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Recent one-dimensional analytical and numerical studies$^{1}$ have demonstrated efficient acceleration of nonthermal positrons by relativistic magnetosonic shock waves in electron-positron-proton plasmas, which are of great relevance to pulsar wind driven nebulae. In the one-dimensional simulations$^{2}$, electron-positron shocks yield a down stream plasma with all the momenta in the plane perpendicular to the background magnetic field $B_0$. Plasmas with such anisotropy are subject to Weibel-like instabilities, and the distribution functions tend to be isotropized by these instabilities. Depending on the time scale of isotropization, the shock acceleration of positrons may be modified significantly when two-dimensional effects are taken into account. With this in mind, we study the Weibel instabilities in relativistic electron-positron plasmas. Detailed stability properties are examined over a wide range of system parameters by linear stability analyses. Nonlinear numerical simulations are carried out to investigate long-term evolution of the instability. Possible applications to the relativistic shock terminating the pulsar wind in the Crab Nebula will be briefly outlined. \vskip.3in
\hrule \begin{description} \item[$^{1}$] Hoshino, M., and Arons, J., 1991, Physics of Fluids, B3, 818. \item[$^{2}$] Gallant, Y., Hoshino, M., Langdon, A.B., Arons, J., and Max, C.E., 1992, Astrophy. J., 391, 73. \end{description}
