The separatrices between trapped and free electrons in phase space, associated with a relativistic electron hole.
Trapping of electromagnetic waves in a REH. (a) The envelope of large-amplitude trapped EM waves, (b) the potential, (c) the electron number density, and (d) the square of the local electron plasma frequency for large-amplitude EMWs with a maximum amplitude of (solid lines) and (dashed lines), and as a comparison a REH with small-amplitude EMWs that have (dotted lines). The nonlinear frequency shift for the case is and for the case it is , to be compared with the small-amplitude case that has . Parameters are , , and . Panel (e) shows small-amplitude trapped EMWs (in arbitrary units) in a REH, corresponding to the dotted lines in (b)–(d). Here three eigenstates of trapped EMWs exist, corresponding to the eigenvalues (solid line), (dashed line), and (dash-dotted line). (After Ref. 40.)
(Color online) Two colliding REHs: (a) phase-space plots of the electron distribution function, and (b) the modulus of the EMW vector potential for , , , and , and the time-dependent dynamics of (c) the modulus of the EMW vector potential, (d) the scalar potential, (e) the squared local plasma frequency, and (f) the electron density. (After Ref. 40.)
The separatrices between trapped and free ions (left) and between reflected and free electrons (right) in phase space, associated with a relativistic ion hole.
(Color online) The ion (left panels) and electron (right panels) distribution functions associated with two colliding nonrelativistic ion holes, at times , , and . (After Ref. 43.)
The electron distribution function at at times , , and . A flat-topped electron distribution function is created after the collision of the ion holes. (After Ref. 43.)
(Color online) The profiles of (a) the potential, (b) the ion density, and (c) the electron density, associated with a RIH with the relativistic factor (solid lines), (dashed lines), and (dash-dotted lines). We used , , , and for all cases, where the hole speed is normalized by the ion thermal speed. Phase-space plots of (d) the ion distribution function and (e) the electron distribution function for ion holes with , , and (upper to lower panels). The phase-space plots are obtained by solving Eq. (14) numerically for and then evaluating and at different and . (After Ref. 41.)
(Color online) Interactions between two RIHs, showing (a) the ion and (b) the electron distribution functions, at times , , , and (top to bottom panels). The parameters used in the initial condition for the two ion holes are , , and , and the left and right holes have the speed and . We used the mass ratio . (After Ref. 41.)
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