<p>This study focuses on the perturbed fractional Chen–Lee–Liu equation, which describes nonlinear wave phenomena in plasma physics and optical fiber communication systems. The main goal is to explore complex wave interactions by employing a newly introduced hyperbolic local fractional derivative, providing a more generalized modeling framework than classical approaches. The new version trial equation method is applied to construct exact wave solutions, including rational, exponential, hyperbolic, and Jacobi elliptic forms, whose structures are significantly influenced by the elliptic modulus. The effects of key parameters on wave dynamics are examined through graphical analysis. The results reveal physically relevant structures, such as bright solitons, kink solitons, and chirped traveling solitary waves, and demonstrate how the elliptic modulus can control periodicity and localization, extending and generalizing previous findings in the literature. This work demonstrates the effectiveness of the proposed method and contributes new insights into the study of fractional-order nonlinear systems.</p>