Soliton, lumps, stability analysis and modulation instability for an extended (2+1)-dimensional Boussinesq model in shallow water

In this work, we study the extended (2+1)-dimensional Boussinesq model, which describes the propagation of waves with small amplitudes in shallow water propagating at a constant speed through a uniformly deep water canal. The governing equation is frequently used in computer simulations for modeling water waves in harbors and shallow seas in ocean engineering. Firstly, we apply the Hirota bilinear technique to establish the bilinear structure of the governing equation. Then, we formulate lump wave solitons and impact of lump wave across single, double strip solitons as well as the impact of lump across periodic waves. Furthermore, some traveling and semi-analytical solitons are developed by applying the unified technique, the hyperbolic ansatz approach and the Adomian decomposition technique. To calculate the absolute error, we have set up a difference table among the exact and approximate results. Moreover, we deliberate the stability analysis and the modulation instability of the governing equation briefly. The physical nature of various solitons is demonstrated by plotting the 3D, contours as well as 2D portraits. The applied techniques have the potential to be very impactful computational tools for efficiently deriving solutions to nonlinear evolution equations, frequently occurring in engineering, sciences and numerous other scientific domains with practical significance. •The Hirota bilinear approach is used to investigate the new extended (2+1)-dimensional Boussinesq model.•A variety of lump, periodic, including the collision among lump and periodic wave solutions are constructed.•Furthermore, some new traveling wave solution are also developed with by employing the unified method and the hyperbolic ansatz method.•Additionally, the stability of the established results is carried out to validate the governing model is a stable structure.•Mathematica 11.0 is used to carry out simulations and to visualize the behavior of solutions.