GREEN'S FUNCTION FOR THE FRACTIONAL KDV EQUATION ON THE PERIODIC DOMAIN VIA MITTAG-LEFFLER FUNCTION

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 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 alpha is an element of (0, 2]. On the other hand, we argue from numerical approximations that in the case of alpha is an element of (2, 4], the Green function is positive and single-lobe for small c and non-positive and non-single lobe for large c.

Automated summary

The study examines the linear operator in the fractional Korteweg-de Vries equation for periodic travelling waves, establishing its relation with the Mittag-Leffler function. It is proven that the Green function is strictly positive and single-lobe for every c > 0 and every alpha in (0, 2], while numerical approximations suggest that for alpha in (2, 4], the Green function is positive and single-lobe for small c and not positive or single-lobe for large c.