Abstract
Two integrable hierarchies are derived from a novel discrete matrix spectral problem by discrete zero curvature equations. They correspond, respectively, to positive power and negative power expansions of Lax operators with respect to the spectral parameter. The bi-Hamiltonian structures of obtained hierarchies are established by a pair of Hamiltonian operators through discrete trace identity. The Liouville integrability of the obtained hierarchies is proved. Through a gauge transformation of the Lax pair, a Darboux–Bäcklund transformation is constructed for the first nonlinear different-difference equation in the negative hierarchy. Ultimately, applying the obtained Darboux–Bäcklund transformation, two exact solutions are given by means of mathematical software.
Highlights
It is well known that the study of nonlinear integrable differential-difference equations (NIDDEs) has attracted much attention in recent decades [1,2,3,4,5,6,7,8,9,10,11,12,13,14]
Many important NIDDEs have been presented such as the Ablowitz–Ladik lattice [1], the Toda lattice [2], the relativistic Toda lattice [3], the modified Toda lattice [4, 5], the Merola–Ragnisco–Tu lattice [6], and the deformed reduced semidiscrete Kaup–Newell lattice [7,8,9,10,11,12,13,14]
Two bi-Hamiltonian forms for the obtained integrable hierarchies are given by the discrete trace identity
Summary
It is well known that the study of nonlinear integrable differential-difference equations (NIDDEs) has attracted much attention in recent decades [1,2,3,4,5,6,7,8,9,10,11,12,13,14]. (3) determines a hierarchy of NIDDEs (or lattice soliton equations): untm K un, un− 1, un+1, . One of the important problems in the lattice soliton theory is hierarchy to of csoenarscehrvefodrdaenHsiatimesiltoHn(miamn)o∞ mp 0ersaototrhaJt1 and a (4) has the following Hamiltonian structures: untm J1δHδu(nnm), m ≥ 0,. We are going to present two integrable hierarchies from a discrete matrix spectral problem. Eory is, respectively, called positive and negative integrable hierarchies. Staring from spectral problem (7), positive and negative integrable hierarchies of NIDDEs are, respectively, presented by discrete zero curvature equations.
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