Riesz potentials for Korteweg-de Vries solitons and Sturm-Liouville problems (Q606223)

Summary: 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 \(\zeta _{+}(s,a)\) and \(\zeta_{-}(s,a)\). 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.