Soliton Dynamics and Modulation Instability in the (3+1)-dimensional ZK equation: A Lie Symmetry Approach

TL;DR

This paper uses Lie symmetry analysis to find invariant solutions and conservation laws of the (3+1)-dimensional ZK equation, revealing wave behaviors, soliton interactions, and stability properties, with validation through numerical methods.

Contribution

It introduces a comprehensive Lie symmetry approach to derive exact solutions, conservation laws, and analyze modulation instability for the higher-dimensional ZK equation, expanding understanding of wave dynamics.

Findings

Exact invariant solutions describing wave propagation and interactions.

Conservation laws and nonlinear self-adjointness properties identified.

Analysis of modulation instability and gain spectrum conducted.

Abstract

The core focus of this research work is to obtain invariant solutions and conservation laws of the (3+1)-dimensional ZK equation, a higher-dimensional generalization of the Korteweg--de Vries (KdV) equation, which describes the phenomenon of wave stability and soliton propagation. Lie symmetry analysis has been applied to derive infinitesimal generators and classify the optimal subalgebras. Utilizing them, we construct exact invariant solutions that reveal how waves retain their shape as they travel, how they interact in space, and the impact of magnetic fields on wave propagation. Further, by implementing the traveling wave transformation, we derive additional exact solutions, including those exhibiting kink-type solitons. It also concludes with the conservation laws and the nonlinear self-adjointness property. Our examination is broadened to cover modulation instability and gain spectrum. To contextualize our results, we compare the solutions obtained from the Lie symmetry method with those derived using the modified simple equation (MSE) method and other symbolic techniques reported in recent literature. To validate the accuracy of the analytical solutions obtained via Lie symmetry, a numerical method is implemented. A review of the ZK equation's physical background and mathematical complexity is explored, emphasizing the limitations of symbolic approaches.