Characteristics of propagation of ultrashort intense laser pulses in a partially stripped pre-formed plasma channel, assumed to be parabolic and unperturbed, are discussed. From Maxwell equations the refractive index equation and Hamilton-Jacobi equation, which describe the evolution of the electric field, are derived including the effects of the diffraction, the third-order intensity-dependent nonlinearity, plasma defocusing, the self-focusing of the plasma channel, and t he relativistic self-focusing. The envelope equation of laser propagating in the plasma channel, and the general expression relating the laser spot size with th e propagation distance and the width of the plasma channel etc., are derived bas ed on the Hamilton-Jacobi equation and the refractive index equation.
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