Bargmann Constraint Research Articles

A Finite-Dimensional Integrable System Related to the Kadometsev–Petviashvili Equation

In this paper, the Kadometsev–Petviashvili equation and the Bargmann system are obtained from a second-order operator spectral problem Lφ=(∂2−v∂−λu)φ=λφx. By means of the Euler–Lagrange equations, a suitable Jacobi–Ostrogradsky coordinate system is established. Using Cao’s method and the associated Bargmann constraint, the Lax pairs of the differential equations are nonlinearized. Then, a new kind of finite-dimensional Hamilton system is generated. Moreover, involutive representations of the solutions of the Kadometsev–Petviashvili equation are derived.

Rogue waves for the Hirota equation on the Jacobi elliptic cn-function background

In this paper, the exact rogue wave solutions for the Hirota equation are investigated on the Jacobi elliptic cn-function background. Under the Bargmann constraint, the differential constraints on the potential are obtained by the nonlinearization of spectral problem. Further, the eigenvalues are determined based on the differential constraint equations. The non-periodic solution of spectral problem is constructed, which can produce rogue periodic wave solutions of the Hirota equation. Moreover, the exact rogue periodic wave solutions are presented via the one- and two-fold Darboux transformation formulas. Finally, the generation mechanism and characteristics of rogue periodic waves are analyzed from the viewpoint of two- and three-dimensional structures.

The new integrable symplectic map and the symmetry of integrable nonlinear lattice equation

A discrete matrix spectral problem is presented and the hierarchy of discrete integrable systems is derived. Their Hamiltonian structures are established. As to the discrete integrable system, nonlinearization of the spatial parts of the Lax pairs and the adjoint Lax pairs generate a new integrable symplectic map. Based on the theory, a new integrable symplectic map and a family of finite-dimension completely integrable systems are given. Especially, two explicit equations are obtained under the Bargmann constraint. Finally, the symmetry of the discrete equation is provided according to the recursion operator and the seed symmetry. Although the solutions of the discrete equations have been gained by many methods, there are few articles that solving the discrete equation via the symmetry. So the solution of the discrete lattice equation is obtained through the symmetry theory.

A Bargmann system and the involutive solutions associated with a new 4-order lattice hierarchy

By means of the nonlinearization technique, a Bargmann constraint associated with a new discrete \(4 \times 4\) matrix eigenvalue problem is proposed, and a new symplectic map of the Bargmann type is obtained through binary nonlinearization of the discrete eigenvalue problem and its adjoint one. Moreover, the generating function of integrals of motion is obtained, by which the symplectic map is further proved to be completely integrable in the Liouville sense. Finally, the involutive representation of solutions are given.

The Integrable System on the Second-Order Eigenvalue Problem with the Potential and Speed

In this paper, the Bargmann system for the second-order eigenvalue problem with the energy is dependent on potential and speed Lϕ −(∂ 2 + λ u+ v)ϕ − λ ϕx is discussed. First of all, some basic concepts are introduced. After that, by means of the auxiliary problem and the isospectral compatible condition, the reasonable bi-Hamilton operator K and J are defined, and the evolution equations related to the spectral problem are obtained. Using the functional gradient and Lenard recursive sequence, the Bargmann constraint is given. By the constraint relation between the potential function and the eigenvector, the associated Lax pairs are nonlin eared, then the Bargmann system of the eigenvalue problem is found. According to the viewpoint of Hamilton mechanics, a reasonable coordinate system has been found. The Bargmann system is transformed into the Hamilton canonical equations in the coordinate system. Finally, based on the Liouville theorem and confocal involutive system, the integrability of the Hamilton system is proved. A new integrable system is found. Moreover, the involutive representations of the solutions for the evolution equations are generated.

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.

Bargmann and Neumann System of the Second-Order Matrix Eigenvalue Problem

This paper discusses the second-order matrix eigenvalue problem by means of the nonlinearization of the Lax pairs, then the author gives the Bargmann and Neumann constraints of this problem. The relation between the potentials and the eigenfunction is set up based on these constraints. By means of the nonlinearization of the Lax pairs, the author found these systems of the eigenvalue problem can be equal to the Hamilton canonical system in real symplectic space. In the end, the infinite-dimensions Dynamical systems can be transformed into the finite-dimensions Hamilton canonical systems in the symplectic space. As well, this paper obtains the representations of the solutions for the evolution equations.

Integrability of the Reduction Fourth-Order Eigenvalue Problem

To study the reduced fourth-order eigenvalue problem, the Bargmann constraint of this problem has been given, and the associated Lax pairs have been nonlineared. By means of the viewpoint of Hamilton mechanics, the Euler-Lagrange function and the Legendre transformations have been derived, and a reasonable Jacobi-Ostrogradsky coordinate system has been found. Then, the Hamiltonian cannonical coordinate system equivalent to this eigenvalue problem has been obtained on the symplectic manifolds. It is proved to be an infinite-dimensional integrable Hamilton system in the Liouville sense. Moreover the involutive representation of the solutions is generated for the evolution equations hierarchy in correspondence with this reduced fourth-order eigenvalue problem.

A New Hierarchy of Evolution Equations and its Integrable System

The second-order matrix eigenvalue problem is discussed by means of the nonlinearization of the Lax pairs,then based on the Bargmann constraint between the potential and the eigenfunctions,a new finite-dimensional Hamilton system is abtained by nonlinearization of the eigenvalue problem and the involutive solutions of the evolution equations are abtained

N-Soliton Solutions of Modified KdV Equation under Bargmann Constraint

By means of Hirota method, N-soliton solutions of the modified KdV equation under the Bargmann constraint are obtained through solving the Bargmann constraint and the related Lax pair and conjugate Lax pair of the modified KdV equation.

Solving the KdV Equation Under Bargmann Constraint via Bilinear Approach

In this paper we consider exact solutions to the KdV equation under the Bargmann constraint. Solutions expressed through exponential polynomials and Wronskians are derived by bilinear approach through solving the Lax pair under the Bargmann constraint. It is also shown that the potential u in the stationary Schrödinger equation can be a summation of squares of wave functions from bilinear point of view.

The N-soliton solutions of the fifth-order KdV equation under Bargmann constraint

We derive the N-soliton solutions for the fifth-order KdV equation under Bargmann constraint through Hirota method and Wronskian technique, respectively. Some novel determinantal identities and properties are presented to finish the Wronskian verifications. The uniformity of these two kinds of N-soliton solutions is proved.

A Hierarchy of Integrable Lattice Soliton Equationsand New Integrable Symplectic Map

Starting from a discrete spectral problem, a hierarchy ofintegrable lattice soliton equations is derived. It is shown thatthe hierarchy is completely integrable in the Liouville sense andpossesses discrete bi-Hamiltonian structure. A new integrablesymplectic map and finite-dimensional integrable systems are givenby nonlinearization method. The binary Bargmann constraint givesrise to a Bäcklund transformation for the resultingintegrable lattice equations. At last, conservation laws of thehierarchy are presented.

A finite‐dimensional integrable system associated with a polynomial eigenvalue problem

M. Antonowicz and A. P. Fordy (1988) introduced the second‐order polynomial eigenvalue problem (∂ = ∂/∂x, α = constant) and discussed its multi‐Hamiltonian structures. For n = 1 and n = 2, the associated finite‐dimensional integrable Hamiltonian systems (FDIHS) have been discussed by Xu and Mu (1990) using the nonlinearization method and Bargmann constraints. In this paper, we consider the general case, that is, n is arbitrary, provide the constrained Hamiltonian systems associated with the above‐mentioned second‐order polynomial ergenvalue problem, and prove them to be completely integrable.

A Lax integrable hierarchy, bi-Hamiltonian structure and finite-dimensional Liouville integrable involutive systems

An eigenvalue problem and the associated new Lax integrable hierarchy of nonlinear evolution equations are presented in this paper. As two reductions, the generalized nonlinear Schrödinger equations and the generalized mKdV equations are obtained. Zero curvature representation and bi-Hamiltonian structure are established for the whole hierarchy based on a pair of Hamiltonian operators (Lenard's operators), and it is shown that the hierarchy of nonlinear evolution equations is integrable in Liouville's sense. Thus the hierarchy of nonlinear evolution equations has infinitely many commuting symmetries and conservation laws. Moreover the eigenvalue problem is nonlinearized as a finite-dimensional completely integrable system under the Bargmann constraint between the potentials and the eigenvalue functions. Finally finite-dimensional Liouville integrable system are found, and the involutive solutions of the hierarchy of equations are given. In particular, the involutive solutions are developed for the system of generalized nonlinear Schrödinger equations.

A Lax integrable hierarchy, bi-Hamiltonian structure and finite-dimensional Liouville integrable involutive systems

Abstract An eigenvalue problem and the associated new Lax integrable hierarchy of nonlinear evolution equations are presented in this paper. As two reductions, the generalized nonlinear Schrodinger equations and the generalized mKdV equations are obtained. Zero curvature representation and bi-Hamiltonian structure are established for the whole hierarchy based on a pair of Hamiltonian operators (Lenard's operators), and it is shown that the hierarchy of nonlinear evolution equations is integrable in Liouville's sense. Thus the hierarchy of nonlinear evolution equations has infinitely many commuting symmetries and conservation laws. Moreover the eigenvalue problem is nonlinearized as a finite-dimensional completely integrable system under the Bargmann constraint between the potentials and the eigenvalue functions. Finally finite-dimensional Liouville integrable system are found, and the involutive solutions of the hierarchy of equations are given. In particular, the involutive solutions are developed for the system of generalized nonlinear Schrodinger equations.

A Hierarchy of Lax Integrable Lattice Equations, Liouville Integrability and a New Integrable Symplectic Map**The project supported by the Scientific Research Award Foundation for Outstanding Young and Middle-Aged Scientists of Shandong Province of China

A discrete matrix spectral problem and the associated hierarchy of Lax integrable lattice equations are presented, and it is shown that the resulting Lax integrable lattice equations are all Liouville integrable discrete Hamiltonian systems. A new integrable symplectic map is given by binary Bargmann constraint of the resulting hierarchy. Finally, an infinite set of conservation laws is given for the resulting hierarchy.

A Hierarchy of Integrable Nonlinear Lattice Equations and New Integrable Symplectic Map**The project supported by Scientific Research Award Foundation for Shandong Provincial Outstanding Young and Middle-Age Scientists

A discrete spectral problem is discussed, and a hierarchy of integrable nonlinear lattice equations related to this spectral problem is devised. The new integrable symplectic map and finite-dimensional integrable systems are given by nonlinearization method. The binary Bargmann constraint gives rise to a Bäcklund transformation for the resulting integrable lattice equations.

A Lax integrable hierarchy, N-Hamiltonian structure, r-matrix, finite-dimensional Liouville integrable involutive systems, and involutive solutions

An eigenvalue problem and the associated new Lax integrable hierarchy of nonlinear evolution equations are presented in this paper. As a reduction, a representative system of the generalized derivative nonlinear Schrödinger equations in the hierarchy is obtained. Zero curvature representation and N-Hamiltonian structures are established for the whole hierarchy based upon N pairs of Hamiltonian operators (Lenard's operators), and it is shown that the hierarchy of nonlinear evolution equations is integrable in Liouville's sense. Thus the hierarchy of equations has infinitely many commuting symmetries and conservation laws. Moreover the eigenvalue problem is nonlinearized as a finite-dimensional completely integrable Hamiltonian system under the Bargmann constraint between the potentials and the eigenvalue functions, and then an associated nondynamical modified r-matrix is constructed. Finally finite-dimensional Liouville integrable involutive systems are found, and the involutive solutions of the hierarchy of equations are given, in particular, the involutive solutions are developed for the system of generalized derivative nonlinear Schrödinger equations.

A hierarchy of the Lax integrable system, its bi-Hamiltonian structure, finite-dimensional integrable system and involutive solution

In this paper an isospectral problem and the associated hierarchy of Lax integrable system are considered. Zero-curvature representation and bi-Hamiltonian structures are established for the whole hierarchy by using trace identity and Lenard's operator pair. Moreover the isospectral problem is nonlinearized as a finite-dimensional, completely integrable Hamiltonian system under the Bargmann constraint between the potentials and the eigenvalue functions, and then an associated Lax representation is constructed. Finally finite-dimensional Liouville integrable involutive systems are found, and the involutive solutions of the hierarchy of equations are given.