On the kink-kink collision problem of for the $ϕ^{6}$ model with low speed
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arXiv
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| Formato: | Preprint |
| Publicado: |
2022
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| _version_ | 1866914027976785920 |
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| author | Moutinho, Abdon |
| author_facet | Moutinho, Abdon |
| contents | We study the elasticity of the collision of two kinks with an incoming low speed $v\in (0,1)$ for the nonlinear wave equation in dimension $1+1$ known as the $ϕ^{6}$ model. We prove for any $k\in\mathbb{N}$ that if the incoming speed $v$ is small enough, then, after the collision, the two kinks will move away with a velocity $v_{f}$ such that $\vert v_{f}-v\vert\leq v^{k}$ and the energy of the remainder will also be smaller than $v^{k}.$ This manuscript is the continuation of our previous paper where we constructed a sequence $ϕ_{k}$ of approximate solutions for the $ϕ^{6}$ model. The proof of our main result relies on the use of the set of approximate solutions from our previous work, modulation analysis, and a refined energy estimate method to evaluate the precision of our approximate solutions during a large time interval. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_09749 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On the kink-kink collision problem of for the $ϕ^{6}$ model with low speed Moutinho, Abdon Analysis of PDEs Mathematical Physics We study the elasticity of the collision of two kinks with an incoming low speed $v\in (0,1)$ for the nonlinear wave equation in dimension $1+1$ known as the $ϕ^{6}$ model. We prove for any $k\in\mathbb{N}$ that if the incoming speed $v$ is small enough, then, after the collision, the two kinks will move away with a velocity $v_{f}$ such that $\vert v_{f}-v\vert\leq v^{k}$ and the energy of the remainder will also be smaller than $v^{k}.$ This manuscript is the continuation of our previous paper where we constructed a sequence $ϕ_{k}$ of approximate solutions for the $ϕ^{6}$ model. The proof of our main result relies on the use of the set of approximate solutions from our previous work, modulation analysis, and a refined energy estimate method to evaluate the precision of our approximate solutions during a large time interval. |
| title | On the kink-kink collision problem of for the $ϕ^{6}$ model with low speed |
| topic | Analysis of PDEs Mathematical Physics |
| url | https://arxiv.org/abs/2211.09749 |