Direct linearisation of the non-commutative Kadomtsev-Petviashvili equations
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| author | Blower, Gordon Malham, Simon J. A. |
| author_facet | Blower, Gordon Malham, Simon J. A. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_01686 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Direct linearisation of the non-commutative Kadomtsev-Petviashvili equations Blower, Gordon Malham, Simon J. A. Exactly Solvable and Integrable Systems Mathematical Physics Analysis of PDEs 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. |
| title | Direct linearisation of the non-commutative Kadomtsev-Petviashvili equations |
| topic | Exactly Solvable and Integrable Systems Mathematical Physics Analysis of PDEs |
| url | https://arxiv.org/abs/2412.01686 |