Analytical characterization of optical solitons and bifurcation analysis for the (2+1)-D Wazwaz-Kaur Boussinesq equation.

In this paper, we address the analytical study of optical soliton solutions and the dynamical behavior of the (2+1)-dimensional Wazwaz-Kaur Boussinesq (WKB) equation, which models nonlinear wave propagation in higher-dimensional physical systems. Understanding such nonlinear structures is essential due to their relevance in modern optical fiber communications and nonlinear dispersive media. To tackle the problem, we employ the improved modified extended tanh function (IMETF) method, a symbolic analytical approach that efficiently generates a wide spectrum of exact solutions. Using this method, we successfully construct bright, dark, and singular soliton solutions, in addition to exponential and periodic wave structures. The analytical findings are further supported by comprehensive 2D, 3D, and contour graphical representations, which confirm the physical relevance and stability of the obtained solutions. Furthermore, we conduct a detailed bifurcation analysis to investigate the qualitative behavior of equilibrium points in the system. Phase portraits under various parameter settings illustrate how key parameters influence the creation, annihilation, and stability of these points. The novelty of this work lies in the combination of soliton solution construction with an in-depth bifurcation analysis for the WKB equation, which has not been previously explored in the literature. Our results extend existing studies by uncovering new types of solutions and providing insights into the system's nonlinear dynamics, thereby contributing to the broader understanding of complex wave behavior in higher-dimensional nonlinear models.