On the kink-kink collision problem for the ϕ6 model with low speed

Moutinho, Abdon

doi:10.2140/apde.2025.18.2145

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

Abdon Moutinho

Vol. 18 (2025), No. 9, 2145–2201

DOI: 10.2140/apde.2025.18.2145

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 $k \in ℕ$ that if the incoming speed $v$ is small enough, then, after the collision, the two solitons move away with a velocity $v_{f}$ such that $| v_{f} - v | \leq v^{k}$ 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.

Keywords

solitons, collision, kinks, nonlinear wave equation, classical scalar fields, dimension $1+1$, nonintegrable model, stability, $\phi^{6}$ model

Mathematical Subject Classification

Primary: 35B35, 35B40, 35L05, 35Q51

Secondary: 35C10, 35C20, 37B25, 37K40

Milestones

Received: 25 October 2023

Revised: 5 July 2024

Accepted: 20 September 2024

Published: 5 September 2025

Authors

Abdon Moutinho
School of Mathematics Georgia Institute of Technology Atlanta, GA United States

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Vol. 18, No. 9, 2025