U. Le and D.E. Pelinovsky

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

Fract. Calc. Appl. Anal. 24 (2021) 1507-1534

Abstract:
The linear operator c + (-Δ)^α/2 with 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'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 α in (0,2]. On the other hand, we argue from numerical approximations that in the case of α in (2,4], the Green's function is positive and single-lobe for small c > 0 and non-positive and non-single lobe for large c > 0.

Keywords:
Fractional Korteweg-de Vries equation, Green's function, Mittag-Leffler function, positivity property, single-lobe property.