Commutative partial differential operators. (English) Zbl 0980.35141
Summary: In one variable, it is possible to describe explicitly the differential operators that commute with a given one, at least when the centralizer of the given operator has rank 1. So far, a generalization of the theory to several variables has been developed (inexplicitly) only for matrices, whose size increases with the number of variables. We propose to develop an algebraic theory of commuting partial differential operators by formulating a generalization of the one-variable techniques, in particular Darboux transformations and differential resultants. In this paper, we present a counterexample to a one-variable feature of maximal-commutative rings and some facts, examples and questions on differential resultants.
MSC:
| 35Q51 | Soliton equations |
| 47F05 | General theory of partial differential operators |
| 37K20 | Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions |
Keywords:
maximal-commutative rings; generalization of the Burchnall-Chaundy problem; Darboux transformations; differential resultantsReferences:
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