Discrete Zero Curvature Equations Research Articles

Algebro-geometric Constructions of a Hierarchy of Integrable Semi-discrete Equations

A hierarchy of integrable semi-discrete equations is deduced in terms of the discrete zero curvature equation as well as its bi-Hamiltonian structure is gotten through the trace identity. The above hierarchy is separated into soluble ordinary differential equations according to the relationship between the elliptic variables and the potentials, from which the continuous flow is straightened out via the Abel–Jacobi coordinates resorting to the algebraic curves theory. Eventually, the meromorphic function and the Baker–Akhiezer function are introduced successively on the hyperelliptic curve and the algebro-geometric solutions which are expressed as Riemann theta function can be obtained through the two functions mentioned above.

A modified Suris hierarchy and N-fold Darboux -Bäcklund transformation

A modified Suris hierarchy is derived by discrete zero curvature equation. Bi-Hamiltonian structure of the whole hierarchy is established through the discrete trace identity. And we prove that the obtained hierarchy is Liouville integrable. Then a one-fold Darboux- Bäcklund transformation for the modified Suris system is established by means of a proper gauge transformation matrix. As application, an explicit solution is given. Finally, as a result of the N times one-fold Darboux–Bäcklund transformation, we derive N -fold Darboux–Bäcklund transformation.

Characters of Explicit Solutions for a Semidiscrete Integrable Coupled Equation

A semidiscrete integrable coupled system is obtained by embedding a free function into the discrete zero-curvature equation. Then, explicit solutions of the first two nontrivial equations in this system are derived directly by the Darboux transformation method. Finally, in order to compare the solutions before and after coupling intuitively, their structure figures are presented and analyzed.

A Family of Integrable Differential-Difference Equations: Tri-Hamiltonian Structure and Lie Algebra of Vector Fields

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.

Positive and Negative Integrable Hierarchies: Bi-Hamiltonian Structure and Darboux–Bäcklund Transformation

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.

A Hierarchy of Integrable Differential-Difference Equations and Darboux Transformation

By embedding a free function into discrete zero curvature equation, we generalize the original hierarchy of integrable differential-difference equations into a new hierarchy which includes the famous relativistic Toda hierarchy. Infinitely many conservation laws and Darboux transformation for the first nontrivial system in the new hierarchy are constructed with the help of its Lax pair. The exact solutions of the system are generated by applying the obtained Darboux transformation. Finally, the figures of one- and two-soliton solutions with proper parameters are presented to illustrate the structures of soliton solutions.

A generating scheme for conservation laws of discrete zero curvature equations and its application

A generating scheme for conservation laws of discrete integrable equations, arising as discrete zero curvature equations, is presented from pairs of associated matrix spectral problems. An illustrative application to the Volterra lattice equation is made through a pair of 2 × 2 matrix spectral problems. The general theory is not limited to 2×2 matrix spectral problems but for arbitrary-order matrix spectral problems.

The Riemann theta function solutions for the hierarchy of Bogoyavlensky lattices

Starting with a discrete 3 × 3 3\times 3 matrix spectral problem, the hierarchy of Bogoyavlensky lattices which are pure differential-difference equations are derived with the aid of the Lenard recursion equations and the stationary discrete zero-curvature equation. By using the characteristic polynomial of Lax matrix for the hierarchy of stationary Bogoyavlensky lattices, we introduce a trigonal curve K m − 1 \mathcal {K}_{m-1} of arithmetic genus m − 1 m-1 and a basis of holomorphic differentials on it, from which we construct the Riemann theta function of the trigonal curve, the related Baker–Akhiezer function, and an algebraic function carrying the data of the divisor. Based on the theory of trigonal curves, the Riemann theta function representations of the Baker–Akhiezer function, the meromorphic function, and in particular, that of solutions of the hierarchy of Bogoyavlensky lattices are obtained.

Algebraic Structure of Discrete Zero Curvature Equations and Master Symmetries of Discrete Evolution Equations**Supported by the National Science Foundation of China under Grant No. 11371244 and the Applied Mathematical Subject of SSPU under Grant No. XXKPY1604

In this paper, based on a discrete spectral problem and the corresponding zero curvature representation, the isospectral and nonisospectral lattice hierarchies are proposed. An algebraic structure of discrete zero curvature equations is then established for such integrable systems. the commutation relations of Lax operators corresponding to the isospectral and non-isospectral lattice flows are worked out, the master symmetries of each lattice equation in the isospectral hierarchyand are generated, thus a τ-symmetry algebra for the lattice integrable systems is engendered from this theory.

Algebro-Geometric Solutions for a Discrete Integrable Equation

With the assistance of a Lie algebra whose element is a matrix, we introduce a discrete spectral problem. By means of discrete zero curvature equation, we obtain a discrete integrable hierarchy. According to decomposition of the discrete systems, the new differential-difference integrable systems with two-potential functions are derived. By constructing the Abel-Jacobi coordinates to straighten the continuous and discrete flows, the Riemann theta functions are proposed. Based on the Riemann theta functions, the algebro-geometric solutions for the discrete integrable systems are obtained.

Quasi-periodic solutions to the hierarchy of four-component Toda lattices

Starting from a discrete 3×3 matrix spectral problem, the hierarchy of four-component Toda lattices is derived by using the stationary discrete zero-curvature equation. Resorting to the characteristic polynomial of the Lax matrix for the hierarchy, we introduce a trigonal curve Km−2 of genus m−2 and present the related Baker–Akhiezer function and meromorphic function on it. Asymptotic expansions for the Baker–Akhiezer function and meromorphic function are given near three infinite points on the trigonal curve, from which explicit quasi-periodic solutions for the hierarchy of four-component Toda lattices are obtained in terms of the Riemann theta function.

New hierarchy of integrable nonlinear differential-difference equations

A hierarchy of integrable nonlinear differential-difference equations associated with a discrete [Formula: see text] matrix spectral problem is proposed based on the discrete zero-curvature equations. Then, Hamiltonian structures for this hierarchy are constructed with the aid of the trace identity. Infinitely many conservation laws of the hierarchy are derived by means of spectral parameter expansions.

Hamiltonian structures and conservation laws of the Belov-Chaltikian lattice hierarchy

A hierarchy of nonlinear differential-difference equations associated with a 3 × 3 matrix spectral problem is proposed through discrete zero curvature equations. One of the nontrivial equations in this hierarchy is the famous Belov-Chaltikian lattice equation. It is shown that the hierarchy possesses infinitely many conservation laws. Hamiltonian structures for this hierarchy are constructed with the aid of the trace identity.

Integrability of the pentagram map

The pentagram map was introduced by R. Schwartz in 1992 for convex planar polygons. Recently, V. Ovsienko, R. Schwartz, and S. Tabachnikov proved Liouville integrability of the pentagram map for generic monodromies by providing a Poisson structure and the sufficient number of integrals in involution on the space of twisted polygons. In this paper we prove algebraic-geometric integrability for any monodromy, i.e., for both twisted and closed polygons. For that purpose we show that the pentagram map can be written as a discrete zero-curvature equation with a spectral parameter, study the corresponding spectral curve, and the dynamics on its Jacobian. We also prove that on the symplectic leaves Poisson brackets discovered for twisted polygons coincide with the symplectic structure obtained from Krichever-Phong's universal formula.

Integrable nonlinear differential-difference hierarchy and Darboux transformation

In this paper, with a free function embedded into a discrete zero-curvature equation, an integrable nonlinear differential-difference hierarchy is derived via the extension of original hierarchy with symbolic computation. Based on the Lax pair, infinitely many conservation laws and Darboux transformations are constructed for the first nonlinear differential-difference equations in the hierarchy. As an application of the Darboux transformation, some explicit solutions of those sample equations are given.

An Integrable Symplectic Map of a Differential-Difference Hierarchy

By choosing a discrete matrix spectral problem, a hierarchy of integrable differential-difference equations is derived from the discrete zero curvature equation, and the Hamiltonian structures are built. Through a higher-order Bargmann symmetry constraint, the spatial part and the temporal part of the Lax pairs and adjoint Lax pairs, which we obtained are respectively nonlinearized into a new integrable symplectic map and a finite-dimensional integrable Hamiltonian system in Liouville sense.

New Lax pairs of the Toda lattice and the nonlinearization under a higher-order Bargmann constraint

By starting from a new discrete spectral problem, the Toda lattice is derived through the discrete zero curvature equation. Applying the discrete variational identity to the spectral problem will also reach to the bi-Hamiltonian structure of the Toda lattice. Under a higher-order Bargmann symmetry constraint, the new Lax pairs and the adjoint Lax pairs are nonlinearized into integrable symplectic maps and finite-dimensional Liouville integrable Hamiltonian systems. Finally, a Bäcklund transformation of the Toda lattice is obtained.

A BLASZAK–MARCINIAK LATTICE HIERARCHY WITH SELF-CONSISTENT SOURCES

In this paper, we consider the discrete soliton hierarchy with self-consistent sources by using Lie algebra sl(3). Then a Blaszak–Marciniak equations hierarchy with self-consistent sources is obtained through the discrete zero curvature equation.

Exact solutions to a mixed Toda lattice hierarchy through the inverse scattering transform

Starting from the Toda spectral problem, a mixed hierarchy of the isospectral equations and the nonisospectral equations is constructed through discrete zero curvature equations. Soliton solutions of the hierarchy are obtained by the inverse scattering transform. Meanwhile, a new kind of time evolution of scattering data for the Toda spectral problem is presented.

A hierarchy of Liouville integrable lattice equations and its integrable coupling systems

A new discrete two-by-two matrix spectral problem with two potentials is introduced, followed by a hierarchy of integrable lattice equations obtained through discrete zero curvature equations. It is shown that the Hamiltonian structures of the resulting integrable lattice equations are established by virtue of the trace identity. Furthermore, based on a discrete four-by-four matrix spectral problem, the discrete integrable coupling systems of the resulting hierarchy are obtained. Then, with the variational identity, the Hamiltonian structures of the obtained integrable coupling systems are established. Finally, the resulting Hamiltonian systems are proved to be all Liouville integrable.