Unveiling pure-quartic solitons for the generalized nonlinear Schrödinger equation with Kerr nonlinearity using the modified extended direct algebraic method

Pure-quartic solitons have recently gained increasing interest due to their unique dispersion landscape, where pulse evolution is governed solely by fourth-order dispersion. This motivates the analytical study of highly dispersive nonlinear Schrödinger-type models that incorporate Kerr nonlinearity and weak nonlocal corrections. However, existing work has largely focused on numerical investigations or simplified cubic forms, leaving a clear gap in closed-form solutions for the pure-quartic, weakly nonlocal regime. To address this gap, the modified extended direct algebraic method (MEDAM) is employed, as it provides a systematic framework capable of generating diverse classes of exact solutions while avoiding the limitations of conventional ansatz-based approaches. In this work, MEDAM is applied to the pure-quartic nonlinear Schrödinger equation with Kerr and weak nonlocal effects for the first time, producing new analytic families that include bright, dark, singular, and hybrid combo dark–bright and dark–singular solitons. These solutions offer deeper insight into the interplay between higher-order dispersion and nonlocal nonlinear response in engineered optical media. The findings not only enrich the analytical landscape of pure-quartic soliton dynamics but also establish a foundation for future studies on stability, parameter sensitivity, and propagation behaviour in advanced photonic structures.