Riesz potentials for Korteweg-de Vries solitons
Abstract
Riesz potentials (also called Riesz fractional derivatives) are defined as fractional powers of Laplacian. They are traditionally used for studying existence and uniqueness for equations of the Korteweg-de Vries type (KdV-type henceforth). Zero mean properties are established for Riesz potentials of solutions of KdV-type equations, $${D_{x}^{\alpha}u(x,t),\, {\rm for}\, \alpha\in(0,3/2)}$$. As an important example Riesz fractional derivatives and their Hilbert transforms are computed for the well-known soliton solution of KdV. Obtained representations involve the Hurwitz Zeta function. Zero mean properties are established and asymptotic expansions are derived. A particular case of the obtained formula provides an algebraic soliton solution for extended KdV.
- Publication:
-
Zeitschrift Angewandte Mathematik und Physik
- Pub Date:
- February 2010
- DOI:
- 10.1007/s00033-009-0003-5
- Bibcode:
- 2010ZaMP...61...41V
- Keywords:
-
- 35Q53;
- 26A33;
- 33E20;
- Riesz potentials;
- Korteweg-de Vries solitons;
- Hilbert transforms