Exploring Nonlinear Dynamics and Chaos in the Modified Korteweg–de Vries–Zakharov–Kuznetsov Equation with NARX Neural Networks

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

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 characterizethe 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.

Demonstration of Soliton Wave Structures in a Higher-Dimensional Nonlinear Physical Model: Dynamic Insights into Sensitivity, Bifurcation, and Chaos

This article presents a comprehensive analytical and dynamical investigation of the (4+1)-dimensional variable-coefficient generalized Kadomtsev Petviashvili equation (vc-gKP). Exact analytical solutions are constructed using the modified Khater method, constructing a diverse class of localized and propagating wave structures that capture the intrinsic nonlinear characteristics of the governing model. To further examine the system, an extensive dynamical analysis is carried out using phase-plane trajectories, temporal evolution, Poincare maps, bifurcation analysis, and Lyapunov exponents, allowing the identification of distinct dynamical states and their transitions. The obtained results reveal a variety of complex nonlinear behaviors, including periodic oscillations, quasiperiodic motion, chaotic dynamics, and multistable responses associated with both softening and hardening nonlinear effects. To further verify these behaviors, power spectral analysis, recurrence plots, return maps, and fractal dimension calculations are incorporated, providing complementary information regarding the spectral distribution, geometric organization, and complexity of the resulting attractors. The corresponding numerical illustrations clearly demonstrate the evolution of the system from regular to chaotic regimes together with the coexistence of multiple stable attractors under different parameter conditions. Overall, the proposed analytical and computational methodology establishes an effective framework for exploring nonlinear wave phenomena in higher-dimensional evolution equations and contributes to a deeper theoretical understanding of models arising in fluid mechanics, plasma physics, nonlinear optics, and related branches of mathematical physics.

Soliton solutions and dynamical bifurcation analysis of the Yajima-Oikawa equations

The Yajima–Oikawa equations describes the resonant interaction between ion sound waves and Langmuir waves in plasma. Despite its importance, the detailed dynamical behavior of this model has not been comprehensively explored. In this work, we present a unified analytical and dynamical framework that combines exact solution construction with qualitative and quantitative analysis. By employing traveling wave transformations together with the first integral method, a class of exact solutions in terms of Jacobi elliptic functions is derived. These solutions are systematically classified, according to bifurcation scenarios, into periodic, quasi-periodic, solitary, and kink-type wave structures. To investigate the global dynamics, an external periodic perturbation is introduced, transforming the system into a higher-dimensional dynamical model. This enables the study of transitions between regular and chaotic regimes. The presence of chaos is confirmed through phase portraits, Poincaré sections, time series analysis, and Lyapunov exponent calculations, while sensitivity to initial conditions is also examined. The results provide a deeper understanding of nonlinear wave interactions and reveal rich dynamical features of the classical Yajima–Oikawa equations. These findings are relevant for applications in plasma physics and related fields, where accurate prediction and control of wave dynamics are essential, including in plasma-based devices such as fusion systems and wave propagation technologies.

Bifurcation Analysis, Chaotic Behavior, and Exact Traveling Wave Solutions for the (2 + 1)-Dimensional Complex Modified Korteweg–de Vries Equation

This paper examines the (2 + 1)-dimensional complex modified Korteweg–de Vries equation, which describes the intricate motion of water particles from the surface to the bottom. By applying an appropriate wave transformation that reduces the governing nonlinear partial differential equations to an ordinary differential system, we conduct a comprehensive bifurcation analysis that identifies equilibrium points and characterizes their phase-space properties. Under periodic perturbations, the system exhibits bifurcations, quasi-periodicity, multistability, and chaotic behavior, supported by Lyapunov exponents, time-series evolution, phase portraits, and Poincaré sections. To obtain exact soliton solutions of various types, we employ both the dynamical system method and the generalized double auxiliary equation method. The physical features of the solutions are depicted using 2D and 3D representations of their real, imaginary, and absolute components. A comparison with related studies shows that the proposed approaches not only reproduce previously reported solutions but also yield broader and more general soliton families.

Revealing hidden nonlinear soliton dynamics, multistable regimes, and chaotic transitions in regularized long-wave equations under complex external forcing.

This paper presents a comprehensive investigation of emerging wave structures associated with the regularized long-wave equation formulated in a nonlinear (2+1)-dimensional framework. By employing advanced analytical techniques, namely the modified Khater method and the Sardar subequation technique, a diverse class of exact solutions is constructed. These solutions encompass bright, dark, singular, and periodic soliton profiles, each demonstrating distinct propagation characteristics. The physical nature of these wave forms is illustrated through detailed two-dimensional plots, three-dimensional surfaces, and projected visual representations to enhance interpretability. To further understand the intrinsic dynamics of the model, qualitative analysis of the corresponding unperturbed planar system is conducted through phase-portrait investigation. When an external periodic forcing term is incorporated, the system exhibits complex nonlinear phenomena, including the onset of chaotic motion. This transition is rigorously examined using phase projections, temporal evolution plots, Poincaré sections, and the computation of Lyapunov exponents to confirm the presence of sensitive dependence on initial conditions. Moreover, an extensive multistability analysis is performed by varying initial states, revealing that slight modifications in system parameters can induce significant transitions between stable and unstable dynamical regimes. Numerical simulations implemented via the fourth-order Runge-Kutta algorithm provide strong computational support for the analytical findings. Overall, the integration of symbolic techniques with high-precision numerical simulations establishes a robust framework for exploring intricate behaviors in higher-dimensional nonlinear dynamical systems.

Collective variable approach to soliton dynamics in the Kuramoto–Sivashinsky equation for nonlinear transport and pattern formation