Optimal Homotopy Asymptotic Treatment of the Generalised KdV-mKdV Equation for Nonlinear Dispersive Waves

An optimal homotopy asymptotic framework is developed for the numerical-semi-analytical treatment of the time-dependent generalized Korteweg-de Vries (KdV)-modified gKdV-mKdV equation, a prototypical nonlinear dispersive model featuring concurrent quadratic and cubic nonlinearities.The equation arises widely in optics, fluid mechanics, plasma physics and condensed-matter systems, where the accurate resolution of solitary waves and complex wave interactions is essential.The Optimal Homotopy Asymptotic Method (OHAM) is formulated without reliance on an artificial small parameter and is equipped with optimally selected convergence-control parameters, thereby overcoming limitations of classical perturbation techniques.Within this formulation, a rapidly convergent approximate analytical solution is constructed, and error dynamics are quantified against benchmark solutions.Comparative assessments indicate that OHAM attains high accuracy with modest computational effort, delivering pointwise errors and global norms that are competitive with, or superior to, those obtained by Homotopy Perturbation and Homotopy Analysis methods.The procedure is straightforward to implement, preserves the dispersive-nonlinear balance intrinsic to the gKdV-mKdV dynamics, and accommodates important special cases (KdV and mKdV limits) within a unified treatment.The approach is thus shown to provide a reliable and easily computable route to soliton-bearing solutions and other nonlinear waveforms, supporting applications in waveguides, shallow-water channels, ion-acoustic media and lattice excitations.The methodological clarity and demonstrated accuracy suggest that OHAM can serve as a practical front-line tool for nonlinear PDEs with mixed nonlinearities and higher-order dispersion, and that its convergence-control strategy can be extended to related integrable and near-integrable models.