Perturbation Analysis of Fractional mKdV and Generalized KdV-Burgers Equations with Cubic Nonlinearity

Abstract

This work deals with an analysis of two singularly perturbed partial differential equations with cubic nonlinearities using the conformable derivative approach. We investigate a modified Korteweg-de Vries type equation and a generalized KdV-Burgers equation, both incorporating conformable fractional derivatives and singular perturbation
parameters. The conformable derivatives enable a novel treatment of fractional-order
effects in these two nonlinear wave problems. Using asymptotic expansion methods,
we derive analytical solutions up to first-order corrections and examine the important
differences between perturbed and unperturbed cases. Our analysis shows how the confromable order and perturbation parameters influence wave propagation characteristics,
and solution behavior. The results demonstrate significant qualitative differences from
classical integer-order models, particularly in soliton dynamics and pattern formation.
Keywords: singular perturbations, Khalil derivatives, mKdV equation, mKdV-Burgers
equation, cubic nonlinearity, asymptotic methods, soliton dynamics.
AMS Classification: 35Q53, 35R11, 35B25, 35C20