Analytical Treatments for a Coupled (2+1)-Dimensional Partial Differential Equations in Nonlinear Physics

The two-dimensional (2D) nonlinear coupled sine-Gordon equations play a significant role in nonlinear physics, describing phenomena in areas such as solid-state physics, fluid dynamics, and nonlinear optics. Fundamentally, the sine-Gordon equation is a nonlinear partial differential equation (PDE) commonly used to model wave propagation, soliton dynamics, and particle interactions across various physical systems. This article focuses on the analytical solution of the 2D nonlinear coupled sine-Gordon equations in nonlinear wave contexts, considering specific boundary and initial conditions. The solution approach combines the triple Sumudu transform (TST) with an iterative method, detailing the analytical techniques and discussing their convergence properties. The exact solution, presented as a convergent series, is visually depicted through graphs. To address the nonlinear part of the equations, a successive iterative method was applied. The efficiency of the proposed method is illustrated through two test problems, with examples from engineering applications demonstrating its applicability. In conclusion, this method proves to be highly effective, efficient, and promising for finding exact solutions. Overall, the 2D nonlinear coupled sine-Gordon equations capture the intricate dynamics of nonlinear wave propagation, soliton interactions, and energy conservation in systems with multiple interacting components, making them a powerful tool for modeling a broad range of physical systems.