Investigating Nonlinear Wave Stability and Dispersive Phenomena in Complex Systems: Dynamical Properties, Bifurcation Analysis, and Chaotic Behaviors

Abstract

We present a systematic investigation of the time–fractional Kundu–Mukherjee–Naskar equation in (2 + 1)–dimensions (FKMN ), formulated with a truncated Mittag–Leffler kernel and a (β)–fractional derivative to incorporate memory effects and nonlocal temporal dynamics. This formulation captures dispersive–nonlinear wave propagation in heterogeneous media with fractional temporal responses, making it pertinent to applications in fluid mechanics, nonlinear optics, and plasma physics, including shallow-water dynamics, optical fiber transmission, and plasma instabilities. A combined fractional-calculus and dynamical-systems framework is employed to characterize
the model across parameter regimes. Bifurcation analysis delineates stability boundaries, while quasiperiodic and chaotic responses are identified and organized. Sensitivity analyses further quantify the impact of key coefficients. Exact traveling-wave solutions are derived using the Khater II (KII) method and an enhanced Kudryashov (EKud) approach, revealing new waveform geometries and dissipation patterns. These solutions are independently validated by the Adomian decomposition method, ensuring consistency between closed-form constructions and numerical approximations. The findings highlight complex waveform structures, dissipative signatures, and parameter-dependent transitions, underscoring the FKMN model’s versatility in representing nonlinear phenomena across disciplines. Methodologically, the study advances a hybrid analytical–numerical pipeline that integrates fractional operators with rigorous dynamical analysis, thereby improving both solution fidelity and interpretability. These contributions establish the (2 + 1)–dimensional FKMN equation as a paradigmatic test case for fractional evolution equations in applied mathematics and physics.