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On weakly commutative triples of partial differential operators. (Russian. English summary) Zbl 1386.35374

Summary: We investigate algebraic properties of weakly commutative triples, appearing in the theory of integrable nonlinear partial differential equations. Algebraic technique of skew fields of formal pseudodifferential operators as well as skew Ore fields of fractions are applied to this problem, relating weakly commutative triples to commuting elements of skew Ore fields of formal fractions of ordinary differential operators. A version of Burchnall-Chaundy theorem for weakly commutative triples is proved by algebraic means avoiding analytical complications typical for its proofs known in the theory of integrable equations.

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
12E15 Skew fields, division rings
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References:

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