Line Soliton Interactions for Shallow Ocean Waves and Novel Solutions with Peakon, Ring, Conical, Columnar, and Lump Structures Based on Fractional KP Equation

Abstract

It is well known that the celebrated Kadomtsev-Petviashvili (KP) equation has many important applications. The aim of this article is to use fractional KP equation to not only simulate shallow ocean waves but also construct novel spatial structures. Firstly, the definitions of the conformable fractional partial derivatives and integrals together with a physical interpretation are introduced and then a fractional integrable KP equation consisting of fractional KPI and KPII equations is derived. Secondly, a formula for the fractional $n$ -soliton solutions of the derived fractional KP equation is obtained and fractional line one-solitons with bend, wavelet peaks, and peakon are constructed. Thirdly, fractional X-, Y- and 3-in-2-out-type interactions in the fractional line two- and three-soliton solutions of the fractional KPII equation are simulated for shallow ocean waves. Besides, a falling and spreading process of a columnar structure in the fractional line two-soliton solution is also simulated. Finally, a fractional rational solution of the fractional KP equation is obtained including the lump solution as a special case. With the development of time, the nonlinear dynamic evolution of the fractional lump solution of the fractional KPI equation can change from ring and conical structures to lump structure.