Analytical Structure and Exact Solutions of a Generalized (2+1)-Dimensional Sasa–Satsuma Equation with Higher-Order Effects

This study presents a unified analytical investigation of the generalized $(2+1)$-dimensional Sasa--Satsuma equation, which models nonlinear wave propagation in optical media. By employing a traveling wave transformation, the governing partial differential equation is reduced to a second-order ordinary differential equation. The intrinsic conservative structure of the reduced system is revealed through the derivation of an explicit first integral, which enables the construction of an effective potential function governing the wave dynamics. Based on this dynamical systems formulation, a fundamental bright soliton solution is obtained directly from the first integral, providing a structural interpretation of localized wave behavior. Furthermore, the $(G'/G,1/G)$-expansion method is applied to construct broader classes of exact analytical solutions, including hyperbolic, trigonometric, and rational forms. Unlike standard approaches where solution methods are applied independently, the obtained solutions are shown to be consistent with the intrinsic dynamical structure of the system. This establishes a direct connection between analytical solution techniques and dynamical systems theory, offering a more comprehensive understanding of nonlinear wave propagation in higher-dimensional Sasa--Satsuma-type models.