Localization of Intense Electromagnetic Waves in a Relativistically Hot Plasma

We consider nonlinear interactions between intense short electromagnetic waves (EMWs) and a relativistically hot electron plasma that supports relativistic electron holes (REHs). It is shown that such EMW-REH interactions are governed by a coupled nonlinear system of equations composed of a nonlinear Schrödinger equation describing the dynamics of the EMWs and the Poisson-relativistic Vlasov system describing the dynamics of driven REHs. The present nonlinear system of equations admits both a linearly trapped discrete number of eigenmodes of the EMWs in a quasistationary REH and a modification of the REH by large-amplitude trapped EMWs. Computer simulations of the relativistic Vlasov and Maxwell-Poisson system of equations show complex interactions between REHs loaded with localized EMWs.

Figure 1

Large-amplitude trapped EMW envelope (upper panel), the potential (second panel), the electron number density (third panel), and the square of the local electron plasma frequency (lower panel) for large-amplitude EMWs with a maximum amplitude of $W_{\mathrm{m}\mathrm{a}\mathrm{x}}=1.5$ (solid lines) and $W_{\mathrm{m}\mathrm{a}\mathrm{x}}=1.0$ (dashed lines), and as a comparison a REH with small-amplitude EMWs which have $W_{\mathrm{m}\mathrm{a}\mathrm{x}}\ll 1$ (dotted lines). The nonlinear frequency shift for the $W_{\mathrm{m}\mathrm{a}\mathrm{x}}=1.5$ case is $\lambda =0.099$ and for the $W_{\mathrm{m}\mathrm{a}\mathrm{x}}=1.0$ case it is $\lambda =0.095$, to be compared with the small-amplitude case which has $\lambda =0.088$. Parameters are: $v_0=0.7$, $\alpha =0.4$, and $\beta =-0.5$. The selected values of $\beta$ are related to the maximum REH potential according to a specific relation similar to one in Ref. 12, which will be presented in a separate paper.

Figure 2

Small-amplitude trapped EMWs (in arbitrary units) in a REH. Three eigenstates of trapped EMWs exist, corresponding to the eigenvalues ${\lambda }_1=0.088$ (solid line), ${\lambda }_2=0.053$ (dashed line), and ${\lambda }_2=0.013$ (dash-dotted line). The parameters used are the same as for the dotted lines in Fig. 1.

Figure 3

Phase space plots of the electron distribution function (left panels) and the modulus of the electromagnetic field (right panels) for $t=0$, $t=50$, $t=125$, and $t=162$.

Figure 4

The electromagnetic field (upper left panel), potential (upper right panel), squared local plasma frequency (lower left panel), and electron density (lower right panel) for two colliding REHs.