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International Journal of Differential Equations
Volume 2010 (2010), Article ID 193893, 18 pages
http://dx.doi.org/10.1155/2010/193893
Research Article

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

Department of Mathematics, The University of Texas-Pan American, Edinburg, TX 78539-2999, USA

Received 9 August 2009; Accepted 9 November 2009

Academic Editor: Mark M. Meerschaert

Copyright © 2010 Vladimir Varlamov. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 𝜁 + ( 𝑠 , 𝑎 ) and 𝜁 ( 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.