Construction of multi solitary waves with symmetry for the damped nonlinear Klein-Gordon equation
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| author | Côte, Raphaël Du, Haiming |
| author_facet | Côte, Raphaël Du, Haiming |
| contents | We are interested in the nonlinear damped Klein-Gordon equation \[ \partial_t^2 u+2α\partial_t u-Δu+u-|u|^{p-1}u=0 \] on $\mathbb{R}^d$ for $2\le d\le 5$ and energy sub-critical exponents $2 < p < \frac{d+2}{d-2}$.
We construct multi-solitons, that is, solutions which behave for large times as a sum of decoupled solitons, in various configurations with symmetry: this includes multi-solitons whose soliton centers lie at the vertices of an expanding regular polygon (with or without a center), of a regular polyhedron (with a center), or of a higher dimensional regular polytope. We give a precise description of these multi-solitons: in particular the interaction between nearest neighbour solitons is asymptotic to $\ln (t)$ as $t \to +\infty$.
We also prove that in any multi-soliton, the solitons can not all share the same sign.
Both statements generalize and precise results from \cite{F98}, \cite{Nak} and are based on the analysis developed in \cite{CMYZ,CMY}. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_11703 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Construction of multi solitary waves with symmetry for the damped nonlinear Klein-Gordon equation Côte, Raphaël Du, Haiming Analysis of PDEs We are interested in the nonlinear damped Klein-Gordon equation \[ \partial_t^2 u+2α\partial_t u-Δu+u-|u|^{p-1}u=0 \] on $\mathbb{R}^d$ for $2\le d\le 5$ and energy sub-critical exponents $2 < p < \frac{d+2}{d-2}$. We construct multi-solitons, that is, solutions which behave for large times as a sum of decoupled solitons, in various configurations with symmetry: this includes multi-solitons whose soliton centers lie at the vertices of an expanding regular polygon (with or without a center), of a regular polyhedron (with a center), or of a higher dimensional regular polytope. We give a precise description of these multi-solitons: in particular the interaction between nearest neighbour solitons is asymptotic to $\ln (t)$ as $t \to +\infty$. We also prove that in any multi-soliton, the solitons can not all share the same sign. Both statements generalize and precise results from \cite{F98}, \cite{Nak} and are based on the analysis developed in \cite{CMYZ,CMY}. |
| title | Construction of multi solitary waves with symmetry for the damped nonlinear Klein-Gordon equation |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2411.11703 |