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

TL;DR

This paper derives a relation between the Green's function of a fractional differential operator on a periodic domain and Mittag-Leffler functions, revealing positivity and shape properties depending on parameters, with implications for fractional KdV equations.

Contribution

It establishes a novel connection between Green's functions of fractional operators and Mittag-Leffler functions, and analyzes their positivity and shape properties across parameter ranges.

Findings

Green's function is positive and single-lobe for all c > 0 and α in (0,2].

Numerical evidence shows Green's function changes shape for α in (2,4] depending on c.

The relation with Mittag-Leffler functions aids in understanding fractional differential operators.

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 $c$.