Darboux Transformations of Nonlinear Coupled Equations and Their Solutions

This study is dedicated to an in-depth exploration of the integrable structure inherent in the coupled KdV equation, with particular emphasis on solving Lax pairs and their application in constructing Darboux Transformation and Bäcklund Transformation. By leveraging these transformations, we have derived a series of superposition formulas that offer novel insights into understanding the interactions among solitary waves within coupled systems. Furthermore, our successful identification of a recursive operator for this system not only enhances our comprehension of the dynamical behaviors exhibited by the coupled KdV equations but also establishes a theoretical cornerstone for future research endeavors in this field.