Abstract
An integrable family of the different-difference equations is derived from a discrete matrix spectral problem by the discrete zero curvature representation. Hamiltonian structure of obtained integrable family is established. Liouville integrability for the obtained family of discrete Hamiltonian systems is proved. Based on the gauge transformation between the Lax pair, a Darboux-Bäcklund transformation of the first nonlinear different-difference equation in obtained family is deduced. Using this Darboux-Bäcklund transformation, an exact solution is presented.
Highlights
During the past decades, study of the integrable differentdifference equations has received considerable attention
Study of the integrable differentdifference equations has received considerable attention. They play the important roles in mathematical physics, lattice soliton theory, cellular automata, and so on
It is easy to see that the matrix spectral problem (7) is equivalent to the following eigenvalue problem: snE−1 the eigenfunction ξn = φn2. Based on this discrete matrix spectral problem, we derive a family of integrable differentdifference equations through the discrete zero curvature representation
Summary
Study of the integrable differentdifference equations (or integrable lattice equations) has received considerable attention. In the lattice soliton theory, the discrete zero curvature representation is a significant way to derive the integrable different-difference equations [18]. It is well known that Darboux-Backlund transformation is an important and very effective method for solving integrable different-difference equations. Based on this discrete matrix spectral problem, we derive a family of integrable differentdifference equations through the discrete zero curvature representation. Some conclusions and remarks are given in the final section
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