Hierarchy Of Integrable Equations Research Articles

A Vector-Product Lie Algebra of a Reductive Homogeneous Space and Its Applications

A new vector-product Lie algebra is constructed for a reductive homogeneous space, which can lead to the presentation of two corresponding loop algebras. As a result, two integrable hierarchies of evolution equations are derived from a new form of zero-curvature equation. These hierarchies can be reduced to the heat equation, a special diffusion equation, a general linear Schrödinger equation, and a nonlinear Schrödinger-type equation. Notably, one of them exhibits a pseudo-Hamiltonian structure, which is derived from a new vector-product identity proposed in this paper.

Finite Genus Solutions to a Hierarchy of Integrable Semi-Discrete Equations

Finite Genus Solutions to a Hierarchy of Integrable Semi-Discrete Equations

A four-component hierarchy of combined integrable equations with bi-Hamiltonian formulations

Upon introducing a specific 4 × 4 matrix eigenvalue problem with four components, we would like to construct a Liouville integrable Hamiltonian hierarchy, within the zero curvature formulation. Bi-Hamiltonian formulations are furnished via the trace identity, through which the Liouville integrability of the resulting hierarchy is explored. The first two nonlinear examples are novel generalized combined nonlinear Schrödinger equations and modified Korteweg–de Vries equations.

On the Propagation Model of Two-Component Nonlinear Optical Waves

Currently, two-component integrable nonlinear equations from the hierarchies of the vector nonlinear Schrodinger equation and the vector derivative nonlinear Schrödinger equation are being actively investigated. In this paper, we propose a new hierarchy of two-component integrable nonlinear equations, which have an important difference from the already known equations. To construct the hierarchical equations, we use the monodromy matrix method, as first proposed by B.A. Dubrovin. The method we use consists of solving the following sequence of problems. First, using the Lax operator, we find the monodromy matrix, which is a polynomial in the spectral parameter. More precisely, we find a sequence of monodromy matrices dependent on the degree of this polynomial. Each Lax operator has its own sequence of monodromy matrices. Then, using the terms from the decomposition of the monodromy matrix, we construct a sequence of second operators from a Lax pair. A hierarchy of evolutionary integrable nonlinear equations follows from the conditions of compatibility of the sequence of Lax pairs. Also, knowledge of the monodromy matrix allows us to find stationary equations that are analogs of the Novikov equations for the Korteweg–de Vries equation. In addition, the characteristic equation of the monodromy matrix corresponds to the spectral curve equation of the relevant multiphase solution for the integrable nonlinear equation. Since the coefficients of the spectral curve equation are integrals of the hierarchical equations, they can be utilized to find the simplest solutions of the constructed integrable nonlinear equations. In this paper, we demonstrate the operation of this method, starting with the assignment of the Lax operator and ending with the construction of the simplest solutions.

Using Vector-Product Loop Algebra to Generate Integrable Systems

A new three-dimensional Lie algebra and its loop algebra are proposed by us, whose commutator is a vector product. Based on this, a positive flow and a negative flow are obtained by introducing a new kind of spectral problem expressed by the vector product, which reduces to a generalized KdV equation, a generalized Schrödinger equation, a sine-Gordon equation, and a sinh-Gordon equation. Next, the well-known Tu scheme is generalized for generating isospectral integrable hierarchies and non-isospectral integrable hierarchies. It is important that we make use of the variational method to create a new vector-product trace identity for which the Hamiltonian structure of the isospectral integrable hierarchy presented in the paper is worded out. Finally, we further enlarge the three-dimensional loop algebra into a six-dimensional loop algebra so that a new isospectral integrable hierarchy which is a type of extended integrable model is produced whose bi-Hamiltonian structure is also derived from the vector-product trace identity. This new approach presented in the paper possesses extensive applications in the aspect of generating integrable hierarchies of evolution equations.

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.

Coverings and nonlocal symmetries as well as fundamental solutions of nonlinear equations derived from the nonisospectral AKNS hierarchy

In the paper, we apply a new scheme to generate nonisospectral integrable hierarchies of evolution equations. First of all, we work out the positive-order and the negative-order nonisospectral AKNS hierarchies. Next we introduce a kind of loop algebra from which an expanding nonisospectral integrable model is derived. Specially, the expanding integrable model reduces to the well-known bond pricing equation and the interest rate modeling. By using the Lie group theory we investigate the Lie representations and characteristic solutions of a generalized bond pricing equation from the positive-order nonisospectral integrable equation, and with the help of the characteristic solutions and the Laplace transformation, we again obtain the fundamental solutions of the generalized bond pricing equation. Besides, we apply the Lie group analysis to deduce the invariant solutions and the nonlinear self-adjointness as well as conservation laws, non-invariant solutions of the generalized bond pricing equation. Some reductions of the negative-order nonisospectral integrable hierarchy are also produced. Two of them are the Sine-Gordan equation and the Sinh-Gordan equation. Finally, we investigate the coverings and the nonlocal symmetries of the generalized bond pricing equation and a generalized KdV equation, respectively, although the bond equation is linear which has many applications in the aspect of finance field, by applying the classical Frobenius theorem and the coordinates of a infinitely-dimensional manifold in the form of Cartesian product.

Several Isospectral and Non-Isospectral Integrable Hierarchies of Evolution Equations

By introducing a 3×3 matrix Lie algebra and employing the generalized Tu scheme, a AKNS isospectral–nonisospectral integrable hierarchy is generated by using a third-order matrix Lie algebra. Through a matrix transformation, we turn the 3×3 matrix Lie algebra into a 2×2 matrix case for which we conveniently enlarge it into two various expanding Lie algebras in order to obtain two different expanding integrable models of the isospectral–nonisospectral AKNS hierarchy by employing the integrable coupling theory. Specially, we propose a method for generating nonlinear integrable couplings for the first time, and produce a generalized KdV-Schrödinger integrable system and a nonlocal nonlinear Schrödinger equation, which indicates that we unite the KdV equation and the nonlinear Schrödinger equation as an integrable model by our method. This method presented in the paper could apply to investigate other integrable systems.

On solutions for some class of integrable difference equations

In this paper we show that an arbitrary solution of one ordinary difference equation is also a solution for a hierarchy of integrable difference equations. We also provide an example of such a solution that is related to a sequence generated by a second-order linear recursion with 2-periodic coefficients.

Riemann–Hilbert problems and inverse scattering of nonlocal real reverse-spacetime matrix AKNS hierarchies

We would like to propose a kind of nonlocal real reverse-spacetime integrable hierarchies of PT-symmetric matrix AKNS equations through nonlocal symmetry reductions on the potential matrix, and formulate their associated Riemann–Hilbert problems to determine generalized Jost solutions of arbitrary-order matrix spectral problems. The Sokhotski–Plemelj formula is applied in transforming the associated Riemann–Hilbert problems into Gelfand–Levitan–Marchenko type integral equations. The Riemann–Hilbert problems corresponding to the reflectionless case are solved explicitly, where eigenvalues could equal adjoint eigenvalues, and thus, soliton solutions are presented for the resulting nonlocal real reverse-spacetime integrable PT-symmetric matrix AKNS equations.

Hamiltonian structures for integrable hierarchies of Lagrangian PDEs

Many integrable hierarchies of differential equations allow a variational description, called a Lagrangian multiform or a pluri-Lagrangian structure. The fundamental object in this theory is not a Lagrange function but a differential $d$-form that is integrated over arbitrary $d$-dimensional submanifolds. All such action integrals must be stationary for a field to be a solution to the pluri-Lagrangian problem. In this paper we present a procedure to obtain Hamiltonian structures from the pluri-Lagrangian formulation of an integrable hierarchy of PDEs. As a prelude, we review a similar procedure for integrable ODEs. We show that exterior derivative of the Lagrangian $d$-form is closely related to the Poisson brackets between the corresponding Hamilton functions. In the ODE (Lagrangian 1-form) case we discuss as examples the Toda hierarchy and the Kepler problem. As examples for the PDE (Lagrangian 2-form) case we present the potential and Schwarzian Korteweg-de Vries hierarchies, as well as the Boussinesq hierarchy.

Kadomtsev–Petviashvili Hierarchy: Negative Times

The Kadomtsev–Petviashvili equation is known to be the leading term of a semi-infinite hierarchy of integrable equations with evolutions given by times with positive numbers. Here, we introduce new hierarchy directed to negative numbers of times. The derivation of such systems, as well as the corresponding hierarchy, is based on the commutator identities. This approach enables introduction of linear differential equations that admit lifts up to nonlinear integrable ones by means of the special dressing procedure. Thus, one can construct not only nonlinear equations, but corresponding Lax pairs as well. The Lax operator of this evolution coincides with the Lax operator of the “positive” hierarchy. We also derive (1 + 1)-dimensional reductions of equations of this hierarchy.

On the modified Poisson–Boltzmann closure for primitive model electrolytes at high concentration

The modified Poisson-Boltzmann closure is applied to the Kirkwood hierarchy of integral equations to investigate high concentration primitive model electrolytes. Two approximations are considered in the two sphere fluctuation potential problem. The derived damped oscillatory mean electrostatic potentials suggest that this closure should be of use in providing a basis for understanding the large experimental decay lengths found at high electrolyte concentrations.

Matrix integral solutions to the discrete and coupled Leznov lattice equations

Matrix integrals used in random matrix theory for the study of eigenvalues of matrix ensembles have been shown to provide τ-functions for several hierarchies of integrable equations. In this paper, we construct the matrix integral solutions to the Leznov lattice equation, semi-discrete and fully-discrete version and the Pfaffianized Leznov lattice systems, respectively. We demonstrate that the partition function of the Jacobi type unitary ensemble is a solution to the semi-discrete Leznov lattice and the partition function of the Jacobi type orthogonal/symplectic ensemble solves the Pfaffianized Leznov lattice.

An Ablowitz-Ladik Integrable Lattice Hierarchy with Multiple Potentials

Within the zero curvature formulation, a hierarchy of integrable lattice equations is constructed from an arbitrary-order matrix discrete spectral problem of Ablowitz-Ladik type. The existence of infinitely many symmetries and conserved functionals is a consequence of the Lax operator algebra and the trace identity. When the involved two potential vectors are scalar, all the resulting integrable lattice equations are reduced to the standard Ablowitz-Ladik hierarchy.

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.

Constructing super D-Kaup-Newell hierarchy and its nonlinear integrable coupling with self-consistent sources

How to construct new super integrable equation hierarchy is an important problem. In this paper, a new Lax pair is proposed and the super D-Kaup-Newell hierarchy is generated, then a nonlinear integrable coupling of the super D-Kaup-Newell hierarchy is constructed. The super Hamiltonian structures of coupling equation hierarchy is derived with the aid of the super variational identity. Finally, the self-consistent sources of super integrable coupling hierarchy is established. It is indicated that this method is a straightforward and efficient way to construct the super integrable equation hierarchy.

A method for generating isospectral and nonisospectral hierarchies of equations as well as symmetries

We have known that the Tu scheme is a powerful tool for generating isospectral integrable hierarchies of evolution equations. However, it is not clear that how to apply it to generate nonisospectral integrable hierarchies along with the time-evolution of the spectral parameter as the polynomial form in the spectral parameter λ. In the paper, we propose a method for generating isospectral and nonisospectral integrable hierarchies by improved the Tu scheme. As one of applications, a spectral–nonisospectral problem is introduced for which a kind of isospectral plus nonisospectral integrable hierarchy is presented, including the generalized KdV equations and the cylinder-KdV equation with variable coefficients. One of the generalized KdV equations generalizes the variable-coefficient integrable equation given by Li Yishen. Therefore, we choose it to discuss and obtain its various symmetries with the help of Lie symmetry analysis which looks complicated, but another two sets of infinite symmetries are produced by the infinite conserved densities of the integrable equation. As the second application, an isospectral–nonisospectral AKNS-Kaup-Newell soliton hierarchy (briefly, AKNS-K-N-SH) is derived from an appropriate spectral problem whose K symmetries and a set of new τ symmetries are generated by applying Lie group analysis. As a reduction of the AKNS-K-N-SH, a generalized sine-Gordon equation is singled out.

Dispersionless limit of the (1+1)-dimensional Fokas-Lenells equation

Nonlinear dispersionless equations can be obtained as dispersionless limits (quasiclassical limits) of integrable hierarchies of equations or by constructing a system of hydrodynamic type. In this paper, the dispersionless limit of the (1+1)-dimensional Fokas-Lenells equation is found. We present Lax pair for this dispersionless equation. As well known, the Fokas-Lenells equation characterizes the propagation of ultrashort nonlinear light pulses in optical fibers. Exact solutions of the dispersionless version of the Fokas-Lenells equation describe nonlinear shock waves in optical fiber systems. By using the obtained results, one can find shock wave solutions of the dispersionless equation, which have different physical applications.

Boundary conditions for General Relativity in three-dimensional spacetimes, integrable systems and the KdV/mKdV hierarchies

We present a new set of boundary conditions for General Relativity on AdS3, where the dynamics of the boundary degrees of freedom are described by two independent left and right members of the Gardner hierarchy of integrable equations, also known as the “mixed KdV-mKdV” hierarchy. This integrable system has the very special property that simultaneously combines both, the Korteweg-de Vries (KdV) and modified Korteweg-de Vries (mKdV) hierarchies in a single integrable structure. This relationship between gravitation in three-dimensional spacetimes and two-dimensional integrable systems is based on an extension of the recently introduced “soft hairy boundary conditions” on AdS3, where the chemical potentials are now allowed to depend locally on the dynamical fields and their spatial derivatives. The complete integrable structure of the Gardner system, i.e., the phase space, the Poisson brackets and the infinite number of commuting conserved charges, are directly obtained from the asymptotic analysis and the conserved surface integrals in the gravitational theory. These boundary conditions have the particular property that they can also be interpreted as being defined in the near horizon region of spacetimes with event horizons. Black hole solutions are then naturally accommodated within our boundary conditions, and are described by static configurations associated to the corresponding member of the Gardner hierarchy. The thermodynamic properties of the black holes in the ensembles defined by our boundary conditions are also discussed. Finally, we show that our results can be naturally extended to the case of a vanishing cosmological constant, and the integrable system turns out to be precisely the same as in the case of AdS3.