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.

在本文中，已考虑缓解Internet瓶颈，其中包括包含\（\ rho \） -导数的二次三次非线性，它描述了Internet流量的控制。利用两种积分方案videlicet，扩展的sinh-Gordon方程扩展技术和改进的\（\ tan（\ Xi / 2）\）分析该方程扩展技术。通过采用这些方案，提出了各种行波解：孤立，拓扑，周期，扭折周期和孤子波解。此外，还提供了有保证条件的大量可用解决方案。列出有效解决方案存在的限制条件。为了更棚光对这些新颖的解决方案，图形特征3D，2D，并用参数值的一些合适的选择密度已描绘。结果表明，缓解的互联网瓶颈在应用科学中被用作放大模型。