Green’s Function for the Fractional KDV Equation on the Periodic Domain via Mittag-Leffler Function

The linear operator c + (−Δ)α/2, where c > 0 and (−Δ)α/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 function of this linear operator with the Mittag-Leffler function, which was previously used in the context of the Riemann-Liouville and Caputo fractional derivatives. By using this relation, we prove that the Green function is strictly positive and single-lobe (monotonically decreasing away from the maximum point) for every c > 0 and every α ∈ (0, 2]. On the other hand, we argue from numerical approximations that in the case of α ∈ (2, 4], the Green function is positive and single-lobe for small c and non-positive and non-single lobe for large c.