Hierarchy Of Lattice Soliton Equations Research Articles

A Hierarchy of Lattice Soliton Equations Associated with a New Discrete Eigenvalue Problem and Darboux Transformations

Abstract By considering a new discrete isospectral eigenvalue problem, a hierarchy of integrable positive and negative lattice models is derived. It is shown that they correspond to positive and negative power expansions of Lax operators with respect to the spectral parameter, respectively. And the equation in the resulting hierarchy is integrable in Liouville sense. Further, a Darboux transformation is established for the typical equations by using gauge transformations of Lax pairs, from which the exact solutions are given.

INTEGRABLE COUPLING SYSTEMS OF HAMILTONIAN LATTICE EQUATIONS BY SEMI-DIRECT SUMS OF LIE ALGEBRAS

By considering a discrete isospectral eigenvalue problem, a hierarchy of lattice soliton equations are derived. The relation to the Toda type lattice is achieved by variable transformation. With the help of Tu scheme, the Hamiltonian structure of the resulting lattice hierarchy is constructed. The Liouville integrability is then demonstrated. Semi-direct sum of Lie algebras is proposed to construct discrete integrable couplings. As applications, two kinds of discrete integrable couplings of the resulting system are worked out.

A Bi-Hamiltonian Lattice System of Rational Type and Its Discrete Integrable Couplings

By considering a new discrete isospectral eigenvalue problem, a hierarchy of lattice soliton equations of rational type are derived. It is shown that each equation in the resulting hierarchy is integrable in Liouville sense and possessing bi-Hamiltonian structure. Two types of semi-direct sums of Lie algebras are proposed, by using of which a practicable way to construct discrete integrable couplings is introduced. As applications, two kinds of discrete integrable couplings of the resulting system are worked out.

A HIERARCHY OF LATTICE SOLITON EQUATIONS AND ITS HIGHER-ORDER SYMMETRY CONSTRAINT

Starting from a discrete matrix spectral problem, a hierarchy of lattice soliton equations is derived through a discrete zero curvature representation. The Hamiltonian structures are established for the resulting hierarchy. Then the higher-order symmetry constraint for the resulting hierarchy is studied. It is shown that under the higher-order symmetry constraint, each lattice soliton equation in the resulting hierarchy can be factored by an integrable symplectic map and a finite-dimensional Liouville integrable Hamiltonian system.

A Liouville integrable lattice soliton equation, infinitely many conservation laws and integrable coupling systems

A lattice soliton equation is proposed as a typical lattice system in the hierarchy of lattice soliton equations, which is derived from a discrete matrix spectral problem. The Liouville integrability for the corresponding lattice system is demonstrated. Moreover, infinitely many conservation laws of corresponding lattice system are obtained by a direct way. Finally, the integrable coupling systems of corresponding lattice systems are deduced through enlarging associated Lax pairs.

A hierarchy of lattice soliton equations and its Darboux transformation

A new discrete spectral problem is introduced and the corresponding hierarchy of the lattice soliton equations are derived by means of the trace identity. We find a new Darboux transformation of the lattice soliton equation, through which the explicit solutions are shown.

A New Integrable Symplectic Map of Neumann Type

The nonlinearization approach is generalized to the case of the Neumann constraint associated with a discrete 3×3 matrix eigenvalue problem. A new symplectic map of the Neumann type is obtained by nonlinearization of the discrete eigenvalue problem and its adjoint one. A scheme for generating the involutive system of conserved integrals of the symplectic map is proposed, by which the symplectic map of the Neumann type is further proved to completely integrable. As an application, the calculation of solutions for the hierarchy of lattice soliton equations connected to the discrete eigenvalue problem is reduced to the solutions of a system of ordinary differential equations plus a simple iterative process of the symplectic map of the Neumann type.

A new integrable symplectic map associated with lattice soliton equations

A method is developed that extends the nonlinearization technique to the hierarchy of lattice soliton equations associated with a discrete 3×3 matrix spectral problem. A new integrable symplectic map and its involutive system of conserved integrals are obtained by the nonlinearization of spatial parts and the time parts of Lax pairs and their adjoint Lax pairs of the hierarchy. Moreover, the solutions of the typical system of lattice equations in the hierarchy are reduced to the solutions of a system of ordinary differential equations and a simple iterative process of the symplectic map.