Conservative finite difference schemes for the Degasperis–Procesi equation

We consider the numerical integration of the Degasperis–Procesi equation, which was recently introduced as a completely integrable shallow water equation. For the equation, we propose nonlinear and linear finite difference schemes that preserve two invariants associated with the bi-Hamiltonian form of the equation at the same time. We also prove the unique solvability of the schemes, and show some numerical examples.     

Integrable discretization and numerical simulations of the generalized coupled integrable dispersionless equations

In this paper, we study the generalized coupled integrable dispersionless (GCID) equations and construct two integrable discrete analogues including a semi-discrete system and a full-discrete one. The results are based on the relations among the GCID equations, the sine-Gordon equation and the two-dimensional Toda lattice equation. We also present the N-soliton solutions to the semi-discrete and fully discrete systems in the form of Casorati determinant. In the continuous limit, we show that the fully discrete GCID equations converge to the semi-discrete GCID equations, then further to the continuous GCID equations. By using the integrable semi-discrete system, we design two numerical schemes to the GCID equations and carry out several numerical experiments with solitons and breather solutions.     

Noncanonical time transformations relating finite-dimensional integrable systems

We consider dual Stäckel schemes related to each other by a noncanonical transformation of the time variable. We prove that this duality of different integrable systems arises from the multivaluedness of the Abel mapping. We construct the Lax matrices and the r-matrix algebras for some integrable systems on a plane. The integrable deformations of the Kepler problem and the Holt-type systems are considered in detail.     

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.     

Symmetries of Integrable Difference Equations on the Quad-Graph

This paper describes symmetries of all integrable difference equations that belong to the famous Adler–Bobenko–Suris classification. For each equation, the characteristics of symmetries satisfy a functional equation, which we solve by reducing it to a system of partial differential equations. In this way, all five-point symmetries of integrable equations on the quad-graph are found. These include mastersymmetries, which allow one to construct infinite hierarchies of local symmetries. We also demonstrate a connection between the symmetries of quad-graph equations and those of the corresponding Toda type difference equations.     

Integrable symplectic maps associated with discrete Korteweg‐de Vries‐type equations

In this paper, we present novel integrable symplectic maps, associated with ordinary difference equations, and show how they determine, in a remarkably diverse manner, the integrability, including Lax pairs and the explicit solutions, for integrable partial difference equations which are the discrete counterparts of integrable partial differential equations of Korteweg‐de Vries‐type (KdV‐type). As a consequence it is demonstrated that several distinct Hamiltonian systems lead to one and the same difference equation by means of the Liouville integrability framework. Thus, these integrable symplectic maps may provide an efficient tool for characterizing, and determining the integrability of, partial difference equations.     

一类差分方程的不变曲线分枝

讨论一类差分方程的不变曲线分枝。由于该差分方程所定义的动力系统是可积的,故该方程的不变曲线的讨论可化为对平面Hamilton系统所定义的轨线的拓扑分类的研究。通过严格的定性分析,获得其不变曲线在参数空间内的分类。     

The discrete Korteweg-de Vries equation

We review the different aspects of integrable discretizations in space and time of the Korteweg-de Vries equation, including Miura transformations to related integrable difference equations, connections to integrable mappings, similarity reductions and discrete versions of Painlevé equations as well as connections to Volterra systems.     

S. V. Kovalevskaya system, its generalization and discretization

We consider an integrable three-dimensional system of ordinary differential equations introduced by S. V. Kovalevskaya in a letter to G. Mittag-Leffler. We prove its isomorphism with the three-dimensional Euler top, and propose two integrable discretizations for it. Then we present an integrable generalization of the Kovalevskaya system, and study the problem of integrable discretization for this generalized system.     

利用李代数构造非线性可积及双可积哈密顿耦合

With the help of a Lie algebra,two kinds of Lie algebras with the forms of blocks are introduced for generating nonlinear integrable and bi-integrable couplings.For illustrating the application of the Lie algebras,an integrable Hamiltonian system is obtained,from which some reduced evolution equations are presented.Finally,Hamiltonian structures of nonlinear integrable and bi-integrable couplings of the integrable Hamiltonian system are furnished by applying the variational identity.The approach presented in the paper can also provide nonlinear integrable and bi-integrable couplings of other integrable system.     

Commutator identities on associative algebras,the non-Abelian Hirota difference equation and its reductions

We show that the non-Abelian Hirota difference equation is directly related to a commutator identity on an associative algebra. Evolutions generated by similarity transformations of elements of this algebra lead to a linear difference equation. We develop a special dressing procedure that results in an integrable non-Abelian Hirota difference equation and propose two regular reduction procedures that lead to a set of known equations, Abelian or non-Abelian, and also to some new integrable equations.     

A unified expressing model of the AKNS hierarchy and the KN hierarchy, as well as its integrable coupling system

A new subalgebra of loop algebra Ã₁ is first constructed. Then a new Lax pair is presented, whose compatibility gives rise to a new Liouville integrable system(called a major result), possessing bi-Hamiltonian structures. It is remarkable that two symplectic operators obtained in this paper are directly constructed in terms of the recurrence relations. As reduction cases of the new integrable system obtained, the famous AKNS hierarchy and the KN hierarchy are obtained, respectively. Second, we prove a conjugate operator of a recurrence operator is a hereditary symmetry. Finally, we construct a high dimension loop algebra to obtain an integrable coupling system of the major result by making use of Tu scheme. In addition, we find the major result obtained is a unified expressing integrable model of both the AKNS and KN hierarchies, of course, we may also regard the major result as an expanding integrable model of the AKNS and KN hierarchies. Thus, we succeed to find an example of expanding integrable models being Liouville integrable.     

The geometry of integrable and superintegrable systems

We consider the automorphism group of the geometry of an integrable system. The geometric structure used to obtain it is generated by a normal-form representation of integrable systems that is independent of any additional geometric structure like symplectic, Poisson, etc. Such a geometric structure ensures a generalized toroidal bundle on the carrier space of the system. Noncanonical diffeomorphisms of this structure generate alternative Hamiltonian structures for completely integrable Hamiltonian systems. The energy-period theorem for dynamical systems implies the first nontrivial obstruction to the equivalence of integrable systems.     

Compatible Nonlocal Poisson Brackets of Hydrodynamic Type and Integrable Hierarchies Related to Them

We construct integrable bi-Hamiltonian hierarchies related to compatible nonlocal Poisson brackets of hydrodynamic type and solve the problem of the canonical form for a pair of compatible nonlocal Poisson brackets of hydrodynamic type. A system of equations describing compatible nonlocal Poisson brackets of hydrodynamic type is derived. This system can be integrated by the inverse scattering problem method. Any solution of this integrable system generates integrable bi-Hamiltonian systems of hydrodynamic type according to explicit formulas. We construct a theory of Poisson brackets of the special Liouville type. This theory plays an important role in the construction of integrable hierarchies.     

Integrable Duffing’s maps and solutions of the Duffing equation

In the numerical integration of nonlinear differential equations, discretization of the nonlinear terms poses extra ambiguity in reducing the differential equation to a discrete difference equation. As for the cubic nonlinear Schrodinger equation, it was well known that there exists the corresponding discrete soliton system. Here, representing the discrete systems by the mappings, we explore structure of the integrable mappings. We introduce the first kind and the second kind of Duffing’s map, and investigate temporal evolution of the orbits. Although the smooth periodic orbits are in accord with the solutions of the Duffing equation, the integrable Duffing’s maps provide much wider variety of orbits.     

UNCONDITIONAL STABLE DIFFERENCE METHODS WITH INTRINSIC PARALLELISM FOR SEMILINEAR PARABOLIC SYSTEMS OF DIVERGENCE TYPE

The general finite difference schemes with intrinsic parallelism for the boundary value problem of the semilinear parabolic system of divergence type with bounded measurable coefficients is studied. By the approach of the discrete functional analysis, the existence and uniqueness of the discrete vector solutions of the nonlinear difference system with intrinsic parallelism are proved. Moreover the unconditional stability of the general difference schemes with intrinsic parallelism justified in the sense of the continuous dependence of the discrete vector solution of the difference schemes on the discrete initial data of the original problems in the discrete W_2~(2,1) (Q△) norms. Finally the convergence of the discrete vector solutions of the certain difference schemes with intrinsic parallelism to the unique generalized solution of the original semilinear parabolic problem is proved.     

GENERAL DIFFERENCE SCHEMES WITH INTRINSIC PARALLELISM FOR SEMILINEAR PARABOLIC SYSTEMS OFDIVERGENCE TYPE

1.IntroductionIn[1]and[2]thegeneralfinitedifferenceschemeshavingtheintrinsiccharacterofparallelismfortheboundaryvalueproblemsofthenonlinearparabolicsystemofgeneralform(i.e.,non-divergencetype)arediscussedundertheassumptionthatthereisanuniquesmoothsol...     

Weierstrass integrability of differential equations

The integrability problem consists of finding the class of functions a first integral of a given planar polynomial differential system must belong to. We recall the characterization of systems which admit an elementary or Liouvillian first integral. We define Weierstrass integrability and we determine some Weierstrass integrable systems which are Liouvillian integrable. Inside this new class of integrable systems there are non-Liouvillian integrable systems.     

Application of the Fourier Transform to the Solution of Singular Integral Equations

A method for solving a certain system of singular integral equations with constant coefficients is proposed. It is based on a procedure for reducing singular equations to equations with continuous difference kernel; the solution of the latter is constructed by using the classical Fourier transform in the class of absolutely integrable functions. Explicit expressions for the solution of the singular integral equations under consideration are obtained.     

Explicit Solutions to an Integrable Lattice

In this paper, we aim to construct the Darboux transformation and explicit solutions for an integrable lattice introduced by Suris. Analysis of properties of the solutions shows that the obtained explicit solutions for this discrete integrable system possess new dynamical characters which are different from the ones of continuous integrable systems.