<p>The dynamics of bright and W-shaped dispersive solitons in birefringent optical fibers described by the coupled Radhakrishnan–Kundu–Lakshmanan equation (cRKL) without four-wave mixing (FWM) is studied. One of the main novelties of the study consists in finding exact analytical solutions with the aid of the Jacobi elliptic cn function method and an explicit detailed study of their behavior in the presence of the physical effects characteristic of ultrafast fiber optics, i.e. self- phase modulation (SPM), third order dispersion (TOD), cross-phase modulation (XPM), and self-steepening (SS). The cRKL equation is a universal nonlinear wave equation for orthogonal polarization modes in birefringent fiber. Graphical analysis shows that the soliton profile changes significantly with various parameter values. Negative TOD results in soliton-broadening and amplitude increase, and positive TOD leads to pulse compression, and provides a mechanism for dispersion-managed optical systems. Self-steepening induces temporal asymmetry and pulse narrowing, useful in the application of femtosecond pulse shaping. XPM controls intermodal energy transfer and affects generating soliton and cross-talk suppression in polarization-division multiplexed systems. SPM leads to pulse shortening and spectral confinement, this being critical for short-pulse compression and supercontinuum generation. Based on exact analytical solutions incorporating nonlinear and dispersion effects, this analytic-and-graphic description gives a coherent picture for engineering soliton propagation in nonlinear birefringent media. Our investigations resulted in W-shaped and bright solitons without FWM with the action of SPM, TOD, XPM, and SS, being calculated. The advancement implies that there are more potential ways of controlling the soliton dynamics by means of ultrafast photonic components, such as femtosecond pulse shaping, dispersion managed amplification, and polarization division multiplexing.</p>