Transverse instability magnetic field thresholds of electron phase-space holes

A detailed comparison is presented of analytical and particle-in-cell (PIC) simulation investigation of the transverse instability, in two dimensions, of initially one-dimensional electron phase-space hole equilibria. Good quantitative agreement is found between the shift-mode analysis and the simulations for the magnetic field ($B$) threshold at which the instability becomes overstable (time-oscillatory) and for the real and imaginary parts of the frequency. The simulation $B$-threshold for full stabilization exceeds the predictions of shift-mode analysis by 20 to 30\%, because the mode becomes substantially narrower in spatial extent than a pure shift. This threshold shift is qualitatively explained by the kinematic mechanism of instability.