the painleve property for partial differential equations in this paper we define the painleve property for partial differential equations and show how it determines in a remarkably simple manner the integrability the backlund transforms the linearizing transforms and the lax pairs of three well known partial differential equations burgers equation kdv equation and the modified kdv equation this indicates that the painleve property may provide a unified description of integrable behavior in dynamical systems ordinary and partial differential equations while at the same time providing an efficient method for determining the integrability of particular systems

lax pair and first integrals of the traveling wave reduction for the kdv hierarchy abstract the traveling wave reduction of the korteweg de vries hierarchy is considered the linear system of equations associated with this hierarchy is found the lax pair is used to obtain the first integrals of the traveling wave reduction for the korteweg de vries hierarchy exact formulas for the first integrals of the hierarchy are given the first three members of the hierarchy are considered in more detail these first integrals are the examples of the integrable nonlinear differential equations with the painleve property exact solutions in the form of the soliton for the traveling wave reduction of the korteweg de vries hierarchy and its first integrals are presented

second and fourth painleve hierarchies and jimbo miwa linear problems the relations between the different linear problems for painleve equations is an intriguing open problem here we consider our previously given second and fourth painleve hierarchies publ res inst math sci kyoto 37 327 347 2001 and show that they could alternatively have been derived using the linear problems of jimbo and miwa that is we give a gauge transformation of our linear problems for these two hierarchies which maps those of the second and fourth painleve equations themselves onto those of jimbo and miwa

non isospectral scattering problems and truncation for hierarchies burgers and dispersive water waves abstract non isospectral scattering problems have been proven useful for several reasons amongst them the information that they provide about painleve truncation for entire hierarchies of integrable partial differential equations pdes we show in this paper how our approach provides in a very straightforward manner truncation results for two hierarchies in 1 1 dimensions namely burgers hierarchy and the dispersive water wave hierarchy burgers equation is well known as a model for turbulence and the dispersive water wave equations as a system governing shallow water waves we also see how these results are easily extended to related hierarchies in 2 1 dimensions our results for the 2 1 dimensional burgers hierarchy and for the 1 1 and 2 1 dimensional dispersive water wave hierarchies are all new

nonisospectral scattering problems a key to integrable hierarchies we show that certain partial differential equations associated to nonisospectral scattering problems in 2 1 dimensions provide a key to associated integrable hierarchies of both ordinary and partial differential equations this is illustrated using an extension of a known second order and two new third order nonisospectral scattering problems these scattering problems allow us to derive new hierarchies of integrable partial differential equations in both 1 1 and 2 1 dimensions together with their underlying linear problems isospectral and nonisospectral and also new hierarchies of integrable ordinary differential equations again with their underlying linear problems

non autonomous he non heiles systems abstract scaling similarity solutions of three integrable pdes namely the sawada kotera fifth order kdv and kaup kupershmidt equations are considered it is shown that the resulting odes may be written as non autonomous hamiltonian equations which are time dependent generalizations of the well known integrable henon heiles systems the time dependent hamiltonians are given by logarithmic derivatives of the tau functions inherited from the original pdes the odes for the similarity solutions also have inherited backlund transformations which may be used to generate sequences of rational solutions as well as other special solutions related to the first painleve transcendent

a note on the generalized symmetries of lattice hierarchies we show that the recursion operators of the integrable lattice equations usually considered in the literature can also be used to generate hierarchies of differential delay equations all members of these hierarchies of lattice and differential delay equations commute it is thus seen that differential delay hierarchies provide a broader context within which to place lattice hierarchies

auto backlund transformations for a differential delay equation discrete painleve equations have over recent years generated much interest one property of such equations that is considered to be particularly important is the existence of auto backlund transformations that is mappings between solutions of the equation in question usually involving changes in the values of parameters appearing as coefficients we have recently presented extensions of discrete painleve equations to equations involving derivatives as well as shifts in the independent variable here we show how auto backlund transformations can also be constructed for such differential delay equations we emphasise that this is the first time that an auto backlund transformation has been given for a differential delay equation

factorization and painleve analysis of a class of nonlinear third order partial differential equations a class of fully nonlinear third order partial differential equations pdes is considered this class contains several examples which have recently appeared in the literature and for which rather unusual travelling wave solutions have been given these solutions consist essentially of sums of exponentials we trace the occurrence of these exponentials back to the existence of a linear subequation which appears as a factor in the travelling wave reduction in addition we consider the painleve analysis of this set of equations both for the original pde and also for reductions to ordinary differential equations odes no equation in the class considered survives the combination of pde and ode tests also an equation in the class considered which is known to be integrable is shown to possess only the weak painleve property our analysis therefore confirms the limitations of the painleve test as a test for complete integrability when applied to fully nonlinear pdes

solving second order ordinary differential equations by extending the prelle singer method we propose a method for solving second order ordinary differential equations odes which is based on the ideas behind the prelle singer ps procedure for first order odes while the ps procedure treats differential equations des of the form y p x y q x y with p and q polynomials whose coefficients lie in the field of complex numbers our method is applicable to des of the form y p x y y q x y y the key to our approach is to focus not on the final solution but on the first order invariants of the equation our method is an attempt to address algorithmically the solution of second order odes with solutions in terms of elementary functions

a nonisospectral extension of the volterra hierarchy to 2 1 dimensions we give a new nonisospectral generalization of the volterra lattice equation to 2 1 dimensions we use this to construct a new nonisospectral lattice hierarchy in 2 1 dimensions along with its underlying linear problem reductions yield a variety of new integrable hierarchies including generalizations of known discrete painleve hierarchies all along with their corresponding linear problems this represents an extension of previously developed techniques to the discrete case

on a new non isospectral variant of the boussinesq hierarchy we give a new non isospectral extension to 2 1 dimensions of the boussinesq hierarchy such a non isospectral extension of the third order scattering problem xxx u x v 0 has not been considered previously this extends our previous results on one component hierarchies in 2 1 dimensions associated to third order non isospectral scattering problems we characterize our entire 2 1 dimensional hierarchy and its linear problem using a single partial differential equation and its corresponding non isospectral scattering problem this then allows an alternative approach to the construction of linear problems for the entire 2 1 dimensional hierarchy reductions of this hierarchy yield new integrable hierarchies of systems of ordinary differential equations together with their underlying linear problems in particular we obtain a fourth painleve hierarchy i e a hierarchy of ordinary differential equations having the fourth painleveequation as its first member we also obtain a hierarchy having as its first member a generalization of an equation defining a new transcendent due to cosgrove

nonisospectral scattering problems and similarity reductions abstract we give mappings between hierarchies having nonisospectral scattering problems and hierarchies having isospectral scattering problems special cases of these changes of variables involve similarity variables and similarity solutions of a hierarchy are seen to correspond to time independent solutions of an equivalent hierarchy we thus explain why the use of nonisospectral scattering problems and similarity reductions yield the same painleve hierarchies as examples we consider the korteweg de vries hierarchy and the dispersive water wave hierarchy

the singular manifold method revisited for many completely integrable partial differential equations pdes the singular manifold method of weiss allows the recovery of the lax pair and darboux transformation dt and so also the backlund transformation from a truncated painleve expansion recently the so called two singular manifold method has been proposed in order to handle pdes such as the modified korteweg de vries mkdv equation here we present a more natural extension of the weiss singular manifold method which makes use of only one singular manifold but is capable of dealing with such pdes in this approach we allow the possibility that the dt might in fact correspond to an infinite painleve expansion for a certain choice of the arbitrary coefficients this then leads us to a new and more consistent definition of singular manifold equation sme this can give smes different from those usually presented the summation of infinite painleve expansions is effected by seeking a truncation in a new riccati variable z the use o

riccati equations in optimal filtering of nonstabilizable systems having singular state transition matrices until recently it was believed that a necessary and sufficient condition for convergence of the riccati difference equation of optimal filtering was that the system be both delectable and stabilizable recently it has been shown that the stabilizability condition can be removed but convergence has only established under restrictive assumptions including the requirement that the state transition matrix be nonsingular the present paper generalizes these results in several directions first properties of the algebraic riccati equation are established for the case of singular state transition matrix second several assumptions previously imposed in establishing convergence of the riccati difference equation for systems with unreachable modes on the unit circle are relaxed including replacing observability by detectability weakening the conditions on the initial covariance and allowing the state transition matrix to be singular third results on the convergence and properties of the riccati equations are expressed as both necessary and sufficient conditions whereas previous results were only sufficient these extensions mean that the results have wider applicability including fixed lag smoothing problems and filtering for systems with time delays the implications of the results in the dual problem of optimal control are also studied

symmetries of a 2 1 dimensional breaking soliton equation infinitely many symmetries for a 2 1 dimensional breaking soliton equation are constructed via the infinitesimal version of the dressing method these symmetries are proved to constitute an infinite dimensional lie algebra which contains some abelian and virasoro subalgebras the hierarchies of equations generated by these symmetries are also considered these hierarchies of equations are proved to be associated with the isospectral and non isospectral deformations of the akns spectral problem

new integrable equations of fourth order and higher degree related to cosgrove s equation we give a general formulation of the algorithm of fokas and ablowitz which then allows us to obtain transformations for nth order ordinary differential equations to equations of the same order but perhaps of higher degree previously this algorithm has been used to obtain transformations for the six second order equations defining new transcendental functions discovered by painleve and co workers either to other equations in the painleve classification or to equations of second order and second degree as an example of our approach we consider a new fourth order ordinary differential equation due to cosgrove which is believed to define a new transcendent we obtain transformations relating this equation to other fourth order ordinary differential equations of degrees 2 all of these transformations as well as the corresponding higher degree differential equations all of which have the painleve property are new

integrable peakon equations with cubic nonlinearity we present a new integrable partial differential equation found by vladimir novikov like the camassa holm and degasperis procesi equations this new equation admits peaked soliton peakon solutions but it has nonlinear terms that are cubic rather than quadratic we give a matrix lax pair for v novikov s equation and show how it is related by a reciprocal transformation to a negative flow in the sawada kotera hierarchy infinitely many conserved quantities are found as well as a bi hamiltonian structure the latter is used to obtain the hamiltonian form of the finite dimensional system for the interaction of n peakons and the two body dynamics n 2 is explicitly integrated finally all of this is compared with some analogous results for another cubic peakon equation derived by zhijun qiao

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

the prelle singer method and painleve hierarchies we consider systems of ordinary differential equations odes of the form bk 0 where b is a hamiltonian operator of a completely integrable partial differential equation hierarchy and k k l t such systems while of quite low order and linear in the components of k may represent higher order nonlinear systems if we make a choice of k in terms of the coefficient functions of b indeed our original motivation for the study of such systems was their appearance in the study of painleve hierarchies where the question of the reduction of order is of great importance however here we do not consider such particular cases instead we study such systems for arbitrary k where they may represent both integrable and nonintegrable systems of ordinary differential equations we consider the application of the prelle singer ps method a method used to find first integrals to such systems in order to reduce their order we consider the cases of coupled second order odes and coupled third order odes as well a

from continuous painleve iv to the asymmetric discrete painleve i we examine the discrete equation that is obtained from the schlesinger transformations of the continuous painleve iv equation the use of the schlesinger transformations naturally establishes the lax pair of the discrete equation which is an asymmetric form of d p and has as continuous limit we analyse the transformations of the discrete equation and show that they lead to a self dual description where the discrete equation and the schlesinger s play the same role finally using the relation between and asymmetric d we examine in detail the special solutions of both equations

auto backlund transformations and integrability of ordinary and partial differential equations we introduce a partial differential equation whose structure in many respects mimics that of the second painleve equation for this equation we give auto backlund transformations of both ordinary differential equation and partial differential equation types our discussion is sufficiently generally phrased to allow us to draw some general conclusions about the connection between auto backlund transformations and the integrability of ordinary and partial differential equations

an alternative perspective on injectivity of modules abstract given modules m and n m is said to be n subinjective if for every extension k of n and every homomorphism n m there exists a homomorphism k m such that n for a module m the subinjectivity domain of m is defined to be the collection of all modules n such that m is n subinjective as an opposite to injectivity a module m is said to be indigent if its subinjectivity domain is smallest possible namely consisting of exactly the injective modules properties of subinjectivity domains and of indigent modules are studied in particular the existence of indigent modules is determined for some families of rings including the ring of integers and artinian serial rings it is also shown that some rings e g artinian chain rings have no middle class in the sense that all modules are either injective or indigent for various classes of modules such as semisimple singular and projective necessary and sufficient conditions for the existence of indigent modules of those types are studied indigent modules are analog to the so called poor modules an opposite of injectivity in terms of injectivity domains recently studied in papers by alahmadi alkan and lopez permouth and by er lopez permouth and sokmez relations between poor and indigent modules are also investigated here

on a construction of general formal solutions for equations of the first painleve hierarchy i abstract we construct general formal solutions containing sufficiently many free parameters for the first painleve hierarchy with a large parameter from a viewpoint of the multiple scale analysis

systematic method of generating new integrable systems via inverse miura maps we provide a new natural interpretation of the lax representation for an integrable system that is the spectral problem is the linearized form of a miura transformation between the original system and a modified version of it on the basis of this interpretation we formulate a systematic method of identifying modified integrable systems that can be mapped to a given integrable system by miura transformations thus this method can be used to generate new integrable systems from known systems through inverse miura maps it can be applied to both continuous and discrete systems in 1 1 dimensions as well as in 2 1 dimensions the effectiveness of the method is illustrated using examples such as the nonlinear schrodinger nls system the zakharov ito system two component kdv the three wave interaction system the yajima oikawa system the ablowitz ladik lattice integrable space discrete nls and two 2 1 dimensional nls systems

2 1 dimensional integrable lattice hierarchies related to discrete fourth order nonisospectral problems in this paper we consider the 2 1 dimensional discrete fourth order nonisospectral problem by using the lax technique three new 2 1 dimensional nonisospectral four field integrable lattice hierarchies are constructed their reductions yield three 1 1 dimensional isospectral four field integrable lattice hierarchies due to mlaszak marciniak we make a comparison between the 1 1 dimensional discrete fourth order nonisospectral problem and the third order nonisospectral problem we found that the integrable lattice hierarchies related to the discrete fourth order nonisospectral problem have new characteristics

generalized scaling reductions and painleve hierarchies we give an alternative derivation of two painleve hierarchies this is done by constructing generalized scaling reductions of the korteweg de vries and dispersive water wave hierarchies we also construct a generalized scaling reduction of burgers hierarchy

a method of reduction of order for discrete systems we show how by establishing a suitable backlund correspondence a reduction in order can be effected for discrete systems having a certain structure in this way we succeed in extending to discrete systems ideas previously developed within the context of ordinary differential equations it is also worth emphasizing that our approach represents an application of properties characteristic of completely integrable systems outside of that context that is to systems of discrete equations integrable or otherwise

on the complete integrability and linearization of certain second order nonlinear ordinary differential equations a method for finding general solutions of second order nonlinear ordinary differential equations by extending the prelle singer ps method is briefly discussed we explore integrating factors integrals of motion and the general solution associated with several dynamical systems discussed in the current literature by employing our modifications and extensions of the ps method we also introduce a novel way of deriving linearizing transformations from the first integrals to linearize the second order nonlinear ordinary differential equations to free particle equations we illustrate the theory with several potentially important examples and show that our procedure is widely applicable

the he non heiles system revisited abstract the known integrable cases of the henon heiles system are shown to be closely related to the stationary flows of the known and only integrable fifth order single component and polynomial nonlinear evolution equations this is further evidence that these are the only integrable cases of the henon heiles system lax pairs are deduced for each of the integrable cases and used to construct the constants of motion a curious lax operator recently found by the painleve method is explained

amalgamations of the painleve equations new hierarchies of nonlinear ordinary differential equations that are generalizations of the painleve equations are presented these hierarchies are shown to contain the painleve equations as special cases emphasis is on the sixth order ordinary differential equations special solutions for one of them are expressed via the general solutions of the p1 p2 equations and special cases of the p3 and p5 equations special solutions of another sixth order ode are determined by the general solutions of the p2 p4 p3 and p5 equations five of six painleve equations can be considered as special cases of these sixth order odes isomonodromic linear problems to solve the cauchy problems of hierarchies by the inverse monodromy transform are given

rings and modules characterized by opposites of injectivity abstract in a recent paper aydo du and lopez permouth have defined a module m to be n subinjective if every homomorphism n m extends to some e n m where e n is the injective hull of n clearly every module is subinjective relative to any injective module their work raises the following question what is the structure of a ring over which every module is injective or subinjective relative only to the smallest possible family of modules namely injectives we show using a dual opposite injectivity condition that such a ring r is isomorphic to the direct product of a semisimple artinian ring and an indecomposable ring which is i a hereditary artinian serial ring with j 2 0 or ii a qf ring isomorphic to a matrix ring over a local ring each case is viable and conversely i is sufficient for the said property and a partial converse is proved for a ring satisfying ii using the above mentioned classification it is also shown that such rings coincide with the fully saturated rings of trlifaj except possibly when von neumann regularity is assumed furthermore rings and abelian groups which satisfy these opposite injectivity conditions are characterized

backlund transformations for fourth painleve hierarchies abstract backlund transformations bts for ordinary differential equations odes and in particular for hierarchies of odes are a topic of great current interest here we give an improved method of constructing bts for hierarchies of odes this approach is then applied to fourth painleve p iv hierarchies recently found by gordoa et al publ res inst math sci kyoto 37 2001 327 347 we show how the known pattern of bts for p iv can be extended to our p iv hierarchies remarkably the bts required to do this are precisely the miura maps of the dispersive water wave hierarchy we also obtain the important result that the fourth painleve equation has only one nontrivial fundamental bt and not two such as is frequently stated

on the integrable variant of the boussinesq system painleve property rational solutions a related many body system and equivalence with the akns hierarchy abstract an infinite family of rational solutions of the integrable version of the boussinesq system u t w x uu x 0 w t u xxx uw x 0 and the associated higher order flows is constructed the differential equations governing the motion of the poles are derived and their complete integrability demonstrated this system embeds into two disjoint calogero moser systems coupled only through a constraint a novel treatment of this constraint is given using the equivalence of to the real version of the nonlinear schrodinger system it is seen that rational solutions of the akns hierarchy have been constructed as well finally it is shown that all solutions of these hierarchies including the rational ones also solve the first modified kp equation

the bi hamiltonian structure of some nonlinear fifth and seventh order differential equations and recursion formulas for their symmetries and conserved covariants using a bi hamiltonian formulation we give explicit formulas for the conserved quantities and infinitesimal generators of symmetries for some nonlinear fifth and seventh order nonlinear partial differential equations among them the caudrey dodd gibbon sawada kotera equation and the kupershmidt equation we show that the lie algebras of the symmetry groups of these equations are of a very special form among the c vector fields they are generated from two given commuting vector fields by a recursive application of a single operator furthermore for some higher order equations those multisoliton solutions which for t asymptotically decompose into traveling wave solutions are characterized as eigenvector decompositions of certain operators

symmetries exact solutions and nonlinear superposition formulas for two integrable partial differential equations we recently introduced two new sixth order partial differential equations pdes associated with third order scattering problems here we extend our study of these pdes by considering the construction of exact solutions both by using the method of symmetry reduction due to lie and by using their darboux transformations dts amongst the ordinary differential equations odes obtained by symmetry reduction is an ode due to cosgrove that is believed to define a new painleve transcendent this ode provides soliton solutions for our integrable pdes that include arbitrary functions of time the dts for our pdes allow the recovery of these solutions and in addition provide other solutions which are not associated with lie symmetries either classical or nonclassical we also consider the iteration of the corresponding backlund transformations bts for these pdes the theorem of permutability allows us to reduce this process of iterating the dt from one of solving a third order linear equation the spatial p

projection and deflation method for partial pole assignment in linear state feedback two projection methods are proposed for partial pole placement in linear control systems these methods are of interest when the system is very large and only a few of its poles must be assigned the first method is based on computing an orthonormal basis of the left invariant subspace associated with the eigenvalues to be assigned and then solving a small inverse eigenvalue problem resulting from projecting the initial problem into that subspace the second method can be regarded as a variant of the weilandt deflation technique used in eigenvalue methods

singular riccati equations stabilizing large scale systems in this paper we discuss the convergence of a stabilization algorithm based on a singular version of the discrete riccati difference equation this method is particularly appealing for large scale linear time invariant dynamical systems since one can nicely exploit the sparsity of such systems in order to reduce the complexity of the algorithm

on classes of integrable systems and the painleve property the caudrey dodd gibbon equation is found to possess the painleve property investigation of the backlund transformations for this equation obtains the kuperschmidt equation a certain transformation between the kuperschmidt and caudrey dodd gibbon equation is obtained this transformation is employed to define a class of p d e s that identically possesses the painleve property for equations within this class backlund transformations and rational solutions are investigated in particular the sequences of higher order kdv caudrey dobb gibbon and kuperschmidt equations are shown to possess the painleve property

numerical aspects of different kalman filter implementations a theoretical analysis is made of the error propagation due to numerical roundoff for four different kalman filter implementations the conventional kalman filter the square root covariance filter the square root information filter and the chandrasekhar square root filter an experimental analysis is performed to validate the new insights gained by the theoretical analysis

new 2 1 dimensional nonisospectral toda lattice hierarchy we give a new 2 1 dimensional nonisospectral generalization of the toda lattice hierarchy reductions yield a variety of new integrable hierarchies along with their underlying linear problems including new 1 1 dimensional differential delay hierarchies nonisospectral and isospectral new ordinary differential delay hierarchies and new discrete painleve hierarchies we also show that a reduction in components yields our previously obtained 2 1 dimensional nonisospectral volterra lattice hierarchy

rings whose modules have maximal or minimal injectivity domains in a recent paper alahmadi alkan and lopez permouth defined a module m to be poor if m is injective relative only to semisimple modules and a ring to have no right middle class if every right module is poor or injective we prove that every ring has a poor module and characterize rings with semisimple poor modules next a ring with no right middle class is proved to be the ring direct sum of a semisimple artinian ring and a ring t which is either zero or of one of the following types i morita equivalent to a right pci domain ii an indecomposable right si ring which is either right artinian or a right v ring and such that soc tt is homogeneous and essential in tt and t has a unique simple singular right module or iii an indecomposable right artinian ring with homogeneous right socle coinciding with the jacobson radical and the right singular ideal and with unique non injective simple right module in case iii either tt is poor or t is a qf ring with j t 2 0 converses of these cases are discussed it is shown in particular that a qf ring r with j r 2 0 and homogeneous right socle has no middle class

pole free solutions of the first painleve hierarchy and non generic critical behavior for the kdv equation we establish the existence of real pole free solutions to all even members of the painleve i hierarchy we also obtain asymptotics for those solutions and describe their relevance in the description of critical asymptotic behavior of solutions to the kdv equation in the small dispersion limit this was understood in the case of a generic critical point and we generalize it here to the case of non generic critical points

the cauchy problem for the second member of a rm p_ iv hierarchy a rigorous method the inverse monodromy transform for studying the riemann hilbert rh problem associated with the classical painleve equations is applied to the second member of a fourth painleve hierarchy we show that the cauchy problem for the second member of this hierarchy admits in general a global meromorphic solution in x moreover for a particular choice of the monodromy data the associated rh problem can be reduced to a set of scalar rh problems and a special solution which can be written in terms of the airy function is obtained

a special solution technique further extensions abstract in a recent paper we gave a technique for identifying special solutions of hierarchies of ordinary differential equations odes and difference equations in particular we considered the phenomenon whereby members of one painleve hierarchy define solutions of members of a different painleve hierarchy where both are related to the same at least bi hamiltonian completely integrable partial differential equation pde or lattice hierarchy we also considered the question of special integrals of further related painleve hierarchies here we consider three extensions of this approach first of all we consider the case of ode hierarchies where the corresponding pde hierarchy is only mono hamiltonian second we show that our approach can also be extended in order to provide special solutions of pdes our third extension is to the non hamiltonian case where we consider hierarchies of odes related to the burgers hierarchy we also make the observation that our results can be phrased quite generally and the systems of equations dealt with do not in fact need to be integrable

wkb analysis of higher order painleve equations with a large parameter local reduction of 0 parameter solutions for painleve hierarchies pj j i ii 1 or ii 2 abstract we generalize the reduction theorem for 0 parameter solutions of a traditional i e second order painleve equation with a large parameter to those of some higher order painleve equation that is each member of the painleve hierarchies p j j i ii 1 or ii 2 thus the scope of applicability of the reduction theorem in kt1 has been substantially enlarged only six equations were covered by our previous result while theorem 3 2 1 of this paper applies to infinitely many equations

construction scheme for discrete miura transformations a direct and elementary scheme for the construction of miura type transformations and discrete differential equations related to them scalar and vector is presented the scheme is illustrated using as examples the volterra and toda models a discrete differential analogue of the calogero degasperis equation is discussed in detail this example is used to show how to construct conservation laws higher symmetries and solutions for an equation obtained with the help of the scheme

lattice modified kdv hierarchy from a lax pair expansion we produce a hierarchy of integrable equations by systematically adding terms to the lax pair for the lattice modified kdv equation the equations in the hierarchy are related to one another by recursion relations these recursion relations are solved explicitly so that every equation in the hierarchy along with its lax pair is known

the discrete first second and thirty fourth painleve hierarchies the discrete first and second painleve equations dp and dp are integrable difference equations which have classical first second or third painleve equations p p or p as continuum limits dp and dp are believed to be integrable because they are discrete isomonodromy conditions for associated single valued linear problems an infinite hierarchy of integrable difference equations that share the same linear deformation problem as dp was shown to exist by cresswell and joshi in this paper we recall the results shown for dp and show how to deduce a hierarchy for dp each member of the respective hierarchies is shown to be generated by difference recursion operators furthermore we show that continuum limits of these difference hierarchies lead to the p p and p hierarchies finally we construct miura transformations of the dp hierarchy and show that these lead to the hierarchy of the discrete thirty fourth painleve equation

backlund transformations for a discrete second painleve hierarchy we present a new general formulation of our approach to obtaining auto backlund transformations this formulation is much more straightforward and can be applied directly to discrete equations as an example we give a derivation of auto backlund transformations for a discrete second painleve hierarchy to the best of our knowledge this is the first time such results have been given for a discrete hierarchy

a new derivation of painleve hierarchies abstract we give a new derivation of two painleve hierarchies this is done by extending the accelerating wave reductions of the korteweg de vries and dispersive water wave equations to their respective hierarchies we also consider the extension of this reduction of burgers equation to the burgers hierarchy