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

Abstract

The linear operator $c+(-\Delta {)}^{\alpha /2}$, where $c>0$ and $(-\Delta {)}^{\alpha /2}$ 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 $c$.

arXiv comment20 pages; 6 figures

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Cited by in corpus (1)

• On the Green function of the killed fractional Laplacian on the periodic domain