The Burgers-KdV limit in one-dimensional plasma with viscous dissipation: A study of dispersion and dissipation effects

The Burgers-KdV equation as a highly nonlinear model, is commonly used in weakly nonlinear analysis to describe small but finite amplitude ion-acoustic waves. In this study, we demonstrate that by considering viscous dissipation, we can derive the Burgers-KdV limit from a one-dimensional plasma system by using the Gardner-Morikawa transformation. This transformation allows us to obtain both homogeneous and inhomogeneous Burgers-KdV equations, which incorporate dissipative and dispersive terms, for the ionic acoustic system. To analyze the remaining system, we employ the energy method in Sobolev spaces to estimate its behavior. As a result, we are able to capture the Burgers-KdV dynamics over a time interval of order $O({\epsilon }^{-1})$, where $\epsilon$ represents a small parameter.