Multi-domain spectral approach for Zakharov-Kuznetsov equations in 3D with cylindrical symmetry

TL;DR

This paper introduces a new numerical framework for 3D Zakharov-Kuznetsov equations with cylindrical symmetry, enabling efficient simulation of traveling waves, fractional nonlinearities, and wave collapse phenomena.

Contribution

The authors develop a domain decomposition spectral method tailored for 3D gZK equations with cylindrical symmetry, handling fractional nonlinearities and analyzing solution thresholds.

Findings

Ground state acts as a sharp threshold for solution behavior.

Method accurately tracks singular solutions and wave collapse.

Efficiently handles fractional nonlinearities in 3D.

Abstract

We present a novel numerical framework for studying nonlinear dispersive equations in higher-dimensional settings, specifically designed for solutions featuring traveling waves along a preferred axis (or field-aligned traveling waves). Using the three-dimensional generalized Zakharov-Kuznetsov (gZK) equation as a model, we convert it into cylindrical coordinates and implement a domain decomposition strategy. By partitioning the computational domain into distinct regions based on expected solution behavior, we significantly reduce computational complexity while maintaining the high resolution necessary for capturing small-scale dynamics. Another key innovation of our method is the ability to efficiently handle fractional nonlinearities, specifically, the critical power $p = 7/3$ in 3D, which typically introduces significant computational overhead and numerical instabilities that compromise simulation accuracy. Using this framework, we are able to investigate the dynamics of solutions (with cylindrical symmetry) close to the ground state soliton and show that for the 3D critical ZK equation, the ground state serves as the sharp threshold for global vs. finite time existence of solutions. Our method successfully tracks the profiles of these singular solutions, providing new insights into the dynamics of wave collapse in three-dimensional magnetized media.