Abstract
In this paper two approaches including the Tanh-expansion method and G'/G(2)-expansion method are introduced to find a hierarchy of traveling wave solutions to the integrable KdV-Burgers-Kuramoto equation (KBKE). The proposed methods offer a large number of new exact analytical solutions. But the most important motive in this study is to obtain a general solution to the non-integrable damped KBKE. As is well-known, the later equation does not have any exact analytical solution due to the presence of the damping term. Accordingly, we will find a general semi-analytical solution to this equation using a new ansatz and with the help of the exact solutions of the integrable KBKE. Also, we will check the accuracy of the semi-analytical solution to ensure its efficiency and suitability for describing many natural phenomena. The most important characteristic of the general solution is its ability to explain the mechanism of the propagation of non-stationary dissipative waves, which can be described by the non-integrable damped KBKE such as dissipative solitons, dissipative cnoidal waves, dissipative periodic waves, dissipative shock waves, etc. Also, the obtained solutions serve a large sector of science, especially the nonlinear structures that propagate in plasma physics, seas, oceans, mechanical fluids, optical fibers, and Bose-Einstein condensates.