In this article, the mitigating Internet bottleneck including quadratic-cubic nonlinearity containing the rho-derivative has been considered that describes the control of Internet traffic. This equation is analyzed utilizing two integration schemes, videlicet, the extended sinh-Gordon equation expansion technique and improved tan(Xi/2)-expansion technique. Various kinds of traveling wave solutions by employing these schemes are presented: solitary, topological, periodic, kink-periodic, and soliton wave solutions. Moreover, the plenty of available solutions with guaranteed conditions are also presented. The restriction conditions for the existence of valid solutions are as well as listed. In order to shed more light on these novel solutions, graphical features 3D, 2D, and density with some suitable choice of parameter values have been depicted. The outcome indicates that the mitigating Internet bottleneck is used as an amplifying model in the applied sciences.