This investigation explores the nonlinear modified generalized Vakhnenko (NMGV) equation, a pivotal model of nonlinear wave interactions within nonlinear dispersive media. The equation represents an extension of the classical Vakhnenko framework, integrating higher-order nonlinearities that significantly impact wave propagation. Comprehending these effects is essential for propelling the advancement of nonlinear wave theory in mathematical physics and engineering contexts. Utilizing the Khater Ⅲ method, we derive exact solutions that encapsulate the salient features of the NMGV equation's wave configurations. The research unveils innovative soliton-like solutions, shedding light on the intricate nonlinear mechanisms at play. These findings underscore the efficacy of the proposed methodology in addressing intricate nonlinear systems and highlight its applicability to analogous nonlinear evolution equations. By introducing novel analytical solutions, this study enriches the comprehension of dispersive wave phenomena and offers valuable insights for future research in nonlinear wave dynamics.

In this paper, we investigated the planar dynamical system and new traveling wave solution of the stochastic Biswas-Milovic equation (BME) with dual-power law nonlinearity and multiplicative white noise in the Itô sense. First, the stochastic BME with dual-power law nonlinearity and multiplicative white noise in the Itô sense was transformed into a nonlinear ordinary differential equation (NLODE) through traveling wave transformation. Second, the dynamical bifurcation conditions and phase diagrams of the equation considered were obtained through the method of planar dynamical systems, and the phase portrait of the dynamical system was given. Moreover, taking into account the periodic disturbances in real-world environments, we extended our analysis to explore the effects of disturbance terms. Meanwhile, two-dimensional (2D) and three-dimensional (3D) phase portraits, sensitivity analyses, and the Poincaré section of its perturbed system were plotted through Maple software. Finally, the new traveling wave solutions of the stochastic BME with dual-power law nonlinearity and multiplicative white noise in the Itô sense were constructed by using the complete discrimination system method.