Sketch of the propagation of a plasma pulse whose group velocity is independent of its amplitude and whose damping rate rapidly decreases with . The shape of the pulse at time is given by the blue dashed line, and at time by the green solid line, while the shape of the damping rate is given by the red dashed-dotted line. Due to damping the pulse maximum, , at time , is not located at time at the new maximum, , but at point on the left of . Hence, for this particular example, the group velocity is less than the speed of propagation of the pulse maximum. Moreover, at time , point was located at , i.e., on the left of point corresponding to the same pulse amplitude as point . Therefore, .
Dimensionless plasma wave amplitude, , as a function of at times (blue solid line) and (green dashed line), obtained from the Vlasov run for the 5 keV case of Table I.
Our theoretical predictions for the group velocity (blue solid line), and , normalized to the thermal velocity (green dashed line) when the electron density is 10% of the critical one and when (a) , (b) , (c) , and (d) .
Our theoretical predictions for (blue solid line) and (green dashed line) for a plasma wave with .
Orbit, calculated between times and of a particle acted upon by the force with and , and whose initial position and velocity are and . The blue solid line is the actual orbit of the trapped electron. The green curve is the symmetric image, with respect to the -axis, of that part of the orbit lying on the half-plane, . The black dashed curve is the virtual separatrix corresponding to the amplitude at .
Values of the nonlinear group velocity, normalized to the thermal one, either calculated theoretically or numerically, and compared to , also normalized to the thermal velocity. All results correspond to a plasma whose electron density is 10% of the critical one.
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