Mukherjee–Choudhury–Chowdhury spectral problem and the semi-discrete integrable system

Starting from the Mukherjee–Choudhury–Chowdhury spectral problem, we derive a semi-discrete integrable system by a proper time spectral problem. A Bäcklund transformation of Darboux type of this system is established with the help of gauge transformation of the Lax pairs. By means of the obtained Bäcklund transformation, an exact solution is given. Moreover, Hamiltonian form of this system is constructed. Further, through a constraint of potentials and eigenfunctions, the Lax pair and the adjoint Lax pair of the obtained semi-discrete integrable system are nonlinearized as an integrable symplectic map and a completely integrable finite-dimensional Hamiltonian system in the Liouville sense. Finally, the involutive representation of solution of the obtained semi-discrete integrable system is presented.

Discrete line complexes and integrable evolution of minors

Based on the classical Plücker correspondence, we present algebraic and geometric properties of discrete integrable line complexes in C P 3 . Algebraically, these are encoded in a discrete integrable system that appears in various guises in the theory of continuous and discrete integrable systems. Geometrically, the existence of these integrable line complexes is shown to be guaranteed by Desargues' classical theorem of projective geometry. A remarkable characterization in terms of correlations of C P 3 is also recorded.

A New Discrete Integrable System Derived from a Generalized Ablowitz-Ladik Hierarchy and Its Darboux Transformation

We find an interesting phenomenon that the discrete system appearing in a reference can be reduced to the old integrable system given by Merola, Ragnisco, and Tu in another reference. Differing from the works appearing in the above two references, a new discrete integrable system is obtained by the generalized Ablowitz-Ladik hierarchy; the Darboux transformation of this new discrete integrable system is established further. As applications of this Darboux transformation, different kinds of exact solutions of this new system are explicitly given. Investigatingthe properties of these exact solutions, we find that these exact solutions are not pure soliton solutions, but their dynamic characteristics are very interesting.

A HIERARCHY OF DISCRETE INTEGRABLE SYSTEM AND ITS COUPLED ENLARGED INTEGRABLE SYSTEM

A hierarchy of nonlinear integrable lattice soliton equations is derived from a discrete spectral problem. The hierarchy is proved to have discrete zero curvature representation. Using an enlarging algebra system [Formula: see text], we construct integrable couplings of the resulting hierarchy.