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

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 ${\varphi }^6$ 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 $|v_f-v|\le v^k$ 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 ${\varphi }_k$ of approximate solutions for the ${\varphi }^6$ 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.

arXiv commentThe second version (Submitted version). This is the sequel of the paper Approximate kink-kink solutions for the ${\varphi }^6$ mode in the low-speed limit

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