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arXiv · 2211.09749

On the kink-kink collision problem of for the $ϕ^{6}$ model with low speed

Abdon Moutinho
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Abstract

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.

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Abdon Moutinho. 2023-05-23. On the kink-kink collision problem of for the $ϕ^{6}$ model with low speed. https://doi.org/10.2140/apde.2025.18.2145

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