal fractions of ordinary differential operators. Aversion of Burchnall–Chaundy theorem for weakly commutative triplesis proved by algebraic means avoiding analytical complications typicalfor its proofs known in the theory of integrable equations.