Fractional memory effects in nonlinear waves: Dynamical study of solitons and chaotic attractors

This paper examines the intricate nonlinear wave behavior characterized by the modified Korteweg–de Vries–Zakharov–Kuznetsov (mKdV–ZK) equation expressed in terms of the M-truncated fractional derivative. To address the involved fractional nonlinear partial differential equation, we use the technique, which efficiently simplifies the nonlinear partial differential equation to an ordinary differential equation, enabling the derivation of exact analytical solutions that exhibit bright, dark, and singular wave structures. The characteristics of these solutions are examined via a series of two-dimensional line plots, three-dimensional surface profiles, and comprehensive density maps, giving transparent insights into their physical dynamics and propagation trends. Beyond wave isolated solutions, the nonlinear dynamics are further investigated by recasting the system into an externally forced nonlinear oscillator so that its chaotic regimes can be investigated through time series analyses, phase space portraits, and poincaré diagrams. Results show the presence of multistability, multiple coexisting attractors in certain regimes of parameters, highlighting the profound impact of memory effects on nonlinear wave evolution. This study presents the first chaos and multistability analysis of the M-truncated fractional Korteweg–de Vries–Zakharov–Kuznetsov (KdV–ZK) equation, revealing memory-induced nonlinear dynamics that have not been reported before.