Positive and negative integrable lattice hierarchies: Conservation laws and [formula omitted]fold Darboux transformations

Abstract

Positive and negative nonlinear integrable lattice hierarchies are established starting from a discrete matrix spectral problem with three potential functions, particularly the corresponding Hamiltonian structures are presented respectively with the help of the trace identity, all these facts show that these two hierarchies are integrable in Liouville sense. By using Lax pair we derive infinite number of conservation laws and N−fold Darboux transformation (DT) for the first nontrivial system in the two hierarchies. Comparing with the usual 1−fold DT, this kind of N−fold DT enables us to generate the multi-soliton solutions without complicated recursive process. As applications, we derive N−fold explicit exact solutions from seed solutions and plot their figures with properly parameters to analyze and illustrate the propagation of solitary waves.