Lie symmetry reductions and exact solutions of Kadomtsev–Petviashvili equation

Abstract

This work intents to obtain symmetry reductions and exact solutions of Kadomtsev–Petviashvili (KP) equation, which describes propagation of long waves on the surface of shallow water. The Lie symmetry method under one-parameter transformation is used to ensure invariance and derive infinitesimal generators. These generators provide similarity variables, which is directed to symmetry reductions of the test equation. This process of reductions recasts test equation into ordinary differential equations (ODEs). These ODEs have finally been solved under various constraints and as a result, exact solutions consisting of arbitrary functions and several arbitrary constants are produced. The solutions are novel and have not yet been published. Due to the existence of arbitrary functions \(f_1(t)\), \(f_2(t)\), \(f_3(t)\) and constants, these solutions present a more generalised form than the existing results and give a wide range of possibilities for the interpretation of various physical phenomena. The physical importance of these solutions is demonstrated by numerical simulation revealing a rich variety of soliton structures including line soliton, doubly soliton, multisoliton, solitons on parabolic surface, soliton fission and annihilation behaviour.