Nonlinear Wave Propagation in Diffusive Systems with Threshold-Modulated Growth

Nonlinear dispersive wave phenomena occupy a central position in mathematical physics, with applications spanning shallow-water hydrodynamics, plasma physics, and nonlinear optics. This study examines the 1+1-dimensional Mikhailov–Novikov–Wang (𝕄ℕ𝕎) integrable equation, a higher-order nonlinear evolution model capturing the balance between nonlinear steepening and dispersive spreading characteristic of such systems. Three independent symbolic methods – the Khater III technique, the enhanced Kudryashov scheme, and the auxiliary equation approach – are employed to construct exact traveling-wave solutions, yielding bright solitons and kink-type profiles together with explicit admissibility and wave-speed constraints. Because symbolic derivations may introduce algebraic artifacts, the accuracy of each solution family is independently verified using He’s Variational Iteration (ℍ𝕍𝕀) method, a semi-analytical scheme based on iterative correction rather than algebraic transformation. Comparison between closed-form expressions and ℍ𝕍𝕀 approximations shows absolute errors on the order of 10 -10 –10 -11 , confirming close agreement between the two approaches. The Khater III method further produces a solution branch containing logarithmic combinations of hyperbolic functions, shown to lie outside the functional class attainable by a commonly used unified auxiliary-equation ansatz, indicating a genuine extension of the solution repertoire relative to recent comparable studies. Three-dimensional surface plots, contour maps, and streamline visualizations illustrate the amplitude structure and internal flow patterns of the derived profiles. The combined symbolic and semi-analytical framework offers a mathematically grounded basis for evaluating solution reliability in higher-order nonlinear evolution equations, with relevance to shallow-water dynamics, plasma physics, and nonlinear optical media exhibiting comparable dispersive-nonlinear balances.