Solution of High-Order Nonlinear Integrable Systems Using Darboux Transformation

The 4x4 trace-free complex matrix set is introduced in this study. By using it, we are able to create a novel soliton hierarchy that is reduced to demonstrate its bi-Hamiltonian structure. Additionally, we give the Darboux matrix T, whose elements are connected to the spectral parameter in accordance with the various positions and numbers of the spectral parameter lambda. The Darboux transformation approach has also been successfully applicated to superintegrable systems.