Darboux transformation and Hamiltonian structure for the Jaulent–Miodek hierarchy

Starting from the Jaulent–Miodek spectral problem, we derive the associated hierarchy of nonlinear evolution equations in this paper. It is shown that this hierarchy is completely integrable in the Liouville sense and possesses the Hamiltonian structure. Moreover, by virtue of symbolic computation, two types of Darboux transformations for the whole hierarchy are explicitly constructed, which enables us to find the new soliton-like, shock and anti-shock solutions for the Jaulent–Miodek hierarchy. Figures are presented to discuss the properties of the new soliton-like solutions.