System In Liouville Sense Research Articles

Bargmann Symmetry Constraint for a Family of Liouville Integrable Differential-Difference Equations

A family of integrable differential-difference equations is derived from a new matrix spectral problem. The Hamiltonian forms of obtained differential-difference equations are constructed. The Liouville integrability for the obtained integrable family is proved. Then, Bargmann symmetry constraint of the obtained integrable family is presented by binary nonliearization method of Lax pairs and adjoint Lax pairs. Under this Bargmann symmetry constraints, an integrable symplectic map and a sequences of completely integrable finite-dimensional Hamiltonian systems in Liouville sense are worked out, and every integrable differential-difference equations in the obtained family is factored by the integrable symplectic map and a completely integrable finite-dimensional Hamiltonian system.

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.

A family of integrable differential–difference equations, its bi-Hamiltonian structure and binary nonlinearization of the Lax pairs and adjoint Lax pairs

A family of integrable differential–difference equations is derived by the method of Lax pairs. A discrete Hamiltonian operator involving two arbitrary real parameters is introduced. When the parameters are suitably selected, a pair of discrete Hamiltonian operators is presented. Bi-Hamiltonian structure of obtained family is established by discrete trace identity. Then, Liouville integrability for the obtained family is proved. Ultimately, through the binary nonlinearization of the Lax pairs and adjoint Lax pairs, every differential–difference equation in obtained family is factored by an integrable symplectic map and a finite-dimensional integrable system in Liouville sense.

A family of Liouville integrable lattice equations with a parameter and its two symmetry constraints

A two-by-two matrix spectral problem with an arbitrary constant is introduced, a family of integrable lattice equations involving a parameter is derived by the technique of Lax pairs. The typical lattice equation in the obtained family is a deformed Toda lattice system. The bi-Hamiltonian structure of the obtained family of integrable lattice equations is established by using the discrete trace identity. And we prove the Liouville integrability of obtained family of integrable lattice equations. Then, Bargmann symmetry constraint and an implicit symmetry constraint of the obtained family are presented by binary nonlinearization method. Under these two symmetry constraints, we obtain two series of integrable symplectic maps and two sequences of completely integrable finite-dimensional Hamiltonian systems in Liouville sense, and every integrable lattice equation in the obtained family is decomposed by an integrable symplectic map and a completely integrable finite-dimensional Hamiltonian system. As a special case, two integrable decompositions of the deformed Toda lattice equation are given.

An integrable coupling family of Merola–Ragnisco–Tu lattice systems, its Hamiltonian structure and related nonisospectral integrable lattice family

Abstract An integrable coupling family of Merola–Ragnisco–Tu lattice systems is derived from a four-by-four matrix spectral problem. The Hamiltonian structure of the resulting integrable coupling family is established by the discrete variational identity. Each lattice system in the resulting integrable coupling family is proved to be integrable discrete Hamiltonian system in Liouville sense. Ultimately, a nonisospectral integrable lattice family associated with the resulting integrable lattice family is constructed through discrete zero curvature representation.

A family of integrable differential-difference equations and its Bargmann symmetry constraint

A family of integrable differential-difference equations is constructed through discrete zero curvature equation. The Hamiltonian structures of the resulting differential-difference equations are established by the discrete trace identity. The Bargmann symmetry constraint of the resulting family is presented. Under this symmetry constraint, every differential-difference equation in the resulting family is factored by an integrable symplectic map and a finite-dimensional integrable system in Liouville sense.

Explicit solutions of a finite-dimensional integrable system

A finite-dimensional Hamiltonian system associated with KdV equation is considered. It is an integrable system in Liouville sense. We present an effective method to construct explicit solutions of the system. Some explicit solutions are obtained.

Factorization of a hierarchy of the lattice soliton equations from a binary Bargmann symmetry constraint

Staring from a discrete spectral problem, a hierarchy of the lattice soliton equations is derived. It is shown that each lattice equation in resulting hierarchy is Liouville integrable discrete Hamiltonian system. The binary nonlinearization of the Lax pairs and the adjoint Lax pairs of the resulting hierarchy is discussed. Each lattice soliton equation in the resulting hierarchy can be factored by an integrable symplectic map and a finite-dimensional integrable system in Liouville sense. Especially, factorization of a discrete Kdv equation is given.

AN UNITED APPROACH TO BOTH THE TM HIERARCHY AND THE COUPLED KdV HIERARCHY

A united model of both the TM hierarchy and the coupled KdV hierarchy is proposed. By using the trace identity, the bi-Hamiltonian structure of the corresponding hierarchy is established. The isospectral problem is nonlinearized as a new completely integrable Hamiltonian system in Liouville sense.

Neumann and Bargmann Systems Associated with an Extension of the Coupled KdV Hierarchy

An eigenvalue problem with a reference function and the corresponding hierarchy of nonlinear evolution equations are proposed. The bi–Hamiltonian structure of the hierarchy is established by using the trace identity. The isospectral problem is nonlinearized as to be finite–dimensional completely integrable systems in Liouville sense under Neumann and Bargmann constraints.

Stationary Harry-Dym's equation and its relation with geodesics on ellipsoid

Harry-Dym's equation (HD) $$r_\tau = (r^{ - \tfrac{1}{2}} )_{xxx} $$ is well-known for its cusp soliton solutions. In this paper, relations are revealed between HD and a completely integrable Hamiltonian system in Liouville sense given by $$H = \frac{1}{2}\left\langle {p,p} \right\rangle - \left\langle {q,q} \right\rangle ^{ - 1} $$ in the symplectic manifold (ℝ2N ,dp/∩). Here 〈ξ,η〉 is the standard inner product in ℝ N .