Higher Hirota Difference Equations and Their Reductions

Theoretical and Mathematical Physics - We previously proposed an approach for constructing integrable equations based on the dynamics in associative algebras given by commutator relations. In the...     

Nonlinear continuous integrable Hamiltonian couplings

Based on a kind of special non-semisimple Lie algebras, a scheme is presented for constructing nonlinear continuous integrable couplings. Variational identities over the corresponding loop algebras are used to furnish Hamiltonian structures for the resulting continuous integrable couplings. The application of the scheme is illustrated by an example of nonlinear continuous integrable Hamiltonian couplings of the AKNS hierarchy of soliton equations.     

Guo族非线性可积耦合的哈密顿结构及其守恒

基于可积耦合的基本理论,我们给出了构造孤子族非线性可积耦合的一般方法,并用相应圈代数上的变分恒等式来求可积耦合的哈密顿结构.作为应用,我们给出了Guo族的非线性可积耦合及其哈密顿结构.最后,给出了Guo族非线性可积耦合的守恒律.     

Commutator identities and integrable hierarchies

Theoretical and Mathematical Physics - The approach based on commutator identities for elements of associative algebras was previously effectively used to investigate $$(2{+}1)$$ -dimensional...     

Coupled systems of nonlinear wave equations and finite-dimensional lie algebras I

We study coupled systems of nonlinear wave equations from the point of view of their formal Darboux integrability. By making use of Vessiot's geometric theory of differential equations, it is possible to associate to each system of nonlinear wave equations a module of vector fields on the second-order jet bundle — the Vessiot distribution. By imposing certain conditions of the structure of the Vessiot distributions, we identify the so-called separable Vessiot distributions. By expressing the separable Vessiot distributions in a basis of singular vector fields, we show that there are, at most, 27 equivalence classes of such distributions. Of these, 14 classes are associated with Darboux integrable nonlinear systems. We take one of these Darboux integrable classes and show that it is in correspondence with the class of six-dimensional simply transitive Lie algebras. Finally, this later result is used to reduce the problem of constructing exact general solutions of the nonlinear wave equations understudy to the integration of Lie systems. These systems were first discovered by Sophus Lie as the most general class of ordinary differential equations which admit nonlinear superposition principles.     

Yang族一个新的非线性可积耦合及其哈密尔顿结构(英文)

Based on a kind of non-semisimple Lie algebras, we establish a way to construct nonlinear continuous integrable couplings. Variational identities over the associated loop algebras are used to furnish Hamiltonian structures of the resulting continuous couplings.As an illustrative example of the scheme is given nonlinear continuous integrable couplings of the Yang hierarchy.     

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.     

Dispersionless integrable equations as coisotropic deformations: Extensions and reductions

We discuss the interpretation of dispersionless integrable hierarchies as equations of coisotropic deformations for certain associative algebras and other algebraic structures. We show that with this approach, the dispersionless Hirota equations for the dKP hierarchy are just the associativity conditions in a certain parameterization. We consider several generalizations and demonstrate that B-type dispersionless integrable hierarchies, such as the dBKP and the dVN hierarchies, are coisotropic deformations of the Jordan triple systems. We show that stationary reductions of the dispersionless integrable equations are connected with dynamical systems on the plane that are completely integrable on a fixed energy level. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 151, No. 3, pp. 439–457, June, 2007.     

Polynomial Identities for Tangent Algebras of Monoassociative Loops

We introduce degree n Sabinin algebras, which are defined by the polynomial identities up to degree n in a Sabinin algebra. Degree 4 Sabinin algebras can be characterized by the polynomial identities satisfied by the commutator, associator, and two quaternators in the free nonassociative algebra. We consider these operations in a free power associative algebra and show that one of the quaternators is redundant. The resulting algebras provide the natural structure on the tangent space at the identity element of an analytic loop for which all local loops satisfy monoassociativity, a ² a ≡ aa ². These algebras are the next step beyond Lie, Malcev, and Bol algebras. We also present an identity of degree 5 which is satisfied by these three operations but which is not implied by the identities of lower degree.     

Quasi-Frobenius Algebras and Their Integrable N-Parameter Deformations Generated by Compatible N×N Metrics of Constant Riemannian Curvature

We prove that the equations describing compatible N×N metrics of constant Riemannian curvature define a special class of integrable N-parameter deformations of quasi-Frobenius (in general, noncommutative) algebras. We discuss connections with open–closed two-dimensional topological field theories, associativity equations, and Frobenius and quasi-Frobenius manifolds. We conjecture that open–closed two-dimensional topological field theories correspond to a special class of integrable deformations of associative quasi-Frobenius algebras.     

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.     

一个新的非等谱可积族和一些相关对称

利用李群$M_nC$的一个子群我们引入一个线性非等谱问题,该问题的相容性条件可导出演化方程的一个非等谱可积族,该可积族可约化成一个广义非等谱可积族.这个广义非等谱可积族可进一步约化成在物理学中具有重要应用的标准非线性薛定谔方程和KdV方程.基于此,我们讨论在广义非等谱可积族等谱条件下的一个广义AKNS族$u_t=K_m(u)$的$K$对称和$\tau$对称.此外,我们还考虑非等谱AKNS族$u_t=\tau_{N+1}^l$的$K$对称和$\tau$对称.最后,我们得到这两个可积族的对称李代数,并给出这些对称和李代数的一些应用,即生成了一些变换李群和约化方程的无穷小算子.     

Coupled systems of nonlinear wave equations and finite-dimensional lie algebras II: A nonlinear system arising from the group G 6.1 and its exact integration

In the first paper of this series a correspondence was established between coupled systems of two-dimensional nonlinear wave equations and the six-dimensional simply transitive Lie algebras. In the present paper we make use of this result to construct a Darboux integrable and exactly integrable nonlinear system associated with the six-parameter nilpotent Lie group G _6,1 and we give its exact general solution in terms of four arbitrary functions. The procedure is shown to be an exact linearization of the nonlinear problem.     

Three nonlinear integrable couplings of the nonlinear Schrödinger equations

Based on two types of expanding Lie algebras of a Lie algebra G, three isospectral problems are designed. Under the framework of zero curvature equation, three nonlinear integrable couplings of the nonlinear Schröding equations are generated. With the help of variational identity, we get the Hamiltonian structure of one of them. Furthermore, we get the result that the hierarchy is also integrable in sense of Liouville.     

Commutator algebras arising from splicing operations

We prove that the variety of Lie algebras arising from splicing operation coincides with the variety CM of centreby-metabelian Lie algebras. Using these Lie algebras we find the minimal dimension algebras generated the variety CM and the variety of its associative envelope algebras. We study the splicing n-ary operation. We show that all n-ary (n > 2) commutator algebras arising from this operation are nilpotent of index 3. We investigate the generalization of the splicing n-ary operation, and we formulate a series of open problems.     

The method of generalized Cole-Hopf substitutions and new examples of linearizable nonlinear evolution equations

We propose a new approach for constructing nonlinear evolution equations in matrix form that are integrable via substitutions similar to the Cole-Hopf substitution linearizing the Burgers equation. We use this new approach to find new integrable nonlinear evolution equations and their hierarchies. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 158, No. 1, pp. 58–71, January, 2009     

Construction of the syzygy module in automaton monomial algebras

In this paper, we consider the problem of algorithmically constructing the left syzygy module for a finite system of elements in an automaton monomial algebra. The class of automaton monomial algebras includes free associative algebras and finitely presented algebras. In such algebras the left syzygy module for a finite system of elements is finitely generated. In general, the left syzygy module in an automaton monomial algebra is not finitely generated. Nevertheless, the generators of the left syzygy module have a recursive specification with the help of regular sets. This allows one to solve many algorithmic problems in automaton monomial algebras. For example, one can solve linear equations, recognize the membership in a left ideal, and recognize zero-divisors. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 2, pp. 101–113, 2005.     

Coupling commutator pairs and integrable systems

According to classification of the matrix Lie algebras, a type of explicit Lie algebras are constructed which can be decomposed into a few Lie subalgebras. These subalgebras constitute several coupling commutator pairs from which some continuous multi-integrable couplings could be generated if the proper isospectral Lax pairs could be set up. Then the above Lie algebras are again decomposed into a kind of Lie algebras which are also closed under the matrix multiplication. From such the Lie algebras, some discrete multi-integrable couplings could be worked out. Finally, a few examples are given. However, the Hamiltonian structures of the (continuous and discrete) integrable couplings obtained by the above Lie algebras cannot be computed by using the trace identity or the quadratic-form identity, which is a strange and interesting problem. The phenomenon indicates that the importance of the Lie-algebra classification. The problem also needs us to try to find an efficient scheme to deal with.     

Integrable ordinary differential equations on free associative algebras

We consider a classification problem for integrable nonlinear ordinary differential equations with an independent variable belonging to a free associative algebra M. Every equation of this type admits an m×m matrix reduction for an arbitrary m. The existence of symmetries or first integrals belonging to M is used as an integrability criterion. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 122, No. 1, pp. 88–101, January, 1999.