Invariance and shallow-water multi-soliton structures with weak surface tension

Abstract Shallow-water solitons, known for maintaining their shapes and velocities over long distances in shallow-water environments, are widely studied in fluid mechanics, applied mathematics and related fields. A (2+1)-dimensional Davey–Stewartson (DS) II system, which models the evolution of two-dimensional wave packets on the surface of shallow water with weak surface tension, is adopted to investigate multi-soliton structures in shallow water. The invariance properties of that system are first discussed. Using the generalized Darboux transformation (DT) method, solitonic dynamics is analysed, including interaction, formation, separation and asymptotic behaviours. Three types of two- and three-soliton structures are considered: soliton interactions, soliton bound states and multi-pole solitons. Compared to the simplified case of (1+1)-dimensional shallow-water multiple solitons, certain (2+1)-dimensional shallow-water solitons display unique generation and separation behaviours. These results provide insights into multi-soliton dynamics in real-world shallow-water environments, such as X-type and Y-type soliton interactions on flat beaches, as well as the experimental observation of multi-soliton fission and O-type solitons.