recursion operator for the stationary nizhnik veselov novikov equation we present a new general construction of a recursion operator from the zero curvature representation using it we find a recursion operator for the stationary nizhnik veselov novikov equation and present a few low order symmetries generated with the help of this operator

darboux transformation with dihedral reduction group we construct the darboux transformation with dihedral reduction group for the 2 dimensional generalisation of the periodic volterra lattice the resulting backlund transformation can be viewed as a nonevolutionary integrable differential difference equation we also find its generalised symmetry and the lax representation for this symmetry using formal diagonalisation of the darboux matrix we obtain local conservation laws of the system

rational recursion operators for integrable differential difference equations in this paper we introduce the concept of prehamiltonian pairs of difference operators demonstrate their connections with nijenhuis operators and give a criteria for the existence of weakly nonlocal inverse recursion operators for differential difference equations we begin with a rigorous setup of the problem in terms of the skew field of rational pseudo difference operators over a difference field with a zero characteristic subfield of constants and the principal ideal ring of matrix rational pseudo difference operators in particular we give a criteria for a rational operator to be weakly nonlocal a difference operator is called prehamiltonian if its image is a lie subalgebra with respect to the lie bracket on the difference field two prehamiltonian operators form a prehamiltonian pair if any linear combination of them is prehamiltonian then we show that a prehamiltonian pair naturally leads to a nijenhuis operator and a nijenhuis operator can be represented in terms of a prehamiltonian pair this provides a systematic method to check whether a rational operator is nijenhuis as an application we construct a prehamiltonian pair and thus a nijenhuis recursion operator for the differential difference equation recently discovered by adler and postnikov the nijenhuis operator obtained is not weakly nonlocal we prove that it generates an infinite hierarchy of local commuting symmetries we also illustrate our theory on the well known examples including the toda the ablowitz ladik and the kaup newell differential difference equations

a conservative free energy dissipating and positivity preserving finite difference scheme for multi dimensional nonlocal fokker planck equation abstract in this work we design and analyze a conservative positivity preserving and free energy dissipating finite difference method for the multi dimensional nonlocal fokker planck fp equation based on a non logarithmic landau transformation a central differencing spatial discretization using harmonic mean approximations is developed both forward and backward euler discretizations in time are employed to derive an explicit scheme and a linearized semi implicit scheme respectively three desired properties that are possessed by analytical solutions i mass conservation ii free energy dissipation and iii positivity are proved to be maintained at discrete level remarkably numerical analysis demonstrates that the semi implicit time discretization ensures the property of positivity preserving unconditionally due to the advantages brought by the harmonic mean approximations an estimate on the upper bound of the condition number of the resulting coefficient matrix is further established for the semi implicit scheme extensive numerical tests are performed to validate aforementioned properties numerically

on a chemotaxis model with degenerate diffusion initial shrinking eventual smoothness and expanding abstract we investigate the propagating profiles of a degenerate chemotaxis model describing the bacteria chemotaxis and consumption of oxygen by aerobic bacteria in particular the effect of the initial attractant distribution on bacterial clustering we prove that the compact support of solutions may shrink if the signal concentration satisfies a special structure and show the finite speed propagating property without assuming the special structure on attractant concentration and obtain an explicit formula of the population spreading speed in terms of model parameters the presented results suggest that bacterial cluster formation can be affected by chemotactic attractants and density dependent dispersal

numerical simulation of nonlinear continuity equations by evolving diffeomorphisms abstract in this paper we present a numerical scheme for nonlinear continuity equations which is based on the gradient flow formulation of an energy functional with respect to the quadratic transportation distance it can be applied to a large class of nonlinear continuity equations whose dynamics are driven by internal energies given external potentials and or interaction energies the solver is based on its variational formulation as a gradient flow with respect to the wasserstein distance positivity of solutions as well as energy decrease of the semi discrete scheme are guaranteed by its construction we illustrate this property with various examples in spatial dimension one and two

integrable equations on time scales integrable systems are usually given in terms of functions of continuous variables on r in terms of functions of discrete variables on z and recently in terms of functions of q variables on kq we formulate the gel fand dikii gd formalism on time scales by using the delta differentiation operator and find more general integrable nonlinear evolutionary equations in particular they yield integrable equations over integers difference equations and over q numbers q difference equations we formulate the gd formalism also in terms of shift operators for all regular discrete time scales we give a method allowing to construct the recursion operators for integrable systems on time scales finally we give a trace formula on time scales and then construct infinitely many conserved quantities casimirs of the integrable systems on time scales

lenard scheme for two dimensional periodic volterra chain we prove that for compatible weakly nonlocal hamiltonian and symplectic operators hierarchies of infinitely many commuting local symmetries and conservation laws can be generated under some easily verified conditions no matter whether the generating nijenhuis operators are weakly nonlocal or not we construct a recursion operator of the two dimensional periodic volterra chain from its lax representation and prove that it is a nijenhuis operator furthermore we show that this system is a generalized bi hamiltonian system rather surprisingly the product of its weakly nonlocal hamiltonian and symplectic operators gives rise to the square of the recursion operator

blow up dynamics for the aggregation equation with degenerate diffusion abstract we study radially symmetric finite time blow up dynamics for the aggregation equation with degenerate diffusion u t u m u k u in r d where the kernel k x is of power law form x depending on m d and the initial data the solution exhibits three kinds of blow up behavior self similar with no mass concentrated at the core imploding shock solution and near self similar blow up with a fixed amount of mass concentrated at the core computations are performed for different values of m d and using an arbitrary lagrangian eulerian method with adaptive mesh refinement

automorphic lie algebras with dihedral symmetry the concept of automorphic lie algebras arises in the context of reduction groups introduced in the early 1980s in the field of integrable systems automorphic lie algebras are obtained by imposing a discrete group symmetry on a current algebra of krichever novikov type past work shows remarkable uniformity between algebras associated to different reduction groups for example if the base lie algebra is sl2 c and the poles of the automorphic lie algebra are restricted to an exceptional orbit of the symmetry group changing the reduction group does not affect the lie algebra structure in this research we fix the reduction group to be the dihedral group and vary the orbit of poles as well as the group action on the base lie algebra we find a uniform description of automorphic lie algebras with dihedral symmetry valid for poles at exceptional and generic orbits

reduction groups and related integrable difference systems of nonlinear schrodinger type we extend the reduction group method to the lax darboux schemes associated with nonlinear schrodinger type equations we consider all possible finite reduction groups and construct corresponding lax operators darboux transformations hierarchies of integrable differential difference equations integrable partial difference systems and associated scalar partial difference equations

the r matrix approach to integrable systems on time scales a general unifying framework for integrable soliton like systems on time scales is introduced the r matrix formalism is applied to the algebra of differential operators in terms of which one can construct an infinite hierarchy of commuting vector fields the theory is illustrated by two infinite field integrable hierarchies on time scales which are differential counterparts of kp and mkp the difference counterparts of akns and kaup broer soliton systems are constructed as related finite field restrictions

factorization of operators ii we extend the methods of a previous paper j math phys 21 2508 1980 factorizing the general scalar third order differential operator and obtain a miura transformation for the boussinesq equation we give a general factorized eigenvalue problem we also give a hamiltonian structure associated with the factorized eigenvalue problem we derive several isospectral flows some of klein gordon type

on matrix painleve hierarchies abstract we define a matrix first painleve hierarchy and a matrix second painleve p ii hierarchy for our matrix p ii hierarchy we also give auto backlund transformations and consider the iteration of solutions this is the first paper to define matrix painleve hierarchies and to give auto backlund transformations for a matrix painleve hierarchy we also consider amongst other results the derivation of sequences of special integrals and autonomous limits until now it has been unknown how to connect the known matrix p ii equation to the obvious candidates for related completely integrable matrix partial differential equations our matrix p ii hierarchy is placed firmly within the context of a matrix modified korteweg de vries mkdv hierarchy in deriving our matrix p ii hierarchy we make use of the hamiltonian structure of this matrix mkdv hierarchy we thus see once again the importance for painleve hierarchies of the integrability structures of related completely integrable equations

hamiltonian theory over noncommutative rings and integrability in multidimensions a generalization of the adler gel fand dikii scheme is used to generate bi hamiltonian structures in two spatial dimensions in order to implement this scheme a hamiltonian theory is built over a noncommutative ring namely the ring of formal pseudodifferential operators bi hamiltonian structures generated in this way can be used for the kadomtsev petviashvili equation as well as other integrable equations in 2 1

novel cloning machine with supplementary information probabilistic cloning was first proposed by duan and guo then pati established a novel cloning machine ncm for copying superposition of multiple clones simultaneously in this paper we deal with the novel cloning machine with supplementary information ncmsi for the case of cloning two states we demonstrate that the optimal efficiency of the ncmsi in which the original party and the supplementary party can perform quantum communication equals that achieved by a two step cloning protocol wherein classical communication is only allowed between the original and the supplementary parties from this equivalence it follows that ncmsi may increase the success probabilities for copying also an upper bound on the unambiguous discrimination of two nonorthogonal pure product states is derived our investigation generalizes and completes the results in the literature

global existence versus blow up in a high dimensional keller segel equation with degenerate diffusion and nonlocal aggregation abstract in this paper we study the degenerate keller segel equation with nonlocal aggregation u t u m u b u in r d r where m 1 d 3 and b u 2 u with 2 d by analyzing the interaction between the degenerate diffusion and the nonlocal aggregation we determine the conditions for initial data under which weak solutions globally exist or blow up in finite time with m 1 d d d furthermore a sharper criterion is given for global existence and finite time blow up of weak solutions with m in the subrange 2 d 2 d d d 1 d d

recursion operators of some equations of hydrodynamic type we give a general method for constructing recursion operators for some equations of hydrodynamic type admitting a nonstandard lax representation we give several examples for n 2 and n 3 containing the equations of shallow water waves and its generalizations with their first two general symmetries and their recursion operators we also discuss a reduction of n 1 systems to n systems of some new equations of hydrodynamic type

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

new interior penalty discontinuous galerkin methods for the keller segel chemotaxis model we develop a family of new interior penalty discontinuous galerkin methods for the keller segel chemotaxis model this model is described by a system of two nonlinear pdes a convection diffusion equation for the cell density coupled with a reaction diffusion equation for the chemoattractant concentration it has been recently shown that the convective part of this system is of a mixed hyperbolic elliptic type which may cause severe instabilities when the studied system is solved by straightforward numerical methods therefore the first step in the derivation of our new methods is made by introducing the new variable for the gradient of the chemoattractant concentration and by reformulating the original keller segel model in the form of a convection diffusion reaction system with a hyperbolic convective part we then design interior penalty discontinuous galerkin methods for the rewritten keller segel system our methods employ the central upwind numerical fluxes originally developed in the context of finite volume methods for hyperbolic systems of conservation laws in this paper we consider cartesian grids and prove error estimates for the proposed high order discontinuous galerkin methods our proof is valid for pre blow up times since we assume boundedness of the exact solution we also show that the blow up time of the exact solution is bounded from above by the blow up time of our numerical solution in the numerical tests presented below we demonstrate that the obtained numerical solutions have no negative values and are oscillation free even though no slope limiting technique has been implemented

integrable systems and their recursion operators in this paper we discuss the structure of recursion operators we show that recursion operators of evolution equations have a nonlocal part that is determined by symmetries and cosymmetries this enables us to compute recursion operators more systematically under certain conditions which hold for all examples known to us nijenhuis operators are well defined i e they give rise to hierarchies of infinitely many commuting symmetries of the operator moreover the nonlocal part of a nijenhuis operator contains the candidates of roots and coroots

a finite volume scheme for the patlak keller segel chemotaxis model a finite volume method is presented to discretize the patlak keller segel pks modeling chemosensitive movements first we prove existence and uniqueness of a numerical solution to the proposed scheme then we give a priori estimates and establish a threshold on the initial mass for which we show that the numerical approximation converges to the solution to the pks system when the initial mass is lower than this threshold numerical simulations are performed to verify accuracy and the properties of the scheme finally in the last section we investigate blow up of the solution for large mass

notes on finite difference schemes to a parabolic elliptic system modelling chemotaxis we consider numerical solutions to a nonlinear parabolic elliptic system in mathematical biology an important property of solutions to the system is the conservation of l 1 norm applying the upwind technique we make a finite difference scheme satisfying the conservation of a discrete l 1 norm we also describe a remark on time discretization from the view point of lyapunov s property which is another important feature of the system several numerical examples clarify the nature of our scheme

integrable coupled kdv systems we give the conditions for a system of n coupled korteweg de vries kdv type of equations to be integrable we find the recursion operators of each subclass and give all examples for n 2

recursion operators and conservation laws for discrete lax equations in this paper a method for constructing recursion operator of discrete lax equations is proposed to illustrate the method recursion operators for discrete boussinesq equation and discrete kdv equation are obtained then the recursion operators of continuous boussinesq equation and kdv equation are successfully recovered by the discrete ones infinitely many conservation laws of discrete boussinesq equation are constructed by the discrete residue formula and the first three conserved quantities are given explicitly and verified numerically existence of recursion operators and infinite number of conservation laws proves the integrability of the discrete lax equations

on construction of recursion operators from lax representation in this work we develop a general procedure for constructing the recursion operators for nonlinear integrable equations admitting lax representation several new examples are given in particular we find the recursion operators for some kdv type systems of integrable equations

darboux transformations finite reduction groups and related yang baxter maps in this paper we construct yang baxter yb maps using darboux matrices which are invariant under the action of finite reduction groups we present six dimensional yb maps corresponding to darboux transformations for the nonlinear schrodinger nls equation and the derivative nonlinear schrodinger dnls equation these yb maps can be restricted to four dimensional yb maps on invariant leaves the former are completely integrable and they also have applications to a recent theory of maps preserving functions with symmetries fordy a and kassotakis a 2013 j phys a math theor 46 205201 we give a six dimensional yb map corresponding to the darboux transformation for a deformation of the dnls equation we also consider vector generalizations of the yb maps corresponding to the nls and dnls equations

noncommutative korteweg de vries and modified korteweg de vries hierarchies via recursion methods here noncommutative hierarchies of nonlinear equations are studied they represent a generalization to the operator level of corresponding hierarchies of scalar equations which can be obtained from the operator ones via a suitable projection a key tool is the application of backlund transformations to relate different operator valued hierarchies indeed in the case when hierarchies in 1 1 dimensions are considered a backlund chart depicts links relating in particular the korteweg de vries kdv to the modified kdv mkdv hierarchy notably analogous links connect the hierarchies of operator equations the main result is the construction of an operator soliton solution depending on an infinite dimensional parameter to start with the potential kdv hierarchy is considered then backlund transformations are exploited to derive solution formulas in the case of kdv and mkdv hierarchies it is remarked that hierarchies of matrix equations of any dimension are also incorporated in the present framework

symmetric singularity formation in lubrication type equations for interface motion fourth order degenerate diffusion equations arise in a lubrication approximation of a thin film or neck driven by surface tension numerical studies of the lubrication equation le h t h n h xxx x 0 with various boundary conditions indicate that singularity formation in which h x t t to 0 occurs for small enough n with anomalous or second type scaling inconsistent with usual dimensional analysis this paper considers locally symmetric or even singularities in the le and in the modified lubrication equation mle h t h n h xxxx 0 both equations have the property that entropy bounds forbid finite time singularities when n is sufficiently large power series expansions for local symmetric similarity solutions are proposed for equation le with n 1 and mle for all n in mathbb r in the latter case special boundary conditions that force singularity formation are required to produce singularities when n is large matching conditions at higher order terms in the

precise computations of chemotactic collapse using moving mesh methods we consider the problem of computing blow up solutions of chemotaxis systems or the so called chemotactic collapse in two spatial dimensions such solutions can have approximate self similar behaviour which can be very challenging to verify in numerical simulations cf betterton and brenner collapsing bacterial cylinders phys rev e 64 2001 061904 we analyse a dynamic scale invariant remeshing method which performs spatial mesh movement based upon equidistribution using a suitably chosen monitor function the numerical solution resolves the fine detail in the asymptotic solution structure such that the computations are seen to be fully consistent with the asymptotic description of the collapse phenomenon given by herrero and velazquez singularity patterns in a chemotaxis model math ann 306 1996 583 623 we believe that the methods we construct are ideally suited to a large number of problems in mathematical biology for which collapse phenomena are expected

on a method for constructing the lax pairs for nonlinear integrable equations we suggest a direct algorithm for searching the lax pairs for nonlinear integrable equations it is effective for both continuous and discrete models the first operator of the lax pair corresponding to a given nonlinear equation is found immediately coinciding with the linearization of the considered nonlinear equation the second one is obtained as an invariant manifold to the linearized equation a surprisingly simple relation between the second operator of the lax pair and the recursion operator is discussed the recursion operator can immediately be found from the lax pair examples considered in the article are convincing evidence that the found lax pairs differ from the classical ones the examples also show that the suggested objects are true lax pairs which allow the construction of infinite series of conservation laws and hierarchies of higher symmetries in the case of the hyperbolic type partial differential equation our algorithm is slightly modified in order to construct the lax pairs from the invariant manifolds we use the cutting off conditions for the corresponding infinite laplace sequence the efficiency of the method is illustrated by application to some equations given in the svinolupov sokolov classification list for which the lax pairs and the recursion operators have not been found earlier

non autonomous svinolupov jordan kdv systems non autonomous svinolupov jordan kdv systems are considered the integrability criteria of such systems are associated with the existence of recursion operators a new non autonomous kdv system and its recursion operator is obtained for all n the examples for n 2 and 3 are studied in detail some possible transformations which map some systems to autonomous ones are also discussed pacs numbers 0230i 0220 0230j

darboux polynomial matrices the classical massive thirring model as a study case one way of constructing explicit expressions of solutions of integrable systems of partial differential equations goes via the darboux method this requires the construction of darboux matrices here we introduce a novel algorithm to obtain such matrices in polynomial form our method is illustrated by applying it to the classical massive thirring model and by combining it with the dihedral group of symmetries possessed by this model

self similar blowup solutions to an aggregation equation in r n we present numerical simulations of radially symmetric finite time blowup for the aggregation equation ut u k u where the kernel k x x the dynamics of the blowup exhibits self similar behavior in which zero mass concentrates at the core at the blowup time computations are performed in r n for n between 2 and 10 using a method based on characteristics in all cases studied the self similarity exhibits second kind anomalous scaling

matrix korteweg de vries and modified korteweg de vries hierarchies noncommutative soliton solutions the present work continues work on kdv type hierarchies presented by s carillo and c schiebold noncommutative korteweg de vries and modified korteweg de vries hierarchies via recursion methods j math phys 50 073510 2009 10 1063 1 3155080 general solution formulas for the kdv and mkdv hierarchies are derived by means of banach space techniques both in the scalar and matrix case a detailed analysis is given of solitons breathers their countable superpositions as well as of multisoliton solutions for the matrix hierarchies

reductions of n wave interactions related to low rank simple lie algebras i z2 reductions the analysis and the classification of all reductions for the nonlinear evolution equations solvable by the inverse scattering method is an interesting and still open problem we show how the second order reductions of the n wave interactions related to low rank simple lie algebras g can be embedded also in the weyl group of g this allows us to display along with the well known ones a number of new types of integrable n wave systems some of the reduced systems find applications to nonlinear optics

twisted hierarchies associated with the generalized sine gordon equation twisted u and twisted u k hierarchies are soliton hierarchies introduced by terng to find higher flows of the generalized sine gordon equation twisted o j j o j o j hierarchies are among the most important classes of twisted hierarchies in this paper we derive explicit interesting first and higher flows of twisted o j j o j o j hierarchies justify that the one dimensional systems of twisted o j j o j o j hierarchies for j iq n q 1 q n 1 called the generalized sinh gordon equations are the gauss codazzi equations for n dimensional timelike submanifolds with constant sectional curvature 1 and index q in pseudo euclidean 2n 1 dimensional space r2q 12n 1 with index 2q 1 furthermore a unified treatment of the inverse scattering theory for twisted o j j o j o j hierarchies is provided

on the classification of automorphic lie algebras the problem of reduction of integrable equations can be formulated in a uniform way using the theory of invariants this provides a powerful tool of analysis and it opens the road to new applications of automorphic lie algebras beyond the con text of integrable systems in this paper it is shown that sl2 c based automorphic lie algebras associated to the icosahedral group i the octahedral group o the tetrahedral group t and the dihedral group dn are isomorphic the proof is based on techniques from classical invariant theory and makes use of clebsch gordan decomposition and transvectants molienfunctionsandthetrace form thisresultprovidesacompleteclas sification of sl2 c based automorphic lie algebras associated to finite groups when the group representations are chosen to be the same and it is a crucial step towards the complete classification of automorphic lie algebras

stochastic particle approximation for measure valued solutions of the 2d keller segel system we construct an approximation to the measure valued global in time solutions to the patlak keller segel model in 2d based on systems of stochastic interacting particles the advantage of our approach is that it reproduces the well known dichotomy in the qualitative behavior of the system and moreover captures the solution even after the possible blow up events we present a numerical method based on this approach and show some numerical results moreover we make a first step toward the convergence analysis of our scheme by proving the convergence of the stochastic particle approximation for the keller segel model with a regularized interaction potential the proof is based on a bbgky like approach for the corresponding particle distribution function

discontinuous galerkin methods for the chemotaxis and haptotaxis models in this work first we formulate and compare three different discontinuous interior penalty galerkin methods for the 2d keller segel chemotaxis model keller segel chemotaxis model is the important starting step in the modeling of the real biological system we show in the numerical tests that two of the proposed methods fail to give accurate oscillation free solutions next we consider the application of the successful method for the keller segel model to the simulation of the more realistic and closely related haptotaxis model of tumor invasion into healthy tissues

variable coefficient third order korteweg de vries type of equations it is shown that the integrable subclasses of the equations q t f x t q 3 h x t q q 1 are the same as the integrable subclasses of the equations q t q 3 f q q 1

a novel noncommutative kdv type equation its recursion operator and solitons a noncommutative kdv type equation is introduced extending the backlund chart in carillo et al symmetry integrability geom methods appl 12 087 2016 this equation called meta mkdv here i

infinite time aggregation for the critical patlak keller segel model in ℝ2 we analyze the two dimensional parabolic elliptic patlak keller segel model in the whole euclidean space r2 under the hypotheses of integrable initial data with finite second moment and entropy we first show local in time existence for any mass of free energy solutions namely weak solutions with some free energy estimates we also prove that the solution exists as long as the entropy is controlled from above the main result of the paper is to show the global existence of free energy solutions with initial data as before for the critical mass 8 actually we prove that solutions blow up as a delta dirac at the center of mass when t keeping constant their second moment at any time furthermore all moments larger than 2 blow up as t if initially bounded

reductions of integrable equations on a iii type symmetric spaces we study a class of integrable nonlinear differential equations related to the a iii type symmetric spaces these spaces are realized as factor groups of the form su n s u n k u k we use the cartan involution corresponding to this symmetric space as an element of the reduction group and restrict generic lax operators to this symmetric space the symmetries of the lax operator are inherited by the fundamental analytic solutions and give a characterization of the corresponding riemann hilbert data

convergence of the mass transport steepest descent scheme for the subcritical patlak keller segel model variational steepest descent approximation schemes for the modified patlak keller segel equation with a logarithmic interaction kernel in any dimension are considered we prove the convergence of the suitably interpolated in time implicit euler scheme defined in terms of the euclidean wasserstein distance associated with this equation for subcritical masses as a consequence we recover the recent result about the global in time existence of weak solutions to the modified patlak keller segel equation for the logarithmic interaction kernel in any dimension in the subcritical case moreover we show how this method performs numerically in dimension one in this particular case this numerical scheme corresponds to a standard implicit euler method for the pseudoinverse of the cumulative distribution function we demonstrate its capabilities to reproduce the blow up of solutions for supercritical masses easily without the need of mesh refinement

nonlocal symmetries of the hirota satsuma coupled korteweg de vries system and their applications exact interaction solutions and integrable hierarchy the nonlocal symmetry is derived from the known darboux transformation dt of the hirota satsuma coupled korteweg de vries hs ckdv system and infinitely many nonlocal symmetries are given by introducing the internal parameters by extending the hs ckdv system to an auxiliary system with five dependent variables the prolongation is found to localize the so called seed nonlocal symmetry related to the dt by applying the general lie point symmetry method to this enlarged system we obtain two main results a new type of finite symmetry transformation is derived which is different from the initial dt and can generate new solutions from old ones some novel exact interaction solutions among solitons and other complicated waves including periodic cnoidal waves and painleve waves are computed through similarity reductions in addition two kinds of new integrable models are proposed from the obtained nonlocal symmetry the negative hs ckdv hierarchy by introducing the internal parameters the integrable models both in lower and higher dimensions by restricting the symmetry constraints

wave fronts and cascades of soliton interactions in the periodic two dimensional volterra system in the paper we develop the dressing method for the solution of the two dimensional periodic volterra system with a period n we derive soliton solutions of arbitrary rank k and give a full classification of rank 1 solutions we have found a new class of exact solutions corresponding to wave fronts which represent smooth interfaces between two nonlinear periodic waves or a periodic wave and a trivial zero solution the wave fronts are non stationary and they propagate with a constant average velocity the system also has soliton solutions similar to breathers which resembles soliton webs in the kp theory we associate the classification of soliton solutions with the schubert decomposition of the grassmannians grr k n and grc k n

the coupled modified korteweg de vries equations similarity reduction lie backlund symmetries and integrability the lie point symmetries of n coupled modified korteweg de vries equations n cmkdv is derived and shown that the similarity reduction associated with symmetries passes the painleve property indicating its integrability the nonclassical symmetry analysis of bluman and cole and the direct method of clarkson and kruskal to n cmkdv equations show that there exists no new similarity reductions a sequence of lie backlund symmetries for n cmkdv equations is derived explicitly establishing its complete integrability the question of constructing the recursion operator of n cmkdv a characteristic of integrable systems is also discussed

reduction groups and automorphic lie algebras we study a new class of infinite dimensional lie algebras which has important applications to the theory of integrable equations the construction of these algebras is very similar to the one for automorphic functions and this motivates the name automorphic lie algebras for automorphic lie algebras we present bases in which they are quasigraded and all structure constants can be written out explicitly these algebras have useful factorisations on two subalgebras similar to the factorisation of the current algebra on the positive and negative parts

recursion operators and bi hamiltonian structures in multidimensions ii the algebraic properties of exactly solvable evolution equations in one spatial and one temporal dimensions have been well studied in particular the factorization of certain operators called recursion operators establishes the bi hamiltonian nature of all these equations recently we have presented the recursion operator and the bi hamiltonian formulation of the kadomtsev petviashvili equation a two spatial dimensional analogue of the korteweg devries equation here we present the general theory associated with recursion operators for bi hamiltonian equations in two spatial and one temporal dimensions as an application we show that general classes of equations which include the kadomtsev petviashvili and the davey stewartson equations possess infinitely many commuting symmetries and infinitely many constants of motion in involution under two distinct poisson brackets furthermore we show that the relevant recursion operators naturally follow from the underlying isospectral eigenvalue problems

on the recursion operators for the gerdjikov mikhailov and valchev system we consider the recursion operator approach to the soliton equations related to an auxiliary linear system introduced recently by gerdjikov mikhailov and valchev gmv system we discuss the recursion operators obtained for the gmv system and show that they are particular cases of the generating operators related to a zakharov shabat type linear system on the algebra sl 3 c in pole gauge

on the use of isospectral eigenvalue problems for obtaining hereditary symmetries for hamiltonian systems we present an algorithmic method for obtaining an hereditary symmetry the generalized squared eigenfunction operator from a given isospectral eigenvalue problem this method is applied to the n n eigenvalue problem considered by ablowitz and haberman and to the eigenvalue problem considered by alonso the relevant hamiltonian formulations are also determined finally an alternative method is presented in the case two evolution equations are related by a miura type transformation and their hamiltonian formulations are known

reductions of integrable equations dihedral group we discuss the algebraic and analytic structure of rational lax operators with algebraic reductions of lax equations we associate a reduction group a group of automorphisms of the corresponding infinite dimensional lie algebra we present a complete study of dihedral reductions for sl 2 c lax operators with simple poles and corresponding integrable equations in the last section we give three examples of dihedral reductions for sl n c lax operators

probabilistic broadcasting of mixed states it is well known that the non broadcasting theorem proved by barnum et al is a fundamental principle of quantum communication as we are aware optimal broadcasting ob is the only method to broadcast noncommuting mixed states approximately in this paper motivated by the probabilistic cloning of quantum states proposed by duan and guo we propose a new way for broadcasting noncommuting mixed states probabilistic broadcasting pb and we present a sufficient condition for pb of mixed states to a certain extent we generalize the probabilistic cloning theorem from pure states to mixed states and in particular we generalize the non broadcasting theorem since the case that commuting mixed states can be exactly broadcast can be thought of as a special instance of pb where the success ratio is 1 moreover we discuss probabilistic local broadcasting plb of separable bipartite states