Abstract
Starting from a novel discrete spectral problem, a family of integrable differential-difference equations is derived through discrete zero curvature equation. And then, tri-Hamiltonian structure of the whole family is established by the discrete trace identity. It is shown that the obtained family is Liouville-integrable. Next, a nonisospectral integrable family associated with the discrete spectral problem is constructed through nonisospectral discrete zero curvature representation. Finally, Lie algebra of isospectral and nonisospectral vector fields is deduced.
Highlights
Over the past three decades, the integrable nonlinear differential-difference systems (INDDEs) have received considerable attention
The Hamiltonian structure of the INDDEs can be established by discrete trace identity or discrete variational identity [4, 5]
We have deduced a family of integrable differential-difference equations through the discrete zero curvature equation
Summary
Over the past three decades, the integrable nonlinear differential-difference systems (INDDEs) have received considerable attention. One of the interesting problems in the theory of lattice soliton and integrable systems is to look for a Hamiltonian operator J and a sequence-conserved functional H (nm), (m ≥ 0) so that equation (1) may be represented as the following Hamiltonian form: un,tm. The research of nonisospectral INDDEs has been widespread concern, the algebraic structure of isospectral and nonisospectral vector fields is established in [16,17,18], and the key of the theory is to derive the corresponding nonisospectral family of INDDEs. is paper is organized as follows.
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