value distribution of meromorphic solutions of certain difference painleve equations abstract the borel exceptional value and the exponents of convergence of poles zeros and fixed points of finite order transcendental meromorphic solutions for difference painleve i and ii equations are estimated and the forms of rational solutions of the difference painleve ii equation and the autonomous difference painleve i equation are also given it is also proved that the non autonomous difference painleve i equation has no rational solution

discrete painleve equations coalescences limits and degeneracies abstract starting from the standard form of the five discrete painleve equations we show how one can obtain through appropriate limits a host of new equations which are also the discrete analogues of the continuous painleve equations a particularly interesting technique is the one based on the assumption that some simplification takes place in the autonomous form of the mapping following which the deautonomization leads to a new n dependence and introduces more new discrete painleve equations

alternate discrete painleve equations from limits of q pvi and d pv abstract we study the special limits of discrete painleve equations belonging to the q p vi and d p v families when the independent variable goes to 1 or 0 respectively we obtain discrete systems which are shown to be either contiguities of solutions of continuous painleve equations usually of p vi but also of the other painleve equations except p i which has no parameters and so no possibility of contiguity or linearisable mappings

degeneration through coalescence of the q painleve vi equation starting from the q discrete form of the painleve vi equation we obtain its degenerate forms by applying the procedure of coalescence of singularities the whole cascade of degenerate forms is thus obtained leading to new forms for the discrete painleve iv and v equations the lax pairs of these discrete painleve equations are explicitly constructed thus confirming their integrability

discretising systematically the painleve equations we apply a method that we have already proposed for the systematic discretisation of differential systems to the construction of discrete analogues of the painleve equations the important point is that in the case of the painleve equations one needs to preserve the integrability we show that this is indeed possible using our systematic procedure applied to the equations obtained from the hamiltonian description of the painleve equations

painleve equations nonlinear special functions the six painleve equations pi pvi were first discovered about a hundred years ago by painleve and his colleagues in an investigation of nonlinear second order ordinary differential equations recently there has been considerable interest in the painleve equations primarily due to the fact that they arise as reductions of the soliton equations which are solvable by inverse scattering consequently the painleve equations can be regarded as completely integrable equations and possess solutions which can be expressed in terms of solutions of linear integral equations despite being nonlinear equations although first discovered from strictly mathematical considerations the painleve equations have arisen in a variety of important physical applications including statistical mechanics plasma physics nonlinear waves quantum gravity quantum field theory general relativity nonlinear optics and fibre optics the painleve equations may be thought of a nonlinear analogues of the classical special functions they possess hierarchies of rational solutions and one parameter families of solutions expressible in terms of the classical special functions for special values of the parameters further the painleve equations admit symmetries under affine weyl groups which are related to the associated backlund transformations in this paper i discuss some of the remarkable properties which the painleve equations possess including connection formulae backlund transformations associated discrete equations and hierarchies of exact solutions in particular the second painleve equation pii is used to illustrate these properties and some of the applications of pii are also discussed

perturbation theory for the effective diffusion constant in a medium of random scatterers we develop perturbation theory and physically motivated resummations of the perturbation theory for the problem of a tracer particle diffusing in a random medium the random medium contains point scatterers of density uniformly distributed throughout the material the tracer is a langevin particle subjected to the quenched random force generated by the scatterers via our perturbative analysis we determine when the random potential can be approximated by a gaussian random potential we also develop a self similar renormalization group approach based on thinning out the scatterers this scheme is similar to that used with success for diffusion in gaussian random potentials and agrees with known exact results to assess the accuracy of this approximation scheme its predictions are confronted with results obtained by numerical simulation

dynamical transition for a particle in a squared gaussian potential we study the problem of a brownian particle diffusing in finite dimensions in a potential given by 2 2 where is gaussian random field exact results for the diffusion constant in the high temperature phase are given in one and two dimensions and it is shown to vanish in a power law fashion at the dynamical transition temperature our results are confronted with numerical simulations where the gaussian field is constructed in a standard way as a sum over random fourier modes we show that when the number of fourier modes is finite the low temperature diffusion constant becomes non zero and has an arrhenius form thus we have a simple model with a fully understood finite size scaling theory for the dynamical transition in addition we analyse the nature of the anomalous diffusion in the low temperature regime and show that the anomalous exponent agrees with that predicted by a trap model

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

perturbation theory for effective diffusivity in random gradient flows we investigate a result for the effective diffusivity of particles in a random gradient flow previously obtained by an intuitively plausible renormalization group argument and very accurately verified by numerical simulation we show that to two loop order the result is consistent with a direct perturbation theory calculation to the same order in perturbation theory we also derive a ward identity which guarantees the equality of the ratio of the effective diffusivity to the renormalized coupling with the ratio of the corresponding bare values the invariance of this ratio under renormalization was an important feature of the successful renormalization group calculation

tri hamiltonian formulation for certain integrable lattice equations a systematic investigation of integrable differential difference equations with two independent variables admitting multi hamiltonian structure is presented considering the volterra vl toda tl relativistic toda rt belov chaltikian bc and blaszak marciniak both three bm3 and four bm4 coupled lattice equations it is shown that they admit a sequence of operators out of which only three are hamiltonian ones and so they are tri hamiltonian systems only it is observed that the constructed third operator for vl and bc lattice equations is hamiltonian only if the field variable is periodic with even period

continuum derrida approach to drift and diffusivity in random media by means of rather general arguments based on an approach due to derrida that makes use of samples of finite size we analyse the effective diffusivity and drift tensors in certain types of random media in which the motion of the particles is controlled by molecular diffusion and a local flow field with known statistical properties the power of the derrida method is that it uses the equilibrium probability distribution which exists for each finite sample to compute asymptotic behaviour at large times in the infinite medium in certain cases where this equilibrium situation is associated with a vanishing microcurrent our results demonstrate the equality of the renormalization processes for the effective drift and diffusivity tensors this establishes for those cases a ward identity previously verified only to two loop order in perturbation theory in certain models the technique can also be applied to media in which the diffusivity exhibits spatial fluctuations we derive a simple relationship between the effective diffusivity in this case and that for an associated gradient drift problem that provides an interesting constraint on previously conjectured results

finite size universe or perfect squash problem we give a physical notion to all self adjoint extensions of the operator id dx in the finite interval it appears that these extensions realize different nonunitary equivalent representations of ccr and are related to the momentum operator viewed from different inertial systems this leads to the generalization of galilei equivalence principle and gives a new insight into the quantum correspondence rule it is possible to get transformation laws of the wave function under galilei transformation for any scalar potential this generalizes the mass superselection rule there is also given a new and general interpretation of a momentum representation of the wave function it appears that consistent treatment of this problem leads to the time dependent interactions and to the abrupt switching off of the interaction

meromorphic solutions of difference equations integrability and the discrete painleve equations the painlev e property is closely connected to differential equations that are integrable via related iso monodromy problems many apparently integrable discrete analogues of the painlev e equations have appeared in the literature the existence of sufficiently many finite order meromorphic solutions appears to be a good analogue of the painlev e property for discrete equations in which the independent variable is taken to be complex a general introduction to nevanlinna theory is presented together with an overview of recent applications to meromorphic solutions of difference equations and the difference and q difference operators new results are presented concerning equations of the form w z 1 w z 1 r z w where r is rational in w with meromorphic coefficients

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

on the solvability of the discrete second painleve equation the inverse monodromy method for studying the riemann hilbert problem associated with classical painlev e equations is applied to the discrete second painlev e equation

discretising the painleve equations a la hirota mickens we present a systematic method for discretising the painleve equations inspired by the method of hirota while extending it and by that of mickens by specifying it to the case at hand we derive various discrete analogues of painleve i and ii we obtain forms that have been previously derived as well as new ones in particular equations with a geometry described by the affine weyl group e8 1 as a by product we obtain also linearisable equations some of which are new

positivity of the self diffusion matrix of interacting brownian particles with hard core we prove the positivity of the self diffusion matrix of interacting brownian particles with hard core when the dimension of the space is greater than or equal to 2 here the self diffusion matrix is a coefficient matrix of the diffusive limit of a tagged particle we will do this for all activities z 0 of gibbs measures in particular for large z the case of high density particles a typical example of such a particle system is an infinite amount of hard core brownian balls

affine linear and d4 symmetric lattice equations symmetry analysis and reductions we consider lattice equations on which are autonomous affine linear and possess the symmetries of the square some basic properties of equations of this type are derived as well as a sufficient linearization condition and a conservation law a systematic analysis of the lie point and the generalized three and five point symmetries is presented it leads to the generic form of the symmetry generators of all the equations in this class which satisfy a certain non degeneracy condition finally symmetry reductions of certain lattice equations to discrete analogs of the painleve equations are considered

second degree discrete painleve equations conceal first degree ones we examine various second degree difference equations which have been proposed over the years and according to their authors claims should be integrable this study is motivated by the fact that we consider that second degree discrete systems cannot be integrable due to the proliferation of the images and pre images of the initial point we show that in the present cases no contradiction exists in all cases examined we show that there exists an underlying integrable first degree mapping which allows us to obtain an appropriate solution of the second degree one

discrete painleve equations from singularity patterns the asymmetric trihomographic case we derive the discrete painleve equations associated with the affine weyl group e8 1 that can be represented by an in the quispel roberts thompson sense asymmetric trihomographic system the method used in this paper is based on singularity confinement we start by obtaining all possible singularity patterns for a general asymmetric trihomographic system and discard those patterns that cannot lead to confined singularities working with the remaining ones we implement the confinement conditions and derive the corresponding discrete painleve equations which involve two variables by eliminating either of these variables we obtain a symmetric equation examining all these equations of a single variable we find that they coincide exactly with those derived in previous works of ours thereby establishing the completeness of our results

special relations for continuous and discrete painleve equations we investigate a relation between solutions of the painleve ii equation corresponding to values of the parameter of pii which cannot be connected through a schlesinger transformation which was first derived by gambier here we present the discrete analogue of this relation relating the discrete pii to the alternate d pii the latter turns out to be a consequence of a quadratic relation existing between two different families of solutions of pv a q discrete analogue of the latter relation is also presented

bilinear discrete painleve equations based on the results of singularity confinement we derive bilinear expressions for the discrete painleve equations in these cases where a bilinear expression is not sufficient we obtain trilinear or higher multilinear expressions we show that the bilinear approach provides a natural framework for the derivation of backlund and schlesinger transforms for the discrete painleve equations

miura transformations for discrete painleve equations coming from the affine e8 weyl group we derive integrable equations starting from autonomous mappings with a general form inspired by the additive systems associated to the affine weyl group e8 1 by deautonomisation we obtain two hitherto unknown systems one of which turns out to be a linearisable one and we show that both these systems arise from the deautonomisation of a non qrt quispel roberts thompson mapping in order to unambiguously prove the integrability of these nonautonomous systems we introduce a series of miura transformations which allows us to prove that one of these systems is indeed a discrete painleve equation related to the affine weyl group e7 1 and to cast it in canonical form a similar sequence of miura transformations allows us to effectively linearise the second system we obtain an interesting off shoot of our calculations is that the series of miura transformations when applied at the autonomous limit allows one to transform a non qrt invariant into a qrt one

the use of field theoretic methods for the study of flow in a heterogeneous porous medium the effects of heterogeneities on the steady state flow of a single fluid in a porous medium are examined it is argued that incomplete knowledge of the permeability requires the use of a stochastic model of the system it is shown that the problem may be written as a field theory which allows a perturbation series to be expressed by diagrammatic means this allows the calculation of effective permeability the mean pressure and the pressure variance the method as well as recovering familiar results gives a formal means of improving the approximation and approaching more complex systems

similarity reduction generalized symmetries and integrability of belov chaltikian and blaszak marciniak lattice equations the lie point symmetries of belov chaltikian bc and blaszak marciniak bm lattice equations is derived using the symmetries similarity reduction for both bc and bm lattice equations is obtained and show that each of the reduced equation possesses lax representation and satisfies the singularity confinement criteria a discrete version of the painleve property indicating their integrability two particular solutions of both the reduced equations are given explicitly a systematic investigation for nonclassical symmetries of bc and bm lattice equations is carried out further a sequence of conserved densities and generalized symmetries of both bc and bm lattice equation are derived explicitly the existence of a sequence of such symmetries is a predictor for integrability

investigating relations between discrete painleve equations the multistep approach we show how starting from a mapping where the independent variable advances one step at a time one can obtain versions of the mapping corresponding to a multi step evolution the same procedure is applied to discrete painleve equations and we proceed to establish miura relations between the single step and the multi step versions in the present study multi referring to double triple and quintuple these miura relations are discrete painleve equations on their own right we show that while in some cases it is impossible to obtain a multi step equation for a single variable deriving a miura system is still possible we perform our analysis for equations associated with the affine weyl groups e8 1 e7 1 e6 1 and a4 1

the effective conductivity of random suspensions of spherical particles the effective conductivity of an infinite random mono disperse hard sphere suspension is reported for particle to matrix conductivity ratios of 10 and 0 01 for sphere volume fractions c up to 0 6 the conductivities are computed with a method previously described by the authors which includes both far and near field interactions and the particle configurations are generated via a monte carlo method the results are consistent with the previous theoretical work of d j jeffrey to o c 2 and the bounds computed by s torquato and f lado it is also found that the clausius mosotti equation is reasonably accurate for conductivity ratios of 10 or less all the way up to 60 by volume the calculated conductivities compare very well with those of experiments in addition percolation like numerical experiments are performed on periodically replicated cubic lattices of n nearly touching spheres with an infinite particle to matrix conductivity ratio where the conductivity is computed as spheres are removed one by one from the lattice under suitable normalization of the conductivity and volume fraction it is found that the initial volume fraction must be extremely close to maximum packing in order to observe a percolation transition indicating that the near field effects must be very large relative to far field effects these percolation transitions occur at the accepted values for simple sc bodycentred bcc and face centred fcc cubic lattices also the vulnerability of the lattices computed here are exactly those of previous investigators due to limited data above the percolation threshold we could not correlate the conductivity with a power law near the threshold however it can be correlated with a power law for large normalized volume fractions in this case the exponents are found to be 1 70 1 75 and 1 79 for sc bcc and fcc lattices respectively

classical diffusion in strong random media we study classical diffusion of particles in random media although many of our results are general we focus on the case of an ion in a three dimensional medium with random quenched charge centers obeying bulk charge neutrality within a functional integral framework we calculate the effective diffusion coefficients by first order and second order self consistent perturbation theory with a gaussian reference in both cases we also carry out a one loop order momentum space renormalization group calculation the self consistent methods are complicated numerically and fail beyond intermediate disorder strengths in contrast the renormalization group calculation gives an analytical result that appears valid even to high disorder strengths the methodology generally applicable to a quantitative calculation of effective diffusion coefficients in disordered media resolves deficiencies in self consistent perturbation theory approaches to this class of problems

on a transcendental equation related to painleve iii and its discrete forms we examine the canonical forms of painlev equations and argue that the equation for piii in which one parameter is taken to be equal to zero should be considered as a canonical form different from the standard piii our argument is based on the fact that the value of this parameter cannot be modified through auto b cklund transformations we investigate the possible discrete forms of this equation and produce two of them one is of a difference type where the independent variable enters linearly while the second one is of q type where the independent variable enters in a multiplicative way the properties of these discrete equations are also studied

from continuous to discrete painleve equations abstract discrete analogues of the painleve equations have recently been derived using either a method based on orthogonal polynomials or the singularity confinement method here we analyze additional methods for deriving integrable non autonomous discrete equations these methods are based on the discrete akns approach on the use of backlund and schlesinger transforms of the continuous painleve equations and on discrete analogues of miura transformations

new contiguity relation of the sixth painleve equation from a truncation for the master painleve equation p6 u we define a consistent method adapted from the weiss truncation for partial differential equations which allows us to obtain the first degree birational transformation of okamoto two new features are implemented to achieve this result the first one is the homography between the derivative of the solution u and a riccati pseudopotential the second one is an improvement of a conjecture by fokas and ablowitz on the structure of this birational transformation we then build the contiguity relation of p6 which yields one new second order non autonomous discrete equation

perturbation schemes for flow in random media we utilize a many body style formalism to compute the long range effective diffusivity kappa e for dispersal of a passive scalar field in a random irrotational flow with molecular diffusivity by comparing the theoretical results with the outcome of computer simulations we show that a renormalization group approach that includes an appropriate vertex renormalization is superior to low order perturbation theory and to the hartree fock approximation and gives a rather accurate method of computing kappa e for a range of parameter values

discrete painleve equations with maximal periodicities we present the discrete painleve equations the coefficients of which have the maximal periodicity allowed by the geometric description in terms of affine weyl groups for each equation of the degeneration pattern obtained by sakai and which starts at the group we give the corresponding discrete painleve equation some of these systems were already known but here we proceed in a systematic way filling in all the gaps in particular we derive the equations with maximal periodicity associated to the affine weyl group which is the richest system in the sakai classification and which has yet to be studied in detail

casorati determinant solutions for the discrete painleve iii equation the discrete painleve iii equation is investigated based on the bilinear formalism it is shown that it admits the solutions expressed by the casorati determinant whose entries are given by the discrete bessel functions moreover based on the observation that these discrete bessel functions are transformed to the q bessel functions by a simple variable transformation we present a q difference analog of the painleve iii equation

on laplace darboux type sequences of generalized weingarten surfaces two novel classes of integrable surfaces generalized weingarten surfaces of class 1 and 2 which may be defined in a coordinate independent manner or equivalently via an extension of the classical lelieuvre formulas are introduced the classes 1 and 2 generalize bianchi surfaces and surfaces of harmonic inverse mean curvature respectively and turn out to be governed by analogs of the ernst weyl equation descriptive of gravitational and neutrino fields in general relativity the latter connection is exploited to derive laplace darboux type transformations for class 2 via the associated loewner konopelchenko rogers representation the laplace darboux type transformations are shown to have remarkable geometric properties in terms of sphere congruences connections with the pohlmeyer lund regge vortex model and an inhomogeneous heisenberg spin equation are recorded a backlund transformation for a particular painleve iii equation is derived in a purely geometric manner and a geometric interpretation of a

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

effective diffusivity in non isotropic gradient flows we study the effective diffusivity tensor for particles in a random gradient flow that statistically lacks rotational symmetry the effective diffusivity tensor is computed up to two loop order in perturbation theory and the ward identity that relates this tensor to the effective coupling is verified to the same order we re examine the renormalization group calculation that produced a very accurate numerical value for the effective diffusivity in the rotationally symmetric case and formulate two versions based on distinct divisions of the random potential field that gives rise to the flow both types of renormalization group calculation give good results when compared with numerical simulations however at two loop order in perturbation theory the two methods differ in detail from each other and from the exact perturbation calculation this is in contrast to the corresponding results in the isotropic case

on the autonomous limit of discrete painleve equations abstract we consider the autonomous limit of the known families of discrete painleve equations we show how these autonomous systems can be written as special cases of the qrt mapping for the latter we present in detail its integration in both the symmetric and asymmetric cases

continuous symmetries of the lattice potential kdv equation in this paper we present a set of results on the integration and on the symmetries of the lattice potential korteweg de vries lpkdv equation using its associated spectral problem we construct the soliton solutions and the lax technique enables us to provide infinite sequences of generalized symmetries finally using a discrete symmetry of the lpkdv equation we construct a large class of non autonomous generalized symmetries

the third painleve equation and associated special polynomials in this paper we are concerned with rational solutions algebraic solutions and associated special polynomials with these solutions for the third painleve equation piii these rational and algebraic solutions of piii are expressible in terms of special polynomials defined by second order bilinear differential difference equations which are equivalent to toda equations the structure of the roots of these special polynomials is studied and it is shown that these have an intriguing highly symmetric and regular structure using the hamiltonian theory for piii it is shown that these special polynomials satisfy pure difference equations fourth order bilinear differential equations as well as differential difference equations further representations of the associated rational solutions in the form of determinants through schur functions are given

growth and integrability in discrete systems we introduce a new discrete integrability criterion inspired from the recent findings of ablowitz and collaborators this criterion is based on the study of the growth of some characteristic of the solutions of a mapping using nevanlinna theory since the practical implementation of the latter does not always lead to a clear cut answer we complement the growth criterion by the singularity confinement property this combination turns out to be particularly efficient its application allows us to recover the known forms of the discrete painleve equations and to show that no new ones may exist within a given parametrization

from the continuous pv to discrete painleve equations we study the discrete transformations that are associated with the auto backlund of the continuous pv equation we show that several two parameter discrete painleve equations can be obtained as contiguity relations of pv among them we find the asymmetric d pii equation which is a well known form of discrete piii the relation between the ternary pi previously obtained through the discrete dressing approach and pv is also established a new discrete painleve equation is also derived

a discrete garnier type system from symmetry reduction on the lattice a symmetry reduction of the lattice modified boussinesq system is studied the full group of lie point symmetries of the relevant system is retrieved and certain group invariant solutions are considered by using an accessional generalized symmetry it is demonstrated that the symmetry reduction leads to a coupled set of second order nonlinear non autonomous ordinary difference equations involving six free parameters generalizing to higher order some of the known discrete analogues of the painleve vi equation the corresponding isomonodromic deformation problem is constructed through the symmetry reduction as well