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 (2023). 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-Pöppe 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-Pöppe product, based on the product rule for semi-additive operators pioneered by Pöppe for the commutative KP equation. Integrability corresponds to establishing a particular polynomial expansion in the respective pre-Pöppe algebra. We also present numerical simulations of soliton-like interactions for the non-commutative KP equation.