Hierarchy Of Integrable Equations Research Articles

A 2-PARAMETER HIERARCHY OF INTEGRABLE LATTICE EQUATIONS

A new discrete matrix spectral problem with two arbitrary constants is introduced, and the corresponding 2-parameter hierarchy of integrable lattice equations is obtained by discrete zero curvature representation. The resulting integrable lattice equations reduce to the hierarchy of relativistic Toda lattice in rational form for a special choice of the parameters. Moreover, a sub-hierarchy of the resulting integrable lattice equations is discussed. It is shown that each lattice equation in the sub-hierarchy is a Liouville integrable discrete Hamiltonian equation.

The multicomponent (2+1)-dimensional Glachette–Johnson (GJ) equation hierarchy and its super-integrable coupling system

This paper presents a set of multicomponent matrix Lie algebra, which is used to construct a new loop algebra ÃM. By using the Tu scheme, a Liouville integrable multicomponent equation hierarchy is generated, which possesses the Hamiltonian structure. As its reduction cases, the multicomponent (2+1)-dimensional Glachette–Johnson (GJ) hierarchy is given. Finally, the super-integrable coupling system of multicomponent (2+1)-dimensional GJ hierarchy is established through enlarging the spectral problem.

New computational formulae concerning the constant [formula omitted] in the trace identity and the quadratic-form identity

The trace identity and the quadratic-form identity are all simple and powerful tools for establishing Hamiltonian structure of integrable hierarchies of soliton equations, the constant γ contained in the two identities are all to be determined. It has been a left problem to seek for computing formulas on γ, which had been specially proposed by Tu [Tu Guizhang. The trace identity, a powerful tool for constructing the Hamiltonian structure of integrable systems. J Math Phys 1989;30(2):330–8]. In this paper, we create an efficient method for obtaining γ by making use of two procedures. First, a few quadratic expressions G(V)’s are discovered from the solvable conditions on Λ, where Λ satisfies the equation [Λ,V]-Vλ=γλV, whereas, G(V) and γ have the clear relations. Second, by means of Vx=[U,V], we prove that G(V) is an one-place function with aspect to λ, but not related to x. It follows from the above two steps that the formula γ=-λ2ddλln|G(V)| is obtained. This technique is verified to be feasible and efficient by applying it to a few examples.

New block matrix spectral problem and Hamiltonian structure of the discrete integrable coupling system

In [W.X. Ma, J. Phys. A: Math. Theor. 40 (2007) 15055], Prof. Ma gave a beautiful result (a discrete variational identity). In this Letter, based on a discrete block matrix spectral problem, a new hierarchy of Lax integrable lattice equations with four potentials is derived. By using of the discrete variational identity, we obtain Hamiltonian structure of the discrete soliton equation hierarchy. Finally, an integrable coupling system of the soliton equation hierarchy and its Hamiltonian structure are obtained through the discrete variational identity.

A hierarchy of Liouville integrable discrete Hamiltonian equations

Based on a discrete four-by-four matrix spectral problem, a hierarchy of Lax integrable lattice equations with two potentials is derived. Two Hamiltonian forms are constructed for each lattice equation in the resulting hierarchy by means of the discrete variational identity. A strong symmetry operator of the resulting hierarchy is given. Finally, it is shown that the resulting lattice equations are all Liouville integrable discrete Hamiltonian systems.

Constructing (2+1)-dimensional nonlinear soliton equations with a generalized zero curvature equation

The generalized zero curvature equation is presented, which is an important tool to construct the (2+1)-dimensional integrable soliton equation hierarchy. A few of new (2+1)-dimensional integrable soliton equation hierarchies are obtained through the generalized zero curvature equation.

Three New Integrable Hierarchies of Equations

A general Lie algebra Vs and the corresponding loop algebraṼs are constructed, from which the linear isospectralLax pairs are established, whose compatibility presents the zerocurvature equation. As its application, a new Lax integrablehierarchy containing two parameters is worked out. It is notLiouville-integrable, however, its two reduced systems areLiouville-integrable, whose Hamiltonian structures are derived bymaking use of the quadratic-form identity and the γformula (i.e. the computational formula on the constant γappeared in the trace identity and the quadratic-form identity).

The multicomponent discrete equation hierarchy with variable spectral parameters and a new integrable coupling system

This Letter reports on a study of the multicomponent discrete integrable equation hierarchy with variable spectral parameters. An instance of new multicomponent lattice equation is obtained. Correspondingly, a feasible way to construct the integrable couplings is presented. It indicates that the study of integrable couplings using the block matrix-form of Lie algebra is an important and effective method.

Special Quasigraded Lie Algebras and Integrable Hamiltonian Systems

We construct a family of special quasigraded Lie algebras \(\widetilde{\mathsf{g}}_{F}\) of functions of one complex variables with values in finite-dimensional Lie algebra \(\mathfrak{g}\) , labeled by the special 2-cocycles F on \(\mathsf{g}\) . The main property of the constructed Lie algebras \(\widetilde{\mathsf{g}}_{F}\) is that they admit Kostant-Adler-Symes scheme. Using them we obtain new integrable finite-dimensional Hamiltonian systems and new hierarchies of soliton equations.

Hierarchies of nonlinear integrable q-difference equations from series of Lax pairs

We present two hierarchies of nonlinear, integrable q-difference equations, one of which includes a q-difference form of each of the second and fifth Painlevé equations, and , the other includes . All the equations have multiple free parameters. A method to calculate a 2 × 2 Lax pair for each equation in the hierarchy is also given.

A hierarchy of integrable partial differential equations in 2+1 dimensions associated with one-parameter families of one-dimensional vector fields

We introduce a hierarchy of integrable partial differential equations in 2+1 dimensions arising from the commutation of one-parameter families of vector fields, and we construct the formal solution of the associated Cauchy problems using the inverse scattering method for one-parameter families of vector fields. Because the space of eigenfunctions is a ring, the inverse problem can be formulated in three distinct ways. In particular, one formulation corresponds to a linear integral equation for a Jost eigenfunction, and another formulation is a scalar nonlinear Riemann problem for suitable analytic eigenfunctions.

Multi-component bi-Hamiltonian Dirac integrable equations

A specific matrix iso-spectral problem of arbitrary order is introduced and an associated hierarchy of multi-component Dirac integrable equations is constructed within the framework of zero curvature equations. The bi-Hamiltonian structure of the obtained Dirac hierarchy is presented be means of the variational trace identity. Two examples in the cases of lower order are computed.

Two types of loop algebras and their expanding Lax integrable models

Though various integrable hierarchies of evolution equations were obtained by choosing proper U in zero-curvature equation Ut−Vx+[U,V] = 0, but in this paper, a new integrable hierarchy possessing bi-Hamiltonian structure is worked out by selecting V with spectral potentials. Then its expanding Lax integrable model of the hierarchy possessing a simple Hamiltonian operator is presented by constructing a subalgebra of the loop algebra A2. As linear expansions of the above-mentioned integrable hierarchy and its expanding Lax integrable model with respect to their dimensional numbers, their (2+1)-dimensional forms are derived from a (2+1)-dimensional zero-curvature equation.

Multi-component Dirac equation hierarchy and its multi-component integrable couplings system

A general scheme for generating a multi-component integrable equation hierarchy is proposed. A simple 3M-dimensional loop algebra is produced. By taking advantage of , a new isospectral problem is established and then by making use of the Tu scheme the multi-component Dirac equation hierarchy is obtained. Finally, an expanding loop algebra M of the loop algebra is presented. Based on the M, the multi-component integrable coupling system of the multi-component Dirac equation hierarchy is investigated. The method in this paper can be applied to other nonlinear evolution equation hierarchies.

A NEW HIERARCHY OF DISCRETE INTEGRABLE MODEL AND ITS DARBOUX TRANSFORMATION

A new isospectral problem is proposed and the corresponding integrable equation hierarchy is given. A Darboux transformation (DT) is derived to obtain the exact solutions for the typical lattice soliton equations.

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.

The computational formula on the constant γ appeared in the equivalently used trace identity and quadratic-form identity

V 3 is a 3-dimensional Lie algebra whose commuting operation involves in nine parameters. Appropriately choosing these parameters makes V 3 become an arbitrary 3-dimensional Lie algebra. Denote the loop algebra V ∼ 3 generated by V 3, from which linear isospectral problems are set up to produce a good many integrable hierarchies of equations. Some of them can be turned into Hamiltonian forms by means of the trace identity, others can be cast into Hamiltonian expressions with the help of the quadratic-form identity. A unified computational formula on the constant γ appeared in the trace identity and the quadratic-form identity is obtained, from which the left problem remarkably proposed in Tu [Tu Guizhang. The trace identity, a powerful tool for constructing the Hamiltonian structure of integrable systems. J Math Phys 1989;30(2):330–8] is completely overcome in the paper.

Constructing a multi-function equation hierarchy and its integrable coupling system with MAPLE

In this article by considering a multi-dimensional Lie algebra system, a new multi-function integrable equation hierarchy is presented with the help of MAPLE. Then the integrable coupling system of the new multi-function integrable equation hierarchy is obtained through enlarging matrix spectral problem.

An extended trace identity and applications

For the loop algebras in the form of non-square matrices, their commuting operations can be used to set up linear isospectral problems. In order to look for the Hamiltonian structures of the corresponding integrable evolution hierarchies of equations, an extended trace identity is obtained by means of commutators, which undoes the constraint on the known trace identity proposed by Tu [Guizhang Tu. The trace identity, a powerful tool for constructing the Hamiltonian structure of integrable systems. J Math Phys 1989;30(2):330–8], and has an obvious simplicity comparing with the quadratic-form identity given by Guo and Zhang [Fukui Guo, Yufeng Zhang. The quadratic-form identity for constructing the Hamiltonian structure of integrable systems. J Phys A 2005;38:8537–48] with the aspect of applications.

Solitary wave interaction and evolution for a higher-order Hirota equation

Solitary wave interaction and evolution for a higher-order Hirota equation is examined. The higher-order Hirota equation is asymptotically transformed to a higher-order member of the NLS hierarchy of integrable equations, if the higher-order coefficients satisfy a certain algebraic relationship. The transformation is used to derive higher-order one- and two-soliton solutions. It is shown that the interaction is asymptotically elastic and the higher-order corrections to the coordinate and phase shifts are derived. For the higher-order Hirota equation resonance occurs between the solitary waves and linear radiation, so soliton perturbation theory is used to determine the details of the evolving wave and its tail. An analytical expression for the solitary wave tail is derived and it is found that the tail vanishes when the algebraic relationship from the asymptotic theory is satisfied. Hence a two-parameter family of higher-order asymptotic embedded solitons exists. A comparison between the theoretical predictions and numerical solutions shows strong agreement for both solitary wave interaction, where the higher-order coordinate and phase shifts are compared, and solitary wave evolution, with comparisons made of the solitary wave tail.