Invariant solutions of the Kaup–Boussinesq equations and their applications in fluid

This research is conducted by utilizing the similarity transformations method (STM) to solve coupled Kaup–Boussinesq equations (cKBEs). All attained solutions are different from those reported [A. M. Wazwaz, Waves Random Complex Media 25, 473 (2015), http://dx.doi.org/10.1080/17455030.2015.1016474 ; R. Ivanov and T. Lyons, J. Nonlinear Math. Phys. 19, 72 (2012), https://doi.org/10.1142/S1402925112400086 ; C. Chen and Y. L. Jiang, Waves Random Complex Media 29, 138 (2019), https://doi.org/10.1080/17455030.2017.1418098 ] earlier, thus all the solutions for cKBEs are novel. The cKBEs describe polydispersive sedimentation. In this paper, we use one-parameter Lie group symmetry analysis to convert the cKBEs to an analogous system of equations. We derive a new variety of invariant analytical solutions to the cKBEs by solving the reduced system. This work aims to get analytical solutions followed by symmetry reductions under one-parameter Lie group transformations. The analytical solutions reflect different types of wave patterns, including progressive, highly progressive, stationary, and kink natures. The inclusion of arbitrary constants gives worthiness and future scope to these solutions. These solutions are critical and may require further research in shallow water waves.