Approximate kink-kink solutions for the ${\phi }^6$ model in the low-speed limit

This manuscript is the first of a series of two papers that study the problem of elasticity and stability of the collision of two kinks with low speed $v$ for the nonlinear wave equation known as the ${\phi }^6$ model in dimension $1+1$. In this paper, we construct a sequence of approximate solutions $({\phi }_k(v,t,x){)}_{k\in {\mathbb{N}}_{\ge 2}}$ for this nonlinear wave equation such that each function ${\phi }_k(v,t,x)$ converges in the energy norm to the traveling kink-kink with speed $v$ when $t$ goes to $+\mathrm{\infty }\mathrm{.}$ The methods used in this paper are not restricted only to the ${\phi }^6$ model.