Decomposing a new nonlinear differential-difference system under a Bargmann implicit symmetry constraint

Firstly, a hierarchy of integrable lattice equations and its bi-Hamilt-onian structures are established by applying the discrete trace identity. Secondly, under an implicit Bargmann symmetry constraint, every lattice equation in the nonlinear differential-difference system is decomposed by an completely integrable symplectic map and a finite-dimensional Hamiltonian system. Finally, the spatial part and the temporal part of the Lax pairs and adjoint Lax pairs are all constrained as finite dimensional Liouville integrable Hamiltonian systems.     

A hierarchy of differential-difference equations,conservation laws and new integrable coupling system

Starting from a discrete spectral problem with two arbitrary parameters, a hierarchy of nonlinear differential-difference equations is derived. The new hierarchy not only includes the original hierarchy, but also the well-known Toda equation and relativistic Toda equation. Moreover, infinitely many conservation laws for a representative discrete equation are given. Further, a new integrable coupling system of the resulting hierarchy is constructed.     

A NEW INTEGRABLE SYMPLECTIC MAP ASSOCIATED WITH A DISCRETE MATRIX SPECTRAL PROBLEM

A hierarchy of lattice soliton equations is derived from a discrete matrix spectral problem. It is shown that the resulting lattice soliton equations are all discrete Liouville integrable systems. A new integrable symplectic map and a family of finite-dimensional integrable systems are given by the binary nonli-nearization method. The binary Bargmann constraint gives rise to a Backlund transformation for the resulting lattice soliton equations.     

The Liouville integrability of integrable couplings of Volterra lattice equation

A hierarchy of integrable couplings of Volterra lattice equations with three potentials is proposed, which is derived from a new discrete six-by-six matrix spectral problem. Moreover, by means of the discrete variational identity on semi-direct sums of Lie algebra, the two Hamiltonian forms are deduced for each lattice equation in the resulting hierarchy. A strong symmetry operator of the resulting hierarchy is given. Finally, we prove that the hierarchy of the resulting Hamiltonian equations are all Liouville integrable discrete Hamiltonian systems.     

Deautonomisation of differential-difference equations and discrete Painlevé equations

We examine a family of integrable differential-difference equations and obtain their non-autonomous extensions using a discrete/continuous integrability criterion.     

Discrete Zakharov-Shabat system and integrable equations

The method of dressing is carried over to the case of discrete spectral problems. As a result one obtains new differential-difference analogues of completely integrable equations: the sine-Gordon equation, the nonlinear Schrö'dinger vector equation, and the system of N-wave equations.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 146, pp. 137–146, 1985.The author is grateful to A. B. Shabat for his constant interest in the paper.     

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.     

On non-isospectral flows,Painlevé equations,and symmetries of differential and difference equations

We identify the Painlevé Lax pairs with those corresponding to stationary solutions of non-isospectral flows, both for partial differential equations and differential-difference equations. We discuss symmetry reductions of integrable differential-difference equations and show that, in contrast with the continuous case, where Painlevé equations naturally arise, in the discrete case the so-called discrete Painlevé equations cannot be obtained in this way. Actually, symmetry reductions of integrable differential-difference equations naturally provide delay Painlevé equations.In Memory of Prof. M. C. PolivanovDipartimento di Fisica, P. le A. Moro 2, 00185 Roma, Istituto Nazionale di Fisica Nucleare, Sezione di Roma, Italy. Departamento de Fisica Teorica, Facultad de Fisicas, Universidad Complutense, 28040 Madrid, Spain. W561@emducm11.bitnet. Published in Teoreticheskaya i Matematicheskaya Fizika, Vol. 93, No. 3, pp. 473–480, December, 1992.     

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 new algebraic procedure to construct exact solutions of nonlinear differential-difference equations

This paper presents a new algebraic procedure to construct exact solutions of selected nonlinear differential-difference equations. The discrete sine-Gordon equation and differential-difference asymmetric Nizhnik-Novikov-Veselov equations are chosen as examples to illustrate the efficiency and effectiveness of the new procedure, where various types of exact travelling wave solutions for these nonlinear differential-difference equations have been constructed. It is anticipated that the new procedure can also be used to produce solutions for other nonlinear differential-difference equations.     

Bianchi Permutability for the Anti‐Self‐Dual Yang‐Mills Equations

The anti‐self‐dual Yang‐Mills equations are known to have reductions to many integrable differential equations. A general Bäcklund transformation (BT) for the anti‐self‐dual Yang‐Mills (ASDYM) equations generated by a Darboux matrix with an affine dependence on the spectral parameter is obtained, together with its Bianchi permutability equation. We give examples in which we obtain BTs of symmetry reductions of the ASDYM equations by reducing this ASDYM BT. Some discrete integrable systems are obtained directly from reductions of the ASDYM Bianchi system.     

The binary nonlinearization of generalized Toda hierarchy by a special choice of parameters

A new discrete matrix spectral problem with two arbitrary constants is introduced. The corresponding 2-parameter hierarchy of integrable lattice equations, which can be reduced to the hierarchy of Toda lattice, is obtained by discrete zero curvature representation. Moreover, the Hamiltonian structure and a hereditary operators are deduced by applying the discrete trace identity. Finally, an integrable symplectic map and a family of finite-dimensional integrable systems are given by the binary nonlinearization for the resulting hierarchy by a special choice of parameters.     

Discrete equation on a square lattice with a nonstandard structure of generalized symmetries

We clarify the integrability nature of a recently found discrete equation on the square lattice with a nonstandard symmetry structure. We find its L-A pair and show that it is also nonstandard. For this discrete equation, we construct the hierarchies of both generalized symmetries and conservation laws. This equation yields two integrable systems of hyperbolic type. The hierarchies of generalized symmetries and conservation laws are also nonstandard compared with known equations in this class.     

MKdV和FPU方程的李点对称

本文研究了离散微分方程的李对称问题.利用差分方程的延拓方法和交换流方法,我们求得了若干重要的差分方程、微分差分方程的李对称,推广了对称性分析法在连续微分方程讨论时的结果.     

A hierarchy of Liouville integrable lattice equations and its integrable coupling systems

A new discrete two-by-two matrix spectral problem with two potentials is introduced, followed by a hierarchy of integrable lattice equations obtained through discrete zero curvature equations. It is shown that the Hamiltonian structures of the resulting integrable lattice equations are established by virtue of the trace identity. Furthermore, based on a discrete four-by-four matrix spectral problem, the discrete integrable coupling systems of the resulting hierarchy are obtained. Then, with the variational identity, the Hamiltonian structures of the obtained integrable coupling systems are established. Finally, the resulting Hamiltonian systems are proved to be all Liouville integrable.     

A differential-difference Kadomtsev–Petviashvili family possesses a common Kac–Moody–Virasoro symmetry algebra

Starting from the Kac–Moody–Virasoro symmetry algebra of the differential-difference Kadomtsev–Petviashvili equation, a differential-difference Kadomtsev–Petviashvili family is constructed and the corresponding invariant solutions are obtained.     

Rational formal solutions of differential-difference equations

In this paper, we study rational formal solutions of differential-difference equations by using a generalized ansätz. With the help of symbolic computation Maple, we obtain many explicit exact solutions of differential-difference equations(DDEs). The solutions contain solitary wave solutions and periodic wave solutions. The (2 + 1)-dimensional Toda lattice equation, relativistic Toda lattice equation and the discrete mKdV equation are chosen to illustrate our algorithm.     

Symmetries of the DΔmKP hierarchy and their continuum limits

In the recent paper [Stud. App. Math. 147 (2021), 752], squared eigenfunction symmetry constraint of the differential-difference modified Kadomtsev–Petviashvili (DΔmKP) hierarchy converts the DΔmKP system to the relativistic Toda spectral problem and its hierarchy. In this paper, we introduce a new formulation of independent variables in the squared eigenfunction symmetry constraint, under which the DΔmKP system gives rise to the discrete spectral problem and a hierarchy of the differential-difference derivative nonlinear Schrödinger equation of the Chen–Lee–Liu type. In addition, by introducing nonisospectral flows, two sets of symmetries of the DΔmKP hierarchy and their algebraic structure are obtained. We then present a unified continuum limit scheme, by which we achieve the correspondence of the mKP and the DΔmKP hierarchies and their integrable structures.     

A discrete generalization of the extended simplest equation method

We modified the so-called extended simplest equation method to obtain discrete traveling wave solutions for nonlinear differential-difference equations. The Wadati lattice equation is chosen to illustrate the method in detail. Further discrete soliton/periodic solutions with more arbitrary parameters, as well as discrete rational solutions, are revealed. We note that using our approach one can also find in principal highly accurate exact discrete solutions for other lattice equations arising in the applied sciences.     

The Euler Equations of Spatial Gasdynamics and the Integrable Heisenberg Spin Equation

The Euler equations for an inviscid and thermally nonconducting gas are investigated with a view to isolating particular integrable structure. A natural physical constraint is imposed and it is demonstrated that, in the generic case, the steady Euler equations are equivalent to an integrable Heisenberg spin equation subject to a volume‐preserving constraint. A limiting case in which this constraint is not present is indicated.