Direct linearisation of the non-commutative Kadomtsev-Petviashvili equations

Abstract

We prove that the non-commutative Kadomtsev-Petviashvili (KP) equation and a `lifted' modified Kadomtsev-Petviashvili (mKP) equation are directly linearisable, and thus integrable in this sense. There are several versions of the non-commutative mKP equations, including the two-dimensional generalisations of the non-commutative modified Korteweg-de Vries (mKdV) equation and its alternative form (amKdV). Herein we derive the `lifted' mKP equation, whose solutions are the natural two-dimensional extension of those for the non-commutative mKdV equation derived in Blower and Malham. We also present the log-potential form of the mKP equation, from which all of these non-commutative mKP equations can be derived. To achieve the integrability results, we construct the pre-Poppe algebra that underlies the KP and mKP equations. This is a non-commutative polynomial algebra over the real line generated by the solution (and its partial derivatives) to the linearised form of the KP and mKP equations. The algebra is endowed with a pre-Poppe product, based on the product rule for semi-additive operators pioneered by Poppe for the commutative KP equation. Integrability corresponds to establishing a particular polynomial expansion in the respective pre-Poppe algebra. We also present numerical simulations of soliton-like interactions for the non-commutative KP equation.