A new lattice hierarchy: Hamiltonian structures,symplectic map and N-fold Darboux transformation

A new lattice hierarchy is constructed from a discrete matrix spectral problem. By the Tu scheme technique, the associated Hamiltonian structures and infinitely many conservation laws of this hierarchy are derived. Then a symplectic map is proposed based on the Lax pair and the adjoint Lax pair. Furthermore, the N-fold Darboux transformation and explicitly exact solutions of the first two equations in the hierarchy are investigated. Finally, the density profiles of these exact solutions are presented to illustrate the overtaking collisions of solitary waves.