The noncommutative KP hierarchy and its solution via descent algebra
Gordon Blower, Simon J. A. Malham
TL;DR
This work furnishes a complete direct linearisation of the noncommutative KP hierarchy via the GLM equation, leveraging semi-additive scattering data to connect two routes: the Sato–Wilson dressing and Poppe’s head-on method. A rich algebraic framework is developed, including the pre-Pöppe algebra, its augmented version, and the descent algebra, together with two representations—the planar binary rooted trees and the left-glue-right encoding—that render the hierarchy explicitly solvable. The central achievement is expressing the Sato coefficients, especially the first nontrivial one $b_{-1}$, as a descent-based sum with coefficients determined by the Sato endomorphism, and proving that the GLM solution $G$ yields the full ncKP hierarchy via $w_k=(-1)^k rac{ ext{d}^{k-1}}{ ext{d} ext{r}^{k-1}}[G](0,0;x,oldsymbol{t})$. The results are constructive, enabling numerical generation of solutions and offering a combinatorial interpretation of the hierarchy through grafting and degrafting on descent algebras, with potential extensions to modified KP, reductive hierarchies, and connections to quasi-determinants and Fredholm Grassmannians.
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 using two approaches. In the first approach we use the standard Sato-Wilson dressing transformation. In the second approach, which was pioneered by Poppe, 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-Poppe 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. 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 second approach are that it is constructive, explicit, highlights the underlying combinatorial structures within the hierarchy, and reveals the mechanisms underlying the solution procedure.