Superconductivity and magnetism in the non-centrosymmetric heavy fermion compound CePt$_3$Si and related materials are theoretically investigated. Based on the randam phase approximation (RPA) analysis for the extended Hubbard model we describe the helical spin fluctuation induced by the Rashba-type anti-symmetric spin-orbit coupling and identify the two stable superconducting phases with either dominantly p-wave ($s$ + $P$-wave) or d-wave ($p$ + $D$ + $f$-wave) symmetry. The influcnce of the coexistent anti-ferromagnetic order is investigated in both states. The SC order parameter, quasiparticle density of state, NMR $1/T_{1}T$, specific heat, anisotropy of $H_{rm c2}$ and possible multiple phase transitions are discussed in details. The comparison with experimental results indicates that the $s$ + $P$-wave superconducting state is likely realized in CePt$_3$Si.
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