On uniqueness of KP soliton structures
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arXiv
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| Format: | Preprint |
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2024
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| author | Alegría, Francisco Chen, Gong Muñoz, Claudio Poblete, Felipe Tardy, Benjamín |
| author_facet | Alegría, Francisco Chen, Gong Muñoz, Claudio Poblete, Felipe Tardy, Benjamín |
| contents | We consider the Kadomtsev-Petviashvili II (KP) model placed in $\mathbb R_t \times \mathbb R_{x,y}^2$, in the case of smooth data that are not necessarily in a Sobolev space. In this paper, the subclass of smooth solutions we study is of ``soliton type'', characterized by a phase $Θ=Θ(t,x,y)$ and a unidimensional profile $F$. In particular, every classical KP soliton and multi-soliton falls into this category with suitable $Θ$ and $F$. We establish concrete characterizations of KP solitons by means of a natural set of nonlinear differential equations and inclusions of functionals of Wronskian, Airy and Heat types, among others. These functional equations only depend on the new variables $Θ$ and $F$. A distinct characteristic of this set of functionals is its special and rigid structure tailored to the considered soliton. By analyzing $Θ$ and $F$, we establish the uniqueness of line-solitons, multi-solitons, and other degenerate solutions among a large class of KP solutions. Our results are also valid for other 2D dispersive models such as the quadratic and cubic Zakharov-Kuznetsov equations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_07125 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On uniqueness of KP soliton structures Alegría, Francisco Chen, Gong Muñoz, Claudio Poblete, Felipe Tardy, Benjamín Analysis of PDEs We consider the Kadomtsev-Petviashvili II (KP) model placed in $\mathbb R_t \times \mathbb R_{x,y}^2$, in the case of smooth data that are not necessarily in a Sobolev space. In this paper, the subclass of smooth solutions we study is of ``soliton type'', characterized by a phase $Θ=Θ(t,x,y)$ and a unidimensional profile $F$. In particular, every classical KP soliton and multi-soliton falls into this category with suitable $Θ$ and $F$. We establish concrete characterizations of KP solitons by means of a natural set of nonlinear differential equations and inclusions of functionals of Wronskian, Airy and Heat types, among others. These functional equations only depend on the new variables $Θ$ and $F$. A distinct characteristic of this set of functionals is its special and rigid structure tailored to the considered soliton. By analyzing $Θ$ and $F$, we establish the uniqueness of line-solitons, multi-solitons, and other degenerate solutions among a large class of KP solutions. Our results are also valid for other 2D dispersive models such as the quadratic and cubic Zakharov-Kuznetsov equations. |
| title | On uniqueness of KP soliton structures |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2405.07125 |