A set of nonlinear coupled equations describing the propagation of picosecond pulses in linearly birefringent optical fibers are derived from the Maxwell equations in the frequency domain. The numerical approach is given, then the collisions of orthogonally polarized fundamental solitons in birefringent fibers are investigated numerically. The effect of initial separation of the two solitons on the soliton collisions is analyzed. In addition, a transition region between trap and separate region is also found. The collisions in the transition region are resonant.