  In this work, we examine the integrable mixed coupled nonlinear Schr\"odinger
(mCNLS) system, which describe the propagation of an optical pulse in a
birefringent optical fiber. By the Riemann-Hilbert(RH) approach, the N-soliton
solutions of the mCNLS system can be expressed explicitly when the jump matrix
of a specific RH problem is a $3\times3$ unit matrix. As a special example, the
expression of one- and two-soliton are displayed explicitly. More generally, as
a promotion, an integrable generalized multi-component NLS system with its
linear spectral problem be discussed. It is hoped that our results can help
enrich the nonlinear dynamical behaviors of the mCNLS.
