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

TL;DR

This paper constructs multi-bump ground-state solutions for the fractional Gierer-Meinhardt system on the real line, revealing how the number and separation of bumps depend on the fractional order and a small parameter.

Contribution

It demonstrates the existence of solutions with any prescribed number of bumps and characterizes their spatial separation in the fractional setting.

Findings

Existence of multi-bump solutions for any positive integer number of bumps.

Bumps are separated by distances depending on the fractional order and small parameter.

Solutions resemble the shape of the unique fractional scalar equation's solution.

Abstract

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{in}~\mathbb{R},\\ (-\Delta)^sv+\varepsilon^{2s}v-u^2=0,\quad &\mathrm{in}~\mathbb{R},\\ u,v>0,\quad u,v\rightarrow0~&\mathrm{as}~|x|\rightarrow+\infty. \end{array}\right. \end{equation*} We prove that given any positive integer $k,$ there exists a solution to this problem for $s\in[\frac12,1)$ exhibiting exactly $k$ bumps in its $u-$component, separated from each other at a distance $O(\varepsilon^{\frac{1-2s}{4s}})$ for $s\in(\frac12,1)$ and $O(|\log\varepsilon|^{\frac12})$ for $s=\frac12$ respectively, whenever $\varepsilon$ 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{as}~|y|\rightarrow\infty. \end{equation*}