Lump Solitons and their Anomalous Interactions in the Bogoyavlenskii-Kadomtsev-Petviashvili Equation

Abstract

This paper mainly investigates lump solitons and their anomalous interactions in the (2+1)-dimensional Bogoyavlenskii-Kadomtsev-Petviashvili (BKP) equation. First, using the truncated Painlevé expansion method, we reveal the relationship among the nonlinear BKP equation, the trilinear BKP equation, and the linear free Schrödinger equations. Subsequently, a special family of generalized solutions is constructed from the solutions to these linear free Schrödinger equations. Second, using the Fourier transform and the generalized linear superposition principle, we derive the N-lump soliton solutions for the (2+1)-dimensional BKP equation, which take the form of a special class of generalized Schur polynomials. Furthermore, we focus on the anomalous interactions of the 2-lump and 3-lump soliton solutions. Through a parameter region analysis, mutually repulsive interactions, fusion–fission interactions, exchange lump soliton interactions, and lump-soliton rotation induced by anomalous two-lump interactions are revealed. In particular, the anomalous interaction patterns of 2- and 3-lump soliton solutions are systematically classified by means of the parameters \(D\) and \(M\). Detailed theoretical analysis and graphical illustrations are presented to interpret these anomalous interaction phenomena. Finally, we explore the long-time asymptotic properties of the N-lump soliton solutions and characterize their superimposed structures composed of N individual single-lump solitons. These results lay a theoretical foundation for understanding the rich dynamics of high-dimensional nonlinear localized waves.