The noncommutative KP hierarchy and its solution via descent algebra

Blower, Gordon and Malham, Simon (2026) The noncommutative KP hierarchy and its solution via descent algebra. Physica D: Nonlinear Phenomena, 490: 135152. ISSN 0167-2789

Abstract

We give the solution to the complete noncommutative Kadomtsev–Petviashvili (KP) hierarchy. We achieve this via direct linearisation which involves the Gelfand–Levitan–Marchenko (GLM) equation. This is a linear integral equation in which the scattering data satisfies the linearised KP hierarchy. The solution to the GLM equation is then shown to coincide with the solution to the noncommutative KP hierarchy. We achieve this via the approach pioneered by Pöppe, Inverse Problems 5(1989), 613–630. We assume the scattering data is semi-additive and by direct substitution, we show that the solution to the GLM equation satisfies the infinite set of field equations representing the noncommutative KP hierarchy. This approach relies on the augmented pre-Pöppe algebra. This is a representative algebra that underlies the field equations representing the hierarchy. It is nonassociative, and isomorphic to a descent algebra equipped with a grafting product. Indeed, it is an example of a weakly nonassociative algebra. While we perform computations in the nonassociative descent algebra, the final result which establishes the solution to the complete hierarchy, resides in the natural associative subalgebra. The advantages of this approach are that it is constructive, explicit, highlights the underlying combinatorial structures within the hierarchy, and reveals the mechanisms underlying the solution procedure.