Effects of Polarization-Dependent Dispersion, Kerr Nonlinearity, and Cross-Phase Modulation on Exact Dark-Bright Vector-Soliton Propagation in Birefringent Optical Fibres

<jats:p>&lt;p dir="ltr"&gt;This study analytically examines how polarization-dependent group-velocity dispersion, Kerr nonlinearity, and incoherent cross-phase modulation govern exact dark-bright vector-soliton propagation in a birefringent optical fibre. A common travelling-wave reduction combined with the lowest-order implicit Bogning-function pair yields closed expressions for the dark-background amplitude, bright-pulse amplitude, common inverse width, carrier-frequency shifts, and propagation constants. For symmetric nonlinear coefficients, a real mixed state exists only when R&lt;sub&gt;β&lt;/sub&gt;&gt;1/σ; at equality, the bright component vanishes continuously, while the equal-dispersion model with the standard cross-phase coefficient does not support the selected tanh-sech pair. The analytical relations separate the roles of the waveguide properties: R&lt;sub&gt;β&lt;/sub&gt; controls branch onset and amplitude partition, γ fixes the absolute power scale, and σ regulates mutual nonlinear trapping. A total-intensity morphology analysis introduces the notch-filling factor F=B&lt;sup&gt;2&lt;/sup&gt;/A&lt;sup&gt;2&lt;/sup&gt; and normalized notch depth D=1-F, showing that stronger dispersion contrast and cross-phase coupling progressively fill the dark notch, whereas the Kerr coefficient does not alter the normalized profile. Polarization-resolved propagation plots, amplitude maps, total-intensity representations, and morphology maps connect the exact solution directly to measurable waveform features. The results provide a coefficient-level framework for determining how the properties of a strongly nonlinear and dispersive optical waveguide shape a locked dark-bright vector state.&lt;/p&gt;</jats:p>