Multi-bump ground states of the fractional Gierer-Meinhardt system in $\mathbb{R}$

In this paper we study ground-states of the fractional Gierer-Meinhardt system on the line, namely the solutions of the problem \begin{equation*} \left{\begin{array}{ll} (-\Delta)^su+u-\frac{u^2}{v}=0,\quad &$\mathrm{i}\mathrm{n}$~$\mathbb{R}$,\ (-\Delta)^sv+\varepsilon^{2s}v-u^2=0,\quad &$\mathrm{i}\mathrm{n}$~$\mathbb{R}$,\ u,v>0,\quad u,v\rightarrow0~&$\mathrm{a}\mathrm{s}$~|x|\rightarrow+\infty. \end{array}\right. \end{equation*} We prove that given any positive integer $k,$ there exists a solution to this problem for $s\in [\frac{1}{2},1)$ exhibiting exactly $k$ bumps in its $u-$component, separated from each other at a distance $O({\epsilon }^{\frac{1-2s}{4s}})$ for $s\in (\frac{1}{2},1)$ and $O(\mathrm{\mid }\log \epsilon {\mathrm{\mid }}^{\frac{1}{2}})$ for $s=\frac{1}{2}$ respectively, whenever $\epsilon$ is sufficiently small. These bumps resemble the shape of the unique solution of \begin{equation*} (-\Delta)^sU+U-U^2=0,\quad 0<U(y)\rightarrow0~$\mathrm{a}\mathrm{s}$~|y|\rightarrow\infty. \end{equation*}