Exploring variable coefficient models: Insights into nonlinear wave behavior and soliton solutions in physical systems

The KdV-Burgers equation, featuring time-varying coefficients is a fundamental equation in the domain of wave propagation for wave propagation, dispersion, and dissipation in different physical systems. This equation is particularly useful for coastal engineering studies, flow analysis in rivers, open channels, and nonlinear phenomena across different physical systems. In this article, we derive feasible soliton solutions to the stated equation with time-varying coefficients through the tanh-expansion, exp (-φ(τ))-expansion, and modified simple equation approaches. The obtained solutions provide insightful information on wave and solitons behavior in diverse physical systems. The wave profiles in soliton shapes, including 3D and 2D waves, are significantly influenced by the variable wave velocity function and associated parameters. Such variations are not visible when the parameters are held constant. This study emphasizes the critical role of nonlinear evolution equations with time-dependent coefficients, leading to an exploration of an intriguing and challenging field of study