New exact solutions of the (3+1)-dimensional defocusing Gardner–KP equation using Lie symmetry analysis

Abstract

In this paper, an attempt is made to present the rigorous and comprehensive group analysis of the (3+1)-dimensional defocusing Gardner–KP equation. According to the Lie invariance condition, the Lie algebra of infinitesimal symmetries spanned by eight vector fields is found. The commutator and adjoint representation tables are derived, and a detailed process for searching the optimal system of one-dimensional subalgebras is shown. Several symmetry reductions and group-invariant solutions with physical or mathematical interests are obtained by using infinitesimal generators in the optimal system. Some new particular solutions are deduced by using effective invariant-solution ansatz and the solutions of a second-order elliptic equation with power-law nonlinearity. Especially, by recasting the reduced two-dimensional counterpart equation into Hirota’s bilinear form, the fundamental solitary waves are found. And a special kind of flat-top soliton excitation is exhibited. It is also shown that this (3+1)-dimensional equation can have only unidirectional multiple-soliton solutions and does not allow soliton resonance to occur. The physical interpretations of resulting solutions are illustrated by three-dimensional graphics through numerical simulation. Different types of two-soliton interactions are also demonstrated in graphical ways.