\(\text{Sech}^{2}\)-type solitary waves and the stability analysis for the KdV–mKdV equation

In this article, we investigated the solitary wave solutions of the KdV–mKdV equation using Hirota’s bilinear method. Closed-form analytical single and multiple solitary wave solutions were obtained. Through qualitative methods and the analysis of solitary waveforms, we discovered that in addition to sech-type solitary waves, the system also contains \(\text{Sech}^{2}\) ${\text{Sech}}^2$ -type solitary waves. By employing the trial functions method, we obtained a single \(\text{Sech}^{2}\) ${\text{Sech}}^2$ -type solitary wave and verified its existence and stability using the split-Step Fourier Transform method. Furthermore, we use the collision of two \(\text{Sech}^{2}\) ${\text{Sech}}^2$ -type single solitary waves to excite a stable \(\text{Sech}^{2}\) ${\text{Sech}}^2$ -type double solitary wave. Similarly, we excite a stable triple solitary wave with three \(\text{Sech}^{2}\) ${\text{Sech}}^2$ -type single solitary waves. This method can also be used to excite stable multiple solitary waves. It is shown that these solitary wave solutions enrich the dynamic behavior of the KdV–mKdV equation and provide methods for solving \(\text{Sech}^{2}\) ${\text{Sech}}^2$ -type solitary waves, which hold significant theoretical value.