Generalization of Gardner’s equation withhigher-order dispersion using the GeneralizedLocal Derivative

Abstract

This work presents a generalization of the Gardner equation withhigher-order dispersion and perturbation terms, replacing the beta derivativewith the Generalized Local Derivative (LGD). By introducing a kernelof weight N(β, t), the model captures the behavior of complex materialsand variable timescales that standard conformable operators cannot describe.Using the modified extended direct algebraic (mEDA) method,we transform the partial differential equation into an ordinary differentialequation to obtain exact soliton-like solutions.A comparative analysis is performed between three specific kernels: thebeta derivative, the Khalil conformable derivative, and the LGD with aMittag-Leffler kernel. The results demonstrate that the use of the Mittag-Leffler kernel introduces a non-local memory effect that alters the groupvelocity and the soliton trajectory. Finally, the linear stability analysisreveals that the LGD framework provides a more robust stability thresholdagainst high-frequency disturbances compared to rigid power models.