Multi-domain spectral approach for Zakharov-Kuznetsov equations in 3D with cylindrical symmetry
Provides a computational tool for studying wave collapse in 3D magnetized plasmas, with implications for nonlinear dispersive equations.
A novel numerical framework for the 3D Zakharov-Kuznetsov equation with cylindrical symmetry is introduced, enabling efficient handling of fractional nonlinearities and domain decomposition. The method shows that the ground state soliton is the sharp threshold for global vs. finite-time existence, successfully tracking singular solution profiles.
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.