Nonlinear Evolution of Obliquely Propagating Langmuir Wave

A wave propagating obliquely with respect to a magnetic field has many resonances, located at $v_{\parallel }=\frac{(\omega -n\Omega )}{k_{\parallel }}$. When they overlap, there is a transition between regular and stochastic particle motion, leading to a strong decrease in the final wave amplitude. Simulations confirm this result. In the regime where particle motion is regular, the usual trapping oscillations disappear as the angle of propagation increases, because of phase mixing of the bounce frequencies in different resonances.