Lie symmetry analysis, optimal system, exact solutions and dynamics of solitons of a ($3+1$)-dimensional generalised BKP–Boussinesq equation

Abstract

The Lie symmetry technique is utilised to obtain three stages of similarity reductions, exact invariant solutions and dynamical wave structures of multiple solitons of a ($3+1$)-dimensional generalised BKP–Boussinesq (gBKP-B) equation. We obtain infinitesimal vectors of the gBKP-B equation and each of these infinitesimals depends on five independent arbitrary functions and two parameters that provide us with a set of Lie algebras. Thenceforth, the commutative and adjoint tables between the examined vector fields and one-dimensional optimal system of symmetry subalgebras are constructed to the original equation. Based on each of the symmetry subalgebras, the Lie symmetry technique reduces the gBKP-B equation into various nonlinear ordinary differential equations through similarity reductions. Therefore, we attain closed-form invariant solutions of the governing equation by utilising the invariance criteria of the Lie group of transformation method. The established solutions are relatively new and more generalised in terms of functional parameter solutions compared to the previous results in the literature. All these exact explicit solutions are obtained in the form of different complex wave structures like multiwave solitons, curved-shaped periodic solitons, strip solitons, wave–wave interactions, elastic interactions between oscillating multisolitons and nonlinear waves, lump waves and kinky waves. The physical interpretation of computational wave solutions is exhibited both analytically and graphically through their three-dimensional postures by selecting relevant values of arbitrary functional parameters and constant parameters.