the 2 1 dimensional sine gordon equation integrability and localized solutions in this paper the 2 1 dimensional sine gordon equation 2dsg introduced by konopelchenko and rogers is investigated and is shown to satisfy the painleve property a variable coefficient hirota bilinear form is constructed by judiciously using the painleve analysis with a non conventional choice of the vacuum solutions first the line kinks are constructed then exact localized coherent structures in the 2dsgi equation are generated by the collision of two non parallel ghost solitons which drive the two non trivial boundaries present in the system also the reason for the difficulty in identifying localized solutions in the 2dsgii equation is indicated we also highlight the significance of the asymptotic values of the boundaries of the system

fft based homogenization for microstructures discretized by linear hexahedral elements summary the fft based homogenization method of moulinec suquet has recently emerged as a powerful tool for computing the macroscopic response of complex microstructures for elastic and inelastic problems in this work we generalize the method to problems discretized by trilinear hexahedral elements on cartesian grids and physically nonlinear elasticity problems we present an implementation of the basic scheme that reduces the memory requirements by a factor of four and of the conjugate gradient scheme that reduces the storage necessary by a factor of nine compared with a naive implementation for benchmark problems in linear elasticity the solver exhibits mesh and contrast independent convergence behavior and enables the computational homogenization of complex structures for instance arising from computed tomography computed tomography ct imaging techniques there exist 3d microstructures involving pores and defects for which the original fft based homogenization scheme does not converge in contrast for the proposed scheme convergence is ensured also the solution fields are devoid of the spurious oscillations and checkerboarding artifacts associated to conventional schemes we demonstrate the power of the approach by computing the elasto plastic response of a long fiber reinforced thermoplastic material with 172 106 displacement degrees of freedom copyright 2016 john wiley sons ltd

a note on the exp function method for traveling waves of nonlinear evolution equations abstract in this note we analyze the paper of noor et al 2010 1 using the exp function method noor et al found the generalized solitary and periodic solutions of zakharov kuznetsov and zakharov kuznetsov modified equal width equations we have checked noor s solutions and proved that seven from ten of them does not satisfy to equations considered the general solution of the zakharov kuznetsov equation in the traveling waves was obtained along ago we give this solutions in this letter for the reference source

a new note on exact complex travelling wave solutions for 2 1 dimensional b type kadomtsev petviashvili equation abstract exact solutions of the 2 1 dimensional kadomtsev petviashvili by zhang huiqun zhang a note on exact complex travelling wave solutions for 2 1 dimensional b type kadomtsev petviashvili equation appl math comput 216 2010 2771 2777 are considered to look for new types of exact solutions travelling wave solutions of equation zhang has used the g g expansion method we demonstrate that there is the general solution for the reduction by zhang from the 2 1 dimensional kadomtsev petviashvili equation and all solutions by zhang are found as partial cases from the general solution

the non local delta problem and 2 1 dimensional soliton equations a method of constructing 2 1 dimensional non linear integrable equations and their solutions by means of the non local delta problem is developed a basic set of equations is obtained by using different normalisations of the non local delta problem and the lagrangian of the set is found other integrable equations which are degenerate cases of the basic set are also lagrangian

reconstructing displacements from the solution to the periodic lippmann schwinger equation discretized on a uniform grid uniform grid solvers of the periodic lippmann schwinger equation have been introduced by moulinec and suquet for the numerical homogenization of heterogeneous materials based on the fast fourier transform these methods use the strain as main unknown and usually do not produce displacement fields while this is generally not perceived as a restriction for homogenization purposes some tasks might require kinematically admissible displacement fields in this paper we show how the numerical solution to the periodic lippmann schwinger equation can be post processed to reconstruct a displacement field our procedure is general it applies to any variant of the moulinec suquet solver the reconstruction is formulated as an auxiliary elastic equilibrium problem of a homogeneous material which is solved with displacement based finite elements taking advantage of the periodicity the uniformity of the grid and the homogeneity of the material the resulting linear system is formulated and solved efficiently in fourier space the cost of our procedure is lower than that of one iteration of the lippmann schwinger solver an application of this post processing procedure is proposed in which the reconstructed displacement field is used to compute a rigorous upper bound on the effective shear modulus of some model microstructure this is the pre peer reviewed version of the following article reconstructing displacements from the solution to the periodic lippmann schwinger equation discretized on a uniform grid which has been published in final form at 10 1002 nme 5263 this article may be used for non commercial purposes in accordance with wiley terms and conditions for self archiving sections 1 introduction and 5 numerical results have been significantly reworked in the published final version of this article the remainder of this article including sections 3 and 4 where the method itself is presented is essentially unchanged

a model order reduction method for computational homogenization at finite strains on regular grids using hyperelastic laminates to approximate interfaces abstract fft based homogenization methods operate on regular voxel grids and cannot resolve interfaces exactly in general in this article we study hyperelastic laminates and associate their effective properties to voxels containing interfaces extending previous ideas successfully applied in the framework of linear elasticity more precisely we show that for strictly rank one convex energy densities of the constituents there is precisely one phase wise affine minimizer of the laminate problem a result of independent interest we demonstrate that furnishing interface voxels with appropriately rotated effective hyperelastic properties of a two phase laminate significantly enhances both the local solution quality and the accuracy of the computed effective elastic properties with only a small computational overhead compared to using classical fft based homogenization in particular for equal accuracy problems with a lower number of degrees of freedom suffice

filtering material properties to improve fft based methods for numerical homogenization fft based solvers introduced in the 1990s for the numerical homogenization of heterogeneous elastic materials have been extended to a wide range of physical properties in parallel alternative algorithms and modified discrete green operators have been proposed to accelerate the method and or improve the description of the local fields in this short note filtering material properties is proposed as a third complementary way to improve fft based methods it is evidenced from numerical experiments that the grid refinement and consequently the computation time and or the spurious oscillations observed on local fields can be significantly reduced in addition if the two filters based on voigt or reuss homogenization rules can improve or deteriorate the method depending on the microstructure a stiff inclusion within a compliant matrix or the reverse the proposed 2 layer filter is efficient in both situations the study is proposed in the context of linear elasticity but similar results are expected in a different physical context thermal electrical

a note on the g g expansion method abstract we demonstrate that the g g expansion method which is often used in finding exact solutions of nonlinear differential equation is equivalent to the well known tanh method and application of these methods gives the same exact solutions of nonlinear differential equations

combining galerkin approximation techniques with the principle of hashin and shtrikman to derive a new fft based numerical method for the homogenization of composites we report on the mathematical analysis of two different fft based numerical schemes for the homogenization of composite media within the framework of linear elasticity the basic scheme of moulinec and suquet 1994 1998 9 and 10 and the energy based scheme of brisard and dormieux 2010 13 casting these two schemes as galerkin approximations of the same variational problem allows us to assert their well posedness and convergence more importantly we extend in this work their domains of application by relieving some stringent conditions on the reference material which were previously thought necessary the origins of the flaws of each scheme are identified and a third scheme is proposed which seems to combine the strengths of the basic and energy based schemes while leaving out their weaknesses finally a rule is proposed for handling heterogeneous pixels voxels a situation frequently met when images of real materials are used as input to these schemes

a note on travelling wave solutions to lax s seventh order kdv equation ganji and abdollahzadeh d d ganji m abdollahzadeh appl math comput 206 2008 438 444 derived three supposedly new travelling wave solutions to lax s seventh order kdv equation each solution was obtained by a different method it is shown that any two of the solutions may be obtained trivially from the remaining solution furthermore it is noted that one of the solutions has been known for many years

exact soliton solutions of the one dimensional complex swift hohenberg equation abstract using painleve analysis the hirota multi linear method and a direct ansatz technique we study analytic solutions of the 1 1 dimensional complex cubic and quintic swift hohenberg equations we consider both standard and generalized versions of these equations we have found that a number of exact solutions exist to each of these equations provided that the coefficients are constrained by certain relations the set of solutions include particular types of solitary wave solutions hole dark soliton solutions and periodic solutions in terms of elliptic jacobi functions and the weierstrass function although these solutions represent only a small subset of the large variety of possible solutions admitted by the complex cubic and quintic swift hohenberg equations those presented here are the first examples of exact analytic solutions found thus far

new kink shaped solutions and periodic wave solutions for the 2 1 dimensional sine gordon equation new exact solutions including the kink shaped solutions bell shaped solutions periodic solutions singular soliton solutions and rational solution for the 2 1 dimensional sine gordon equation are obtained using the tanh method the g g expansion method and the auxiliary function method respectively

symmetry group and non propagating solitons in the 2 1 dimensional sine gordon equation starting from the compact version of the 2 1 dimensional sine gordon equation the symmetry group and then the lie symmetries and the related algebra can be obtained via a simple combination of a gauge transformation of the spectral function and the transformations of the space time variables thereout applying the group transformation theorem on a two straight line soliton solution one can derive two types of meaningful non propagating dromion lattice excitations

complex manifolds in q convex boundaries we consider a c infty boundary b omega subset mathbb c n which is q convex in the sense that its levi form has positive trace on every complex q plane we prove that b omega is tangent of infinite order to the complexification of each of its submanifolds which is complex tangential and of finite bracket type this generalizes diederich and fornaess ann math 107 371 384 1978 from pseudoconvex to q convex domains we also readily prove that the rows of the levi form are frac 1 2 subelliptic multipliers for the bar partial neumann problem on q forms cf ho in math ann 290 3 18 1991 this allows to run the kohn algorithm of acta math 142 79 122 1979 in the chain of ideals of subelliptic multipliers for q forms if b omega is real analytic and the algorithm gets stuck on q forms then it produces a variety of holomorphic dimension q and in fact by our result above a complex q manifold which is not only tangent but indeed contained in b omega altogether the absence of complex q manifolds in b omega produces a subelliptic estimate on q forms

a unified approach to infinitesimal loewner and geroch transformations and the ernst and einstein maxwell equations it is demonstrated that the dual ernst equation of general relativity constitutes a stationary loewner system in an analogous manner it is shown that the einstein maxwell equations for stationary axisymmetric space times and their extension to einstein s equations coupled with an arbitrary number of u 1 gauge fields may be interpreted as generalized loewner systems moreover it is recorded that the base geroch transformation for the dual ernst equation may be located in a class of infinitesimal backlund transformations introduced by loewner in 1952 a geroch type transformation for a generic class of 2 1 dimensional loewner systems is set down and it is shown how the base geroch and hoenselaers kinnersley xanthopoulos transformations are naturally retrieved

resolution locale du bar partial avec regularite gevrey au bord d un domainer convexeavec regularite gevrey au bord d un domainer convexe

exact solitary wave solutions of nonlinear evolution and wave equations using a direct algebraic method the authors present a systematic and formal approach toward finding solitary wave solutions of nonlinear evolution and wave equations from the real exponential solutions of the underlying linear equations the physical concept is one of the mixing of these elementary solutions through the nonlinearities in the system the emphasis is however on the mathematical aspects i e the formal procedure necessary to find single solitary wave solutions by means of examples it is shown how various cases of pulse type and kink type solutions are to be obtained by this method an exhaustive list of equations so treated is presented in tabular form together with the particular intermediate relations necessary for deriving their solutions the extension of the technique to construct n soliton solutions and indicate connections with other existing methods is outlined

letter to the editor comment on new types of exact solutions for nonlinear schrodinger equation with cubic nonlinearity in this comment we analyze the paper abdelhalim ebaid s m khaled new types of exact solutions for nonlinear schrodinger equation with cubic nonlinearity j comput appl math 235 2011 1984 1992 using the traveling wave ebaid and khaled have found new types of exact solutions for nonlinear schrodinger equation with cubic nonlinearity we demonstrate that the authors studied the well known nonlinear ordinary differential equation with the well known general solution we illustrate that ebaid and khaled have looked for some exact solution for the reduction of the nonlinear schrodinger equation taking the general solution of the same equation into account

short note accelerating a fft based solver for numerical homogenization of periodic media by conjugate gradients in this short note we present a new technique to accelerate the convergence of a fft based solver for numerical homogenization of complex periodic media proposed by moulinec and suquet 1 the approach proceeds from discretization of the governing integral equation by the trigonometric collocation method due to vainikko 2 to give a linear system which can be efficiently solved by conjugate gradient methods computational experiments confirm robustness of the algorithm with respect to its internal parameters and demonstrate significant increase of the convergence rate for problems with high contrast coefficients at a low overhead per iteration

on generalized loewner systems novel integrable equations in 2 1 dimensions a reinterpretation and generalization of a class of infinitesimal backlund transformations originally introduced in a gas dynamics context by loewner in 1952 leads to a linear representation for a novel class of 2 1 dimensional nonlinear equations the latter may be parametrized in terms of a triad of eigenfunctions moreover like the well known integrable davey stewartson and nizhnik novikov veselov equations the nonlinear systems typically contain the two spatial variables on an equal footing the dressing method is outlined for these generalized loewner systems it is noted that basic reductions lead to 2 1 dimensional versions of the principal chiral fields model toda lattice system and notably the classical sine gordon equation loewner systems on grassmannian the projective cpn and rpn manifolds are considered in particular the 2 1 dimensional integrable sine gordon system in which x and y occur in a symmetric manner arises naturally in this context a gauge equivalent system likewise eme

the diederich fornaess index and the global regularity of the bar partial neumann problem we describe along the guidelines of kohn quantitative estimates for global regularity analysis and geometry in several complex variables pp 97 128 trend math birkhauser boston 1999 the constant mathcal e s which is needed to control the commutator of a totally real vector field t mathcal e with bar partial in order to have h s a priori estimates for the bergman projection b k k ge q 1 on a smooth q convex domain d subset subset mathbb c n this statement not explicit in kohn quantitative estimates for global regularity analysis and geometry in several complex variables pp 97 128 trend math birkhauser boston 1999 yields regularity of b k in specific sobolev degree s next we refine the pseudodifferential calculus at the boundary in order to relate for a defining function r of d the operators t frac delta 2 and r frac delta 2 we are thus able to extend to general degree k ge 0 of b k the conclusion of quantitative estimates for global regularity analysis and geometry in several complex variables pp 97 128 trend math birkhauser boston 1999 which only holds for q 1 and k 0 if for the diederich fornaess index delta of d we have 1 delta frac 1 2 le mathcal e s then b k is h s regular

field dislocation mechanics for heterogeneous elastic materials a numerical spectral approach spectral methods using fast fourier transform fft algorithms have recently seen a surge in interest in the mechanics of materials community the present contribution addresses the critical question of determining accurate local mechanical fields using fft methods without artificial fluctuations arising from materials and defects induced discontinuities precisely the present work introduces a numerical approach based on intrinsic discrete fourier transforms for the simultaneous treatment of material discontinuities arising from the presence of dislocations and from elastic stiffness heterogeneities to this end the elasto static equations of the field dislocation mechanics theory for periodic heterogeneous materials are numerically solved with fft in the case of dislocations in proximity of inclusions of varying stiffness an optimal intrinsic discrete fourier transform method is sought based on two distinct schemes a centered finite difference scheme for differential rules are used for numerically solving the poisson type equation in the fourier space while centered finite differences on a rotated grid is chosen for the computation of the modified fourier green s operator associated with the lippmann schwinger type equation by comparing different methods with analytical solutions for an edge dislocation in a composite material it is found that the present spectral method is accurate devoid of any numerical oscillation and efficient even for an infinite phase elastic contrast like a hole embedded in a matrix containing a dislocation the present fft method is then used to simulate physical cases such as the elastic fields of dislocation dipoles located near the matrix inclusion interface in a 2d composite material and the ones due to dislocation loop distributions surrounding cubic inclusions in 3d composite material in these configurations the spectral method allows investigating accurately the elastic interactions and image stresses due to dislocation fields in the presence of elastic inhomogeneities

a computational scheme for linear and non linear composites with arbitrary phase contrast a numerical method making use of fast fourier transforms has been proposed in moulinec and suquet 1994 1998 to investigate the effective properties of linear and non linear composites this method is based on an iterative scheme the rate of convergence of which is proportional to the contrast between the phases composites with high contrast typically above 104 or infinite contrast those containing voids or rigid inclusions or highly non linear materials cannot be handled by the method this paper presents two modified schemes the first one is an accelerated scheme for composites with high contrast which extends to elasticity a scheme initially proposed in eyre and milton 1999 its rate of convergence varies as the square root of the contrast the second scheme adequate for composites with infinite contrast is based on an augmented lagrangian method the resulting saddle point problem involves three steps the first step consists of solving a linear elastic problem using the fast fourier transform method in the second step a non linear problem is solved at each individual point in the volume element the third step consists of updating the lagrange multiplier applications of this scheme to rigidly reinforced and to voided composites are shown copyright 2001 john wiley sons ltd

ermakov systems of arbitrary order and dimension structure and linearization ermakov systems of arbitrary order and dimension are constructed these inherit an underlying linear structure based on that recently established for the classical ermakov system as an application alignment of a 2 1 dimensional ermakov and integrable ernst system is shown to produce a novel integrable hybrid of a 2 1 dimensional sinh gordon system and of a conventional ermakov system

a comparative study on low memory iterative solvers for fft based homogenization of periodic media in this paper we assess the performance of four iterative algorithms for solving non symmetric rank deficient linear systems arising in the fft based homogenization of heterogeneous materials defined by digital images our framework is based on the fourier galerkin method with exact and approximate integrations that has recently been shown to generalize the lippmann schwinger setting of the original work by moulinec and suquet from 1994 it follows from this variational format that the ensuing system of linear equations can be solved by general purpose iterative algorithms for symmetric positive definite systems such as the richardson the conjugate gradient and the chebyshev algorithms that are compared here to the eyre milton scheme the most efficient specialized method currently available our numerical experiments carried out for two dimensional elliptic problems reveal that the conjugate gradient algorithm is the most efficient option while the eyre milton method performs comparably to the chebyshev semi iteration the richardson algorithm equivalent to the still widely used original moulinec suquet solver exhibits the slowest convergence besides this we hope that our study highlights the potential of the well established techniques of numerical linear algebra to further increase the efficiency of fft based homogenization methods

guaranteed upper lower bounds on homogenized properties by fft based galerkin method abstract guaranteed upper lower bounds on homogenized coefficients arising from the periodic cell problem are calculated in a scalar elliptic setting our approach builds on the recent variational reformulation of the moulinec suquet 1994 fast fourier transform fft homogenization scheme by vond ejc et al 2014 which is based on the conforming galerkin approximation with trigonometric polynomials upper lower bounds are obtained by adjusting the primal dual finite element framework developed independently by dvo ak 1993 and wi ckowski 1995 to the fft based galerkin setting we show that the discretization procedure differs for odd and non odd number of grid points thanks to the helmholtz decomposition inherited from the continuous formulation the duality structure is fully preserved for the odd discretizations in the latter case a more complex primal dual structure is observed due to presence of the trigonometric polynomials associated with the nyquist frequencies these theoretical findings are confirmed with numerical examples to conclude the main advantage of the fft based approach over conventional finite element schemes is that the primal and the dual problems are treated on the same basis and this property can be extended beyond the scalar elliptic setting

on the geometry of an integrable 2 1 dimensional sine gordon system it is recorded that darboux9s method of linking the classical lame system governing triply orthogonal systems of surfaces with an integrable 2 1 dimensional sine gordon equation may be extended and applied to the integrable two component generalization of the latter introduced by konopelchenko and rogers thus in a reinterpretation this 2 1 dimensional sine gordon system is shown to define particular integrable motions of surfaces

determination of the size of the representative volume element for random composites statistical and numerical approach the representative volume element rve plays a central role in the mechanics and physics of random heterogeneous materials with a view to predicting their effective properties a quantitative definition of its size is proposed in this work a rve size can be associated with a given precision of the estimation of the wanted overall property and the number of realizations of a given volume v of microstructure that one is able to consider it is shown to depend on the investigated morphological or physical property the contrast in the properties of the constituents and their volume fractions the methodology is applied to a specific random microstructure namely a two phase three dimensional voronoi mosaic finite element simulations of volumes of different sizes are performed in the case of linear elasticity and thermal conductivity the volumes are subjected to homogeneous strain stress or periodic boundary conditions the effective properties can be determined for large volumes and a small number of realizations conversely smaller volumes can be used providing that a sufficient number of realizations are considered a bias in the estimation of the effective properties is observed for too small volumes for all types of boundary conditions the variance of computed apparent properties for each volume size is used to define the precision of the estimation the key notion of integral range is introduced to relate this error estimation and the definition of the rve size for given wanted precision and number of realizations one is able to provide a minimal volume size for the computation of effective properties the results can also be used to predict the minimal number of realizations that must be considered for a given volume size in order to estimate the effective property for a given precision the rve sizes found for elastic and thermal properties but also for a geometrical property like volume fraction are compared

l2 estimates on p convex riemannian manifolds abstract in this paper we establish various l 2 estimates for the exterior differential operator on p convex riemannian manifolds in the sense of harvey and lawson as applications we establish a carleman type estimate which is uniform with respect to both weight functions and domains and we also obtain topological restrictions for a riemannian manifold to be p convex

combining fft methods and standard variational principles to compute bounds and estimates for the properties of elastic composites abstract in this paper we develop a computational approach based on variational principles combined with fft technique to determine the effective elastic properties of composite materials the unit cell problem of the composite is recast in a weak and integral form by making use of classic variational approaches based on the strain and the stress potential and the hashin shtrikman variational principles the problem is discretized with fourier series and the stationary point is computed numerically by means of an iterative scheme these algorithms use a representation of the local elastic tensor on the double grid twice the size of the discretization which is introduced by the integral formulation it advantageously accounts for the exact geometry of the microstructure and allows the derivation of rigorous bounds of the effective elasticity coefficients in the second part of the paper we establish a hierarchy between the different fft solutions the strain the stress based solutions coming from the classic variational formulations and the polarization based solution derived from the hashin shtrikman principle which depends on the choice of the elastic moduli of the reference material it is proved that the strain and the stress based solutions deliver optimal bounds since they are always better than those obtained from the hashin shtrikman variational principle when the same discretization is used alternatively when the elastic moduli of the reference material is negative the polarization based solution provides an estimate of the effective properties which is comprised between the strain based upper bound and the stress based lower bound

observations on the tanh coth expansion method for finding solutions to nonlinear evolution equations abstract the tanh coth expansion method for finding solitary travelling wave solutions to nonlinear evolution equations has been used extensively in the literature it is a natural extension to the basic tanh function expansion method which was developed in the 1990s it usually delivers three types of solution namely a tanh function expansion a coth function expansion and a tanh coth expansion it is known that for every tanh function expansion solution there is a corresponding coth function expansion solution it is shown that there is a tanh coth expansion solution that is merely a disguised version of the coth solution in many papers such tanh coth solutions are erroneously claimed to be new however other tanh coth solutions may be delivered that are genuinely new in the sense that they would not be delivered via the basic tanh function method similar remarks apply to tan cot and tan cot expansion solutions

a numerical spectral approach for solving elasto static field dislocation and g disclination mechanics a spectral approach is developed to solve the elasto static equations of field dislocation and g disclination mechanics in periodic media given the spatial distribution of nye s dislocation density and or g disclination density tensors in heterogeneous or homogenous linear elastic media the incompatible and compatible elastic distortions are respectively obtained from the solutions of poisson and navier type equations in the fourier space intrinsic discrete fourier transforms solved by the fast fourier transform fft method which are consistent with the pixel grid for the calculation of first and second order spatial derivatives are preferred and compared to the classical discrete approximation of continuous fourier transforms when deriving elastic fields of defects numerical examples are provided for homogeneous linear elastic isotropic solids for various defects a regularized defect density in the core is considered and smooth elastic fields without gibbs oscillations are obtained when using intrinsic discrete fourier transforms the results include the elastic fields of single screw and edge dislocations standard wedge disclinations and associated dipoles as well as twinning g disclinations in order to validate the present spectral approach comparisons are made with analytical solutions using the riemann graves integral operator and with numerical simulations using the finite element approximation

four phase checkerboard composites two dimensional periodic rectangular checkerboard media are considered in the situation where mean fluxes are prescribed across the structure the closed form solution is obtained in the general case where the checkerboard is constructed using four rectangular cells each having a different constant resistivity this four cell structure then repeats doubly periodically to cover the whole plane this general solution is then used to calculate the effective properties thus this four phase checkerboard encapsulates many limiting and special cases as a starting point we develop a concise closed form solution to a basic problem involving four joined quarter planes each of a different resistivity subsequent manipulations yield solutions to problems posed in increasingly convoluted domains while retaining the essentially simple structure found for joined quarter planes

extended multilinear variable separation approach and multivalued localized excitations for some 2 1 dimensional integrable systems the multilinear variable separation approach and the related universal formula have been applied to many 2 1 dimensional nonlinear systems starting from the universal formula abundant 2 1 dimensional localized excitations have been found in this paper the universal formula is extended in two different ways one is obtained for the modified nizhnik novikov veselov equation such that two universal terms can be combined linearly and this type of extension is also valid for the 2 1 dimensional symmetric sine gordon system the other is for the dispersive long wave equation the broer kaup kupershmidt system the higher order broer kaup kupershmidt system and the burgers system where arbitrary number of variable separated functions can be involved because of the existence of the arbitrary functions in both the original universal formula and its extended forms the multivalued functions can be used to construct a new type of localized excitations folded solitary waves fsws and foldons the fsws

fft based homogenisation accelerated by low rank tensor approximations abstract fast fourier transform fft based methods have turned out to be an effective computational approach for numerical homogenisation in particular fourier galerkin methods are computational methods for partial differential equations that are discretised with trigonometric polynomials their computational effectiveness benefits from efficient fft based algorithms as well as a favourable condition number here these kinds of methods are accelerated by low rank tensor approximation techniques for a solution field using canonical polyadic tucker and tensor train formats this reduced order model also allows to efficiently compute suboptimal global basis functions without solving the full problem it significantly reduces computational and memory requirements for problems with a material coefficient field that admits a moderate rank approximation the advantages of this approach against those using full material tensors are demonstrated using numerical examples for the model homogenisation problem that consists of a scalar linear elliptic variational problem defined in two and three dimensional settings with continuous and discontinuous heterogeneous material coefficients this approach opens up the potential of an efficient reduced order modelling of large scale engineering problems with heterogeneous material

the double f expansion approach and novel nonlinear wave solutions of soliton equation in 2 1 dimension abstract the aim of this paper is to construct the double nonlinear wave solutions of the 2 1 dimensional asymmetric version of the nizhnik novikov veselov annv equation by means of double f expansion approach as a result families of double jacobi elliptic function like solutions and double solitary wave like solutions of the 2 1 dimensional annv equation are successfully constructed this method is very valid to some other 2 1 dimensional soliton equations

a full field image conversion method for the inverse conductivity problem with internal measurements this article investigates a fourier based algorithm for computing heterogeneous material parameter distributions from internal measurements of physical fields within the framework of the periodic scalar conductivity model a pair of dual lippmann schwinger integral equations is derived for the sought constitutive parameters based on full intensity or current density field measurements a numerical method based on the fast fourier transform and fixed point iterations is proposed convergence stability and approximation quality of the method are analysed for materials with small contrast a first order born like approximation is also obtained overall the proposed reconstruction approach enables a direct conversion of full field measurement images possibly noisy into maps of material conductivity a set of numerical results is presented to illustrate the performance of the method

energy based comparison between the fourier galerkin method and the finite element method abstract the fourier galerkin method in short ffth has gained popularity in numerical homogenisation because it can treat problems with a huge number of degrees of freedom because the method incorporates the fast fourier transform fft in the linear solver it is believed to provide an improvement in computational and memory requirements compared to the conventional finite element method fem here we systematically compare these two methods using the energetic norm of local fields which has the clear physical interpretation as being the error in the homogenised properties this enables the comparison of memory and computational requirements at the same level of approximation accuracy we show that the methods effectiveness relies on the smoothness regularity of the solution and thus on the material coefficients thanks to its approximation properties fem outperforms ffth for problems with jumps in material coefficients while ambivalent results are observed for the case that the material coefficients vary continuously in space ffth profits from a good conditioning of the linear system independent of the number of degrees of freedom but generally needs more degrees of freedom to reach the same approximation accuracy more studies are needed for other fft based schemes non linear problems and dual problems which require special treatment in fem but not in ffth

a note on a capillarity model and the nonlinear schrodinger equation a system of two equations governing the irrotational flow of a capillary fluid is shown for a particular class of free energy functions to reduce to a nonlinear schrodinger equation

fourier based schemes with modified green operator for computing the electrical response of heterogeneous media with accurate local fields a modified green operator is proposed as an improvement of fou rier based numerical schemes commonly used for computing the electrical or thermal response of heterogeneous media contrary to other methods the number of iterations necessary to achieve convergence tends to a finite value when the contrast of properties between the phases becomes infinite furtherm ore it is shown that the method produces much more accurate local field s inside highly conducting and quasi insulating phases as well as in the vicinity of the phases interfaces these good propert ies stem from the discretization of green s function which is consistent with the pixel grid while retaining the local nature of the op erator that acts on the polarization field finally a fast im plementation of the direct scheme of moulinec et al 1994 that allows for parcimonious memory use is proposed

extended simplest equation method for nonlinear differential equations abstract the modified simplest equation method to look for the exact solutions of the nonlinear differential equations is presented our approach is applied to search for the exact solutions of the sharma tasso olver and the burgers huxley equations the new exact solutions of these equations are obtained

localized excitations of the 2 1 dimensional sine gordon system the 2 1 dimensional sine gordon system is investigated via a generalized bilinear operator representation plateau basin bowl and saddle type ring solitons are thereby constructed it is indicated how iterated moutard transformations may be employed to extend the range of the method

virasoro structure and localized excitations of the lkr system a symmetry analysis is conducted for a master 2 1 dimensional soliton system the classical symmetries are shown to constitute an infinite dimensional kac moody virasoro algebra finite symmetry group transformations are then used to construct localized excitations of the system

novel integrable reductions in nonlinear continuum mechanics via geometric constraints the nonlinear equations that describe solitonic behavior in physical systems have to date typically been derived by approximation or expansion methods here by contrast hidden integrable structure is revealed in diverse areas of nonlinear continuum mechanics through natural geometric constraints

a note on new travelling wave solutions to the ostrovsky equation in a recent paper by yasar e yasar new travelling wave solutions to the ostro vsky equation appl math comput 216 2010 3191 3194 new travelling wave solutions to the transformed reduced ostrovsky equation are presented in this note it is shown that some of these solutions are disguised versions of known solutions

revisitation of the localized excitations of the 2 1 dimensional kdv equation in a previous paper lou s y 1995 j phys a math gen 28 7227 a generalized dromion structure was revealed for the 2 1 dimensional kdv equation which was first derived by boiti et al boiti m leon j j p manna m and pempinelli f 1986 inverse problems 2 271 using the idea of the weak lax pair in this paper using a backlund transformation and the variable separation approach we find there exist much more abundant localized structures for the 2 1 dimensional kdv equation the abundance of the localized structures of the model is introduced by the entrance of an arbitrary function of the seed solution some special types of dromion solution lumps breathers instantons and the ring type of soliton are discussed by selecting the arbitrary functions appropriately the dromion solutions can be driven by sets of straight line and curved line ghost solitons the dromion solutions may be located not only at the cross points of the lines but also at the closed points of the curves the breathers may breathe both in amplitude and in shape

a note on new kink shaped solutions and periodic wave solutions for the 2 1 dimensional sine gordon equation exact solutions of the nizhnik novikov veselov equation by li new kink shaped solutions and periodic wave solutions for the 2 1 dimensional sine gordon equation appl math comput 215 2009 3777 3781 are analyzed we have observed that fourteen solutions by li from 30 do not satisfy the equation the other 16 exact solutions by li can be found from the general solutions of the well known solution of the equation for the weierstrass elliptic function

efficient fft based upscaling of the permeability of porous media discretized on uniform grids with estimation of rve size abstract an efficient numerical scheme at the crossroad of fourier based methods and boundary element methods is derived to upscale the permeability of porous media from stokes flow within the pore space the method relies on a variational framework which involves trial force fields and the green function the discretization of the trial force fields is carried out on a uniform grid which allows for the use of fast fourier transforms to evaluate the convolution product arising from the green function a discretization of the green function consistent with the variational framework is introduced to ensure the upper bound status of the homogenized permeability the energy consistent discretization is compared to six alternative discretizations arising from truncation of fourier series or from various finite difference schemes in terms of accuracy of the homogenized permeability and of the local velocity fields as well as on the number of iterations required for convergence of the iterative solver the method is combined to a strategy to estimate the representative volume element rve size at a target accuracy level from a single simulation on a large sample to study the permeability and associated rve size for the voronoi cell model over the whole range of porosity

an fft based galerkin method for homogenization of periodic media in 1994 moulinec and suquet introduced an efficient technique for the numerical resolution of the cell problem arising in homogenization of periodic media the scheme is based on a fixed point iterative solution to an integral equation of the lippmann schwinger type with action of its kernel efficiently evaluated by the fast fourier transform techniques the aim of this work is to demonstrate that the moulinec suquet setting is actually equivalent to a galerkin discretization of the cell problem based on approximation spaces spanned by trigonometric polynomials and a suitable numerical integration scheme for the latter framework and scalar elliptic problems we prove convergence of the approximate solution to the weak solution including a priori estimates for the rate of convergence for sufficiently regular data and the effects of numerical integration moreover we also show that the variational structure implies that the resulting non symmetric system of linear equations can be solved by the conjugate gradient method apart from providing a theoretical support to fast fourier transform based methods for numerical homogenization these findings significantly improve on the performance of the original solver and pave the way to similar developments for its many generalizations proposed in the literature

the implementation of b splines to hashin and shtrikman variational principle based fft method for the homogenization of composite abstract fast fourier transform fft has been successfully used to estimate the effective properties of composites with meso periodic structure for more than two decades numerous improvements have been done to make it adequate for infinite contrast bound elastic energy and composite voxels the hashin and shtrikman variational principle based fft method handles these three problems simultaneously and is very attractive in this paper b splines were adopted to discretize the polarized field the approximate polarized field was substituted into the hashin and shtrikman variational principle to obtain the discretized system the constitutive matrix of a composite voxel was calculated by both the energetically consistent way and the laminate mixing rule numerical example of a square matrix with a circular inclusion shows that the results are more accurate than the remaining cases when the first and second order b splines as well as the laminate mixing constitutive matrix are used for higher order b spline the summation to compute the discrete green operator for periodic boundary conditions converges faster while the discretized polarized field needs more iterations to converge only when the polarized field is sufficiently smooth high order b spline improves the numerical accuracy and efficiency

two dimensional schro dinger operator inverse scattering transform and evolutional equations abstract the inverse problem for the two dimensional schrodinger operator on the data from one energy level is solved in a special class of finite zone or algebraic operators this class seems to be dense among all smooth periodic schrodinger operators evolutional equations associated with this problem are constructed

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

doubly periodic and multiple pole solutions of the sinh poisson equation application of reciprocal transformations in subsonic gas dynamics vortex patterns associated with the sinh poisson equation arise in a remarkable manner as relaxation states of the navier stokes equations here doubly periodic and multiple pole solutions of the sinh poisson equation are generated via the hirota bilinear operator formalism and exploitation of the phenomenon of coalescence of wave numbers it is then shown how the multi parameter reciprocal transformations of gas dynamics may be applied to a seed doubly periodic solution of the sinh poisson equation to generate associated periodic vortex structures valid in the subsonic flow of a generalized karman tsien gas

necessary conditions for subellipticity of the overline partial neumann problem let a be a a closed form of type p q with l2 coefficients on a smoothly bounded domain q in c the problem of determining both the existence and regularity properties of the solution u of au a where u is orthogonal to the null space of a on p q l forms is known as the a neumann problem one of the principal methods used in the investigation of this problem is the proof of certain a priori subelliptic estimates let u be a neighborhood of a point z0 in the boundary of q a subelliptic estimate is said to hold in u if the estimate

local solvability of the equation with boundary regularity on weakly q convex domains we consider weakly q convex domains with smooth boundary and show that the open image in new window equation is locally solvable with regularity up to the boundary for smooth forms of degree p s for s q

real hypersurfaces orders of contact and applications a fundamental problem in the modem theory of several complex variables concerns the boundary behavior of the cauchy riemann equations suppose that d is an open domain in cn and that its boundary m is a smooth real submanifold of cn how does the geometry of m influence the function theory on d one approach to this question is the a neumann problem 6 there one studies the inhomogeneous cauchy riemann equations au a where a is a 0 q form with distribution coefficients on the closed domain d u m satisfying the necessary compatibility condition a a 0 the a neumann problem constructs a particular solution u which is orthogonal to the null space of the complex laplacian kohn 10a has solved this problem and has shown that good local regularity properties for u in terms of a follow from so called subelliptic estimates it is natural to seek geometric conditions on m that are necessary and sufficient for these estimates in this paper we will be concerned with a certain geometric condition on m that we feel is the right one for subellipticity although the motivation for the questions discussed here comes from partial differential equations the techniques come from algebraic geometry to see why we state theorems of kohn greiner and catlin concerning subellipticity kohn 1oa has discovered a sufficient condition for a subelliptic estimate near a fixed point p in m in case d is pseudoconvex and is contained in c2 this condition is equivalent to the finiteness of order of tangency of every complex analytic one dimensional manifold with m at p greiner 13 established the necessity of this condition in

letter to the editor a note on new exact solutions for the kawahara equation using exp function method exact solutions of the kawahara equation by assas l m b assas new exact solutions for the kawahara equation using exp function method j comput appl math 233 2009 97 102 are analyzed it is shown that all solutions do not satisfy the kawahara equation and consequently all nontrivial solutions by assas are wrong

the classical korteweg capillarity system geometry and invariant transformations a class of invariant transformations is presented for the classical korteweg capillarity system the invariance is an extension of a kind originally introduced in an anisentropic gasdynamics context in a particular instance application of the invariant transformation leads to a deformed one parameter class of karman tsien type capillarity laws associated with a deformation of an integrable nonlinear schrodinger type equation which incorporates a de broglie bohm potential the latter and another integrable case associated with the classical boussinesq equation may be linked to the motion of curves in euclidean and projective space so that both the invariant transformation and the galilean invariance of the capillarity system may be interpreted in a geometric and soliton theoretic manner the work is set in the broader context of other connections of invariant transformations in gasdynamics with soliton theory

new exact homoclinic wave and periodic wave solutions for the ginzburg landau equation abstract new exact solutions including homoclinic wave and periodic wave solutions for the 2d ginzburg landau equation are obtained using the auxiliary function method and the g g expansion method respectively the solutions are expressed by the hyperbolic functions and the trigonometric functions there result shows that there exists a kink wave solution which tends to one and the same periodic wave solution as time tends to infinite

backlund transformations for the nizhnik novikov veselov equation the backlund transformations for the nizhnik novikov veselov equation are presented it is shown that these transformations can be iterated and that the resulting sequence can be described by the volterra equations the relationships between the nizhnik novikov veselov equation and the volterra hierarchy are discussed

the bilinear negative kadomtsev petviashvili system and its kac moody virasoro symmetry group the symmetry transformation group of the bilinear negative kadomtsev petviashvili system is studied by means of a direct method the kac moody virasoro type lie point symmetry algebra is found to be a special infinitesimal form of the symmetry group

menelaus theorem clifford configurations and inversive geometry of the schwarzian kp hierarchy it is shown that the integrable discrete schwarzian kp dskp equation which constitutes an algebraic superposition formula associated with for instance the schwarzian kp hierarchy the classical darboux transformation and quasi conformal mappings encapsulates nothing but a fundamental theorem of ancient greek geometry thus it is demonstrated that the connection with menelaus theorem and more generally clifford configurations renders the dskp equation a natural object of inversive geometry on the plane the geometric and algebraic integrability of dskp lattices and their reductions to lattices of menelaus darboux schwarzian kdv schwarzian boussinesq and schramm type are discussed the dskp and discrete schwarzian boussinesq equations are shown to represent discretizations of families of quasi conformal mappings

global solvability and regularity for on an annulus between two weakly pseudoconvex domains let m1 and m2 be two bounded pseudo convex domains in c with smooth boundaries such that m1 c m2 we consider the cauchy riemann operators 8 on the annulus m m2 m1 the main result of this paper is the following given a a closed p q form a 0 3 a has l2 coefficients and satisfies 0 2 and a is orthogonal to a finite dimensional space i e the harmonic p q forms then there exists a p q 1 form u such that 0 1 holds furthermore if a is smooth up to the boundary of m then we can find a u that is smooth up to the boundary also and u satisfies 0 1 theorems 1 2 and 3 our method is to use the a neumann problem with weights which was used by hormander 3 and kohn 4 to solve the equation 0 1 on weakly pseudo convex domains the a neumann problem was a method to solve the equation a with solutions smooth up to the boundary if one can show that the subelliptic estimate holds for the a neumann problem then one can conclude that the harmonic forms are finite dimensional and one can solve 0 1 provided a has l2 coefficients and satisfies 0 2 and a is orthogonal to the harmonic space one can find a solution u received by the editors october 26 1984 1980 mathematics subject classification primary 35b45 35b65 c1985 american mathematical society 0002 9947 85 1 00 25 per page

computational homogenization of elasticity on a staggered grid summary in this article we propose to discretize the problem of linear elastic homogenization by finite differences on a staggered grid and introduce fast and robust solvers our method shares some properties with the fft based homogenization technique of moulinec and suquet which has received widespread attention recently because of its robustness and computational speed these similarities include the use of fft and the resulting performing solvers the staggered grid discretization however offers three crucial improvements firstly solutions obtained by our method are completely devoid of the spurious oscillations characterizing solutions obtained by moulinec suquet s discretization secondly the iteration numbers of our solvers are bounded independently of the grid size and the contrast in particular our solvers converge for three dimensional porous structures which cannot be handled by moulinec suquet s method thirdly the finite difference discretization allows for algorithmic variants with lower memory consumption more precisely it is possible to reduce the memory consumption of the moulinec suquet algorithms by 50 we underline the effectiveness and the applicability of our methods by several numerical experiments of industrial scale copyright 2015 john wiley sons ltd

exact travelling solutions for the lax s seventh order kdv equation by sech method and rational exp function method in this paper we establish exact solutions for a nonlinear evolution equation the sech method and the exp function method are used to construct the solitary travelling wave solutions of lax s seventh order kdv equation these solutions may be important of significance for the explanation of some practical physical problem