a new cloud based hp finite element method a hybrid computational method for solving boundary value problems is introduced which combines features of the meshless hp cloud methods with features of conventional finite elements the method admits straightforward nonuniform hp type approximations easy implementation of essential boundary conditions is robust under severe distortions of the mesh and can deliver exponential rates of convergence results of numerical experiments are presented

spectral theory for the periodic sine gordon equation a concrete viewpoint a summary of the spectral theory for quasiperiodic sine and sinh gordon equations is given analogies with whole line solitons and scattering theory motivates the discussion the relation between the ingredients in the inverse spectral solution of the periodic sine gordon equation and physical characteristics of sine gordon waves is emphasized the explicit topics covered are summarized in the table of contents in the introduction

on the stability of time harmonic localized states in a disordered nonlinear medium we study the problem of localization in a disordered one dimensional nonlinear medium modeled by the nonlinear schrodinger equation devillard and souillard have shown that almost every time harmonic solution of this random pde exhibits localization we consider the temporal stability of such time harmonic solutions and derive bounds on the location of any unstable eigenvalues by direct numerical determination of the eigenvalues we show that these time harmonic solutions are typically unstable and find the distribution of eigenvalues in the complex plane the distributions are distinctly different for focusing and defocusing nonlinearities we argue further that these instabilities are connected with resonances in a schrodinger problem and interpret the earlier numerical simulations of caputo newell and shelley and of shelley in terms of these instabilities finally in the defocusing case we are able to construct a family of asymptotic solutions which includes the stable limiting time harmonic state observed in the simulations of shelley

the generalized finite element method for helmholtz equation theory computation and open problems in this paper we address the generalized finite element method for the helmholtz equation we obtain our method by employing the finite element method on cartesian meshes which may overlap the boundaries of the problem domain and by enriching the approximation by plane waves pasted into the finite element basis at each mesh vertex by the partition of unity method here we address the q convergence of the method where q is the number of plane waves added at each vertex for the class of smooth analytic solutions for which we get better than exponential convergence for sufficiently small h depending on p an important observation is that we can monitor the accuracy in any computed solution quantity of interest at negligible cost by using q extrapolation our results assume exact integration of all the employed integrals further studies are needed to analyze the effects of the numerical integrations and also the effect of the roundoff errors

numerical chaos symplectic integrators and exponentially small splitting distances two discrete versions of duffing s equation are derived as reductions of discretizations of the nonlinear schrodinger nls equation inheriting the properties of the nsl discretizations one is shown to be integrable and using a mel nikov type analysis the other discretization is shown to be nonintegrable arguments are given why it is possible for the nonintegrable scheme to restore the homoclinic orbit at an exponentially fast rate which is faster than the order of the approximation as h 0 this suggests why there can be fundamental differences between symplectic and nonsymplectic discretizations of certain continuous hamiltonian systems

ccs with hennessy s merge has no finite equational axiomatization this paper confirms a conjecture of bergstra and klop s from 1984 by establishing that the process algebra obtained by adding an auxiliary operator proposed by hennessy in 1981 to the recursion free fragment of milner s calculus of communicating systems is not finitely based modulo bisimulation equivalence thus hennessy s merge cannot replace the left merge and communication merge operators proposed by bergstra and klop at least if a finite axiomatization of parallel composition modulo bisimulation equivalence is desired

an approximate sine gordon equation and its traveling wave solution in n 1 dimensional space in this paper we apply the painleve analysis to the study of an approximate sine gordon equation and obtain the traveling solitary wave solution in n 1 dimensional space explicitly which is significantly different from one which is found in our last work the technique described herein can also be applied to other nonlinear evolution equations such as the kdv type equation and the nonlinear schrodinger equation in higher dimensional space

the piecewise polynomial partition of unity functions for the generalized finite element methods a partition of unity pu function is an essential component of the generalized finite element method gfem the popular shepard pu functions which are rational functions are easy to construct but have difficulties in dealing with essential boundary conditions and require lengthy computing time for reasonable accuracy in numerical integration in this paper we introduce two simple pu functions the first is a highly regular piecewise polynomial consisting of two distinct polynomials that is effective for uniformly partitioned patches the second is a highly regular piecewise polynomial consisting of three distinct polynomials which is for arbitrary partitioned patches

dynamic properties of cubic nonlinear schrodinger equation with varying nonlinear parameter the dynamic properties of the cubic nonlinear schrodinger equation are investigated numerically using the symplectic scheme euler centred scheme we discuss the dynamic behaviour of the cubic nonlinear schrodinger equation with varying nonlinear parameter the results show that the system exhibits regular recurrence for weakly nonlinearity we also illustrate that the system will exhibit varying dynamic behaviour with increasing nonlinear parameter i e the system will show the homoclinic orbit hmo crossing quasi recurrence pseudorecurrence irregular motion or stochastic motion for a strongly nonlinear constants

on the approximability and the selection of particle shape functions particle methods also known as meshless or meshfree methods have become popular in approximating solutions of partial differential equations especially in the engineering community these methods do not employ a mesh or use it minimally in the construction of shape functions there is a wide variety of classes of shape functions that can be used in particle methods in this paper we primarily address the issue of selecting a class of shape functions among this wide variety that would yield efficient approximation of the unknown solution we have also made several comments and observations on the order of convergence of the interpolation error when these shape functions are used specifically we have shown that the interpolation error estimate for certain classes of shape functions may not indicate the actual order of convergence of the approximation error

influence of solitons in the initial state on chaos in the driven damped sine gordon system abstract the appearance of chaos in the a c driven damped sine gordon equation is studied numerically several transitions from periodic to chaotic behaviour are investigated in detail for flat initial conditions spatial structures breather kink in the initial conditions smooth out many of these transitions and give rise to an interesting symbiosis of time and spatial intermittency this symbiosis appears to be due to the competition between the background tendency towards chaos and the system s preference to maintain a spatial pattern the way that this competition is relieved is also found to depend very strongly on symmetry in the initial conditions

generalized p fem in homogenization summary a new finite element method for elliptic problems with locally periodic microstructure of length varepsilon 0 is developed and analyzed it is shown that the method converges as varepsilon rightarrow 0 to the solution of the homogenized problem with optimal order in varepsilon and exponentially in the number of degrees of freedom independent of varepsilon 0 the computational work of the method is bounded independently of varepsilon numerical experiments demonstrate the feasibility and confirm the theoretical results

a note on the singular linear system of the generalized finite element methods we characterize the kernel of the global stiffness matrix in the singular linear system of the generalized finite element methods gfem which uses the classical finite element fe shape functions and local approximation space of harmonic polynomials

a method to compute statistics of large noise induced perturbations of nonlinear schrodinger solitons we describe in detail the application of importance sampling to numerical simulations of large noise induced perturbations in soliton based optical transmission systems governed by the nonlinear schrodinger equation the method allows one to concentrate the samples in monte carlo simulations around those noise realizations that are most likely to produce the large pulse deformations connected with errors and it yields computational speedups of several orders of magnitude over standard monte carlo simulations we demonstrate the method by using it to calculate the probability density functions associated with pulse amplitude frequency timing and phase fluctuations in a prototypical soliton based communication system

discretizations integrable systems and computation discretizations and associated numerical computation of solutions of certain integrable systems such as the nonlinear schrodinger equation nls and the sine gordon sg equations with periodic boundary values can lead to instabilities chaotic and spurious results the chaos can be due to truncation errors or even roundoff errors and can be traced to the fact that these integrable systems are strongly unstable when the initial values are in the neighbourhood of homoclinic manifolds by using the associated nonlinear spectral transform of the nls equation and tracking the evolution of relevant eigenvalues one can observe and relate crossing of homoclinic manifolds to the temporal chaos in the waveforms when the initial data is even for general initial values even though there is no crossing of the unperturbed homoclinic manifolds the waveforms still exhibit chaotic phenomena which can be related to the evolution of the spectrum this paper reviews the current understanding of this intriguing phenomena and also compares the implementation of certain symplectic integrators and runge kutta algorithms for the nls and sg equations in regions of phase space proximate to the homoclinic manifolds

orbits homoclinic to resonances with an application to chaos in a model of the forced and damped sine gordon equation abstract in this paper we develop new global perturbation techniques for detecting homoclinic and heteroclinic orbits in a class of four dimensional ordinary differential equations that are perturbations of completely integrable two degree of freedom hamiltonian systems our methods are fundamentally different than other global perturbation methods e g standard melnikov theory in that we are seeking orbits homoclinic and heteroclinic to fixed points that are created in a resonance resulting from the perturbation our methods combine the higher dimensional melnikov theory with geometrical singular perturbation theory and the theory of foliations of invariant manifolds we apply our methods to a modified model of the forced and damped sine gordon equation developed by bishop et al we give explicit conditions in terms of the system parameters for the model to possess a symmetric pair of homoclinic orbits to a fixed point of saddle focus type chaotic dynamics follow from a theorem of silnikov this provides a mechanism for chaotic dynamics geometrically similar to that observed by bishop et al namely a random jumping between two spatially dependent states with an intermediate passage through a spatially independent state however in order for this type of silnikov dynamics to exist we require a different and unphysical type of damping compared to that used by bishop et al

a numerical study of the large period limit of a zakharov shabat eigenvalue problem with periodic potentials deconinck and kutz 2006 j comput phys 219 296 321 developed an efficient algorithm for solving the zakharov shabat eigenvalue problem with periodic potentials numerically it is natural to use the same algorithm for solving the problem for non periodic potential decaying potentials defined over the whole real line using large periods in this paper we propose the use of a specific value of the floquet exponent our numerical results indicate that it can produce accurate results long before the period becomes large enough for the analytical convergence results of gardner 1997 j reine angew math 491 149 81 to be valid we also illustrate the rather complicated path to convergence of some nonlinear schr dinger potentials

mesh based construction of flat top partition of unity functions a novel idea to construct flat top partition of unity functions in a closed form on a general structured or unstructured finite element mesh is presented an efficient and practical construction method of a flat top partition of unity function is important in the generalized finite element method gfem details on how to construct flat top partition of unity functions on a provided mesh are given the generalized finite element approximation with the use of the new flat top partition of unity function is presented with various numerical examples that demonstrate the effectiveness of proposed flat top partition of unity functions

chaos and coherent structures in partial differential equations abstract this paper addresses the possible connections between chaos the unpredictable behavior of solutions of finite dimensional systems of ordinary differential and difference equations and turbulence the unpredictable behavior of solutions of partial differential equations it is dedicated to martin kruskal on the occasion of his 60th birthday

effects of the smoothness of partitions of unity on the quality of representation of singular enrichments for gfem xfem stress approximations around brittle cracks abstract the convergence rates of the conventional generalized extended finite element method gfem xfem in crack modeling are similar to the convergence rates of the finite element method fem laborde et al 2005 and bechet et al 2005 unless the crack tip enrichment functions are applied in a subdomain with fixed dimension independently of the mesh parameter h and also demanding some special care for blending along the transition zone chahine et al 2006 and taracon et al 2009 thus to improve convergence rates more degrees of freedom dof are generated due to the larger quantity of enriched nodes this work seeks to identify and understand the advantages of better capturing the information provided by singular enrichments over mesh based smooth partitions of unity pou such pou with higher regularity can be built through the so called c k gfem framework following duarte et al 2006 based on a moving least square of degree zero and considering mesh based smooth weighting functions associated with arbitrary polygonal clouds the purpose herein is to investigate some possible advantages of mesh based smooth pou for modeling discontinuities and singularities in two dimensional problems of linear elastic fracture mechanics in such a fashion that the discretization error associated to stress discontinuities inherent in standard c 0 continuous gfem xfem approximations is eliminated the procedure shares features similar to the standard fem regarding domain partition and numerical integration but as neither the pou nor the enrichment functions are defined in natural domains the integrations are performed using only global coordinates the approximation capabilities of c k gfem discretizations with different patterns of singular enrichment distribution are investigated by analyzing the convergence rates of the h and p versions considering global measures in terms of strain energy and l 2 norm of displacements the effects on stability are also verified by analyzing the evolution of the condition number the effect of smoothness on conditioning is investigated and the eigenvalues distributions are used to identify the several aspects involved the smoothness of the pou the different types of enrichment functions and the pattern of enrichment the performance of the smooth approximations is compared to the c 0 counterparts built using conventional c 0 fem based pou it is shown that smoothness in the presence of extrinsically applied singular enrichments is important as the information provided by such enrichment is better captured in addition there are no stress jumps around singularities reducing error propagation beyond the neighborhood of the singularity

a quasi periodic route to chaos in a near integrable pde abstract pattern formation and transitions to chaos are described for the damped ac driven one dimensional periodic sine gordon equation in a nonlinear schrodinger regime a generic quasi periodic route to intermittent chaos is exhibited in detail using a range of dynamical systems diagnostics in addition a nonlinear spectral transform is exploited i to identify and quantify coordinates of space time attractors in terms of a small number of soliton modes of the underlying integrable system ii to use these analytic coordinates to identify homoclinic orbits as possible sources of chaos and iii to demonstrate the significance of energy transfer between coherent and extended states in this chaotic system

nonlinear normal modes for the toda chain abstract the toda chain is a nonlinear mass spring chain which can in principle be integrated analytically by the action angle theory of classical mechanics we show that the transformation from physical variables to action variables can be implemented very simply on a computer we study the correspondence between action variables and the motions of the chain and find the role of the action variables to be very similar to the role of normal mode amplitudes in the harmonic chain this similarity leads to a partly rigorous partly heuristic normal mode analysis for the toda chain new results are obtained when this normal mode analysis is applied to certain perturbations of the toda chain such as the fermi pasta ulam chain or the toda chain with a mass impurity

coherence and chaos in the driven damped sine gordon equation measurement of the soliton spectrum abstract a numerical procedure is developed which measures the sine gordon soliton and radiation content of any field t which is periodic in space the procedure is applied to the field generated by a damped driven sine gordon equation this field can be either temporally periodic locked to the driver or chaotic in either case the numerical measurement shows that the spatial structure can be described by only a few spatially localized soliton wave train modes the numerical procedure quantitatively identifies the presence number and properties of these soliton wave trains for example an increase of spatial symmetry is accompanied by the injection of additional solitons into the field

free shuffle algebras in language varieties we give simple concrete descriptions of the free algebras in the varieties generated by the shuffle semirings l p 0 1 or the semirings r r 0 1 where p is the collection of all subsets of the free monoid and r is the collection of all regular subsets the operation x y is the shuffle product

breakup of a multisoliton state of the linearly damped nonlinear schrodinger equation we address the breakup splitting of multisoliton solutions of the nonlinear schrodinger equation nlse occurring due to linear loss two different approaches are used for the study of the splitting process the first one is based on the direct numerical solution of the linearly damped nlse and the subsequent analysis of the eigenvalue drift for the associated zakharov shabat spectral problem the second one involves the multisoliton adiabatic perturbation theory applied for studying the evolution of the solution parameters with the linear loss taken as a small perturbation we demonstrate that in the case of strong nonadiabatic loss the evolution of the zakharov shabat eigenvalues can be quite nontrivial we also demonstrate that the multisoliton breakup can be correctly described within the framework of the adiabatic perturbation theory and can take place even due to small linear loss eventually we elucidate the occurrence of the splitting and its dependence on the phase mismatch between the solitons forming a two soliton bound state

the generalized finite element method this paper describes a pilot design and implementation of the generalized finite element method gfem as a direct extension of the standard finite element method sfem or fem which makes possible the accurate solution of engineering problems in complex domains which may be practically impossible to solve using the fem the development of the gfem is illustrated for the laplacian in two space dimensions in domains which may include several hundreds of voids and or cracks for which the construction of meshes used by the fem is practically impossible the two main capabilities are 1 it can construct the approximation using meshes which may overlap part or all of the domain boundary 2 it can incorporate into the approximation handbook functions which are known analytically or are generated numerically and approximate well the solution of the boundary value problem in the neighborhood of corner points voids cracks etc the main tool is a special integration algorithm which we call the fast remeshing approach which is robust and works for any domain with arbitrary complexity the incorporation of the handbook functions into the gfem is done by employing the partition of unity method pum the presented formulations and implementations can be easily extended to the multi material medium where the voids are replaced by inclusions of various shapes and sizes and to the case of the elasticity problem this work can also be understood as a pilot study for the feasibility and demonstration of the capabilities of the gfem which is needed before analogous implementations are attempted in the three dmensional and nonlinear cases which are the cases of main interest for future work

the generalized product partition of unity for the meshless methods the partition of unity is an essential ingredient for meshless methods named by gfem pufem partition of unity fem xfem extended fem rkpm reproducing kernel particle method rppm reproducing polynomial particle method the method of hp clouds in the literature there are two popular choices for partition of unity a piecewise linear fem mesh and the shepard type partition of unity however the partition of unity pu by a fem mesh leads to the singular or nearly singular matrices and non smooth approximation functions the shepard type partition of unity requires lengthy computing time and its implementation is difficult in order to alleviate these difficulties oh et al introduced the smooth piecewise polynomial pu functions with flat top that lead to small matrix condition numbers and almost everywhere partition of unity that can handle essential boundary conditions nevertheless we could not have the smooth closed form pu functions with flat top for general polygonal patches 2d and general polyhedral patches 3d in this paper we introduce one of the most simple and efficient partition of unity called the generalized product partition of unity the product pu functions constructed by this method are the closed form smooth piecewise polynomials with flat top and could handle background meshes general polygonal patches as well as general polyhedral patches arising in practical applications of meshless methods

arbitrary branched and intersecting cracks with the extended finite element method summary extensions of a new technique for the nite element modelling of cracks with multiple branches multiple holes and cracks emanating from holes are presented this extended nite element method x fem allows the representation of crack discontinuities and voids independently of the mesh a standard displacementbased approximation is enriched by incorporating discontinuous elds through a partition of unity method a methodology that constructs the enriched approximation based on the interaction of the discontinuous geometric features with the mesh is developed computation of the stress intensity factors sif in dierent examples involving branched and intersecting cracks as well as cracks emanating from holes are presented to demonstrate the accuracy and the robustness of the proposed technique copyright 2000 john wiley sons ltd

the recognition of series parallel digraphs we present a linear time algorithm to recognize the class of vertex series parallel vsp digraphs our method is based on the relationship between vsp digraphs and the class of edge series paralle

generalized finite element method using mesh based handbooks application to problems in domains with many voids this paper describes a new version of the generalized finite element method originally developed int j numer methods engrg 47 2000 1401 comput methods appl mech engrg 181 2000 43 the design and implementation of the generalized finite element method ph d thesis texas am comput methods appl mech engrg 190 2001 4081 which is well suited for problems set in domains with a large number of internal features e g voids inclusions cracks etc the main idea is to employ handbook functions constructed on subdomains resulting from the mesh discretization of the problem domain the proposed new version of the gfem is shown to be robust with respect to the spacing of the features and is capable of achieving high accuracy on meshes which are rather coarse relative to the distribution of the features

linear dependence problems of partition of unity based generalized fems a known problem of partition of unity based generalized finite element methods referred to as gfem is the linear dependence problem which leads to singular global stiffness matrices thus far attempts to eliminate the linear dependence problem have been unsuccessful numerical experiments are carried out among several gfems to investigate the problem based on the numerical experiments simple but effective approaches to eliminating the linear dependence problem are suggested

numerical algorithms for the direct spectral transform with applications to nonlinear schrodinger type systems we implement two different algorithms for computing numerically the direct zakharov shabat eigenvalue problem on the infinite line the first algorithm replaces the potential in the eigenvalue problem by a piecewise constant approximation which allows one to solve analytically the corresponding ordinary differential equation the resulting algorithm is of second order in the step size the second algorithm uses the fourth order runge kutta method we test and compare the performance of these two algorithms on three exactly solvable potentials we find that even though the runge kutta method is of higher order this extra accuracy can be lost because of the additional dependence of its numerical error on the eigenvalue this limits the usefulness of the runge kutta algorithm to a region inside the unit circle around the origin in the complex plane of the eigenvalues for the computation of the continuous spectrum density this limitation is particularly severe as revealed by the spectral decomposition of thel2 norm of a solution to the nonlinear schrodinger equation we show that no such limitations exist for the piecewise constant algorithm in particular this scheme converges uniformly for both continuous and discrete spectrum components

extended finite element method for three dimensional crack modelling an extended finite element method x fem for three dimensional crack modelling is described a discontinuous function and the two dimensional asymptotic crack tip displacement fields are added to the finite element approximation to account for the crack using the notion of partition of unity this enables the domain to be modelled by finite elements with no explicit meshing of the crack surfaces computational geometry issues associated with the representation of the crack and the enrichment of the finite element approximation are discussed stress intensity factors sifs for planar three dimensional cracks are presented which are found to be in good agreement with benchmark solutions copyright 2000 john wiley sons ltd

superconvergence in the generalized finite element method in this paper we address the problem of the existence of superconvergence points of approximate solutions obtained from the generalized finite element method gfem of a neumann elliptic boundary value problem gfem is a galerkin method that uses non polynomial shape functions and was developed in babuska et al in siam j numer anal 31 945 981 1994 babuska et al in int j numer meth eng 40 727 758 1997 melenk and babuska in comput methods appl mech eng 139 289 314 1996 in particular we show that the superconvergence points for the gradient of the approximate solution are the zeros of a system of non linear equations this system does not depend on the solution of the boundary value problem for approximate solutions with second derivatives we have also characterized the superconvergence points of the second derivatives of the approximate solution as the roots of a system of non linear equations we note that smooth generalized finite element approximation is easy to construct

the design and analysis of the generalized finite element method abstract in this paper we introduce the generalized finite element method gfem as a combination of the classical finite element method fem and the partition of unity method pum the standard finite element spaces are augmented by adding special functions which reflect the known information about the boundary value problem and the input data e g the singular functions obtained from the local asymptotic expansion of the exact solution in the neighborhood of a corner point etc the special functions are multiplied with the partition of unity corresponding to the standard linear vertex shape functions and pasted together with the existing finite element basis to construct an augmented conforming finite element space in this way the local approximability afforded by the special functions is included in the approximation while maintaining the existing infrastructure of finite element codes the major features of the gfem are 1 the essential boundary conditions can be imposed exactly as in the standard fem unlike other partition of unity based methods where this is a major issue 2 the accuracy of the numerical integration of the entries of the stiffness matrix and load vector is controlled adaptively so that the errors in integration of the special functions do not affect the accuracy of the constructed approximation this issue also has not been sufficiently addressed in other implementations of partition of unity or meshless methods and 3 linear dependencies in the system of equations are resolved by employing an easy modification of the direct linear solver the power of the gfem for solving problems in domains with complex geometry with less error and less computer resources than the standard fem is illustrated by numerical examples

subharmonic and homoclinic bifurcations in the driven and damped sine gordon system abstract chaotic responses induced by an applied biharmonic driven signal on the sine gordon sg system influenced by a constant dc driven and the damping fields are investigated using a collective coordinate approach for the motion of the breather in the system for this biharmonic signal one term has a large amplitude at low frequency thus the classical melnikov method does not apply to such a system however we use the modified version of the melnikov method to homoclinic bifurcations of the perturbed sg system additionally resonant breathers are studied using the modified subharmonic melnikov theory this dynamic behavior is illustrated by some numerical computations

parametric instabilities in the discrete sine gordon equation abstract the one dimensional sine gordon equation is an example of an exactly solvable nonlinear partial differential equation when discretized on a periodic lattice in space and in time it corresponds to a lattice of pendula coupled by linear springs we show that the discretized system is unstable to a parametric mode when the frequency of time discretization is sufficiently small and we obtain the instability condition and growth rate by considering the effects of two finite amplitude modes we also obtain the nonlinear saturation of the instability we also examine how solutions of the exact sine gordon equation behave under this map both in and away from the parametrically unstable regime

mel nikov analysis of numerically induced chaos in the nonlinear schro dinger equation abstract certain hamiltonian discretizations of the periodic focusing nonlinear schrodinger equation nls have been shown to be responsible for the generation of numerical instabilities and chaos in this paper we undertake a dynamical systems type of approach to modeling the observed irregular behavior of a conservative discretization of the nls using heuristic mel nikov methods the existence of a pair of isolated homoclinic orbits is indicated for the perturbed system the structure of the persistent homoclinic orbits that are predicted by the mel nikov theory possesses the same features as the wave form observed numerically in the perturbed system after the onset of chaotic behavior and appears to be the main structurally stable feature of this type of chaos the mel nikov analysis implemented in the pde context appears to provide relevant qualitative information about the behavior of the pde in agreement with the numerical experiments in a neighborhood of the persistent homoclinic orbits the existence of a horseshoe is investigated and related with the onset of chaos in the numerical study

piecewise polynomial spaces and geometric continuity of curves we say that a curve has geometric continuity if its curvatures and frenet frame are continuous in this paper we introduce spaces of piecewise polynomials which can be used to model space curves which have geometric continuity we show that the basic theoretical properties of ordinary spline functions also hold for these spaces these results extend and unify recent work on beta splines and nu splines which are used as a design tool in computer aided geometric design of free form curves and surfaces

numerical inverse spectral transform for the periodic sine gordon equation theta function solutions and their linearized stability abstract the inverse spectral transform for integrable nonlinear ordinary and partial differential equations such as the toda lattice korteweg de vries sine gordon and nonlinear schrodinger equations provides explicit algorithms to generate exact solutions under periodic or quasiperiodic boundary conditions these oscillatory wavetrains may be prescribed a priori to consist of a nonlinear superposition of n phases j x t j x j t 0 j j 1 n where the wave is 2 periodic independently in each phase this paper exhibits the numerical implementation of the inverse spectral solution of the sine gordon equation the general construction is outlined and then implemented for n 1 2 and 3 we compute 1 the exact theta function solutions 2 the floquet spectrum of x periodic solutions 3 the labelling of linearized instabilities of n phase solutions in terms of spectral data and 4 the linearized growth rate in each unstable mode the associated surfaces q n x t are displayed to illustrate a variety of spatial and dynamical phenomena in the oscillatory solution space of this integrable system

the equational theory of pomsets abstract pomsets have been introduced as a model of concurrency since a pomset is a string in which the total order has been relaxed to be a partial order in this paper we view them as a generalization of strings and investigate their algebraic properties in particular we investigate the axiomatic properties of pomsets sets of pomsets and ideals of pomsets under such operations as concatenation parallel composition union and their associated closure operations we find that the equational theory of sets pomsets under concatenation parallel composition and union is finitely axiomatizable whereas the theory of languages under the analogous operations is not a similar result is obtained for ideals of pomsets which incorporate the notion of subsumption which is also known as augmentation finally we show that the addition of any closure operation parallel or serial leads to nonfinite axiomatizability of the resulting equational theory

extracting solitons from noisy pulses we describe an iterative method that extracts the underlying soliton from a noisy pulse the method is formulated as a functional iteration at each step the soliton component of the difference between the noisy pulse and the current underlying soliton is determined via soliton perturbation theory this is then added to the soliton and the process is repeated we show that this iteration converges if the perturbation is not too large and we give the specific types of deviations which most easily cause the iteration to fail to converge as an example of the method s use we apply it to obtain improved statistics of the amplitude phase frequency and position of a soliton propagating in an optical fiber in the presence of amplifier noise

a unified enrichment approach addressing blending and conditioning issues in enriched finite elements abstract we present a combination of techniques to improve the convergence and conditioning properties of partition of unity pu enriched finite element methods by applying these techniques to different types of enrichment functions namely polynomial discontinuous and singular higher order convergence rates can be obtained while keeping condition number growth rates similar to the ones corresponding to standard finite elements

a method to compute statistics of large noise induced perturbations of nonlinear schrodinger solitons we demonstrate in detail the application of importance sampling to the numerical simulation of large noise induced perturbations in soliton based optical transmission systems governed by the nonlinear schrodinger equation the method allows one to concentrate numerical monte carlo simulations around the noise realizations that are most likely to produce the large pulse deformations connected with errors and yields computational speedups of several orders of magnitude over standard monte carlo simulations we demonstrate the method by using it to calculate the probability density functions associated with pulse amplitude frequency and timing fluctuations in a prototypical soliton based communication system

axiomatizing shuffle and concatenation in languages abstract we consider the varietylanggenerated by all language structures p 0 1 and the varietylg of ordered algebras generated by the structures p 0 1 wherep is the powerset of and whereb cis the complex concatenation of the languagesb c b cis their shuffle product andb cis their union we prove that for each finite seteof equations valid inlangthere is a finite modelseofein which some inequation valid inlg fails it follows that neither variety is finitely axiomatizable

characterization of strange attractors a new measure of strange attractors is introduced which offers a practical algorithm to determine their character from the time series of a single observable the relation of this new measure to fractal dimension and information theoretic entropy is discussed

computation of the direct scattering transform for the nonlinear schroedinger equation abstract the cubic nonlinear schroedinger equation nls describes the space time evolution of narrow banded wave trains in one space and one time 1 1 dimensions the richness of nonlinear wave motions described by nls is exemplified by the fully nonlinear envelope soliton and breather solutions which are fully understood only in terms of the general solution of the equation as described by the inverse scattering transform 1st the method may be viewed as a nonlinear generalization of the linear fourier transform herein we develop a numerical algorithm for determining the scattering transform spectrum of a nonlinear wave train described by the nls equation the analysis of space or time series data obtained from computer simulations of nonlinear narrow band wave trains or from experimental measurements is thus a central point of discussion in particular we develop a numerical algorithm for computing the direct scattering transform dst which may be interpreted as the nonlinear fourier spectrum of the complex envelope function of a wave train the fact that the nonlinear fourier modes are constants of the motion for all time provides a physical basis for the analysis of data while the nonlinear fourier method is specifically applied to the nls equation the approach is easily generalized to include the class of spectral problems due to ablowitz kaup newell and segur akns this class includes several other nonlinear wave equations of physical interest in 1 1 including the kortewegde vries kdv modified kdv sine gordon and sinh gordon equations

combustion waves in hydraulically resistant porous media in a special parameter regime abstract in this paper we study the stability of fronts in a reduction of a well known pde system that is used to model the combustion in hydraulically resistant porous media more precisely we consider the original pde system under the assumption that one of the parameters of the model the lewis number is chosen in a specific way and with initial conditions of a specific form for a class of initial conditions then the number of unknown functions is reduced from three to two for the reduced system the existence of combustion fronts follows from the existence results for the original system the stability of these fronts is studied here by a combination of energy estimates and numerical evans function computations and nonlinear analysis when applicable we then lift the restriction on the initial conditions and show that the stability results obtained for the reduced system extend to the fronts in the full system considered for that specific value of the lewis number the fronts that we investigate are proved to be either absolutely unstable or convectively unstable on the nonlinear level