Fractional Dynamics of Nonlinear Kersten-Krasil'shchik Coupled KdV-mKdV Systems: Analytical Insights Through Analytical Methods

Abstract

This study examines the fractional order formulations of the nonlinear coupled Korteweg-de Vries type systems in the wave interaction phenomena, especially the Kersten-Krasil'shchik KdV-mKdV system and a two-component coupled KdV model. The φ-Caputo fractional derivative is used to describe the time evolution of the systems and offers a versatile framework for integrating memory and nonlocal time effects. Two effective semi-analytical methods, the φ-Caputo fractional q-homotopy analysis method, and the φ-Caputo fractional variational iteration method, are proposed to derive analytical approximations. Clear series solutions are developed and demonstrated to converge very fast when appropriate auxiliary parameters are chosen. The effect of the fractional order on the dynamical behavior of the solutions is explored in detail using two and three dimensional graphs. It has been shown that as the fractional order is lowered the wave profile becomes smoother with a reduction in amplitude, which reflects an increase in the effect of memory. The close similarity of the approximate solutions obtained and the exact solutions in the case of integer-order validates the precision and accuracy of the proposed methods. The fractional models in this paper provide a better understanding of the nonlinear propagation of waves in complex media where classical integer-order models might fail.