Exact Solitary and Singular Wave Solutions for Generalized Zakharov–Kuznetsov Z K ( m , n , k ) Equations via the Modified Variational Iteration Method (MVIM)
Fadlallah Mustafa Mosa
Source abstract
This paper presents exact analytical solitary and singular wave solutions for the generalized two‐dimensional Zakharov–Kuznetsov ZK( m , n , k ) equations, which model nonlinear ion‐acoustic waves in magnetized plasma. We apply the modified variational iteration method (MVIM) to solve the spatio‐temporal differential models without using perturbation, discretization, or restrictive spatial assumptions. We analyze the standard ZK(1, 1, 1) model alongside generalized nonlinear forms ZK(2, 2, 2) and ZK(3, 3, 3) under regular hyperbolic secant (sech) and singular hyperbolic sine (sinh) initial profiles. By combining optimal Lagrange multipliers with an accelerated decomposition step, MVIM avoids unneeded polynomial expansions and simplifies iteration steps compared to standard VIM. Convergence of the scheme is proved using the Banach fixed‐point theorem. Numerical evaluations show that exact solutions for these generalized equations are obtained within a few iterations with small absolute errors. The results show that MVIM is an efficient and reliable tool for solving nonlinear partial differential equations in mathematical physics.
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