Lie Symmetry Analysis for Soliton Solutions of Generalised Kadomtsev-Petviashvili-Boussinesq Equation in (3 + 1)-dimensions

The Lie group of infinitesimal transformations technique and similarity reduction is performed for obtaining an exact invariant solution to generalized Kadomstev-PetviashviliBoussinesq (gKPB) equation in (3+1)-dimensions. We obtain generators of infinitesimal transformations, which provide us a set of Lie algebras. In addition, we get geometric vector fields, a commutator table of Lie algebra, and a group of symmetries. A detailed geometrical framework related to the nature of the solutions possessing traveling wave, bright and dark soliton, standing wave with multiple breathers, and one-dimensional kink, for the appropriate values of the parameters involved. It is observed that there is a qualitative difference between dark soliton and bright soliton. Multiple breathers are detected, a breather is a nonlinear wave wherein energy concentrates in a local and oscillatory manner. A partially standing and partially traveling wave with changing amplitude is also observed and it is seen that breathers are localized solutions with varying amplitude.