The Integrability of a New Fractional Soliton Hierarchy and Its Application

Abstract

Two fractional soliton equations are presented generated from the same spectral problem involved in a fractional potential by the zero-curvature representations. They are a kind of special reductions of the famous AKNS system. The two equations are integrable for they both possess explicit soliton solutions constructed by the $N-$ fold Darboux transformation. As an application of the obtained solutions, new soliton solutions of the classic $\left(2+1\right)$ -dimensional Kadometsev-Petviashvili (KP) equation are soughed out by a cubic polynomial relation. Dynamic properties are analyzed in detail.