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

on homogenization and scaling limit of some gradient perturbations of a massless free field we study the continuum scaling limit of some statistical mechanical models defined by convex hamiltonians which are gradient perturbations of a massless free field by proving a central limit theorem for these models we show that their long distance behavior is identical to a new homogenized continuum massless free field we shall also obtain some new bounds on the 2 point correlation functions of these models

correlation at low temperature ii asymptotics the present paper is a continuation of ref 4 where the truncated two point correlation function for a class of lattice spin systems was proved to have exponential decay at low temperature under a weak coupling assumption in this paper we compute the asymptotics of the correlation function as the temperature goes to zero this paper thus extends ref 3 in two directions the hamiltonian function is allowed to have several local minima other than a unique global minimum and we do not require translation invariance of the hamiltonian function we are in particular able to handle spin systems on a general lattice

a review of the decomposition method in applied mathematics abstract the decomposition method can be an effective procedure for analytical solution of a wide class of dynamical systems without linearization or weak nonlinearity assumptions closure approximations perturbation theory or restrictive assumptions on stochasticitiy

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

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

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

on the correlation for kac like models in the convex case the aim of this paper is to stu the behavior asm tends to of a family of measures exp m x dx m on m where m is a potential on m which is a perturbation in a suitable sense of the harmonic potential j x j 2

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

motion by mean curvature from the ginzburg landau interface model we consider the scalar field t with a reversible stochastic dynamics which is defined by the standard dirichlet form relative to the gibbs measure with formal energy the potential v is even and strictly convex we prove that under a suitable large scale limit the t field becomes deterministic such that locally its normal velocity is proportional to its mean curvature except for some anisotropy effects as an essential input we prove that for every tilt there is a unique shift invariant ergodic gibbs measure for the field

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

the structure of fluctuations in stochastic homogenization four quantities are fundamental in homogenization of elliptic systems in divergence form and in its applications the field and the flux of the solution operator applied to a general deterministic right hand side and the field and the flux of the corrector homogenization is the study of the large scale properties of these objects in case of random coefficients these quantities fluctuate and their fluctuations are a priori unrelated depending on the law of the coefficient field and in particular on the decay of its correlations on large scales these fluctuations may display different scalings and different limiting laws if any in this contribution we identify another crucial intrinsic quantity motivated by h convergence which we refer to as the homogenization commutator and is related to variational quantities first considered by armstrong and smart in the simplified setting of the random conductance model we show what we believe to be a general principle namely that the homogenization commutator drives at leading order the fluctuations of each of the four other quantities in a strong norm in probability which is expressed in form of a suitable two scale expansion and reveals the pathwise structure of fluctuations in stochastic homogenization in addition we show that the rescaled homogenization commutator converges in law to a gaussian white noise and we analyze to which precision the covariance tensor that characterizes the latter can be extracted from the representative volume element method this collection of results constitutes a new theory of fluctuations in stochastic homogenization that holds in any dimension and yields optimal rates extensions to the non symmetric continuum setting are also discussed the details of which are postponed to forthcoming works

grad o perturbations of massless gaussian fields we investigate weak perturbations of the continuum massless gaussian measure by a class of approximately local analytic functionals and use our general results to give a new proof that the pressure of the dilute dipole gas is analytic in the activity

on the spectrum of semi classical witten laplacians and schrodinger operators in large dimension we investigate the low lying spectrum of witten laplacians on forms of arbitrary degree in the semi classical limit and uniformly in the space dimension we show that under suitable assumptions implying that the phase function has a unique local minimum one obtains a number of clusters of discrete eigenvalues at the bottom of the spectrum moreover we are able to count the number of eigenvalues in each cluster we apply our results to certain sequences of schrodinger operators having strictly convex potentials and show that some well known results of semi classical analysis hold also uniformly in the dimension

on green s function in higher order stochastic differential equations abstract in adomian s solution of linear or nonlinear deterministic or stochastic differential equations the decomposition of the linear part into l r can be made so as to simplify green s function and consequently the resulting computations

scaling limit for a class of gradient fields with nonconvex potentials we consider gradient fields x x d whose law takes the gibbs boltzmann form z 1 exp x y v v x where the sum runs over nearest neighbors we assume that the potential v admits the representation v log d exp 1 2 2 where is a positive measure with compact support in 0 hence the potential v is symmetric but nonconvex in general while for strictly convex v s the translation invariant ergodic gradient gibbs measures are completely characterized by their tilt a nonconvex potential as above may lead to several ergodic gradient gibbs measures with zero tilt still every ergodic zero tilt gradient gibbs measure for the potential v above scales to a gaussian free field

a characterisation of the gaussian free field we prove that a random distribution in two dimensions which is conformally invariant and satisfies a natural domain markov property is a multiple of the gaussian free field this result holds subject only to a fourth moment assumption

a strong central limit theorem for a class of random surfaces this paper is concerned with d 2 dimensional lattice field models with action v nabla phi cdot where v mathbf r d rightarrow mathbf r is a uniformly convex function the fluctuations of the variable phi 0 phi x are studied for large x via the generating function given by g x mu ln langle e mu phi 0 phi x rangle a in two dimensions g x mu partial 2g x mu partial mu 2 is proportional to ln vert x vert the main result of this paper is a bound on g x mu partial 3 g x mu partial mu 3 which is uniform in vert x vert for a class of convex v the proof uses integration by parts following helffer sjostrand and witten and relies on estimates of singular integral operators on weighted hilbert spaces

a review of the decomposition method and some recent results for nonlinear equations abstract the decomposition method can be an effective procedure for solution of nonlinear and or stochastic continous time dynamical systems without usual restrictive assumptions this paper is intended as a convenient tutorial review of the method 1

the diffusion limit for reversible jump processes on z d with ergodic random bond conductivities we consider a reversible jump process on d whose jump rates themselves are random we show mean square convergence of this process under diffusion scaling to a limiting brownian motion with a certain diffusion matrix characterizing effective conductivity

large deviations and concentration properties for ϕ interface models we consider the massless field with zero boundary conditions outside d n d d n n d a suitable subset of d i e the continuous spin gibbs measure n on d n with hamiltonian given by h x y x y 1 v x y and x 0 for x d n c the interaction v is taken to be strictly convex and with bounded second derivative this is a standard effective model for a d 1 dimensional interface represents the height of the interface over the base d n due to the choice of scaling of the base we scale the height with the same factor by setting n n we study various concentration and relaxation properties of the family of random surfaces n and of the induced family of gradient fields n n as the discretization step 1 n tends to zero n in particular we prove a large deviation principle for n and show that the corresponding rate function is given by d u x dx where is the surface tension of the model this is a multidimensional version of the sample path large deviation principle we use this result to study the concentration properties of n under the volume constraint i e the constraint that 1 n d x dn n x stays in a neighborhood of a fixed volume v 0 and the hard wall constraint i e n x 0 for all x this is therefore a model for a droplet of volume v lying above a hard wall we prove that under these constraints the field n of rescaled heights concentrates around the solution of a variational problem involving the surface tension as it would be predicted by the phenomenological theory of phase boundaries our principal result however asserts local relaxation properties of the gradient field n n to the corresponding extremal gibbs states thus our approach has little in common with traditional large deviation techniques and is closer in spirit to hydrodynamic limit type of arguments the proofs have both probabilistic and analytic aspects essential analytic tools are p estimates for elliptic equations and the theory of young measures on the side of probability tools a central role is played by the helffer sjostrand 31 pde representation for continuous spin systems which we rewrite in terms of random walk in random environment and by recent results of t funaki and h spohn 25 on the structure of gradient fields

adomian s method of decomposition critical review and examples of failure abstract adomian s method of decomposition is considered in application to initial boundary value problems for the one space dimensional spatially homogeneous heat conduction equation it is shown that the fundamental equation of the method is well defined only for certain restricted types of boundary conditions within the class of such boundary conditions examples are given such that the fundamental equation fails to have a unique solution and such that the sequence produced by iteration of this equation is divergent the latter is a counterexample to a published assertion of convergence

quantification of ergodicity in stochastic homogenization optimal bounds via spectral gap on glauber dynamics we study quantitatively the effective large scale behavior of discrete elliptic equations on the lattice mathbb z d with random coefficients the theory of stochastic homogenization relates the random stationary and ergodic field of coefficients with a deterministic matrix of effective coefficients this is done via the corrector problem which can be viewed as a highly degenerate elliptic equation on the infinite dimensional space of admissible coefficient fields in this contribution we develop new quantitative methods for the corrector problem based on the assumption that ergodicity holds in the quantitative form of a spectral gap estimate w r t a glauber dynamics on coefficient fields as it is the case for independent and identically distributed coefficients as a main result we prove an optimal decay in time of the semigroup associated with the corrector problem i e of the generator of the process called random environment as seen from the particle as a corollary we recover existence of stationary correctors in dimensions d 2 and prove new optimal estimates for regularized versions of the corrector in dimensions d ge 2 we also give a self contained proof of a new estimate on the gradient of the parabolic variable coefficient green s function which is a crucial analytic ingredient in our approach as an application of these results we prove the first and optimal estimates for the approximation of the homogenized coefficients by the popular periodization method in case of independent and identically distributed coefficients

annealed estimates on the green function we consider a random uniformly elliptic coefficient field a x on the d dimensional integer lattice mathbb z d we are interested in the spatial decay of the quenched elliptic green function g a x y next to stationarity we assume that the spatial correlation of the coefficient field decays sufficiently fast to the effect that a logarithmic sobolev inequality holds for the ensemble langle cdot rangle we prove that all stochastic moments of the first and second mixed derivatives of the green function that is langle nabla x g x y p rangle and langle nabla x nabla y g x y p rangle have the same decay rates in x y gg 1 as for the constant coefficient green function respectively this result relies on and substantially extends the one by delmotte and deuschel probab theory relat fields 133 358 390 2005 which optimally controls second moments for the first derivatives and first moments of the second mixed derivatives of g that is langle nabla x g x y 2 rangle and langle nabla x nabla y g x y rangle as an application we are able to obtain optimal estimates on the random part of the homogenization error even for large ellipticity contrast

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

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

on the averaged green s function of an elliptic equation with random coefficients we consider a divergence form elliptic difference operator on the lattice mathbb z d with a coefficient matrix that is an i i d perturbation of the identity matrix recently bourgain introduced novel techniques from harmonic analysis to prove the convergence of the feshbach schur perturbation series related to the averaged green s function of this model our main contribution is a refinement of bourgain s approach which improves the key decay rate from 2d epsilon to 3d epsilon the optimal decay rate is conjectured to be 3d as an application we derive estimates on higher derivatives of the averaged green s function which go beyond the second derivatives considered by delmotte deuschel and related works

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