Exact Traveling Wave Solutions of Nonlinear Coupled Wave Systems via Analytical Methods

This research explores the construction of exact traveling wave solutions for significant nonlinear wave equations, specifically focusing on the coupled Korteweg–de Vries (KdV) equations and the coupled Hirota–Satsuma system. These systems are vital in mathematical physics for modeling complex wave interactions in fields such as fluid dynamics and plasma physics. To address the inherent complexity of these nonlinear partial differential equations (NLPDEs), the study employs two robust analytical techniques: the tanh--coth method and the generalized $\exp(-\phi(\xi))$-expansion (GEE) method. The core of the methodology involves applying a traveling wave transformation, $\xi = x - ct$, which reduces the original NLPDEs into more manageable nonlinear ordinary differential equations (ODEs). By assuming specific expansion forms for the solutions—hyperbolic functions for the tanh--coth approach and an exponential-based auxiliary equation for the GEE method—the researchers are able to convert the differential problems into systems of nonlinear algebraic equations. The study leverages the symbolic computation power of Maple 17 to solve these intricate algebraic systems, leading to the derivation of a diverse array of exact solutions. These results encompass various wave structures, including solitary waves, kink-type solutions, periodic waveforms involving trigonometric functions, and rational wave profiles. The comparative application of these two methods reveals that while the tanh--coth method is exceptionally efficient for capturing solitonic and kink behaviors, the GEE method offers greater flexibility by generating a broader class of solutions. Ultimately, the work demonstrates that the integration of these analytical frameworks with symbolic computing provides a powerful toolkit for understanding nonlinear phenomena. The derived solutions not only validate the effectiveness of the chosen methods but also provide deeper physical insights into how waves propagate and interact within complex nonlinear media, reinforcing the importance of exact solutions in the ongoing study of nonlinear dynamics.