Nonlinear waves in (3+1)-dimensions: Multi-soliton solutions of a partial differential equation

In this paper, we use the Lie symmetry analysis to obtain the closed-form solutions of the (3+1)-dimensional generalized Kadomtsev-Petviashvili Benjamin-Bona-Mahony (gKP-BBM) equation. We obtain the infinitesimal generators, commutator table of Lie algebra, adjoint representation of subalgebras, symmetry group, and similarity reduction for the gKP-BBM equation. Also, we evaluated the interactions of soliton solutions of the gKP-BBM equation. It is observed that multi-soliton complexes are self-localized in state wherein, several fundamental solitons including bright solitons and dark solitons are nonlinearly superimposed. A notable observation is the existence of topological defects in soliton interactions, where two solitons appear to be separated, with the adjoining structure being "out of phase" with each other. The kink, acting as a twist in the soliton's value, overcomes topological defects, leading to a transition from one phase to another. This comprehensive analysis contributes to a deeper understanding of the dynamics and interactions of solitons in the (3+1)-dimensional gKP-BBM equation.