Formal diagonalization of a discrete lax operator and conservation laws and symmetries of dynamical systems

We consider the problem of constructing a formal asymptotic expansion in the spectral parameter for an eigenfunction of a discrete linear operator. We propose a method for constructing an expansion that allows obtaining conservation laws of discrete dynamical systems associated with a given linear operator. As illustrative examples, we consider known nonlinear models such as the discrete potential Kortewegde Vries equation, the discrete version of the derivative nonlinear Schrödinger equation, the Veselov-Shabat dressing chain, and others. We describe the infinite set of conservation laws for the discrete Toda chain corresponding to the Lie algebra A ₁ ⁽¹⁾ . We find new examples of integrable systems of equations on a square lattice.     

Applications of a few Lie algebras

Two isomorphic groups R ² andM are firstly constructed. Then we extend them into the differential manifold R ²ⁿ and n products of the group M for which four kinds of Lie algebras are obtained. By using these Lie algebras and the Tu scheme, integrable hierarchies of evolution equations along with multi-component potential functions can be generated, whose Hamiltonian structures can be worked out by the variational identity. As application illustrations, two integrable Hamiltonian hierarchies with 4 component potential functions are obtained, respectively, some new reduced equations are followed to present. Specially remark that the integrable hierarchies obtained by taking use of the approach presented in the paper are not integrable couplings. Finally, we generalize an equation obtained in the paper to introduce a general nonlinear integrable equation with variable coefficients whose bilinear form, B¨acklund transformation, Lax pair and infinite conserved laws are worked out, respectively, by taking use of the Bell polynomials.     

Generations of integrable hierarchies and exact solutions of related evolution equations with variable coefficients

We first propose a way for generating Lie algebras from which we get a few kinds of reduced 6 6 Lie algebras, denoted by R6, R8 and R1,R6/2, respectively. As for applications of some of them, a Lax pair is introduced by using the Lie algebra R6 whose compatibility gives rise to an integrable hierarchy with 4- potential functions and two arbitrary parameters whose corresponding Hamiltonian structure is obtained by the variational identity. Then we make use of the Lie algebra R6 to deduce a nonlinear integrable coupling hierarchy of the mKdV equation whose Hamiltonian structure is also obtained. Again,via using the Lie algebra R62, we introduce a Lax pair and work out a linear integrable coupling hierarchy of the mKdV equation whose Hamiltonian structure is obtained. Finally, we get some reduced linear and nonlinear equations with variable coefficients and work out the elliptic coordinate solutions, exact traveling wave solutions, respectively.     

Monodromy of Rational KZ Equations and Some Related Topics

A special class of integrable Fuchsian systems on C ⁿ related to KZ equations is considered. We survey the construction of such systems and the list of the structural properties their monodromy representations. The relation of the Fuchsian systems obtained by the Veselov construction assosiated with a deformation of the A _n–1-type root system and the Gauss–Manin connection of the natural projection C ⁿ C ^n–1 is described. In this case, we prove that the monodromy representation is equivalent to the Burau representation of the Artin braid group. For a deformations of the other root system, we introduce generalized Burau representations. We conjecture that the integrable Fuchsian systems related to essential new finite sets of the vectors described by Veselov and Chalykh are the result of the Klares–Schlesinger isomonodromic deformations (or transformation) of the integrable Fuchsian system related to the Coxeter root systems.     

Fast Construction of Optimal Circulant Preconditioners for Matrices from the Fast Dense Matrix Method

In this paper, we consider solving non-convolution type integral equations by the preconditioned conjugate gradient method. The fast dense matrix method is a fast multiplication scheme that provides a dense discretization matrix A approximating a given integral equation. The dense matrix A can be constructed in O(n) operations and requires only O(n) storage where n is the size of the matrix. Moreover, the matrix-vector multiplication A xcan be done in O(n log n) operations. Thus if the conjugate gradient method is used to solve the discretized system, the cost per iteration is O(n log n) operations. However, for some integral equations, such as the Fredholm integral equations of the first kind, the system will be ill-conditioned and therefore the convergence rate of the method will be slow. In these cases, preconditioning is required to speed up the convergence rate of the method. A good choice of preconditioner is the optimal circulant preconditioner which is the minimizer of C – A _F in Frobenius norm over all circulant matrices C. It can be obtained by taking arithmetic averages of all the entries of A and therefore the cost of constructing the preconditioner is of O(n ²) operations for general dense matrices. In this paper, we develop an O(n log n) method of constructing the preconditioner for dense matrices A obtained from the fast dense matrix method. Application of these ideas to boundary integral equations from potential theory will be given. These equations are ill-conditioned whereas their optimal circulant preconditioned equations will be well-conditioned. The accuracy of the approximation A, the fast construction of the preconditioner and the fast convergence of the preconditioned systems will be illustrated by numerical examples.     

Generalized self-dual Yang-Mills flows,explicit solutions and reductions

An evolution equation is added to the generalized self-dual Yang-Mills equations. The evolution equation contains terms of negative powers of the spectral parameter as well as terms of positive powers. The Darboux matrix method is used to obtain explicit solutions, especially single and multiple solitons. All integrable soliton equations in the framework of AKNS system (inR ⁿ⁺¹ orR ¹⁺¹) can be derived from the generalized Yang-Mills flows by reduction.Supported by Chinese research project Nonlinear Science and K.C. Wong Education Foundation, Hong Kong.     

The depth invariant for group actions on countable phase spaces

The construction of the depth of a countable symbolic system, which was studied earlier by the author for actions of the group , is generalized to the case of an arbitrary finitely generated Abelian group action. A new property, which is called the uniformity of the depth, is studied. The depth, which takes values in the set of at most countable ordinals, is a topological invariant of countable symbolic systems. In the paper we describe the set of possible values of the depth invariant and the method of constructing dynamical systems with an arbitrary admissible depth. Translated fromMatematicheskie Zametki, Vol. 65, No. 6, pp. 893–907, June, 1999.     

Dynamical systems that admit normal displacement

The classical geometrical construction of Bianchi—Lie, Bäcklund, and Darboux transformations is considered and generalized for dynamical systems. For a transformation that generalizes normal displacement, a class of dynamical systems that admit this transformation is found. A differential equation that distinguishes dynamical systems inR ² that belong to this class is derived, and some solutions of it are considered.State University, Bashkir. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 97, No. 3, pp. 386–395, December, 1993.     

A new polynomial involutive system and a classical completely integrable system

Under the constrained condition induced by the eigenfunctions and the potentials, the Lax systems of nonlinear evolution equations in relation to a matrix eigenvalue problem are nonlinearized to be a completely integrable system (R ^zN ,dpdq,H), while the time part of it is nonlinearized to be itsN-involutive system {F _m}. the involutive solution of the compatible system (F ₀), (F _m) is transformed into the solution of them-th nonlinear evolution equation.     

Infinite-dimensional Lie algebras determined by the space of symmetric squares of hyperelliptic curves

We construct Lie algebras of vector fields on universal bundles of symmetric squares of hyperelliptic curves of genus g = 1, 2,.. For each of these Lie algebras, the Lie subalgebra of vertical fields has commuting generators, while the generators of the Lie subalgebra of projectable fields determines the canonical representation of the Lie subalgebra with generators L _2q , q = ?1, 0, 1, 2,.., of the Witt algebra. As an application, we obtain integrable polynomial dynamical systems.     

Hamiltonian Hierarchies on Semisimple Lie Algebras

A recursion formula is described which generates infinite hierarchies of completely integrable Hamiltonian systems of nonlinear partial differential equations. These equations govern the evolution of a function u of x, t which takes its values in a semisimple Lie algebra. A Hamiltonian for the hierarchy is given in terms of a meromorphic connection matrix.     

Universal minimal topological dynamical systems

Rokhlin (1963) showed that any aperiodic dynamical system with finite entropy admits a countable generating partition. Krieger (1970) showed that aperiodic ergodic systems with entropy < log a, admit a generating partition with no more than a sets. In Symbolic Dynamics terminology, these results can be phrased— ℕ^ℤ is a universal system in the category of aperiodic systems, and [a]^ℤ is a universal system for aperiodic ergodic systems with entropy < log a. Weiss ([We89], 1989) presented a Minimal system, on a Compact space (a subshift of ) which is universal for aperiodic systems. In this work we present a joint generalization of both results: given ɛ, there exists a minimal subshift of [a]^ℤ, universal for aperiodic ergodic systems with entropy < log a − ɛ.     

Toroidal Lie algebras and vertex representations

The paper describes the theory of the toroidal Lie algebra, i.e. the Lie algebra of polynomial maps of a complex torus ^××^× into a finite-dimensional simple Lie algebra g. We describe the universal central extension t of this algebra and give an abstract presentation for it in terms of generators and relations involving the extended Cartan matrix of g. Using this presentation and vertex operators we obtain a large class of integrable indecomposable representations of t in the case that g is of type A, D, or E. The submodule structure of these indecomposable modules is described in terms of the ideal structure of a suitable commutative associative algebra.To Professor J. Tits for his sixtieth birthday     

Groupes Moyennables, Dimension Topologique Moyenne et sous-décalages

Let G be an infinite countable residually finite amenable group. In this paper we construct a continuous action of G on a compact metrisable space X such that the dynamical system (X, G) cannot be embedded in the G-shift on [0,1] ^G . This result generalizes a construction due to E. Lindenstrauss and B. Weiss (Mean topological dimension, Israel J. Math. 115 (2000), 1–24) for .     

Soliton surfaces in the generalized symmetry approach

We investigate some features of generalized symmetries of integrable systems aiming to obtain the Fokas–Gel’fand formula for the immersion of two-dimensional soliton surfaces in Lie algebras. We show that if there exists a common symmetry of the zero-curvature representation of an integrable partial differential equation and its linear spectral problem, then the Fokas–Gel’fand immersion formula is applicable in its original form. In the general case, we show that when the symmetry of the zero-curvature representation is not a symmetry of its linear spectral problem, then the immersion function of the two-dimensional surface is determined by an extended formula involving additional terms in the expression for the tangent vectors. We illustrate these results with examples including the elliptic ordinary differential equation and the CP^N?1 sigma-model equation.     

Complete involutive algebras of functions on cotangent bundles of homogeneous spaces

Homogeneous spaces of all compact Lie groups admit Riemannian metrics with completely integrable geodesic flows by means of C –smooth integrals [9, 10]. The purpose of this paper is to give some constructions of complete involutive algebras of analytic functions, polynomial in velocities, on the (co)tangent bundles of homogeneous spaces of compact Lie groups. This allows us to obtain new integrable Riemannian and sub-Riemannian geodesic flows on various homogeneous spaces, such as Stiefel manifolds, flag manifolds and orbits of the adjoint actions of compact Lie groups. Mathematics Subject Classification (2000): 70H06, 37J35, 53D17, 53D25     

The integrable case of Adler–van Moerbeke. Discriminant set and bifurcation diagram

The Adler–van Moerbeke integrable case of the Euler equations on the Lie algebra so(4) is investigated. For the L–A pair found by Reyman and Semenov-Tian-Shansky for this system, we explicitly present a spectral curve and construct the corresponding discriminant set. The singularities of the Adler–van Moerbeke integrable case and its bifurcation diagram are discussed. We explicitly describe singular points of rank 0, determine their types, and show that the momentum mapping takes them to self-intersection points of the real part of the discriminant set. In particular, the described structure of singularities of the Adler–van Moerbeke integrable case shows that it is topologically different from the other known integrable cases on so(4).     

On extensions of countable filterbases to ultrafilters and ultrafilter compactness

We show in the Zermelo-Fraenkel set theory ZF without the axiom of choice:

1. Given an infinite set X, the Stone space S(X) is ultrafilter compact.

2. For every infinite set X, every countable filterbase of X extends to an ultra-filter i? for every infinite set X, S(X) is countably compact.

3. ω has a free ultrafilter i? every countable, ultrafilter compact space is countably compact.

   We also show the following:

4. There are a permutation model 𝒩 and a set X ∈ 𝒩 such that X has no free ultrafilters and S(X) is not compact but S(X) is countably compact and every countable filterbase of X extends to an ultrafilter.

5. It is relatively consistent with ZF that every countable filterbase of ω extends to an ultrafilter but there exists a countable filterbase of ? which does not extend to an ultrafilter. Hence, it is relatively consistent with ZF that ? has free ultrafilters but there exists a countable filterbase of ? which does not extend to an ultrafilter.

     

Extremal Projector and Dynamical Twist

We describe a relation between the dynamical twist J() and the extremal projector for simple Lie algebras. This correspondence finds two obvious applications: first, the solution of the Arnaudon–Buffenoir–Ragoucy–Roche equation can be obtained from the known multiplicative expression for the extremal projector; second, the structure constants are determined by the matrix coefficients of the dynamical twist.     

A numerical algorithm for solving the matrix equation AX + XTB = C1

An algorithm of the Bartels-Stewart type for solving the matrix equation AX + X ^T B = C is proposed. By applying the QZ algorithm, the original equation is reduced to an equation of the same type having triangular matrix coefficients A and B. The resulting matrix equation is equivalent to a sequence of low-order systems of linear equations for the entries of the desired solution. Through numerical experiments, the situation where the conditions for unique solvability are “nearly” violated is simulated. The loss of the quality of the computed solution in this situation is analyzed.