Green's function for the fractional KdV equation on the periodic domain via Mittag-Leffler's function

The linear operator $c+(-\mathrm{\Delta }{)}^{\alpha \mathrm{/}2}$, where $c>0$ and $(-\mathrm{\Delta }{)}^{\alpha \mathrm{/}2}$ is the fractional Laplacian on the periodic domain, arises in the existence of periodic travelling waves in the fractional Korteweg–de Vries equation. We establish a relation of the Green’s function of this linear operator with the Mittag–Leffler function, which was previously used in the context of Riemann–Liouville’s and Caputo’s fractional derivatives. By using this relation, we prove that Green’s function is strictly positive and single-lobe (monotonically decreasing away from the maximum point) for every $c>0$ and every $\alpha \in (0,2]$. On the other hand, we argue from numerical approximations that in the case of $\alpha \in (2,4]$, the Green’s function is positive and single-lobe for small $c$ and non-positive and non-single lobe for large $c$.