The dynamics of nonlinear molecular waves in a (3+1)-dimensional nonlinear evolution equation in fluid mediums

Under investigation in this paper is a (3+1)-dimensional nonlinear wave equation, which has potential applications in understanding the evolution of waves in bodies of water such as seas and oceans. Based on the velocity resonance condition, certain interesting nonlinear molecular waves related to solitons, breathers and transformed waves are derived. These waves have the fixed spacing during the propagation, as evidenced by the characteristic lines of each wave component. First, we delve into the dynamical evolutions of two- and three-soliton molecules, achieved through manipulation of the associated parameters. Further, we analytically study the breather and transformed molecular waves by adjusting specific parameters, resulting in various types of molecular waves such as the breather-breather, M-M-shaped, and oscillatory M (OM)-OM-shaped molecular waves. It is highlighted that molecular waves with transformed wave atoms exhibit distinct changes in their shapes. In addition, we explore the hybrid molecules, including the soliton-M-shaped, soliton-breather, and multi-peaks-breather molecules. These findings of hybrid molecules not only contribute to the understanding of soliton and breather molecules but also expand the knowledge of various shape-changed transformed molecular waves. Our results offer valuable insights into the behavior of molecular waves and their various forms, shedding light on the intricate dynamics of waves in fluids.