Nonlinear waves and conversion mechanisms for a (3+1)-dimensional shallow water wave equation

This paper delves into the nonlinear wave dynamics and conversion mechanisms within a (3+1)-dimensional shallow water wave equation. By employing the Hirota bilinear method, the $N$-soliton solutions of the equation are constructed first. Through complex conjugate constraints, the first-order breather solution is subsequently generated. Further analysis utilizing characteristic line methods reveals the conversion mechanism between the breather and the soliton. It demonstrates that the breather can transform into various types of nonlinear transformed waves under specific conditions, including quasi-kink soliton, M-shaped soliton, oscillating M-shaped soliton, multi-peak soliton and quasi-periodic soliton. The intrinsic mechanisms of the time-varying characteristics of these transformed waves are investigated by analyzing the distance between characteristic lines. Additionally, based on multi-soliton solutions, the interactions among a soliton, a breather and the transformed waves are explored. These findings not only enrich the theoretical framework of nonlinear wave dynamics in high-dimensional nonlinear equations but also provide a theoretical foundation for predicting nonlinear waves in practical systems.