a local wave tracking strategy for efficiently solving mid and high frequency helmholtz problems abstract we propose a procedure for selecting basis function orientation to improve the efficiency of solution methodologies that employ local plane wave approximations the proposed adaptive approach consists of a local wave tracking strategy each plane wave basis set within considered elements of the mesh partition is individually or collectively rotated to best align one function of the set with the local propagation direction of the field systematic determination of the direction of the field inside the computational domain is formulated as a minimization problem as the resultant system is nonlinear with respect to the directions of propagation the newton method is employed with exact characterization of the jacobian and hessian to illustrate the salient features and evaluate the performance of the proposed wave tracking approach we present error estimates as well as numerical results obtained by incorporating the procedure into a prototypical plane wave based approach the least squares method lsm developed by monk and wang 1999 1 the numerical results obtained for the case of a two dimensional rigid scattering problem indicate that a convergence was achievable to a prescribed level of accuracy even upon initial application of the tracking wave strategy outside the pre asymptotic convergence region and b the proposed approach reduced the size of the resulting system by up to two orders of magnitude depending on the frequency range with respect to the size of the standard lsm system

unified hybridization of discontinuous galerkin mixed and continuous galerkin methods for second order elliptic problems we introduce a unifying framework for hybridization of finite element methods for second order elliptic problems the methods fitting in the framework are a general class of mixed dual finite element methods including hybridized mixed continuous galerkin nonconforming and a new wide class of hybridizable discontinuous galerkin methods the distinctive feature of the methods in this framework is that the only globally coupled degrees of freedom are those of an approximation of the solution defined only on the boundaries of the elements since the associated matrix is sparse symmetric and positive definite these methods can be efficiently implemented moreover the framework allows in a single implementation the use of different methods in different elements or subdomains of the computational domain which are then automatically coupled finally the framework brings about a new point of view thanks to which it is possible to see how to devise novel methods displaying very localized and simple mortaring techniques as well as methods permitting an even further reduction of the number of globally coupled degrees of freedom

genus 4 trigonal reduction of the benney equations it was shown by gibbons and tsarev 1996 phys lett a 211 19 1999 phys lett a 258 263 that n parameter reductions of the benney equations correspond to particular n parameter families of conformal maps in recent papers baldwin and gibbons 2003 j phys a math gen 36 8393 417 baldwin and gibbons 2004 j phys a math gen 37 5341 54 the present authors have constructed examples of such reductions where the mappings take the upper half p plane to a polygonal slit domain in the plane in those cases the mapping function was expressed in terms of the derivatives of kleinian functions of hyperelliptic curves restricted to the one dimensional stratum 1 of the divisor this was done using an extension of the method given in enolskii et al 2003 j nonlinear sci 13 157 extended to a genus 3 curve enolski v z and gibbons j addition theorems on the strata of the theta divisor of genus three hyperelliptic curves in preparation here we use similar ideas but now applied to a trigonal curve of genus 4 fundamental to this approach is a family of differential relations which satisfies on the divisor again it is shown that the mapping function is expressible in terms of quotients of derivatives of on the divisor 1 one significant by product is an expansion of the leading terms of the taylor series of for the given family of 3 5 curves to the best of the authors knowledge this is new

finite element solution of the helmholtz equation with high wave number part i the h version of the fem abstract the paper addresses the properties of finite element solutions for the helmholtz equation the h version of the finite element method with piecewise linear approximation is applied to a one dimensional model problem new results are shown on stability and error estimation of the discrete model in all propositions assumptions are made on the magnitude of hk only where k is the wavelength and h is the stepwidth of the fe mesh previous analytical results had been shown with the assumption that k 2 h is small for medium and high wavenumber these results do not cover the meshsizes that are applied in practical applications the main estimate reveals that the error in h 1 norm of discrete solutions for the helmholtz equation is polluted when k 2 h is not small the error is then not quasioptimal i e the relation of the fe error to the error of best approximation generally depends on the wavenumber k it is noted that the pollution term in the relative error is of the same order as the phase lead of the numerical solution in the result of this analysis thorough and rigorous understanding of error behavior throughout the range of convergence is gained numerical results are presented that show sharpness of the error estimates and highlight some phenomena of the discrete solution behavior the h p version of the fem is studied in part ii

echocardiogram analysis using fuzzy sets and relations abstract this paper describes part of a larger project whose objective is to obtain completely automated analysis of a set of time sequenced echocardiogram images to yield such heart function parameters as chamber and stroke volumes ejection fraction and indices of regional myocardial function the research reported here is concerned with identifying collections of local regions in different frames which have similar characteristics and hence comprise a global region and those global regions which add together to form other known global regions the local regions are described by a fuzzy vector whose components are fuzzy numbers an equality like relation is defined on the set of local regions and equivalence classes are used to define the global regions a similar technique is used to decide which global regions add to larger global regions the end result is used by a fuzzy expert system which classifies the global regions according to their anatomical significance or as artifacst the results here reported suggest the utility of the fuzzy set techniques employed the possible application of our methods to other pattern recognition problems is discussed

a first order system least squares method for the helmholtz equation we present a first order system least squares fosls method for the helmholtz equation at high wave number k which always leads to a hermitian positive definite algebraic system by utilizing a non trivial solution decomposition to the dual fosls problem which is quite different from that of the standard finite element methods we give an error analysis to the h p version of the fosls method where the dependence on the mesh size h the approximation order p and the wave number k is given explicitly in particular under some assumption of the boundary of the domain the l 2 norm error estimate of the scalar solution from the fosls method is shown to be quasi optimal under the condition that k h p is sufficiently small and the polynomial degree p is at least o log k numerical experiments are given to verify the theoretical results

sasa satsuma complex modified korteweg de vries ii and the complex sine gordon ii equation revisited recursion operators nonlocal symmetries and more we present a new symplectic structure and a hereditary recursion operator for the sasa satsuma equation which is widely used in nonlinear optics using an integrodifferential substitution relating this equation to a third order symmetry flow of the complex sine gordon ii equation enabled us to find a hereditary recursion operator and higher hamiltonian structures for the latter equation we also show that both the sasa satsuma equation and the third order symmetry flow for the complex sine gordon ii equation are bi hamiltonian systems and we construct several hierarchies of local and nonlocal symmetries for these systems

the fuzzy mathematics of finance this paper develops fuzzy analogues of the elementary compound interest problems in the mathematics of finance we find the fuzzy present value and fuzzy future value of fuzzy cash amounts using fuzzy interest rates over n periods where n may be crisp or fuzzy this paper also develops a method of comparing fuzzy net cash flows in order to rank fuzzy investment alternatives from best to worst

integrability and bright soliton solutions to the coupled nonlinear schrodinger equation with higher order effects in this paper we study the integrability and exact bright soliton solutions to the coupled nonlinear schrodinger equation with higher order effects arising from nonlinear fiber medium firstly the lax integrability of this equation is investigated by prolongation technique and an integrable generalized coupled higher order nonlinear schrodinger equation is proposed then the general n bright bright soliton solutions of this integrable equation are obtained by riemann hilbert formulation and the collision dynamics between two solitons is analyzed finally an integrable generalized n coupled higher order nonlinear schrodinger equation together with its linear spectral problem are given

wave solutions of evolution equations and hamiltonian flows on nonlinear subvarieties of generalized jacobians the algebraic geometric approach is extended to study evolution equations associated with the energy dependent schr dinger operators having potentials with poles in the spectral parameter in connection with hamiltonian flows on nonlinear subvarieties of jacobi varieties the general approach is demonstrated by using new parametrizations for constructing quasi periodic solutions of the shallow water and dym type equations in terms of theta functions a qualitative description of real valued solutions is provided

bright n soliton solutions in terms of the triple wronskian for the coupled nonlinear schrodinger equations in optical fibers the coupled nonlinear schrodinger nls equations are usually used to describe the dynamics of two component solitons in optical fibers via the darboux transformation the coupled nls equations of the manakov type are found to have triple wronskian solutions proof is finished by virtue of some new triple wronskian identities by solving the zero potential lax pair the triple wronskian solutions give the bright n soliton solutions which are characterized by 3n complex parameters to obtain an understanding of the asymptotic behavior of the bright n soliton solutions with arbitrary n some algebraic properties of the triple wronskian are analyzed and an algebraic procedure is presented to derive the explicit expressions of the asymptotic solitons as t

an extension of the wronskian technique for the multicomponent wronskian solution to the vector nonlinear schrodinger equation in this paper the wronskian technique is applied to the vector nonlinear schrodinger equation with arbitrary m components which arises from some applications in the multimode fibers photorefractive materials and bose einstein condensates via the iterative algorithm based on the darboux transformation the m 1 component wronskian solution is generated from the zero solution the verification of the solution is finished by using the m 1 component wronskian notation and new determinantal identities with a set of n linearly independent solutions of the zero potential lax pair the m 1 component wronskian solution is found to be the representation of the bright n soliton solution which contains m 1 n parameters for characterizing the asymptotic behavior of the generic bright n soliton solution an algebraic procedure is derived to obtain the explicit expressions of asymptotic solitons as t

backlund transformations soliton solutions and wave functions of kaup newell and wadati konno ichikawa systems using the bargmann darboux method the backlund transformations n soliton solutions and corresponding wave functions of the kaup newell and wadati konno ichikawa systems are obtained these results culminate in an algebraic recursive procedure for the determination of multisoliton solutions and their wave functions of the derivative and mixed derivative nonlinear schrodinger equations iqt qxx i q 2q x q 2q 0 0 0

discontinuous galerkin methods for the helmholtz equation with large wave number this paper develops and analyzes some interior penalty discontinuous galerkin ipdg methods using piecewise linear polynomials for the helmholtz equation with the first order absorbing boundary condition in two and three dimensions it is proved that the proposed discontinuous galerkin methods are stable hence well posed without any mesh constraint for each fixed wave number k optimal order with respect to h error estimate in the broken h 1 norm and suboptimal order estimate in the l 2 norm are derived without any mesh constraint the latter estimate improves to optimal order when the mesh size h is restricted to the preasymptotic regime i e k 2 h gtrsim 1 numerical experiments are also presented to gauge the theoretical result and to numerically examine the pollution effect with respect to k in the error bounds the novelties of the proposed ipdg methods include the following first the methods penalize not only the jumps of the function values across the element edges but also the jumps of the normal and tangential derivatives second the penalty parameters are taken as complex numbers of positive imaginary parts so essentially and practically no constraint is imposed on the penalty parameters since the helmholtz problem is a non hermitian and indefinite linear problem as expected the crucial and the most difficult part of the whole analysis is to establish the stability estimates i e a priori estimates for the numerical solutions to this end the cruxes of our analysis are to establish and to make use of a local version of the rellich identity for the laplacian and to mimic the stability analysis for the pde solutions given in p cummings analysis of finite element based numerical methods for acoustic waves elastic waves and fluid solid interactions in the frequency domain ph d thesis the university of tennessee knoxville tn 2001 p cummings and x feng math models methods appl sci 16 2006 pp 139 160 u hetmaniuk commun math sci 5 2007 pp 665 678

symmetrically coupled higher order nonlinear schrodinger equations singularity analysis and integrability the integrability of a system of two symmetrically coupled higher order nonlinear schrodinger equations with parameter coefficients is tested by means of singularity analysis it is proven that the system passes the painleve test for integrability only in ten distinct cases of which two are new for one of the new cases a lax pair and a multi field generalization are obtained for the other one the equations of the system are uncoupled by a nonlinear transformation

darboux transformation and rogue wave solutions for the variable coefficients coupled hirota equations abstract we construct the lax pair and darboux transformation for the variable coefficients coupled hirota equations based on modulation instability and by taking the limit approach we derive two types of nth order rogue wave solutions with different dynamic structures in compact determinant representations the explicit first order rogue wave solution is presented prolific vector rogue wave patterns such as the dark bright composite three sister quadruple and sextuple rogue waves with multiple compression points are demonstrated in particular in contrast to the standard peregrine combs unusual vector rogue wave combs such as the dark bright and composite rogue wave combs are revealed by choosing sufficiently large periodic modulation amplitudes further some wave characteristics such as the difference between light intensity and continuous wave background and pulse energy evolution of the dark rogue wave solution that features multiple compression points are discussed in detail

algebro geometric solutions to the modified sawada kotera hierarchy based on solving the lenard recursion equation and the zero curvature equation we derive the modified sawada kotera sk hierarchy associated with a 3 3 matrix spectral problem resorting to the characteristic polynomial of lax matrix for the modified sk hierarchy we introduce a trigonal curve km 1 and present the corresponding baker akhiezer function and meromorphic function on it the abel map is introduced to straighten out the modified sk flows with the aid of the property of the baker akhiezer function and the meromorphic function and their asymptotic expansions we arrive at their explicit riemann theta function representations algebro geometric solutions of the entire modified sk hierarchy are obtained by using asymptotic expansion of the meromorphic function and its riemann theta function representation as an application we give two simple examples

methodology of linguistics evaluation in risk situations using fuzzy techniques abstract in this article the mathematical expression which determines the fuzzy weighted mean of a system in terms of a determined or specific configuration of estimates is studied its objective lies in obtaining fuzzy sets that are as close as possible to the natural expressions in systems which need an estimative calculation of risk whether it be of failure access or of any other type with incomplete and or unprecise data to do this the calculation fuzzy weighted mean of the fuzzy restriction associated with a system has been established as the proposed evaluation technique increasing the number of natural expressions and thus the number of fuzzy processes that are involved in the said calculation this increase has been achieved by analyzing all the possible results on the basis of the number of fuzzy sets in the numerator and denominator of the expression that denotes the fuzzy weighted mean of a system taking into account to what point of the calculation such an increase is feasible without impairing the resulting fuzzy set the importance of the resulting fuzzy set is the motivation behind effecting an enhancement process in this analytic phase since it determines a linguistic meaning that is taken as an element of comparison and reference for the obtention of the final natural expression the results of the proposed method have been verified by means of an interpreter which has been designed to deal with all the fuzzy operations which intervene in the different processes

on the application of a generalized dressing method to the integration of variable coefficient coupled hirota equations based on the generalized dressing method we propose two variable coefficient coupled hirota equations and derive their lax pairs the generalized dressing method is applied to study these variable coefficient coupled hirota equations from which explicit solutions of the two equations and their reductions are constructed

integrability aspects with optical solitons of a generalized variable coefficient n coupled higher order nonlinear schrodinger system from inhomogeneous optical fibers for describing the long distance communication and manufacturing problems of n fields propagation in inhomogeneous optical fibers we consider a generalized variable coefficient n coupled nonlinear schrodinger system with higher order effects such as the third order dispersion self steepening and self frequency shift using the painleve singularity structure analysis we obtain two cases for this system to admit the painleve property then for case 1 we derive the optical dark solitons via solving the hirota bilinear equations and based on the obtained 2n 1 2n 1 lax pair we construct the darboux transformation to obtain the optical bright solitons including the multisoliton profiles for case 2 finally the features of optical solitons both dark and bright ones in inhomogeneous optical fibers are analyzed and graphically discussed

hyperelliptic quasi periodic and soliton solutions of the nonlinear schrodinger equation on considere la construction de solutions par des methodes de geometrie algebriques et en particulier par les fonctions theta hyperelliptiques

conformal maps and integrable hierarchies we show that conformal maps of simply connected domains with an analytic boundary to a unit disk have an intimate relation to the dispersionless 2d toda integrable hierarchy the maps are determined by a particular solution to the hierarchy singled out by the conditions known as string equations the same hierarchy locally solves the 2d inverse potential problem i e reconstruction of the domain out of a set of its harmonic moments this is the same solution which is known to describe 2d gravity coupled to c matter we also introduce a concept of the function for analytic curves

higher genus hyperelliptic reductions of the benney equations it was shown by gibbons and tsarev 1996 phys lett a 211 19 1999 phys lett a 258 263 that n parameter reductions of the benney equations correspond to n parameter families of conformal maps here we consider a specific set of these the hyperelliptic reductions the mapping function for this is calculated explicitly by inverting a second kind abelian integral on the stratum 1 of the jacobi variety of a genus g g 3 hyperelliptic curve this is done using a method based on the result of jorgenson 1992 isr j math 77 273

effective integration of the nonlinear vector schrodinger equation abstract a comprehensive algebro geometric integration of the two component nonlinear vector schrodinger equation manakov system is developed the allied spectral variety is a trigonal riemann surface which is described explicitly and the solutions of the equations are given in terms of functions of the surface the final formulae are effective in the sense that all entries like transcendental constants in exponentials winding vectors etc are expressed in terms of the prime form of the curve and well algorithmized operations on them that made the result available for direct calculations in applied problems implementing the manakov system the simplest solutions in jacobian functions are given as a particular case of general formulae and are discussed in detail

a hybridizable discontinuous galerkin method for the helmholtz equation with high wave number this paper analyzes the error estimates of the hybridizable discontinuous galerkin hdg method for the helmholtz equation with high wave number in two and three dimensions the approximation piecewise polynomial spaces we deal with are of order p geq 1 through choosing a specific parameter and using the duality argument it is proved that the hdg method is stable without any mesh constraint for any wave number kappa by exploiting the stability estimates the dependence of convergence of the hdg method on kappa h and p is obtained numerical experiments are given to verify the theoretical results

fuzzy sets and type 2 under algebraic product and algebraic sum abstract the concept of fuzzy sets of type 2 has been proposed by l a zadeh as an extension of ordinary fuzzy sets a fuzzy set of type 2 can be defined by a fuzzy membership function the grade or fuzzy grade of which is taken to be a fuzzy set in the unit interval 0 1 rather than a point in 0 1 this paper investigates the algebraic properties of fuzzy grades that is fuzzy sets of type 2 under the operations of algebraic product and algebraic sum which can be defined by using the concept of the extension principle and shows that fuzzy grades under these operations do not form such algebraic structures as a lattice and a semiring moreover the properties of fuzzy grades are also discussed in the case where algebraic product and algebraic sum are combined with the well known operations of join and meet for fuzzy grades and it is shown that normal convex fuzzy grades form a lattice ordered semigroup under join meet and algebraic product

computing solitary wave solutions of coupled nonlinear hirota and helmholtz equations abstract in this article we obtain the exact solutions two coupled models one integrable system namely coupled nonlinear hirota cnhi equation and another non integrable system namely coupled nonlinear helmholtz cnhe equation via the exp e expansion method the obtained travelling wave solutions are structured in rational trigonometric and hyperbolic functions these solutions lead to diverse types of solitary optical waves for free choices of parameters that guarantee the sustainability of such solutions also 3d illustrations for the free choices of the physical parameters is provided to understand the physical explanation of the problems these results further enrich and deepen the understanding of the dynamics of higher dimensional soliton propagation

efficient dg like formulation equipped with curved boundary edges for solving elasto acoustic scattering problems summary a discontinuous galerkin dg based approach is proposed for computing the scattered field from an elastic bounded object immersed in an infinite homogeneous fluid medium the proposed method possesses two distinctive features first it employs higher order polynomial shape functions needed to address the high frequency propagation regime second it is equipped with curved boundary edges to provide an accurate representation of the fluid structure interface the most salient benefits resulting from the latter feature as demonstrated by the numerical investigation are the following i an improvement by at least two orders of magnitude on the relative error and ii the disappearance of spurious resonance frequencies in the surrounding fluid medium in addition the reported numerical results reveal that when using cubic polynomials with less than three elements per wavelength the proposed dg method computes the scattered field with a relative error below 1 for an elastic scatterer of about 30 wavelengths this observation highlights the potential of the proposed solution methodology for efficiently solving mid frequency to high frequency elasto acoustic scattering problems copyright 2014 john wiley sons ltd

pre asymptotic error analysis of h p interior penalty discontinuous galerkin methods for the helmholtz equation with large wave number in this paper we shall consider to improve the pre asymptotic stability and error estimates of some h p interior penalty discontinuous galerkin h p ipdg methods for the helmholtz equation with the first order absorbing boundary condition in two and three dimensions given in feng and wu 2011 the proposed h p ipdg methods are defined using a sesquilinear form which is not only mesh dependent or h dependent but also degree dependent or p dependent by using a modified duality argument given in zhu and wu 2013 pre asymptotic error estimates are improved for the proposed h p ipdg methods under the condition of k h p c 0 p k 1 p 1 in this paper where c 0 is some constant independent of k h p and the penalty parameters it is shown that the pollution error of the method in the broken h 1 norm is o k 2 p 1 h 2 p if p o 1 which coincides with existent dispersion analyses for the dg method on cartesian grids numerical tests are provided to verify the theoretical findings and to illustrate great capability of the ipdg method in reducing the pollution effect

quasi periodic and periodic solutions for coupled nonlinear schrodinger equations of manakov type we consider travelling periodic and quasi periodic wave solutions in coupled nonlinear schrodinger equations in fibre optics these equations can be used to model single mode fibres with strong birefringence and two mode optical fibres recently these equations appear as a model describing pulse pulse interactions in wavelength division multiplexed channels of optical fibre transmission systems in some cases this model reduces to the integrable manakov system ims two phase quasi periodic solutions for the ims are given in terms of two dimensional kleinian functions the reduction of quasi periodic solutions to elliptic functions is discussed new solutions are found in terms of generalized hermite polynomials which are associated with two gap treibich verdier potentials

isospectral hamiltonian flows in finite and infinite dimensions the approach to isospectral hamiltonian flow introduced in part i is further developed to include integration of flows with singular spectral curves the flow on finite dimensional ad invariant poisson submanifolds of the dual widetilde gl r of the positive part of the loop algebra widetilde gl r is obtained through a generalization of the standard method of linearization on the jacobi variety of the invariant spectral curves these curves are embedded in the total space of a line bundlet p1 c allowing an explicit analysis of singularities arising from the structure of the image of a moment map tilde j r m n r times m n r to widetilde gl r from the space of rank r deformations of a fixedn n matrixa it is shown how the linear flow of line bundles e t to tilde s over a suitably desingularized curve tilde s may be used to determine both the flow of matricial polynomialsl and the hamiltonian flow in the spacemn r mn r in terms of functions the resulting flows are proved to be completely integrable the reductions to subalgebras developed in part i are shown to correspond to invariance of the spectral curves and line bundles e t to tilde s under certain linear or anti linear involutions the integration of two examples from part i is given to illustrate the method the rosochatius system and the cnls coupled non linear schrodinger equation

soliton solutions to coupled higher order nonlinear schrodinger equations a set of coupled higher order nonlinear schrodinger equations which describe electromagnetic pulse propagation in coupled optical waveguides is formulated in terms of an eigenvalue problem using that result the inverse scattering problem is solved and explicit soliton solutions are found additionally linear coupling terms are studied systematically

genus two finite gap solutions to the vector nonlinear schrodinger equation a recently published article presents a technique used to derive explicit formulae for odd genus solutions to the vector nonlinear schrodinger equation in another article solutions of genus two are derived using a different approach which assumes a separable ansatz in this communication the extension of the first technique to the even genus case is discussed and this extension is carried out explicitly for genus two furthermore a birational mapping is found between the spectral curves that arise in the two approaches

absolutely stable local discontinuous galerkin methods for the helmholtz equation with large wave number two local discontinuous galerkin ldg methods using some non standard numerical uxes are developed for the helmholtz equation with the rst order absorbing boundary condition in the high frequency regime it is shown that the proposed ldg methods are absolutely stable hence well posed with respect to both the wave number and the mesh size optimal order with respect to the mesh size error estimates are proved for all wave numbers in the preasymptotic regime to analyze the proposed ldg methods they are recasted and treated as non conforming mixed nite element methods the crux of the analysis is to establish a generalized inf sup condition which holds without any mesh constraint for each ldg method the generalized inf sup conditions then easily infer the desired absolute stability of the proposed ldg methods in return the stability results not only guarantee the well posedness of the ldg methods but also play a crucial role in the derivation of the error estimates numerical experiments which conrm the theoretical results and compare the proposed two ldg methods are also presented in the paper

a phase based hybridizable discontinuous galerkin method for the numerical solution of the helmholtz equation we introduce a new hybridizable discontinuous galerkin hdg method for the numerical solution of the helmholtz equation over a wide range of wave frequencies our approach combines the hdg methodology with geometrical optics in a fashion that allows us to take advantage of the strengths of these two methodologies the phase based hdg method is devised as follows first we enrich the local approximation spaces with precomputed phases which are solutions of the eikonal equation in geometrical optics second we propose a novel scheme that combines the hdg method with ray tracing to compute multivalued solution of the eikonal equation third we utilize the proper orthogonal decomposition to remove redundant modes and obtain locally orthogonal basis functions which are then used to construct the global approximation spaces of the phase based hdg method and fourth we propose an appropriate choice of the stabilization parameter to guarantee stability and accuracy for the proposed method numerical experiments presented show that optimal orders of convergence are achieved that the number of degrees of freedom to achieve a given accuracy is independent of the wave number and that the number of unknowns required to achieve a given accuracy with the proposed method is orders of magnitude smaller than that with the standard finite element method

staggered discontinuous galerkin methods for the helmholtz equation with large wave number abstract in this paper we investigate staggered discontinuous galerkin method for the helmholtz equation with large wave number on general polygonal meshes the method is highly flexible by allowing rough grids such as the trapezoidal grids and highly distorted grids and at the same time is numerical flux free furthermore it allows hanging nodes which can be simply treated as additional vertices by exploiting a modified duality argument the stability and convergence can be proved under the condition that h c 0 1 1 m 1 where m is the polynomial order is the wave number h is the mesh size and c 0 is a positive constant independent of h error estimates for both the scalar and vector variables in l 2 norm are established several numerical experiments are tested to verify our theoretical results and to present the capability of our method for capturing singular solutions

the vector nonlinear schrodinger hierarchy abstract an article published recently by one of the authors jne presents closed form solutions for zero curvature representations of the vector nonlinear schrodinger hierarchy several of these results are confirmed computationally but left unproven in this article we begin by providing strict algebraic proofs of these results the forms of the hierarchy s associated spectral curves are investigated and proven to have only odd genus without the introduction of integration constants we admit specific non zero constants of integration to the hierarchy and show that even genus curves may be introduced in a very straightforward manner thereby accessing the full family of finite gap solutions to the vector nonlinear schrodinger equation finally an original construction of the infinite set of conserved densities of the hierarchy is given

generalized and extended fuzzy sets with applications abstract this paper introduces generalized fuzzy sets extended fuzzy sets and generalized extended fuzzy sets and shows how these fuzzy sets may arise naturally in some applications the applications presented are chosen from the areas of decision theory hierarchical analysis and expert systems zadeh s method of reducing a level 2 fuzzy set to a regular fuzzy set is generalized to reduce some of these new fuzzy sets to simpler fuzzy sets what is needed then for applications is an algebra of irreducible generalized extended fuzzy sets

adaptive numerical solution of a discontinuous galerkin method for a helmholtz problem in low frequency regime we develop an a posteriori error analysis for helmholtz problem using the local discontinuous galerkin ldg for short approach for the sake of completeness we give a description of the main a priori results of this method indeed under some assumptions on regularity of the solution of an adjoint problem we prove that a the corresponding indefinite discrete scheme is well posed b the approach is convergent with the expected convergence rates as long as the meshsize h is small enough we give precise information on how small h has to be in terms of the size of the wave number and its distance to the set of eigenvalues for the same boundary value problem for the laplacian after that we present a reliable and efficient a posteriori error estimator with detailed information on the dependence of the constants on the wave number we finish presenting extensive numerical experiments which illustrate the theoretical results proven in this paper and suggest that stability and convergence may occur under less restrictive assumptions than those taken in the present work we derive an a posteriori error estimator that is reliable and locally quasi efficient the efficiency and reliability constants have been characterized with respect to the physical parameter we include numerical examples that validate our theoretical results recognizing singularities and inner layers for moderate values of

preasymptotic error analysis of cip fem and fem for helmholtz equation with high wave number part ii hp version in this paper which is the second in a series of two the preasymptotic error analysis of the continuous interior penalty finite element method cip fem and the fem for the helmholtz equation in two and three dimensions is continued while part i contained results on the linear cip fem and fem the present part deals with approximation spaces of order p ge 1 by using a modified duality argument preasymptotic error estimates are derived for both methods under the condition of frac kh p le c 0 frac p k frac 1 p 1 where k is the wave number h is the mesh size and c 0 is a constant independent of k h p and the penalty parameters it is shown that the pollution errors of both methods in h 1 norm are o k 2p 1 h 2p if p o 1 and are o frac k p 2 frac kh sigma p 2p if the exact solution u in h 2 omega which coincide with existent dispersion analyses for the fem on cartesian grids here sigma is a constant independent of k h p and the penalty par

ranking alternatives using fuzzy numbers we investigate the problem of employing expert opinion to rank alternatives across a set of criteria the experts use fuzzy numbers to express their preferences and we employ fuzzy arithmetic to compute an issue s fuzzy ranking this leads to a partition of the alternatives into sets h1 h2 where h1 contains the highest ranked issues h2 has all the second highest ranked alternatives etc the total ranking process is shown to possess a number of important properties an example is presented to illustrate the method

second generation fuzzy expert system abstract we describe our approach to designing a fuzzy expert systems shell which is decision theoretic in nature we wish to determine the interval operating characteristics in order to maximize the information in relevant working memory our solution is a shell which generalizes and contains our present shell flops major additions are 1 1 upper and lower confidence values 2 2 allowing rules to fire on either or both confidence values and 3 3 a memory update algorithm designed to maximize information

quasiperiodic solutions for matrix nonlinear schrodinger equations the adler kostant symes theorem yields isospectral hamiltonian flows on the dual erm g of a lie subalgebra erm g of a loop algebra erm g a general approach relating the method of integration of krichever novikov and dubrovin to such flows is used to obtain finite gap solutions of matrix nonlinear schrodinger equations in terms of quotients of functions

computing riemann matrices of algebraic curves a black box program for the explicit calculation of riemann matrices of arbitrary compact connected riemann surfaces is presented all such riemann surfaces are represented as plane algebraic curves these algebraic curves are allowed to have arbitrary singularities the method of calculation of the riemann matrix is essentially its definition we numerically integrate the holomorphic differentials of the riemann surface over the cycles of a canonical basis of the homology of the riemann surface both the holomorphic differentials and the canonical basis of the homology of the riemann surface are obtained exactly through symbolic calculations this program is included in maple 6 as part of the algcurves package

abelian functions associated with a cyclic tetragonal curve of genus six we develop the theory of abelian functions defined using a tetragonal curve of genus six discussing in detail the cyclic curve y 4 x 5 4x 4 3x 3 2x 2 1x 0 we construct abelian functions using the multivariate function associated with the curve generalizing the theory of the weierstrass function we demonstrate that such functions can give a solution to the kp equation outlining how a general class of solutions could be generated using a wider class of curves we also present the associated partial differential equations satisfied by the functions the solution of the jacobi inversion problem a power series expansion for u and a new addition formula

on the backlund gauge transformation and homoclinic orbits of a coupled nonlinear schrodinger system abstract the backlund gauge transformation for a system of coupled nls nonlinear schrodinger equations with a degenerate associated spectral operator is derived from an algebraic perspective extending aspects of other results m boiti tu guizhang il nuovo cimento 71b 1982 253 264 d h sattinger v d zurkowski physica d 26 1 3 1987 225 250 that apply in the context of non degenerate spectral operators moreover we demonstrate how the backlund gauge transformation can be used to explicitly construct the entire unstable manifold via superpositions of homoclinic orbits of a plane wave solution with both self phase instabilities and coupling instabilities this work builds on the results of ercolani et al n ercolani m g forest d w mclaughlin physica d 18 1986 472 474 n ercolani m g forest d w mclaughlin physica d 43 2 3 1990 349 384 for the sine gordon equation and forest et al m g forest d w mclaughlin d j muraki o c wright j nonlinear sci in press m g forest s p sheu o c wright phys lett a in press for the integrable coupled nls system

an explicit hybridizable discontinuous galerkin method for the acoustic wave equation abstract we present an explicit hybridizable discontinuous galerkin hdg method for numerically solving the acoustic wave equation the method is fully explicit high order accurate in both space and time and coincides with the classic discontinuous galerkin dg method with upwinding fluxes for a particular choice of its stabilization function this means that it has the same computational complexity as other explicit dg methods however just as its implicit version it provides optimal convergence of order k 1 for all the approximate variables including the gradient of the solution and when the time stepping method is of order k 2 it displays a superconvergence property which allow us by means of local postprocessing to obtain new improved approximations of the scalar field variables at any time levels for which an enhanced accuracy is required in particular the new approximations converge with order k 2 in the l 2 norm for k 1 these properties do not hold for all numerical fluxes indeed our results show that when the hdg numerical flux is replaced by the lax friedrichs flux the above mentioned superconvergence properties are lost although some are recovered when the lax friedrichs flux is used only in the interior of the domain finally we extend the explicit hdg method to treat the wave equation with perfectly matched layers we provide numerical examples to demonstrate the performance of the proposed method

genus n algebraic reductions of the benney hierarchy within a schottky model by exploiting a new theoretical connection between reductions of the benney hierarchy and the dirichlet problem for laplace s equation the solution to a spectral problem associated with n parameter algebraic reductions of the benney hierarchy is found explicitly the solutions can be written in terms of the modified green s function associated with reflectionally symmetric n connected planar domains whose holes are all centred on the symmetry axis explicit formulae for the modified green s function in a canonical class of circular domains are constructed using a schottky model of the schottky double of these domains uniformizations of the spectral problem associated with two different types of reductions then follow from these formulae

matrix nonlinear schrodinger equations and moment maps into loop algebras it is shown how darboux coordinates on a reduced symplectic vector space may be used to parametrize the phase space on which the finite gap solutions of matrix nonlinear schrodinger equations are realized as isospectral hamiltonian flows the parametrization follows from a moment map embedding of the symplectic vector space reduced by suitable group actions into the dual g of the algebra g of positive frequency loops in a lie algebra g the resulting phase space is identified with a poisson subspace of g consisting of elements that are rational in the loop parameter reduced coordinates associated to the various hermitian symmetric lie algebras g k corresponding to the classical lie algebras are obtained

the full positive flows of manakov hierarchy hamiltonian structures and conservation laws based on four sets of lenard recursion sequences and zero curvature equation we derive the full positive flows of the manakov hierarchy associated with a 3x3 matrix spectral problem from which some new nonlinear evolution equations are proposed with the help of the darboux transformation soliton solutions of two new nonlinear evolution equations in the manakov hierarchy are constructed as two special reductions the full positive flows of the coupled modified korteweg de vries hierarchy and the sasa satsuma hierarchy are deduced in which some new nonlinear evolution equations are included and then we construct the hamiltonian structures of the manakov hierarchy and infinite conservation laws of several nonlinear evolution equations

effective integration of ultra elliptic solutions of the focusing nonlinear schrodinger equation abstract an effective integration method based on the classical solution of the jacobi inversion problem using kleinian ultra elliptic functions and riemann theta functions is presented for the quasi periodic two phase solutions of the focusing cubic nonlinear schrodinger equation each two phase solution with real quasi periods forms a two real dimensional torus modulo a circle of complex phase factors expressed as a ratio of theta functions associated with the riemann surface of the invariant spectral curve the initial conditions of the dirichlet eigenvalues satisfy reality conditions which are explicitly parametrized by two physically meaningful real variables the squared modulus and a scalar multiple of the wavenumber simple new formulas for the maximum modulus and the minimum modulus are obtained in terms of the imaginary parts of the branch points of the riemann surface

the stationary equations of a coupled nonlinear schrodinger system abstract the integrable coupled nonlinear schrodinger cnls equations under periodic boundary conditions are known to possess linearized instabilities in both the focussing and defocussing cases m g forest d w mclaughlin d muraki o c wright non focussing instabilities in coupled integrable nonlinear schrodinger pdes in preparation d j muraki o c wright d w mclaughlin birefringent optical fibers modulational instability in a near integrable system nonlinear processes in physics proceedings of iii postdam v kiev workshop 1991 pp 242 245 o c wright modulational stability in a defocussing coupled nonlinear schrodinger system physica d 82 1995 1 10 whereas the scalar nls equation is linearly unstable only in the focussing case m g forest j e lee geometry and modulation theory for the periodic schrodinger equation in dafermas et al eds oscillation theory computation and methods of compensated compactness i m a math appl 2 1986 35 70 these instabilities indicate the presence of crossed homoclinic orbits similar to those in the phase plane of the unforced duffing oscillator y li d w mclaughlin morse and melnikov functions for nls pde s commun math phys 162 1994 175 214 d w mclaughlin e a overman whiskered tori for integrable pde s chaotic behaviour in near integrable pde s in keller et al eds surveys in applied mathematics vol 1 chapter 2 plenum press new york 1995 the homoclinic orbits and the near homoclinic tori that are connected to the unstable wave trains of the nls and the cnls reside in the finite dimensional phase space of certain stationary equations s p novikov funct anal prilozen 8 3 1974 54 66 of the infinite hierarchy of integrable commuting flows the correct stationary equations must be matched to the unstable torus through the analytic structure of the spectral curves o c wright near homoclinic orbits of the focussing nonlinear schrodinger equation preprint thus in this paper the stationary equations of the cnls are derived and the analytic structure of the trigonal spectral curve is examined providing a basis for further study of the near homoclinic orbits of the cnls system

hyperelliptic reduction of the benney moment equations we consider n parameter reductions of the benney moment equations these were shown in gibbons and tsarev 1996 phys lett a 211 19 1999 phys lett a 258 263 to correspond to n parameter families of conformal maps and to satisfy a particular system of pde a specific known example of this the n 2 elliptic reduction l yu and j gibbons 2000 inverse probl 16 605 is described we then consider an analogous reduction for a genus 2 hyperelliptic curve n 3 the mapping function is given by the inversion of a second kind abelian integral on the divisor this is found explicitly following a method given by enolskii et al 2003 j nonlinear sci 13 157

integrable discretizations of derivative nonlinear schrodinger equations we propose integrable discretizations of derivative nonlinear schrodinger dnls equations such as the kaup newell equation the chen lee liu equation and the gerdjikov ivanov equation by constructing lax pairs the discrete dnls systems admit the reduction of complex conjugation between two dependent variables and possess bi hamiltonian structure through transformations of variables and reductions we obtain novel integrable discretizations of the nonlinear schrodinger nls modified kdv mkdv mixed nls matrix nls matrix kdv matrix mkdv coupled nls coupled hirota coupled sasa satsuma and burgers equations we also discuss integrable discretizations of the sine gordon equation the massive thirring model and their generalizations

a reduced order dg formulation based on pod method for the time domain maxwell s equations in dispersive media abstract in this work a proper orthogonal decomposition pod method is applied to time domain maxwell s equations coupled to a drude dispersion model which are discretized in space by a discontinuous galerkin dg method an auxiliary differential equation ade method is used to represent the constitutive relation for the dispersive medium a pod dgtd formulation with lower dimension and sufficiently high accuracy is established together with the description of the pod reduced order basis its construction from a snapshot set and its application to the solution of the time domain maxwell s equations the overall goal is to reduce the computational time while maintaining an acceptable level of accuracy in order to obtain an efficient time domain solver to be used as a starting point for an optimization strategy we provide results from numerical experiments for two dimensional problems that illustrate the capabilities of the proposed pod dgtd formulation and assess its efficiency

mixed type soliton solutions for the n coupled hirota system in an optical fiber under investigation in this paper is the n coupled hirota system which describes the ultrashort pulse simultaneous propagation of the n field components in an optical fiber through the hirota method and symbolic computation using some auxiliary functions we obtain the improved bilinear forms and two mixed type soliton solutions for the 3 coupled hirota system while the system admits two mixed type solitons for all the cases of 1 2 and 3 which the represent the defocusing or focusing nonlinearities of the 3 components with the help the analytic and graphic analysis we study the properties and interactions for the two mixed type solitons for the 3 coupled hirota system amplitudes for the two mixed type one solitons are not affected by but their velocities are linearly related to with as the higher order effect which represents the relative width of the spectrum that arises due to the quasi monochromocity for the 2 bright 1 dark two solitons there exist the elastic and inelastic interactions while for the 1 bright 2 dark two solitons there is only the elastic interaction the 2 bright 1 dark and 1 bright 2 dark bound state two solitons are also presented finally we extend those results to the n coupled hirota system

multicomponent integrable wave equations i darboux dressing transformation the darboux dressing transformations are applied to the lax pair associated with systems of coupled nonlinear wave equations in the case of boundary values which are appropriate to both bright and dark soliton solutions the general formalism is set up and the relevant equations are explicitly solved several instances of multicomponent wave equations of applicative interest such as vector nonlinear schrodinger type equations and three resonant wave equations are considered

a discontinuous galerkin finite element method for swelling model of polymer gels abstract an attractive feature of discontinuous galerkin dg finite element schemes is that this concept offers a unified and versatile discretization platform for various types of partial differential equations the locality of the trial functions not only supports local mesh refinements but also offers a framework for comfortably varying the order of the discretization in this paper we propose and analyze a mixed dg finite element method for a displacement pressure model which describes swelling dynamics of polymer gels under mechanical constraints by introducing a flux variable we first present a reformulation of the governing equations of polymer gels we then approximate the pressure and flux variables by a mixed finite element space and the displacement by dg finite element method existence and uniqueness are proved and error estimates are derived for mixed dg finite element scheme finally numerical experiments are presented to show the performance of the mixed dg approximation of polymer gels

propagation of dark solitons in a system of coupled higher order nonlinear schrodinger equations in this paper we consider the higher order linear and nonlinear effects in optics and analyse a coupled system of higher order nonlinear schrodinger equations to identify the conditions for dark soliton propagation through painleve analysis to supplement the known bright soliton conditions we also construct the explicit lax pair and the hirota bilinear form is used to generate one and two dark solitons

on elliptic solutions of a coupled nonlinear schrodinger system abstract an explicit formula is obtained for single phase bounded elliptic solutions of the manakov system of integrable coupled nonlinear schrodinger equations in terms of the weierstrass sigma function with a real quasiperiod the parametrization is effective in the sense that the reality conditions are completely characterized for each of the three possible couplings focusing focusing defocusing defocusing and focusing defocusing the manakov soliton is recovered in the soliton limit and the small wave modulation limit is shown to satisfy the linearized dispersion relation of planewave solutions

on a coupled system of equations describing pulse propagation in quadratic media in the slowly varying envelope approximation we derive the basic equations that describe the propagation of ultrashort pulses in quadratically nonlinear media in which a wave at a fundamental frequency interacts with its second harmonic in the governing equations we keep linear terms that account for both second and third order dispersion and nonlinear terms describing both nonlinear dispersion and self steepening of the pulse edge we then perform the painleve singularity structure analysis of the most general system of coupled partial differential equations we derived in a specific case when third order dispersion is negligible by using a hirota like method we found zero and one parameter families of bright fundamental frequency and dark second harmonic solitary waves which travel at a locked group velocity