Investigating nonlinear wave structures via auto-Bäcklund transformation and Hirota bilinear method in the coupled Boussinesq system

We investigate various nonlinear wave structures of the coupled Boussinesq system (CBS) using two efficient techniques: the auto-Bäcklund transformation and the Hirota bilinear method (HBM). Various travelling wave solutions, including kink waves, solitary waves with lump excitation and multi-shock waves, are obtained for the CBS using the auto-Bäcklund transformation. We also obtain the interaction between two-soliton solutions utilising this technique. Initially, we have shown that the CBS passes the Painlevé test for integrability. Then, we derive the auto-Bäcklund transformation for CBS through the truncated Painlevé expansion. Next, we observe the interaction and propagation behaviours of multi-soliton solutions for CBS using the HBM. The existence of multi-soliton solutions establishes that the CBS is also integrable in the Hirota sense. Interestingly, we observe that the HBM offers a more comprehensive framework for exploring diverse interactions between bright solitons and dark solitons, compared to the auto-Bäcklund approach. The multi-solitons and solitary waves with lump excitation obtained in this study can be used to describe surface water waves, while the kink wave and multi-shock waves represent topological changes in the water surface modelled by the CBS. Our findings on nonlinear structures for the CBS, provide valuable insights into the behaviour of nonlinear waves in water dynamics and are important for analysing various phenomena, such as ocean waves and waves in coastal areas.