Linear systems, determinants and solutions of the Kadomtsev-Petviashvili equation
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| Format: | Preprint |
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2025
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| author | Blower, Gordon Malham, Simon J. |
| author_facet | Blower, Gordon Malham, Simon J. |
| contents | Let $(-A,B,C)$ be a linear system in continuous time $t>0$ with input and output space ${\mathbb C}$ and state space $H$. The scattering (or impulse response) functions $ϕ_{(x)}(t)=Ce^{-(t+2x)A}B$ determines a Hankel integral operator $Γ_{ϕ_{(x)}}$; if $Γ_{ϕ_{(x)}}$ is trace class, then the Fredholm determinant $τ(x)=\det (I+Γ_{ϕ_{(x)}})$ determines the tau function of $(-A,B,C)$. The paper establishes properties of algebras including $R_x = \int_x^\infty e^{-tA}BCe^{-tA}\,dt$ on $H$, and obtains solutions of the Kadomtsev-Petviashvili PDE. Pöppe's semi-additive operators are identified with orbits of a shift action on integral kernels, and Pöppe's bracket operation is expressed in terms of the Fedosov product. The paper shows that the Fredholm determinant $\det (I+R_x)$ gives an effective method for numerical computation of solutions of $KP$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_15245 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Linear systems, determinants and solutions of the Kadomtsev-Petviashvili equation Blower, Gordon Malham, Simon J. Analysis of PDEs 47B35, 47A48, 35Q53, 65M99 Let $(-A,B,C)$ be a linear system in continuous time $t>0$ with input and output space ${\mathbb C}$ and state space $H$. The scattering (or impulse response) functions $ϕ_{(x)}(t)=Ce^{-(t+2x)A}B$ determines a Hankel integral operator $Γ_{ϕ_{(x)}}$; if $Γ_{ϕ_{(x)}}$ is trace class, then the Fredholm determinant $τ(x)=\det (I+Γ_{ϕ_{(x)}})$ determines the tau function of $(-A,B,C)$. The paper establishes properties of algebras including $R_x = \int_x^\infty e^{-tA}BCe^{-tA}\,dt$ on $H$, and obtains solutions of the Kadomtsev-Petviashvili PDE. Pöppe's semi-additive operators are identified with orbits of a shift action on integral kernels, and Pöppe's bracket operation is expressed in terms of the Fedosov product. The paper shows that the Fredholm determinant $\det (I+R_x)$ gives an effective method for numerical computation of solutions of $KP$. |
| title | Linear systems, determinants and solutions of the Kadomtsev-Petviashvili equation |
| topic | Analysis of PDEs 47B35, 47A48, 35Q53, 65M99 |
| url | https://arxiv.org/abs/2512.15245 |