Riesz Potentials for Korteweg-de Vries Solitons and Sturm-Liouville Problems

Abstract

Riesz potentials (also called Riesz fractional derivatives) and their Hilbert transforms are computed for the Korteweg-de Vries soliton. They are expressed in terms of the full-range Hurwitz Zeta functions ${\zeta }_{+}\left(s,a\right)$ and ${\zeta }_{-}\left(s,a\right)$. It is proved that these Riesz potentials and their Hilbert transforms are linearly independent solutions of a Sturm-Liouville problem. Various new properties are established for this family of functions. The fact that the Wronskian of the system is positive leads to a new inequality for the Hurwitz Zeta functions.