A bivariate extension of a flexible univariate family is proposed. The new family is called bivariate q-extended Weibull Morgenstern family of distributions which can be constructed based on the Farlie-Gumbel-Morgenstern (FGM) copula technique. After introducing the new family, four sub-models are discussed in detail from the theoretical and numerical coefficient of correlation point of view with pointing to the effect of the $q$ parameter.

Optical solitons in nonlinear fiber optics represent fundamental wave phenomena that maintain their shape during propagation, yet existing analytical methods often fail to capture the complex dynamics of models incorporating dual nonlocal nonlinearity effects. Recent studies have shown significant research gaps in solving Kudryashov's law with simultaneous refractive index variations and multiple nonlinear terms, particularly when conformable derivatives are involved. In this study, we investigate the analytical solutions of Kudryashov's equation with dual nonlocal nonlinearity and refractive index effects in optical fibers using the improved modified Sardar Sub-equation expansion method (IMSSEM). By applying systematic traveling wave transformations and rigorous mathematical analysis, we derive an extensive collection of novel optical soliton solutions, including rational, hyperbolic, and trigonometric function families. The solutions exhibit diverse wave structures encompassing bright, dark, kink, and W-shaped soliton profiles. Through comprehensive 2D and 3D graphical representations, we demonstrate the wave propagation characteristics under various parameter conditions. Furthermore, we analyze the influence of different conformable derivative orders ( $\tau$) on soliton behavior, revealing significant variations in wave amplitude, width, and propagation velocity. These findings provide crucial insights for understanding nonlinear wave dynamics in optical communication systems, particularly in wavelength-division multiplexing, ultrashort pulse propagation, and nonlinear photonic devices. The derived solutions offer practical applications in designing optical amplifiers, mode-locked lasers, and soliton-based communication protocols. This work contributes significantly to the mathematical physics community by providing a comprehensive analytical framework for solving complex nonlinear optical models and advances the field of integrable systems theory with direct applications to modern fiber optic technology.