Accurate demonstrating of the interactions of two long waves with different dispersion relations: Generalized Hirota–Satsuma couple KdV equation

In this study, the generalized formula of the Hirota–Satsuma coupled KdV equation derived by Hirota and Satsuma in 1981 [Hirota and Satsuma, Phys. Lett. A 85, 407−408 (1981)] is analytically and semi-analytically investigated. This model is formulated to describe the interaction of two long undulations with diverse dispersion relations; that is why it is also known with a generalized model of the well-known KdV equation. The generalized Kudryashov and Adomian decomposition methods construct novel soliton wave and semi-analytical solutions. These solutions are represented in some distinct graphs to show the waves’ interactions. In addition, the accuracy of solutions is verified by comparing the obtained analytical and semi-analytical solutions that show the matching between them. All solutions are checked by putting them back into the original model through Mathematica 12.