Riesz potentials for Korteweg-de Vries solitons

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.