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

recursion operators and non local symmetries a new formulation of recursion operators is presented which eliminates diffi culties associated with integro differential operators this interpretation treats recursion operators and their inverses on an equal footing efficient techniques for constructing non local symmetries of differential equations result

why nonlocal recursion operators produce local symmetries new results and applications it is well known that integrable hierarchies in 1 1 dimensions are local while the recursion operators that generate these hierarchies usually contain nonlocal terms we resolve this apparent discrepancy by providing simple and universal sufficient conditions for a nonlocal recursion operator in 1 1 dimensions to generate a hierarchy of local symmetries these conditions are satisfied by virtually all recursion operators known today and are much easier to verify than those found in earlier work we also give explicit formulae for the nonlocal parts of higher recursion hamiltonian and symplectic operators of integrable systems in 1 1 dimensions using these two results we prove under some natural assumptions the maltsev novikov conjecture stating that higher hamiltonian symplectic and recursion operators of integrable systems in 1 1 dimensions are weakly nonlocal i e the coefficients of these operators are local and these operators contain at most one integration operator in each term

interacting elliptic inclusions by the method of complex potentials abstract an accurate series solution has been obtained for a piece homogeneous elastic plane containing a finite array of non overlapping elliptic inclusions of arbitrary size aspect ratio location and elastic properties the method combines standard muskhelishvili s representation of general solution in terms of complex potentials with the superposition principle and newly derived re expansion formulae to obtain a complete solution of the many inclusion problem by exact satisfaction of all the interface conditions a primary boundary value problem stated on a complicated heterogeneous domain has been reduced to an ordinary well posed set of linear algebraic equations a properly chosen form of potentials provides a remarkably simple form of solution and thus an efficient computational algorithm the theory developed is rather general and can be applied to solve a variety of composite mechanics problems the advanced models of composite involving up to several hundred inclusions and providing an accurate account for the microstructure statistics and fiber fiber interactions can be considered in this way the numerical examples are given showing high accuracy and numerical efficiency of the method developed and disclosing the way and extent to which the selected structural parameters influence the stress concentration at the matrix inclusion interface

on transformation strains and uniform fields in multiphase elastic media the effect of local eigenstrain and eigenstress fields or transformation fields on the local strains and stresses is explored in multiphase elastic solids of arbitrary geometry and material symmetry the residual local fields caused by such transformation fields are sought in terms of certain transformation influence functions and transformation concentration factor tensors general properties of these functions and concentration factors and their relation to the analogous mechanical influence functions and concentration factors are established in part with the help of uniform strain fields in multiphase media specific estimates of the transformation concentration factor tensors are evaluated by the self consistent and mori tanaka methods it is found here that although the two methods use different constraint tensors in solutions of the respective dilute problems their estimates of the mechanical thermal and transformation concentration factor tensors and of the overall stiffness of multiphase media have a similar structure proofs that guarantee that these methods comply with the general properties of the transformation influence functions and provide diagonally symmetric estimates of the overall elastic stiffness are given for two phase and multiphase systems consisting of or reinforced by inclusions of similar shape and alignment one of the possible applications of the results in analysis of overall instantaneous properties and local fields in inelastic composite materials is described in the following paper

on recursion operators abstract we observe that application of a recursion operator of the burgers equation does not produce the expected symmetries this is explained by the incorrect assumption that d x 1 d x 1 we then proceed to give a method to compute the symmetries using the recursion operator as a first order approximation

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

elastic equilibrium of a medium containing a finite number of aligned spheroidal inclusions abstract the strict solution in series is obtained of the elasticity theory problem for an unbounded domain containing some aligned spheroidal inhomogeneities under uniform far field loads the essence of the method used is the representation of the displacement field in a multiply connected domain as a sum of general solutions for corresponding single connected domains each term of this sum in turn is expanded into series on vectorial partial solutions of lame s equation in a local spheroidal basis in order to satisfy exactly all interfacial boundary conditions the re expansion formulae addition theorems for external partial solutions are used as a result the primary boundary value problem of elasticity theory is reduced to an infinite set of linear algebraic equations the convergence rate of the proposed solution procedure is evaluated numerically some numerical results demonstrating the influence on stress distribution of material properties spatial position of inclusions and external load are presented

the boundary due terms in the green operator of inclusion patterns from distant to contact and to connected situations using radon transforms illustration for spheroid alignments in isotropic media abstract we examine the boundary due components in the mean modified green operator integral green operator for short of an inclusion pattern in distant contact and contact connection transitions the direct rt and inverse irt radon transforms which allow specification of the different contributions to the mean green operator of the pattern in simple geometrical terms are used the already well documented case of axially symmetric alignments of equidistant identical oblate spheroids in an infinite matrix of isotropic elastic like or dielectric like properties is treated up to infinite alignments and for any aspect ratio from unity spheres to infinitesimal platelets simple closed forms for this mean green operator and for its different parts are newly obtained these closed forms allow an easy parametric study of the operator variations in terms of the alignment characteristics from distant to contact situations from contact to connection of the inclusions the changes in the green operator s contributions are pointed what provides relevant operator forms for the connected patterns these results are of interest in problems where phase percolation connectivity inversions or co continuity are implied

the short pulse hierarchy we study a new hierarchy of equations containing the short pulse equation which describes the evolution of very short pulses in nonlinear media and the elastic beam equation which describes nonlinear transverse oscillations of elastic beams under tension we show that the hierarchy of equations is integrable we obtain the two compatible hamiltonian structures we construct an infinite series of both local and nonlocal conserved charges a lax description is presented for both systems for the elastic beam equations we also obtain a nonstandard lax representation

on the nonlocal equations and nonlocal charges associated with the harry dym hierarchy a large class of nonlocal equations and nonlocal charges for the harry dym hierarchy is exhibited they are obtained from nonlocal casimirs associated with its bi hamiltonian structure the lax representation for some of these equations is also given

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

a strange recursion operator demystified we show that a new integrable two component system of kdv type studied by karasu kalkanli et al 2004 acta appl math 83 85 94 is bi hamiltonian and its recursion operator which has a highly unusual structure of nonlocal terms can be written as a ratio of two compatible hamiltonian operators found by us using this we prove that the system in question possesses an infinite hierarchy of local commuting generalized symmetries and conserved quantities in involution and the evolution systems corresponding to these symmetries are bi hamiltonian as well we also show that upon introduction of suitable nonlocal variables the nonlocal terms of the recursion operator under study can be written in the usual form with the integration operator d 1x appearing in each term at most once

interaction of two elliptic inclusions abstract the paper presents the solution of a boundary value problem involving two interacting elliptical inhomogeneities in an infinite elastic body the loading cases considered include a far field biaxial tension and a thermally induced residual field that is modeled by uniform eigenstrains sustained by the inhomogeneities the mathematical formulation is based upon the papkovich neuber displacement approach certain interesting aspects of the solution and the effects of perfectly bonded and slipping interfaces are discussed in some detail

a class of nonautonomous coupled kdv systems a class of nonautonomous coupled korteweg de vries kdv systems in 1 1 dimensions are considered for integrability classification integrability of the systems is associated with the existence of a certain recursion operator some new integrable nonautonomous two component kdv systems are found

partially commutative context free processes expressibility and tractability bisimulation equivalence is decidable in polynomial time for both sequential and commutative normed context free processes known as bpa and bpp respectively despite apparent similarity between the two classes different algorithmic techniques were used in each case we provide one polynomial time algorithm that works in a superclass of both normed bpa and bpp it is derived in the setting of partially commutative context free processes a new process class introduced in the paper it subsumes both bpa and bpp and seems to be of independent interest expressibility issue of the new class in comparison with the normed pa class is also tackled in the paper

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

integrable nonlinear evolution equations with time dependent coefficients a simple and straightforward method for generating completely integrable nonlinear evolution equations with time dependent coefficients is presented herein for the equations under consideration the solutions to those given by vector fields which are independent of time are given thus explicit links between equations are obtained as an application of the proposed method it is shown that the linear superposition with arbitrary time dependent coefficients of different members of an integrable hierarchy is again integrable furthermore it turns out that for some integrable equations like the korteweg de vries kdv the benjamin ono bo or the kadomtsev petviashvili kp the resolvent operator of lower order flows can be explicitly obtained from that of any higher order flow those flows demonstrated for the kdv which can be generated by lie homomorphisms coming from first order problems are completely classified many well known equations which can be found in the literature are of that type as an application of such a first order link a direct link is given from kdv to the cylindrical kdv and from there to the kp with nontrivial dependence on the second spatial variable

symmetries and integrability of the cylindrical korteweg de vries equation the direct scheme to test integrability of a given nonlinear equation proposed by chen lee and liu is tested on the cylindrical korteweg de vries equation the explicit dependence on t of this equation does not present any real difficulties constants of motion and symmetries are found readily and the lax operators for the scattering problem constructed accordingly

semi analytic solution for multiple interacting three dimensional inhomogeneous inclusions of arbitrary shape in an infinite space this paper develops a semi analytic solution for multiple arbitrarily shaped three dimensional inhomogeneous inclusions embedded in an infinite isotropic matrix under external load all interactions between the inhomogeneous inclusions are taken into account in this solution the inhomogeneous inclusions are discretized into small cuboidal elements each of which is treated as a cuboidal inclusion with initial eigenstrain plus unknown equivalent eigenstrain according to the equivalent inclusion method all the unknown equivalent eigenstrains are determined by solving a set of simultaneous constitutive equations established for each equivalent cuboidal inclusion the final solution is obtained by summing up the closed form solutions for each individual equivalent cuboidal inclusion in an infinite space the solution evaluation is performed by application of the fast fourier transform algorithm which greatly increases the computational efficiency finally the solution is validated by taking eshelby s analytic solution of an ellipsoidal inhomogeneous inclusion as a benchmark and by the finite element analysis a few sample results are also given to demonstrate the generality of the solution the solution may have potentially significant applications in solving a wide range of inhomogeneity related problems copyright 2011 john wiley sons ltd

effect of crack orientation statistics on effective stiffness of mircocracked solid this paper addresses the problem of calculating effective elastic properties of a solid containing multiple cracks with prescribed orientation statistics to do so the representative unit cell approach has been used the microgeometry of a cracked solid is modeled by a periodic structure with a unit cell containing multiple cracks a sufficient number is taken to account for the microstructure statistics the developed method combines the superposition principle the technique of complex potentials and certain new results in the theory of special functions a proper choice of potentials provides reducing the boundary value problem to an ordinary well posed set of linear algebraic equations the exact finite form expression of the effective stiffness tensor has been obtained by analytical averaging the strain and stress fields the convergence study has been performed the statistically meaningful results obtained show dependence of the effective elastic stiffness on angular scattering of cracks comparison has been made with the selected simple micromechanical models namely non interaction approximation differential scheme and modified differential scheme it is found that among these models the differential scheme provides the best fit of the numerical data

on the equivalence of linearization and formal symmetries as integrability tests for evolution equations gurses integrability test 1991 consists of the compatibility of the linearized equation with an eigenvalue equation and leads to the recursion operator this test is applied to quasilinear fifth order equations and the same classification as the formal symmetry method of mikhailov et al 1990 is obtained the same classification for polynomial equations is obtained using fokas test 1987 i e the existence of one higher order symmetry it is shown that the recursion operators of a specific form can be constructed using symmetries and conserved covariants and the recursion operators for polynomial equations are obtained with this method

symmetries and a hierarchy of the general kdv equation two groups old and new of symmetries and their lie algebra properties for the kdv and cylindrical kdv equations are unified and extended to the general kdv equation