Department of Mathematics, Chittagong University of Engineering & Technology, Chattogram 4349, Bangladesh.
10.22034/cmde.2025.68401.3312
Abstract
This study investigates a generalized Zakharov–Kuznetsov (ZK) equation that incorporates mixed nonlinearity (quadratic and square-root terms), anisotropic dispersion, linear damping, and external periodic forcing. The model is motivated by the need to describe nonlinear wave phenomena in dispersive media, where directional dependence, energy dissipation, and external driving play crucial roles in applications ranging from space plasma physics to laboratory plasma confinement and nonlinear optics. Through a traveling wave reduction, the system is transformed into a second-order nonlinear ODE, enabling comprehensive analysis. The onset of chaotic dynamics is rigorously confirmed using Melnikov’s method, establishing conditions for transverse homoclinic intersections. We further extend the new mapping method to derive exact solutions for non-polynomial nonlinearities, overcoming limitations of classical approaches. Additional exact solutions via Jacobi elliptic functions are constructed for the quadratic case, providing a complete spectrum of nonlinear waveforms. Numerical simulations reveal transitions between solitonic behavior, weak chaos, and broadband turbulence across parameter regimes, with direct implications for understanding wave instability and energy transfer in driven dissipative systems. These results provide new insights into the complex dynamics of forced-damped nonlinear systems and represent a significant extension of the classical ZK framework, offering both analytical advances and practical tools for predicting nonlinear wave behavior in physical applications.
Rahman, A. , Hafez, M. Golam and Islam, M. Jahidul (2026). Dynamical Analysis and Exact Solutions of Generalised Zakharov Kuznetsov Equation with Composite Nonlinearity. Computational Methods for Differential Equations, (), -. doi: 10.22034/cmde.2025.68401.3312
MLA
Rahman, A. , , Hafez, M. Golam, and Islam, M. Jahidul. "Dynamical Analysis and Exact Solutions of Generalised Zakharov Kuznetsov Equation with Composite Nonlinearity", Computational Methods for Differential Equations, , , 2026, -. doi: 10.22034/cmde.2025.68401.3312
HARVARD
Rahman, A., Hafez, M. Golam, Islam, M. Jahidul (2026). 'Dynamical Analysis and Exact Solutions of Generalised Zakharov Kuznetsov Equation with Composite Nonlinearity', Computational Methods for Differential Equations, (), pp. -. doi: 10.22034/cmde.2025.68401.3312
CHICAGO
A. Rahman , M. Golam Hafez and M. Jahidul Islam, "Dynamical Analysis and Exact Solutions of Generalised Zakharov Kuznetsov Equation with Composite Nonlinearity," Computational Methods for Differential Equations, (2026): -, doi: 10.22034/cmde.2025.68401.3312
VANCOUVER
Rahman, A., Hafez, M. Golam, Islam, M. Jahidul Dynamical Analysis and Exact Solutions of Generalised Zakharov Kuznetsov Equation with Composite Nonlinearity. Computational Methods for Differential Equations, 2026; (): -. doi: 10.22034/cmde.2025.68401.3312
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