Exact Soliton Solutions to the Variable-Coefficient Korteweg–de Vries System with Cubic–Quintic Nonlinearity

In this manuscript, we investigate the (2+1)-dimensional variable-coefficient Korteweg–de Vries (KdV) system with cubic–quintic nonlinearity. Based on different methods, we also obtain different solutions. Under the help of the wave ansatz method, we obtain the exact soliton solutions to the variable-coefficient KdV system, such as the dark and bright soliton solutions, Tangent function solution, Secant function solution, and Cosine function solution. In addition, we also obtain the interactions between dark and bright soliton solutions, between rogue and soliton solutions, and between lump and soliton solutions by using the bilinear method. For these solutions, we also give their three dimensional plots and density plots. This model is of great significance in fluid. It is worth mentioning that the research results of our paper is different from the existing research: we not only use different methods to study the solutions to the variable-coefficient KdV system, but also use different values of parameter t to study the changes in solutions. The results of this study will contribute to the understanding of nonlinear wave structures of the higher dimensional KdV systems.