The combined KdV-mKdV equation: Bilinear approach and rational solutions with free multi-parameters

In this paper, we investigate the combined KdV-mKdV equation which serves as a valuable tool in the study of water waves, enabling researchers to understand and predict the behaviour of various wave phenomena, including solitary waves, wave breaking, turbulence, wave-structure interactions, and tsunamis. Using the bilinear approach, its rational solutions with free multi-parameters and N-soliton solutions in the determinant form are obtained. The novel idea in this paper is that we develop the bilinear approach allowing free multi-parameters in rational solutions. The dynamics of rational solutions are presented by selecting appropriate parameters. By the bilinear approach, different to the Darboux transformation method, there are no limiting techniques involved on the spectrum parameters to obtain rational solutions. The approach can be extended to obtain other types of solutions if outside the framework of polynomials.