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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2411.11703 |
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Table of 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}.