conservation laws for integrable difference equations this paper deals with conservation laws for all integrable difference equations that belong to the famous adler bobenko suris classification all inequivalent three point conservation laws are found as are three five point conservation laws for each equation we also describe a method of generating conservation laws from known ones this method can be used to generate higher order conservation laws from those that are listed here

extension of the adler bobenko suris classification of integrable lattice equations the classification of lattice equations that are integrable in the sense of higher dimensional consistency is extended by allowing directed edges we find two cases that are not transformable via the admissible transformations to the lattice equations in the existing classification

linearizable mappings and the low growth criterion we examine a family of discrete second order systems which are integrable through reduction to a linear system these systems were previously identified using the singularity confinement criterion here we analyse them using the more stringent criterion of nonexponential growth of the degrees of the iterates we show that the linearizable mappings are characterized by a very special degree growth the ones linearizable by reduction to projective systems exhibit zero growth i e they behave like linear systems while the remaining ones derivatives of riccati and gambier mapping lead to linear growth this feature may well serve as a detector of integrability through linearization

exchange dynamical quantum groups for any simple lie algebra and any complex number q which is not zero or a nontrivial root of unity but may be equal to 1 we construct a dynamical quantum group hopf algebroid whose representation theory is essentially the same as the representation theory of the quantum group uq this dynamical quantum group is obtained from the fusion and exchange relations between intertwining operators in representation theory of uq and is an algebraic structure standing behind these relations

lie bialgebroids and poisson groupoids lie bialgebras arise as infinitesimal invariants of poisson lie groups a lie bialgebra is a lie algebra g with a lie algebra structure on the dual g which is compatible with the lie algebra g in a certain sense for a poisson group g the multiplicative poisson structure induces a lie algebra structure on the lie algebra dual g which makes g g into a lie bialgebra in fact there is a one one correspondence between poisson lie groups and lie bialgebras if the lie groups are assumed to be simply connected 7 16 19 the importance of poisson lie groups themselves arises in part from their role as classical limits of quantum groups 8 and in part because they provide a class of poisson structures for which the realization problem is tractable 15 poisson groupoids were introduced by weinstein 24 as a generalization of both poisson lie groups and the symplectic groupoids which arise in the integration of arbitrary poisson manifolds 4 11 he noted that the lie algebroid dual a g of a poisson groupoid g itself has a lie algebroid structure but did not develop the infinitesimal structure further in this paper we introduce and study a natural infinitesimal invariant for poisson groupoids the lie bialgebroids of the title our ultimate purpose is to develop a lie theory for poisson groupoids parallel to that for poisson groups in this paper we are primarily concerned with the first half of this process that is with the construction of the lie bialgebroid of a poisson groupoid after the preliminary 2 in which we describe the generalization to arbitrary lie algebroids of the exterior calculus and schouten calculus in 3 we define a lie bialgebroid to be a lie algebroid a whose dual a is also equipped with a lie algebroid structure such that the coboundary operator d a a associated to a satisfies a cocycle equation with respect to a the lie algebra of sections of a this is clearly a straightforward extension of the concept of a lie bialgebra 16 but cannot be formalized in lie algebroid cohomological terms since there is no satisfactory adjoint representation for a general lie algebroid most of 3 is devoted to proving that this definition is self dual if a a is a lie bialgebroid then a a is also in 4 we briefly consider the special case of lie bialgebroids satisfying a triangularity condition which include some important examples such as the usual triangular lie bialgebras and lie bialgebroids associated to poisson manifolds the techniques used in 2 4 are similar to those known for lie bialgebras it would be possible by suitably generalizing the proof for poisson groups to prove

large time behavior of the heat equation on complete manifolds with non negative ricci curvature sur une variete de riemann complete m on considere le noyau de la chaleur h x y t qui resout l equation de la chaleur t f x t 0 sur mx 0 avec la donnee initiale f x 0 f x en posant f x t m h x y t f y dy

linear integral equations and nonlinear difference difference equations abstract in this paper we present a systematic method to obtain various integrable nonlinear difference difference equations and the associated linear integral equations from which their solutions can be inferred it is argued that these difference difference equations can be regarded as arising from bianchi identities expressing the commutativity of backlund transformations applying an appropriate continuum limit we first obtain integrable nonlinear differential difference equations together with the associated linear integral equations and after a second continuum limit we can obtain the corresponding integrable nonlinear partial differential equations and their linear integral equations as special cases we treat the difference difference versions and the differential difference versions of the korteweg de vries equation the modified korteweg de vries equation the nonlinear schrodinger equation the isotropic classical heisenberg spin chain and the complex and real sine gordon equation

integrable symplectic maps abstract in this paper we first give a terse survey of symplectic maps their canonical formulation and integrability then we introduce a rigorous procedure to construct integrable symplectic maps starting from integrable evolution equations on lattices a number of illustrative examples are provided

expansions over the squared solutions and difference evolution equations the completeness relation for the system of squared solutions of the discrete analog of the zakharov shabat problem is derived it allows one to rederive the known statements concerning the class of difference evolution equations related to this linear problem and to obtain additional results these include i the expansion of the potential and its variations over the system of squared solutions the expansion coefficients being the scattering data and their variations respectively thus the interpretation of the inverse scattering transform ist as a generalized fourier transform becomes obvious ii compact expressions for the trace identities through the operator for which the squared solutions are eigenfunctions iii brief exposition of the spectral theory of the operator iv direct calculation of the action angle variables based on the symplectic form of the completeness relation v the generating functional of the m operators in the lax representation vi the quantum ve

superposition principles associated with the moutard transformation an integrable discretization of a 2 1 dimensional sine gordon system superposition principles both linear and nonlinear associated with the moutard transformation are found on suitable reinterpretation these constitute an integrable discrete nonlinear system and its associated linear system further it is shown that in a particular form this system is an integrable discretization of a 2 1 dimensional sine gordon system solutions of the discrete nonlinear system are constructed by means of a discrete analogue of the moutard transformation included in these solutions are discrete analogues of the kink solutions of the continuous system

integrable three dimensional lattices following the work of zakharov and shabat 1974 the authors derive nonlinear equations which are integrable through a discrete gel fand levitan integral equation in two resp one resp zero continuous and one resp two resp three discrete variables

quasi bialgebras and dynamical r matrices abstract we study the relationship between general dynamical poisson groupoids and lie quasi bialgebras for a class of lie quasi bialgebras g naturally compatible with a reductive decomposition we extend the description of the moduli space of classical dynamical r matrices of etingof and schiffmann we construct in each gauge orbit an explicit analytic representative l can we translate the notion of duality for dynamical poisson groupoids into a duality for lie quasi bialgebras it is shown that duality maps the dynamical poisson groupoid for l can and g to the dynamical poisson groupoid for l can and the dual quasi bialgebra g

lattice equations hierarchies and hamiltonian structures iii the 2d toda and kp hierarchies abstract we continue the investigation of the continuum limit properties of discrete versions of the kadomtsev petviashvili kp equation using a multiple time scale expansion in which one of the discrete variables is replaced by an infinite number of continous variables we can derive a hierarchy of integrable equations in this paper hierarchies associated with the 2d toda equation and the kp equation are treated the direct linearization of the hierarchy is obtained applying the expansion mentioned above to the free wave function in the integral equation the hamiltonian structure of the hierarchy is established via a relation between the generator of the hierarchy and an appropriate monodromy factor of the scattering problem

integrable systems without the painleve property we examine whether the painleve property is a necessary condition for the integrability of nonlinear ordinary differential equations we show that for a large class of linearizable systems this is not the case in the discrete domain we investigate whether the singularity confinement property is satisfied for the discrete analogues of the non painleve continuous linearizable systems we find that while these discrete systems are themselves linearizable they possess non confined singularities

deconstructing an integrable lattice equation we show that an integrable lattice equation obtained by j hietarinta using the consistency around a cube method without the tetrahedron assumption is indeed solvable by linearization we also present its nonautonomous extension

integrability conditions for nonautonomous quad graph equations this paper presents a systematic investigation of the integrability conditions for nonautonomous quad graph maps using the lax pair approach the ultra local singularity confinement criterion and direct construction of conservation laws we show that the integrability conditions derived from each of the methods are the one and the same suggesting that there exists a deep connection between these techniques for partial difference equations

triangular dynamical r matrices and quantization abstract we study some general aspects of triangular dynamical r matrices using poisson geometry we show that a triangular dynamical r matrix r h 2 g always gives rise to a regular poisson manifold using the fedosov method we prove that non degenerate triangular dynamical r matrices i e those such that the corresponding poisson manifolds are symplectic are quantizable and that the quantization is classified by the relative lie algebra cohomology h2 g h

baklund transformations and singular integral equations abstract a systematic method for deriving backlund transformations for singular linear integral equations is presented the method leads in a natural way to the backlund transformations for the corresponding integrable nonlinear partial differential equations by repeated use of the method a hierarchy of singular integral equations and backlund transformations can be derived corresponding to so called multi modified partial differential equations specific examples include the korteweg de vries equation and the first second and third modified korteweg de vries equation the nonlinear schrodinger equation and the anisotropic heisenberg spin chain the sine gordon equation and the modified sine gordon equation

on ricci curvature and volume growth in dimension three we prove that any complete metric on mathbb r 3 minus an open ball with non negative ricci curvature and quadratic ricci curvature decay has cubic volume growth

universal exchange algebra for bloch waves and liouville theory we derive a universal formula for the exchange algebra in the bloch wave basis the main tool we use is a lattice version of the coulomb gas picture of conformal field theory making its quantum group structure explicit from the very beginning calulations are then reduced to a factorization problem inuq sl2

generalized symmetry classification of discrete equations of a class depending on twelve parameters we carry out the generalized symmetry classification of polylinear autonomous discrete equations defined on the square which belong to a twelve parametric class the direct result of this classification is a list of equations containing no new examples however as an indirect result of this work we find a number of integrable examples pretending to be new one of them has a nonstandard symmetry structure the others are analogues of the liouville equation in the sense that they are darboux integrable we also enumerate all equations of the class which are linearizable via a two point first integral and specify the nature of integrability of some known equations

3 manifolds with nonnegative ricci curvature for a complete noncompact 3 manifold with nonnegative ricci curvature we prove that either it is diffeomorphic to 3 or the universal cover splits this confirms milnor s conjecture in dimension 3

linear quadrilateral lattice equations and multidimensional consistency it is shown that every scalar linear quadrilateral lattice equation lies within a family of similar equations members of which are compatible between one another on a higher dimensional lattice there turn out to be two such families a natural parametrization is given for each

wronskian and rational solutions of the differential difference kp equation we present the wronskian form of the n soliton solutions of the differential difference kadomtsev petviashvilli dkp equation also a wider class of rational solutions are derived using semi discrete analogue of schur polynomials and its generalization our approach is based on sato theory formalism which gives all these solutions naturally

classification of integrable equations on quad graphs the consistency approach a classification of discrete integrable systems on quad graphs i e on surface cell decompositions with quadrilateral faces is given the notion of integrability laid in the basis of the classification is the three dimensional consistency this property yields among other features the existence of the discrete zero curvature representation with a spectral parameter for all integrable systems of the obtained exhaustive list the so called three leg forms are found this establishes lagrangian and symplectic structures for these systems and the connection to discrete systems of the toda type on arbitrary graphs generalizations of these ideas to the three dimensional integrable systems and to the quantum context are also discussed

dynamical yang baxter equation and quantum vector bundles we develop a categorical approach to the dynamical yang baxter equation dybe for arbitrary hopf algebras in particular we introduce the notion of a dynamical extension of a monoidal category which provides a natural environment for quantum dynamical r matrices dynamical twists etc in this context we define dynamical associative algebras and show that such algebras give quantizations of vector bundles on coadjoint orbits we build a dynamical twist for any pair of a reductive lie algebra and its levi subalgebra using this twist we obtain an equivariant star product quantization of vector bundles on semisimple coadjoint orbits of reductive lie groups

quantization of formal classical dynamical r matrices the reductive case abstract in this paper we prove the existence of a formal dynamical twist quantization for any triangular and non modified formal classical dynamical r matrix in the reductive case the dynamical twist is constructed as the image of the dynamical r matrix by a l quasi isomorphism this quasi isomorphism also allows us to classify formal dynamical twist quantizations up to gauge equivalence

gerstenhaber algebras and bv algebras in poisson geometry the purpose of this paper is to establish an explicit correspondence between various geometric structures on a vector bundle with some well known algebraic structures such as gerstenhaber algebras and bv algebras some applications are discussed in particular we find an explicit connection between the koszul brylinski operator and the modular class of a poisson manifold as a consequence we prove that poisson homology is isomorphic to poisson cohomology for unimodular poisson structures

integrable mappings and soliton equations ii abstract some simple solutions of discrete soliton equations are shown to satisfy 2d mappings we show that these belong to a recently introduced 18 parameter family of integrable reversible mappings of the plane thus lending weight to a previous conjecture we also give an example of an integrable mapping occuring in an exactly solvable model in statistical mechanics finally we discuss the notion of generalized reversibility

integrable mappings derived from soliton equations abstract we derive a hierarchy of ibtegrable mappings integrable ordinary difference equations corresponding to solutions of the initial value problem of an integrable partial difference equation with periodic initial data for each n n this hierarchy contains at least one integrable mapping r n r n the integrals of these mappings are constructed using the lax pair of the underlying partial difference equation our approach is illustrated for the integrable partial difference analogues of the sine gordon and the modified korteweg de vries equations

quantization of symplectic dynamical r matrices and the quantum composition formula in this paper we quantize symplectic dynamical r matrices over a possibly nonabelian base the proof is based on the fact that the existence of a star product with a nice property called strong invariance is sufficient for the existence of a quantization we also classify such quantizations and prove a quantum analogue of the classical composition formula for coboundary dynamical r matrices

on some linear integral equations generating solutions of nonlinear partial differential equations abstract two types of linear inhomogeneous integral equations which yield solutions of a broad class of nonlinear evolution equations are investigated one type is characterized by a two fold integration with an arbitrary measure and contour over a complex variable k and thier complex conjugates whereas the other one has a two fold integration over one and the same contour the inhomogeneous term which may contain an arbitrary function of k makes it possible to define a matrix structure on the solutions of the integral equations the elements of these matrices are shown to obey a system of partial differential equations the special form of which depends on the choice of the dispersion relation occurring in the integral equations for special elements of the matrices closed partial differential equations can be derived such as e g the nonlinear schrodinger equation and the real and complex modified korteweg de vries and sine gordon equations the relations between the matrix elements are shown to lead to miura transformations between the various partial differential equations

solutions of the quantum dynamical yang baxter equation and dynamical quantum groups the quantum dynamical yang baxter qdyb equation is a useful generalization of the quantum yang baxter qyb equation this generalization was introduced by gervais neveu and felder unlike the qyb equation the qdyb equation is not an algebraic but a difference equation with respect to a matrix function rather than a matrix the qdyb equation and its quasiclassical analogue the classical dynamical yang baxter equation arise in several areas of mathematics and mathematical physics conformal field theory integrable systems representation theory the most interesting solution of the qdyb equation is the elliptic solution discovered by felder in this paper we prove the first classification results for solutions of the qdyb equation these results are parallel to the classification of solutions of the classical dynamical yang baxter equation obtained in our previous paper all solutions we found can be obtained from felder s elliptic solution by a limiting process and gauge transformations fifteen years ago the quantum yang baxter equation gave rise to the theory of quantum groups namely it turned out that the language of quantum groups hopf algebras is the adequate algebraic language to talk about solutions of the quantum yang baxter equation in this paper we propose a similar language originating from felder s ideas which we found to be adequate for the dynamical yang baxter equation this is the language of dynamical quantum groups or hopf algebroids which is the quantum counterpart of the language of dynamical poisson groupoids introduced in our previous paper

lattice equations hierarchies and hamiltonian structures ii kp type of hierarchies on 2d lattices abstract we investigate the continuum limit properties of an integrable lattice version of the kadomtsev petviashvili kp equation by applying continuum limits with vertex operators involving an infinite number of continous variables to the lattice kp we obtain hierarchies of integrable equations for fields depending on one continous variable at the sites of a two dimensional lattice the direct linearization of the hierarchies is obtained applying the same limit to the free wave function in the integral equation for lattice kp on the basis of the relation between the generator of the hierarchy and the monodromy factor for the scattering problem on a line which is not involved in the continuum limit we establish the hamiltonian structure of the equations of the hierarchy

brst quantization of quasisymplectic manifolds and beyond we consider a class of factorizable poisson brackets which includes almost all reasonable poisson structures a particular case of the factorizable brackets are those associated with symplectic lie algebroids the brst theory is applied to describe the geometry underlying these brackets as well as to develop a deformation quantization procedure in this particular case this can be viewed as an extension of the fedosov deformation quantization to a wide class of irregular poisson structures in a more general case the factorizable poisson brackets are shown to be closely connected with the notion of n algebroid a simple description is suggested for the geometry underlying the factorizable poisson brackets based on construction of an odd poisson algebra bundle equipped with an abelian connection it is shown that the zero curvature condition for this connection generates all the structure relations for the n algebroid as well as a generalization of the yang baxter equation for the symplectic structure

symmetries and conservation laws of the abs equations and corresponding differential difference equations of volterra type a sequence of canonical conservation laws for all the adler bobenko suris abs equations is derived and is employed in the construction of a hierarchy of master symmetries for equations h1 h3 q1 q3 for the discrete potential and schwarzian kdv equations it is shown that their local generalized symmetries and nonlocal master symmetries in each lattice direction form centerless virasoro type algebras in particular for the discrete potential kdv the structure of its symmetry algebra is explicitly given interpreting the hierarchies of symmetries of equations h1 h3 q1 q3 as differential difference equations of yamilov s discretization of krichever novikov equation corresponding hierarchies of master symmetries along with isospectral and nonisospectral zero curvature representations are derived for all of them

a completeness study on discrete 2 2 lax pairs we propose a method to identify and examine every partial difference lax pair that consists of two 2 2 discrete linear problems where the matrices contain one separable term in each entry we thereby derive new higher order versions of the lattice sine gordon and lattice modified korteweg de vries equations while showing that there can be no other partial difference equations associated with this type of lax pair

explicit quantization of dynamical r matrices for finite dimensional semisimple lie algebras 1 1 classical r matrices in the early eighties belavin and drinfeld bd classified nonskewsymmetric classical r matrices for simple lie algebras it turned out that such r matrices up to isomorphism and twisting by elements from the exterior square of the cartan subalgebra are classified by combinatorial objects which are now called belavin drinfeld triples by definition a belavin drinfeld triple for a simple lie algebra g is a triple 1 2 t where 1 2 are subsets of the dynkin diagram of g and t 1 2 is an isomorphism which preserves the inner product and satisfies the nilpotency condition if 1 then there exists k such that t k 1 1 but t 1 the r matrix corresponding to such a triple is given by a certain explicit formula these results generalize in a straightforward way to semisimple lie algebras in s the third author generalized the work of belavin and drinfeld and classified classical nonskewsymmetric dynamical r matrices for simple lie algebras it turns out that they have an even simpler classification up to gauge transformations they are classified by generalized belavin drinfeld triples which are defined as the usual belavin drinfeld triples but without any nilpotency condition the dynamical rmatrix corresponding to such a triple is given by a certain explicit formula as before these results can be generalized to semisimple lie algebras

manifolds of positive ricci curvature with almost maximal volume 10 in this note we consider complete riemannian manifolds with ricci curvature bounded from below the well known theorems of myers and bishop imply that a manifold mn with ric n 1 satisfies diam mn diam sn l vol mn vol sn l it follows from ch that equality in either of these estimates can be achieved only if mn is isometric to sn 1 the natural conjecture is that a manifold mn with almost maximal diameter or volume must be a topological equivalent to sn with respect to diameter this is true only if mn satisfies some additional assumptions see an 0 gp e with respect to volume however no extra restriction is necesary

fedosov products and quantum momentum maps the purpose of this paper is to study various aspects of star products on a symplectic manifold related to the fedosov method by introducing the notion of quantum exponential maps we give a characterization of fedosov connections as an application a geometric realization is obtained for the equivalence between an arbitrary product and a fedosov one every fedosov product is shown to be a vey product consequently we find that every product is equivalent to a vey product a classical result of lichnerowicz quantization of a hamiltonian g space and in particular quantum momentum maps are studied lagrangian submanifolds are also studied under a deformation quantization

sharp geometric inequalities for closed hypersurfaces in manifolds with nonnegative ricci curvature in this paper we consider complete noncompact riemannian manifolds m g with nonnegative ricci curvature and euclidean volume growth of dimension n ge 3 for every bounded open subset omega subset m with smooth boundary we prove that begin aligned int limits partial omega left frac mathrm h n 1 right n 1 mathrm d sigma ge mathrm avr g big mathbb s n 1 big end aligned where mathrm h is the mean curvature of partial omega and mathrm avr g is the asymptotic volume ratio of m g moreover the equality holds true if and only if m setminus omega g is isometric to a truncated cone over partial omega an optimal version of huisken s isoperimetric inequality for 3 manifolds is obtained using this result finally exploiting a natural extension of our techniques to the case of parabolic manifolds we also deduce an enhanced version of kasue s non existence result for closed minimal hypersurfaces in manifolds with nonnegative ricci curvature

backlund transformations for integrable lattice equations we give new backlund transformations bts for some known integrable in the sense of being multidimensionally consistent quadrilateral lattice equations as opposed to the natural auto bt inherent in every such equation these bts are of two other kinds specifically it is found that some equations admit additional auto bts with backlund parameter whilst some pairs of apparently distinct equations admit a bt which connects them

linearizing integral transform for the multicomponent lattice kp abstract we present a generalization of the direct linearization method for the kadomtsev petviashvili kp equation and its lattice analogues such as the two dimensional toda equation the generalization consists of a riemann hilbert type of integral transform which relates solutions of the associated spectral problem to any given solution of the spectral problem by dimensional reduction we obtain the integral transform for the korteweg de vries kdv equation and its lattice analogues the integral transform is well suited for the investigation of the finite matrix generalizations of the above mentioned equations such finite matrix generalizations may be of use in connection with partial difference equations for spin correlation functions in the two dimensional ising model

continuous symmetries of difference equations lie group theory was originally created more than 100 years ago as a tool for solving ordinary and partial differential equations in this article we review the results of a much more recent program the use of lie groups to study difference equations we show that the mismatch between continuous symmetries and discrete equations can be resolved in at least two manners one is to use generalized symmetries acting on solutions of difference equations but leaving the lattice invariant the other is to restrict them to point symmetries but to allow them to also transform the lattice

a new two dimensional lattice model that is consistent around a cube for two dimensional lattice equations one definition of integrability is that the model can be naturally and consistently extended to three dimensions i e that it is consistent around a cube cac as a consequence of cac one can construct a lax pair for the model recently adler bobenko and suris conducted a search based on this principle and certain additional assumptions one of those assumptions was the tetrahedron property which is satisfied by most known equations we present here one lattice equation that satisfies the consistency condition but does not have the tetrahedron property its lax pair is also presented and some basic properties are discussed

covariant and equivariant formality theorems abstract we give a proof of kontsevich s formality theorem for a general manifold using fedosov resolutions of algebras of polydifferential operators and polyvector fields the main advantage of our construction of the formality quasi isomorphism is that it is based on the use of covariant tensors unlike kontsevich s original proof which is based on jets of polydifferential operators and polyvector fields using our construction we prove that if a group g acts smoothly on a manifold m and m admits a g invariant affine connection then there exists a g equivariant quasi isomorphism of formality this result implies that if a manifold m is equipped with a smooth action of a finite or compact group g or equipped with a free action of a lie group g then m admits a g equivariant formality quasi isomorphism in particular this gives a solution of the deformation quantization problem for an arbitrary poisson orbifold

collapsing riemannian manifolds to ones of lower dimensions on considere l effondrement d une suite de varietes de riemann mi vers un espace de dimension inferieure x on etudie le cas ou l espace limite x est une variete de riemann et les courbures sectionnelles de mi sont bornees

the direct linearization of a class of nonlinear evolution equations the paper deals with the direct linearization an approach used to generate particular solutions of the partial differential equations that can be solved through the inverse scattering transform linear integral equations are presented which enable one to find broad classes of solutions to certain nonlinear evolution equations in 1 1 and 2 1 dimensions

conservation laws for nqc type difference equations this paper presents a classification of all three point conservation laws for a large class of integrable difference equations that has been described by nijhoff quispel and capel we show that every nonlinear equation from this class has at least two nontrivial conservation laws most of the conservation laws that are found are new

the gervais neveu felder equation and the quantum calogero moser systems we quantize the spin calogero moser model in ther matrix formalism the quantumr matrix of the model is dynamical thisr matrix has already appeared in gervais neveu s quantization of toda field theory and in felder s quantization of the knizhnik zamolodchikov bernard equation

singularity confinement and chaos in discrete systems we present a number of second order maps which pass the singularity confinement test commonly used to identify integrable discrete systems but which nevertheless are non integrable as a more sensitive integrability test we propose the analysis of the complexity algebraic entropy of the map using the growth of the degree of its iterates integrability is associated with polynomial growth while the generic growth is exponential for chaotic systems

geometry and classificatin of solutions of the classical dynamical yang baxter equation the classical yang baxter equation cybe is an algebraic equation central in the theory of integrable systems its nondegenerate solutions were classified by belavin and drinfeld quantization of cybe led to the theory of quantum groups a geometric interpretation of cybe was given by drinfeld and gave rise to the theory of poisson lie groups

exact solutions of quantum mappings from the lattice kdv as multi dimensional operator difference equations the lattice kdv equation is defined on an associative algebra and a new initial value problem on the lattice kdv is considered this leads to two new families of iterative mappings discrete equations of motion which can be solved for all discrete times imposing a certain commutation relation on the algebra makes it possible to prove using a yang baxter structure that these mappings are quantum integrable maps in the classical limit the maps become a new class of integrable mappings in the sense of liouville arnold veselov lav

lattice equations hierarchies and hamiltonian structures by now many integrable systems with soliton solutions are known in two dimensions for fields u n1 n2 defined at the sites nln2 of a two dimensional 2d lattice for time dependent fields u n t defined at the sites n of a one dimensional chain and for fields u x t depending on two continuous variables x and t the relation between an integrable partial differential equation and its integrable discrete versions can be treated in the framework of the direct linearization method 1 2 which is based on a linear integral equation with arbitrary measure and contour 3 4 in the treatment use is made of backlund transformations bt s which are generated by scalar multiplications of the free wave function and or measure in the integral equation 5 the integrable lattice equations are obtained in the form of bianchi identities expressing the commutativity of bt s the integrable equations with one or more continuous variables and their direct linearizations are obtained applying suitable continuum limits to the lattice equation and the integral equation at the same time

on the conservation laws for nonlinear partial difference equations in this paper a direct method is proposed to construct conservation laws for nonautonomous nonlinear partial difference equations p es the identified p es can be classified into qrt type kdv type mkdv type and sg type the integrability of the identified partial difference equations is also discussed

linearization through symmetries for discrete equations we show that one can devise through the symmetry approach a procedure to check the linearizability of a difference equation via a point or a discrete cole hopf transformation if the equation is linearizable then the symmetry provides the linearizing transformation at the end we present a few examples of applications for equations defined on four lattice points

discrete reductive perturbation technique we expand a partial difference equation p e on multiple lattices and obtain the p e which governs its far field behavior the perturbative reductive approach is here performed on well known nonlinear p es both integrable and nonintegrable we study the cases of the lattice modified korteweg de vries mkdv equation the hietarinta equation the lattice volterra kac van moerbeke equation and a nonintegrable lattice kdv equation such reductions allow us to obtain many new p es of the nonlinear schrodinger type

direct linearization approach to discrete integrable systems associated with ℤ graded lax pairs fordy and xenitidis j phys a math theor 50 2017 165205 doi 10 1088 1751 8121 aa639a recently proposed a large class of discrete integrable systems which include a number of novel integr

quantization of classical dynamical r matrices with nonabelian base we construct some classes of dynamical r matrices over a nonabelian base and quantize some of them by constructing dynamical pseudo twists in the sense of xu this way we obtain quantizations of r matrices obtained in earlier work of the second author with schiffmann and varchenko a part of our construction may be viewed as a generalization of the donin mudrov nonabelian fusion construction we apply these results to the construction of equivariant star products on poisson homogeneous spaces which include some homogeneous spaces introduced by de concini