On Lie symmetries and invariant solutions of Broer–Kaup–Kupershmidt equation in shallow water of uniform depth

Abstract

The dynamics of atmosphere and ocean can be examined under different circumstances of shallow water waves like shallow water gravity waves, Kelvin waves, Rossby waves and inertio-gravity waves. The influences of these waves describe the climate change adaptation on marine environment and planet. Therefore, the present work aims to derive symmetry reductions of Broer–Kaup–Kupershmidt equation in shallow water of uniform depth and then a variety of exact solutions are constructed. It represents the propagation of nonlinear and dispersive long gravity waves in two horizontal directions in shallow water. The invariance of test equations under one parameter transformation leads to reduction of independent variable. Therefore, twice implementations of symmetry method result into equivalent system of ordinary differential equations. Eventually, the exact solutions of these ODEs are computed under parametric constraints. The derive results entail several arbitrary constants and functions, which make the findings more admirable. Based on the appropriate choice of existing parameters, these solutions are supplemented numerically and show parabolic nature, intensive and non-intensive behavior of solitons.