Multi-Soliton Solutions for the Combined KdV–mKdV Equation in Terms of Wronskian with Multi-Wave and Periodic Cross-Kink Dynamics

This article examines the integrability of the combined KdV-mKdV equation, which provides an effective framework for modeling coherent structures in turbulent flows. We generate the explicit Darboux solutions for the combined KdV-mKdV equation using Wronskians. These results are further generalized to the K-th order and supplemented as the logarithmic derivative of the K-th order Wronskian that provides us with the multi-soliton solutions. We generate the exact explicit solution for one-, two-, and three-solitons. Graphical depictions of the soliton formations’ interactions, dynamical characteristics, and temporal evolution are used to support these conclusions. Furthermore, we generate the multi-wave and periodic cross-kink wave solutions by employing bilinear formulism. The graphical representations of these nonlinear excitations highlight their extensive dynamical activity and structural complexity.