Lie Symmetry Analysis, Parametric Reduction and Conservation Laws of (3+1) Dimensional Nonlinear Dispersive Soliton Equation

Abstract: The core focus of this research work is to obtain invariant solutions and conservation laws of the (3+1)-dimensional ZK equation which describes the phenomenon of wave stability and solitons propagation. ZK equation is significant in fluid dynamics, nonlinear optics, and acoustics. ZK equation plays an important role in understanding the behavior of nonlinear, dispersive waves especially in the presence of the magnetic field. Lie symmetry analysis has been applied to the (3+1)-dimensional ZK equation to derive invariant solutions. These solutions reveal how waves retain their shape as they travel, how they interact in space, and the impact of magnetic fields on wave propagation. Using the Lie symmetry, the (3+1) dimensional ZK equation is reduced to various ordinary differential equations. It also concludes with the conservation laws and non-linear self-adjoint ness property. Additionally, we align our findings with real-life problems, namely plasma physics, electromagnetic wave propagation, and water wave dynamics, providing the applicability of our theoretical results.