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Direct linearisation of the non-commutative Kadomtsev–Petviashvili equations

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Direct linearisation of the non-commutative Kadomtsev–Petviashvili equations. / Blower, G.; Malham, S.J.A.
In: Physica D: Nonlinear Phenomena, Vol. 481, 134745, 30.11.2025.

Research output: Contribution to Journal/Magazine › Journal article › peer-review

Harvard

Blower, G & Malham, SJA 2025, 'Direct linearisation of the non-commutative Kadomtsev–Petviashvili equations', Physica D: Nonlinear Phenomena, vol. 481, 134745. https://doi.org/10.1016/j.physd.2025.134745

APA

Blower, G., & Malham, S. J. A. (2025). Direct linearisation of the non-commutative Kadomtsev–Petviashvili equations. Physica D: Nonlinear Phenomena, 481, Article 134745. Advance online publication. https://doi.org/10.1016/j.physd.2025.134745

Vancouver

Blower G, Malham SJA. Direct linearisation of the non-commutative Kadomtsev–Petviashvili equations. Physica D: Nonlinear Phenomena. 2025 Nov 30;481:134745. Epub 2025 Jun 6. doi: 10.1016/j.physd.2025.134745

Author

Blower, G. ; Malham, S.J.A. / Direct linearisation of the non-commutative Kadomtsev–Petviashvili equations. In: Physica D: Nonlinear Phenomena. 2025 ; Vol. 481.

Bibtex

@article{33c1d308d9f7428fb2aac184ecc9245f,
title = "Direct linearisation of the non-commutative Kadomtsev–Petviashvili equations",
abstract = "We prove that the non-commutative Kadomtsev–Petviashvili (KP) equation and a {\textquoteleft}lifted{\textquoteright} 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 {\textquoteleft}lifted{\textquoteright} 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-P{\"o}ppe 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-P{\"o}ppe product, based on the product rule for semi-additive operators pioneered by P{\"o}ppe for the commutative KP equation. Integrability corresponds to establishing a particular polynomial expansion in the respective pre-P{\"o}ppe algebra. We also present numerical simulations of soliton-like interactions for the non-commutative KP equation.",
author = "G. Blower and S.J.A. Malham",
year = "2025",
month = jun,
day = "6",
doi = "10.1016/j.physd.2025.134745",
language = "English",
volume = "481",
journal = "Physica D: Nonlinear Phenomena",
issn = "0167-2789",
publisher = "Elsevier",

}

RIS

TY - JOUR

T1 - Direct linearisation of the non-commutative Kadomtsev–Petviashvili equations

AU - Blower, G.

AU - Malham, S.J.A.

PY - 2025/6/6

Y1 - 2025/6/6

N2 - 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.

AB - 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.

U2 - 10.1016/j.physd.2025.134745

DO - 10.1016/j.physd.2025.134745

M3 - Journal article

VL - 481

JO - Physica D: Nonlinear Phenomena

JF - Physica D: Nonlinear Phenomena

SN - 0167-2789

M1 - 134745

ER -