a parameter method for computing highly oscillatory integrals this paper presents a new efficient parameter method for integration of the highly oscillatory integral 0 1f x e i w g x dx with an irregular oscillator the effectiveness and accuracy are tested by means of numerical examples for the case where g x has stationary points

on the stability of perfect viscoelastic columns abstract the stability problem associated with an euler bernoulli beam made of an arbitrary linear viscoelastic material is formulated the three parameter and the kelvin voigt models are analyzed in the presence of constant as well as periodic loads the application of a finite time stability concept is shown for the constant loading case when the traditional stability criterion fails to make sense for the case of a periodic loading the stability diagrams are obtained through an application of floquet theory it is found that the addition of periodic loads may significantly alter the stability behavior of a column which is originally subjected to a constant load only

a comparative study of three boundary element approaches to calculate the transient response of viscoelastic solids with unbounded domains abstract as an alternative to domain discretization methods the boundary element method bem provides a powerful tool for the calculation of dynamic structural response in frequency and time domain field equations of motion and boundary conditions are cast into boundary integral equations bie which are discretized only on the boundary fundamental solutions are used as weighting functions in the bie which fulfil the sommerfeld radiation condition i e the energy radiation into a surrounding medium is modelled correctly therefore infinite and semi infinite domains can be effectively treated by the method the soil represents such a semi infinite domain in soil structure interaction problems the response to vibratory loads superimposed to static pre loads can often be calculated by linear viscoelastic constitutive equations conventional viscoelastic constitutive equations can be generalized by taking fractional order time derivatives into account in the present paper two time domain bem approaches including generalized viscoelastic behaviour are compared with the laplace domain bem approach and subsequent numerical inverse transformation one of the presented time domain approaches uses an analytical integration of the elastodynamic bie in a time step viscoelastic constitutive properties are introduced after laplace transformation by means of an elastic viscoelastic correspondence principle the transient response is obtained by inverse transformation in each time step the other time domain approach is based on the so called convolution quadrature method in this formulation the convolution integral in the bie is numerically approximated by a quadrature formula whose weights are determined by the same laplace transformed fundamental solutions used in the first method and a linear multistep method a numerical study of wave propagation problems in 3 d viscoelastic continuum is performed for comparing the three bem formulations

wronskian solutions of the constrained kadomtsev petviashvili hierarchy the integrable kadomtsev petviashvili kp hierarchy is compatible with generalized k constraints of the type lk i qi 1xri a large class of solutions among them solitons can be represented by wronskian determinants of functions satisfying a set of linear equations in this paper we shall obtain additional conditions for these functions imposed by the constraints

gmres for the differentiation operator we investigate using the gmres method with the differentiation operator this operator is unbounded and thus does not fall into the framework of existing krylov subspace theory we establish conditions under which a function can be approximated by its own derivatives in a domain of the complex plane these conditions are used to determine when gmres converges this algorithm outperforms traditional quadrature schemes for a large class of highly oscillatory integrals even when the kernel of oscillations is unknown

numerical evaluation of highly oscillatory bessel transforms abstract in this paper we mainly focus on numerical evaluation of highly oscillatory bessel transforms based on the multiple integral the schemes for computing this class of the transform are presented by means of the bounds of special functions we give the corresponding errors of the schemes to demonstrate the efficiency and accuracy of the proposed methods some test examples are shown

backlund transformations and loop group actions we construct a local action of the group of rational maps from 2 to gl n on local solutions of flows of the zs akns sl n hierarchy we show that the actions of simple elements linear fractional transformations give local backlund transformations and we derive a permutability formula from different factorizations of a quadratic element we prove that the action of simple elements on the vacuum may give either global smooth solutions or solutions with singularities however the action of the subgroup of the rational maps that satisfy the u n reality condition on the space of global rapidly decaying solutions of the flows in the u n hierarchy is global and the action of a simple element gives a global backlund transformation the actions of certain elements in the rational loop group on the vacuum give rise to explicit time periodic multisolitons multibreathers we show that this theory generalizes the classical backlund theory of the sine gordon equation the group structures of backlund transformations for various hierarchies are determined by their reality conditions we identify the reality conditions the group structures for the sl n u k n k kdv kupershmidt wilson and gel fand dikii hierarchies the actions of linear fractional transformations that satisfy a reality condition modulo the center of the group of rational maps give backlund and darboux transformations for the hierarchy defined by the reality condition since the factorization cannot always be carried out under this reality condition the action is again local and backlund transformations only generate local solutions for these hierarchies unless singular solutions are allowed 2000 john wiley sons inc

rogue waves rational solutions the patterns of their zeros and integral relations the focusing nonlinear schrodinger equation which describes generic nonlinear phenomena including waves in the deep ocean and light pulses in optical fibres supports a whole hierarchy of recently discovered rational solutions we present recurrence relations for the hierarchy the pattern of zeros for each solution and a set of integral relations which characterizes them

fast fourier galerkin methods for solving singular boundary integral equations numerical integration and precondition we develop a fast fully discrete fourier galerkin method for solving a class of singular boundary integral equations we prove that the number of multiplications used in generating the compressed matrix is o nlog 3n and the solution of the proposed method preserves the optimal convergence order o n t where n is the order of the fourier basis functions used in the method and t denotes the degree of regularity of the exact solution moreover we propose a preconditioning which ensures the numerical stability when solving the preconditioned linear system numerical examples are presented to confirm the theoretical estimates and to demonstrate the approximation accuracy and computational efficiency of the proposed algorithm

a two dimensional bem fem coupling applied to viscoelastic analysis of composite domains a coupling between the boundary element and finite element methods is studied for the viscoelastic analysis of reinforced media the viscous behaviour of the composed body is taken into account by an alternative bem methodology developed for the boltzmann model this methodology is based on differential constitutive relations for viscoelasticity the reinforcements are modelled by finite elements and are considered elastic the coupling is based on the sub region technique due to its generality and easy implementation the resulting time marching process is able to represent both the instantaneous and the time dependent behaviour of a body subjected to general boundary conditions the method is validated by an experimental result and its accuracy tested by comparing numerical results with analytical solutions the generality of the method is proved by an infinite domain application copyright 2003 john wiley sons ltd

other 2n 2 parameters solutions of the nls equation and 2n 1 highest amplitude of the modulus of the n th order ap breather in this paper we construct new deformations of the akhmediev peregrine ap breather of order n or apn breather with real parameters other families of quasirational solutions of the nonlinear schrodinger nls equation are obtained we evaluate the highest amplitude of the modulus of the ap breather of order n we give the proof that the highest amplitude of the apn breather is equal to we get new formulas for the solutions of the nls equation which are different from these already given in previous works new solutions for the order 8 and their deformations according to the parameters are explicitly given we simultaneously get triangular configurations and isolated rings moreover the appearance for certain values of the parameters and of new configurations of concentric rings are underscored

computing the hilbert transform and its inverse we construct a new method for approximating hilbert transforms and their inverse throughout the complex plane both problems can be formulated as riemann hilbert problems via plemelj s lemma using this framework we re derive existing approaches for computing hilbert transforms over the real line and unit interval with the added benefit that we can compute the hilbert transform in the complex plane we then demonstrate the power of this approach by generalizing to the half line combining two half lines we can compute the hilbert transform of a more general class of functions on the real line than is possible with existing methods

properties of solutions of the kadomtsev petviashvili i equation the kadomtsev petviashvili i kpi equation is considered as a useful laboratory for experimenting with new theoretical tools able to handle the specific features of integrable models in 2 1 dimensions the linearized version of the kpi equation is first considered by solving the initial value problem for different classes of initial data properties of the solutions in different cases are analyzed in details the obtained results are used as a guideline for studying the properties of the solution u t x y of the kadomtsev petviashvili i kpi equation with given initial data u 0 x y belonging to the schwartz space the spectral theory associated to kpi is studied in the space of the fourier transform of the solutions the variables p p1 p2 of the fourier space are shown to be the most convenient spectral variables to use for spectral data spectral data are shown to decay rapidly at large p but to be discontinuous at p 0 direct and inverse problems are solved with special attention to the behavior of

the cauchy problem of coupled elliptic sine gordon equations with noise analysis of a general kernel based regularization and reliable tools of computing abstract developments in numerical methods for problems governed by nonlinear partial differential equations underpin simulations with sound arguments in diverse areas of science and engineering in this paper we explore the regularization method for the coupled elliptic sine gordon equations along with cauchy data the system of equations originates from the static case of the coupled hyperbolic sine gordon equations modeling the coupled josephson junctions in superconductivity and so far it addresses the josephson junctions in general the cauchy problem is not well posed and herein the hadamard instability occurs drastically generalizing the kernel based regularization method we propose a stable approximate solution confirmed by the error estimate this solution strongly converges to the exact solution in l 2 norm the main concern of this paper is also with the way to compute the regularized solution formed by an alike integral equation we employ the proposed techniques that successfully approximated the highly oscillatory integral and apply the picard like iteration to organize an efficient and reliable tool of computations the results are viewed as the improvement as well as the generalization of many previous works the paper is also accompanied by a numerical example that demonstrates the potential of this idea

rcms right correction magnus series approach for oscillatory odes we consider rcms a method for integrating differential equations of the form y a a1 t y with highly oscillatory solution it is shown analytically and numerically that rcms can accurately integrate problems using stepsizes determined only by the characteristic scales of a1 t typically much larger than the solution wavelength in fact for a given t grid the error decays with or is independent of increasing solution oscillation rcms consists of two basic steps a transformation which we call the right correction and solution of the right correction equation using a magnus series with suitable methods of approximating the highly oscillatory integrals appearing therein rcms has high order of accuracy with little computational work moreover rcms respects evolution on a lie group we illustrate with application to the 1d schrodinger equation and to frenet serret equations the concept of right correction integral series schemes is suggested and right correction neumann schemes are discussed asymptotic analysis for a large class of odes is included which gives certain numerical integrators converging to exact asymptotic behaviour

on evaluation of bessel transforms with oscillatory and algebraic singular integrands in this paper we study efficient methods for computing the integrals of the form 0 1x a 1 x bf x j v wx dx where a b v w are the given constants and w 1 j v is the bessel function of the first kind and of order v f is a sufficiently smooth function on 0 1 firstly we express the moments in a closed form with the aid of special functions secondly we induce the filon type method based on the taylor interpolation polynomial at two endpoints and the hermite interpolation polynomial at clenshaw curtis points on evaluating the highly oscillatory bessel integrals with algebraic singularity theoretical results and numerical experiments perform that the methods are very efficient in obtaining very high precision approximations if w is sufficiently large

analysis of inhomogeneous anisotropic viscoelastic bodies described by multi parameter fractional differential constitutive models the response to static loads of plane inhomogeneous anisotropic bodies made of linear viscoelastic materials is investigated multi parameter differential viscoelastic constitutive equations are employed which are generalized using fractional order time derivatives the governing equations which are derived by considering the equilibrium of the plane body element are two coupled linear fractional evolution partial differential equations in terms of the displacement components using the analog equation method aem in conjunction with the boundary element method bem these equations are transformed into a system of multi term ordinary fractional differential equations fdes which are solved using a numerical method for fdes developed recently by katsikadelis numerical examples are presented which not only demonstrate the efficiency of the solution procedure and validate its accuracy but also permit a better understanding of the response of plane bodies described by different viscoelastic models

degenerate determinant representation of solutions of the nonlinear schrodinger equation higher order peregrine breathers and multi rogue waves we present a new representation of solutions of the focusing nls equation as a quotient of two determinants this work is based on a recent paper in which we have constructed a multi parametric family of solutions of this equation in terms of wronskians this formulation was written in terms of a limit involving a parameter here we give a very compact formulation without presence of a limit this is a completely new result which gives a very efficient procedure to construct families of quasi rational solutions of the nls equation with this method we construct peregrine breathers of orders n 4 6 and multi rogue waves associated by deformation of parameters

abundant lump type solutions of the jimbo miwa equation in 3 1 dimensions abstract based on the hirota bilinear form of the 3 1 dimensional jimbo miwa equation ten classes of its lump type solutions are generated via maple symbolic computations whose analyticity can be easily achieved by taking special choices of the involved parameters those solutions supplement the existing lump type solutions presented previously in the literature

filon clenshaw curtis rules for a class of highly oscillatory integrals with logarithmic singularities in this work we propose and analyse a numerical method for computing a family of highly oscillatory integrals with logarithmic singularities for these quadrature rules we derive error estimates in terms of n the number of nodes k the rate of oscillations and a sobolev like regularity of the function we prove that the method is not only robust but the error even decreases for fixed n as k increases practical issues about the implementation of the rule are also covered in this paper by a writing down ready to implement algorithms b analysing the numerical stability of the computations and c estimating the overall computational cost we finish by showing some numerical experiments which illustrate the theoretical results presented in this paper

boundary elements and the analog equation method for the solution of elastic problems in 3 d non homogeneous bodies abstract the application of the boundary element method bem to elastostatic problems involving 3d non homogeneous materials like functionally graded materials fgms is presented in this paper the analog equation method aem is used to transform the original problem into a new problem with unknown fictitious source but known fundamental solution by means of this transformation a system of uncoupled domain boundary integral equations d bies is first obtained combining standard boundary element discretization and radial basis functions rbfs approximation for the fictitious source the application of the original differential operator to the displacement bie provides the extra equations to compute the unknown fictitious source the boundary character of the method is maintained since the integrals involved in the equations are limited only to the boundary since the rbfs are selected in such a way that the corresponding analog equation could be solved analytically numerical examples for three dimensional problems in continuously non homogeneous isotropic and linear elastic fgms are presented and discussed

wkb based schemes for the oscillatory 1d schrodinger equation in the semiclassical limit an efficient and accurate numerical method is presented for the solution of highly oscillatory differential equations while standard methods would require a very fine grid to resolve the oscillations the presented approach uses first an analytic second order wkb type transformation which filters out the dominant oscillations the resulting ode is much smoother and can hence be discretized on a much coarser grid with significantly reduced numerical costs in many practically relevant examples the method is even asymptotically correct w r t the small parameter varepsilon that identifies the oscillation wavelength indeed in these cases the error then vanishes for varepsilon to0 even on a fixed spatial mesh applications to the stationary schrodinger equation are presented

classifying the hierarchy of nonlinear schrodinger equation rogue wave solutions we present a systematic classification for higher order rogue wave solutions of the nonlinear schrodinger equation constructed as the nonlinear superposition of first order breathers via the recursive darboux transformation scheme this hierarchy is subdivided into structures that exhibit varying degrees of radial symmetry all arising from independent degrees of freedom associated with physical translations of component breathers we reveal the general rules required to produce these fundamental patterns consequently we are able to extrapolate the general shape for rogue wave solutions beyond order 6 at which point accuracy limitations due to current standards of numerical generation become non negligible furthermore we indicate how a large set of irregular rogue wave solutions can be produced by hybridizing these fundamental structures

rational solutions of the kadomtsev petviashvili hierarchy and the dynamics of their poles i new form of a general rational solution a new approach to the construction of the rational solutions to the hierarchy of the kadomtsev petviashvili equation is presented the generalization of the superposition formula is found to permit the construction of a new general solution from some partial ones the features of the general polynomial factorization and the types of scattering in the many body calogero moser problem are investigated

fast integration of rapidly oscillatory functions a collocation method for approximating integrals of rapidly oscillatory functions is presented the method is efficient for integrals involving bessel functions jv rx with large oscillation parameter r as well as for many other one and multi dimensional integrals of functions with rapid irregular oscillations

deformations of third order peregrine breather solutions of the nonlinear schrodinger equation with four parameters we present a new representation of solutions of the one dimensional nonlinear focusing schr odinger equation nls as a quotient of two determinants this formulation gives in the case of the order 3 new solutions with four parameters this gives a very efficient procedure to construct families of quasirational solutions of the nls equation and to describe the apparition of multirogue waves with this method we construct analytical expressions of four parameters solutions when all these parameters are equal to 0 we recover the peregrine breather of order 3 it makes possible with this four parameters representation to generate all the types of patterns for the solutions like the triangular configurations or the ring structures

numerical integration of oscillatory airy integrals with singularities on an infinite interval abstract this work is devoted to the quadrature rules and asymptotic expansions for two classes of highly oscillatory airy integrals on an infinite interval we first derive two important asymptotic expansions in inverse powers of the frequency then based on structure characteristics of the two asymptotic expansions in inverse powers of the frequency both the so called filon type method and the more efficient clenshaw curtis filon type method are introduced and analyzed the required moments in the former can be explicitly expressed by the meijer g functions the latter can be implemented in o n log n operations based on fast fourier transform fft and fast computation of the modified moments here we can construct two useful recurrence relations for computing the required modified moments accurately with the help of the airy s equation and some properties of the chebyshev polynomials particularly we also provide their error analyses in inverse powers of the frequency furthermore the presented error analysis shows the advantageous property that the accuracy improves greatly as increases numerical examples are provided to illustrate the efficiency and accuracy of the proposed methods

extreme superposition rogue waves of infinite order and the painleve iii hierarchy we study the fundamental rogue wave solutions of the focusing nonlinear schrodinger equation in the limit of large order using a recently proposed riemann hilbert representation of the rogue wave solution of arbitrary order k we establish the existence of a limiting profile of the rogue wave in the large k limit when the solution is viewed in appropriate rescaled variables capturing the near field region where the solution has the largest amplitude the limiting profile is a new particular solution of the focusing nonlinear schrodinger equation in the rescaled variables the rogue wave of infinite order which also satisfies ordinary differential equations with respect to space and time the spatial differential equations are identified with certain members of the painleve iii hierarchy we compute the far field asymptotic behavior of the near field limit solution and compare the asymptotic formulas with the exact solution using numerical methods for solving riemann hilbert problems in a certain transitional region for the asymptotics the near field limit function is described by a specific globally defined tritronquee solution of the painleve ii equation these properties lead us to regard the rogue wave of infinite order as a new special function

evaluation of cauchy principal value integrals of oscillatory kind abstract this paper studies the evaluation of cauchy principal value c p v integrals of oscillatory kind a b f x x e i x d x where a b the singularity of the c p v integral is transferred to an individual oscillatory integral that can be analytically calculated the remaining integral which is non singular can be well calculated with a stable and fast quadrature method namely the improved levin method numerical examples confirm the advantages of the proposed method

on the convergence of filon quadrature we analyze the convergence behavior of filon type quadrature rules by making explicit the dependence on both k the parameter that controls the oscillatory behavior of the integrand and n the number of function evaluations we provide explicit conditions on the domain of analyticity of the integrand to ensure convergence for n

an incremental relaxation finite element analysis of viscoelastic problems with contact and friction abstract based on the principle of virtual work a general but effective finite element technique associated with an incremental relaxation procedure is established for the analysis of viscoelastic contact problems with friction a generalized maxwell model is used to model the viscoelastic constitutive equations of which the relaxation function is represented by the sum of a series of decaying exponential functions of time the contact behaviors are accurately performed through a transformation matrix method by the incremental relaxation procedure developed the stress of each element of the generalized maxwell model at instant t n 1 includes two parts 1 the part decaying from the stress at instant t n during the time interval t n t n 1 and 2 the incremental stress induced from the incremental strain applied at instant t n the total stress at instant t n 1 can thus be computed by the sum of the stress of the elements of the generalized maxwell model and is only dependent on the previous stress level and the incremental strain applied at instant t n hence both the total strain and displacement at each instant are not used in the total stress calculation in addition the numerical errors caused by selecting larger time increments can also be avoided finally the influence of friction on the contact traction distributed on the contact surface at various time instants is presented

method for numerical integration of rapidly oscillating functions in diffraction theory the quadrature of general highly oscillatory integrals is a relatively complicated computational problem that occurs in a wide range of practical applications e g in physics chemistry and image analysis it is often necessary to use a high number of nodal points with classical quadrature formulas in order to achieve a required accuracy of numerical integration of rapidly oscillating functions however an increase in integration points leads to a larger computational time our work describes and analyses a method for numerical integration of rapidly oscillating functions which enables to obtain the required accuracy with a smaller number of nodal points than classical integration rules and with a relatively simple integration formula the proposed method is generally suitable for integration of rapidly oscillating functions in various applications the method is demonstrated in examples of calculation of the diffraction integral in optics where the integrand is often a rapidly oscillatory function the results can be readily adapted to similar problems from other fields copyright 2009 john wiley sons ltd

a simple and efficient bem implementation of quasistatic linear visco elasticity abstract a simple yet efficient procedure to solve quasistatic problems of special linear visco elastic solids at small strains with equal rheological response in all tensorial components utilizing boundary element method bem is introduced this procedure is based on the implicit discretisation in time the so called rothe method combined with a simple algebraic transformation of variables leading to a numerically stable procedure proved explicitly by discrete energy estimates which can be easily implemented in a bem code to solve initial boundary value visco elastic problems by using the kelvin elastostatic fundamental solution only it is worth mentioning that no inverse laplace transform is required here the formulation is straightforward for both 2d and 3d problems involving unilateral frictionless contact although the focus is to the simplest kelvin voigt rheology a generalization to maxwell boltzmann jeffreys and burgers rheologies is proposed discussed and implemented in the bem code too a few 2d and 3d initial boundary value problems one of them with unilateral frictionless contact are solved numerically

the construction of cubature rules for multivariate highly oscillatory integrals we present an efficient approach to evaluate multivariate highly oscillatory integrals on piecewise analytic integration domains cubature rules are developed that only require the evaluation of the integrand and its derivatives in a limited set of points a general method is presented to identify these points and to compute the weights of the corresponding rule the accuracy of the constructed rules increases with increasing frequency of the integrand for a fixed frequency the accuracy can be improved by incorporating more derivatives of the integrand the results are illustrated numerically for fourier integrals on a circle and on the unit sphere and for more general oscillators on a rectangular domain

localized waves and interaction solutions to a 3 1 dimensional generalized kp equation abstract based on the hirota bilinear method and long wave limit four kinds of localized waves namely solitons lumps breathers and rogue waves are constructed for the 3 1 dimensional generalized kp equation n soliton solutions are obtained by employing bilinear method then breathers two breathers and interaction breather solutions are obtained by selecting appropriate parameters on two soliton solution and four soliton solution these breathers own different dynamic behaviors in the different planes taking a long wave limit on the two and four soliton solutions under special parameter constraints one order lumps and rogue waves two order lumps and rogue waves and interaction solutions between lumps and rogue waves are derived applying the same method on the three soliton solution interaction solutions between kink solitons with periodic solutions lumps and rogue waves are constructed respectively the influence of parameters on the solution is analyzed the propagation directions phase shifts energies and shapes for these solutions can be affected and controlled by the parameters moreover graphics are presented to demonstrate the properties of the explicit analytical localized wave solutions

soliton solution to bkp equation in wronskian form we find out a group of identities named wronskian identities of bilinear kp hierarchy here as well as two useful properties of d operator this makes it possible to easily search for some new wronskian solutions for pde which owns bilinear form and to simplify the process of the proof as an application of this new method we propose the first wronskian condition for the bkp i equation and for the bkp ii equation respectively in addition we also discuss several generalized bkp equations and obtain their wronskian conditions

new methodology for the treatment of two dimensional viscoelastic coupling problems abstract in this work is described an efficient procedure to analyse two dimensional viscoelastic coupling problems the coupling between the finite element method and the boundary element method is based on the sub regions technique the viscous behaviour is taken into account by rheological constitutive relations properly imposed in the development of the integral representations after imposing the space approximations applying kinematical relations the time integration is carried out using appropriate operators the kelvin voigt viscoelastic model is implemented in order to validate the idea the approach allows applying in an easy way external loads and boundary condition which change along time in order to demonstrate the robustness of the technique examples are presented in the end of this work

general high order rogue waves and their dynamics in the nonlinear schrodinger equation general high order rogue waves in the nonlinear schrodinger equation are derived by the bilinear method these rogue waves are given in terms of determinants whose matrix elements have simple algebraic expressions it is shown that the general n th order rogue waves contain n 1 free irreducible complex parameters in addition the specific rogue waves obtained by akhmediev et al akhmediev et al 2009 phys rev e 80 026601 doi 10 1103 physreve 80 026601 correspond to special choices of these free parameters and they have the highest peak amplitudes among all rogue waves of the same order if other values of these free parameters are taken however these general rogue waves can exhibit other solution dynamics such as arrays of fundamental rogue waves arising at different times and spatial positions and forming interesting patterns

delaminating quadrature method for multi dimensional highly oscillatory integrals a delaminating quadrature method for multi dimensional highly oscillatory integrals is put forward in comparison with the levin method the new method has the benefits of being numerically stable and having a high computational speed numerical experiments testified the advantages of this method

a three dimensional finite element formulation for thermoviscoelastic orthotropic media summary this paper is concerned with the development of a numerical algorithm for the solution of the uncoupled quasistatic initial boundary value problem involving orthotropic linear viscoelastic media undergoing thermal and or mechanical deformation the constitutive equations expressed in integral form involving the relaxation moduli are transformed into an incremental algebraic form prior to development of the nite element formulation this incrementalization is accomplished in closed form and results in a recursive relationship which leads to the need of solving a simple set of linear algebraic equations only for the extraction of the nite element solution use is made of a dirichlet prony series representation of the relaxation moduli in order to derive the recursive relationship and thereby eliminate the storage problem that arises when dealing with materials possessing memory three illustrative example problems are included to demonstrate the method 1997 by john wiley sons ltd

letter to the editor on the filon and levin methods for highly oscillatory integral this paper shows that for any suitably smooth function f x and arbitrarily selected interpolation nodes c1 c2 cv in a b the filon method and the levin method for with the polynomial interpolation approach are identical when g x is a linear function based on this result a new efficient levin quadrature for is presented

two parameters wronskian representation of solutions of nonlinear schrodinger equation eighth peregrine breather and multi rogue waves in this paper we present a representation of solutions of the one dimensional focusing nonlinear schrodinger equation as a quotient of two wronskians depending on two parameters here we give the complete proof of this representation we have already constructed peregrine breathers and their two parameter deformations until order 7 with this method the construction of the explicit analytical expressions of peregrine breather of order 8 was made for the first time when parameters a or b are equal to 0 we recover the peregrine breather of order 8 we obtain multi rogue waves by deformation of parameters a and b these expressions enable us to understand the evolution of the solutions in the case of order 8 it is shown for high values of parameters a or b the appearance of peregrine breather of order 6

on van der corput type lemmas for bessel and airy transforms and applications abstract this paper presents van der corput type lemmas for bessel and airy transforms with shifted jacobi weight functions these results on the asymptotic orders of the highly oscillatory integrals on the frequency are optimal furthermore from these estimates the convergence rates on filon type methods are easily derived

simple determinant representation for rogue waves of the nonlinear schrodinger equation we present a simple representation for arbitrary order rogue wave solution and a study on the trajectories of them explicitly we find that the trajectories of two valleys on whole temporal spatial distribution all look x shaped for rogue waves additionally we present different types of high order rogue wave structures which could be helpful towards realizing the complex dynamics of rogue waves

fredholm and wronskian representations of solutions to the kpi equation and multi rogue waves we construct solutions to the kadomtsev petviashvili equation kpi in terms of fredholm determinants we deduce solutions written as a quotient of wronskians of order 2n these solutions called solutions of order n depend on 2n 1 parameters when one of these parameters tends to zero we obtain n order rational solutions expressed as a quotient of two polynomials of degree 2n n 1 in x y and t depending on 2n 2 parameters so we get with this method an infinite hierarchy of solutions to the kpi equation

six parameters deformations of fourth order peregrine breather solutions of the nonlinear schrodinger equation we construct solutions of the focusing nonlinear schrodinger nls equation as a quotient of two determinants this formulation gives in the case of the order 4 new deformations of the peregrine breather with 6 real parameters we construct families of quasi rational solutions of the nls equation and describe the apparition of multi rogue waves with this method we construct the analytical expressions of deformations of the peregrine breather of order 4 with 6 real parameters and plot different types of rogue waves

computation of oscillatory integrals with an exponential kernel and jacobi type singularities abstract in this communication two rules are considered for computing oscillatory integrals with an exponential kernel and jacobi type singularities an asymptotic rule is presented with the aid of integration by parts an asymptotic filon type quadrature rule is constructed on the basis of a two point taylor interpolation polynomial error analysis on the proposed rules in inverse powers of large frequency is provided numerical examples are provided to illustrate that the two rules are very efficient in obtaining very high precision approximations if is sufficiently large

periodic multiphase solutions of the kadomtsev petviashvili equation n phase solutions of the kadomtsev petviashvili kp equation 1970 that are periodic in space variables x and y were obtained and effectively investigated using the schottky uniformisation of which a short description is given many wave patterns are represented graphically as contour plots and as isometric projections for different parameter values of two three and four phase solutions of the kp equation

boundary integral equation method for general viscoelastic analysis from basic assumptions of viscoelastic constitutive relations and weight residual techniques a boundary element procedure is achieved for both kelvin and boltzmann models imposing spatial approximations and adopting convenient kinematical relations for strain velocities a system of time differential equations is achieved this system is solved adopting linear approximations for displacements resulting in a time marching methodology this approach avoids the use of relaxation functions and makes easier changes in boundary conditions along time natural or essential an important feature of the resulting technique is the absence of domain discretizations which simplify the treatment of problems involving infinite domains tunnels and cavities inside the soil some examples are shown in order to demonstrate the accuracy and stability of the technique when compared to analytical solutions

darboux theorems and wronskian formulas for integrable systems i constrained kp flows abstract generalizations of the classical darboux theorem are established for pseudo differential scattering operators of the form l i 0 n u i i i 1 m i 1 i i iteration of the darboux transformations leads to a gauge transformed operator with coefficients given by wronskian formulas involving a set of eigenfunctions of l nonlinear integrable partial differential equations are associated with the scattering operator l which arise as a symmetry reduction of the multicomponent kp hierarchy with a suitable linear time evolution for the eigenfunctions the darboux transformation is used to obtain solutions of the integrable equations in terms of wronskian determinants

families of quasi rational solutions of the nls equation and multi rogue waves we construct a multi parametric family of the solutions of the focusing nonlinear schr dinger equation nls from the known results describing the multi phase almost periodic elementary solutions given in terms of riemann theta functions we give a new representation of their solutions in terms of wronskians determinants of order 2n composed of elementary trigonometric functions when we perform a special passage to the limit when all the periods tend to infinity we obtain a family of quasi rational solutions this leads to efficient representations for the peregrine breathers of orders n 1 2 3 first constructed by akhmediev and his co workers and also allows us to obtain a simpler derivation of the generic formulas corresponding the three or six rogue wave formation in the frame of the nls model first explained by v b matveev in 2010 our formulation allows us to isolate easily the second or third order peregrine breathers from generic solutions and also to compute the peregrine breathers of orders 2 and 3 easily with respect to other approaches in the cases n 2 3 we obtain the comfortable formulas to study the deformation of a higher peregrine breather of order 2 to the three rogue wave or order 3 to the six rogue wave solutions via the variation of the free parameters of our construction

algebraic construction of the darboux matrix revisited we present algebraic construction of darboux matrices for 1 1 dimensional integrable systems of nonlinear partial differential equations with a special stress on the nonisospectral case we discuss different approaches to the darboux backlund transformation based on different dependences of the darboux matrix polynomial sum of partial fractions or the transfer matrix form we derive symmetric n soliton formulae in the general case the matrix spectral parameter and dressing actions in loop groups are also discussed we describe reductions to twisted loop groups unitary reductions the matrix lax pair for the kdv equation and reductions of chiral models harmonic maps to su n and to grassmann spaces we show that in the kdv case the nilpotent darboux matrix generates the binary darboux transformation the paper is intended as a review of known results usually presented in a novel context but some new results are included as well e g general compact formulae for n soliton surfaces and linear and bilinear constraints on the nonisospectral lax pair matrices which are preserved by darboux transformations

on the use of somigliana s formula and fourier series for elasticity problems with circular boundaries this paper considers the problem of an infinite isotropic elastic plane containing an arbitrary number of non overlapping circular holes and isotropic elastic inclusions the holes and inclusions are of arbitrary size and the elastic properties of all of the inclusions can if desired be different the analysis is based on the two dimensional version of somigliana s formula which gives the displacements at a point inside a region v in terms of integrals of the tractions and displacements over the boundary s of this region we take v to be the infinite plane and s to be an arbitrary number of circular holes within this plane any or all of the holes can contain an elastic inclusion and we assume for simplicity that all inclusions are perfectly bonded to the material matrix the displacements and tractions on each circular boundary are represented as truncated fourier series and all of the integrals involved in somigliana s formula are evaluated analytically an iterative solution algorithm is used to solve the resulting system of linear algebraic equations several examples are given to demonstrate the accuracy and efficiency of the numerical method copyright 2003 john wiley sons ltd

meshless methods for two dimensional oscillatory fredholm integral equations abstract in this paper a meshless solution procedure incorporating delaminating quadrature method for two dimensional highly oscillatory fredholm integral equation is put forward the proposed method is an extension of our earlier findings of meshless methods for two dimensional oscillatory fredholm integral equation having kernel function free of stationary phase point s siraj ul islam et al 2015 the current method deals not only with the kernels having no stationary phase point s but also with the kernels having stationary phase point s in the context of highly oscillatory integral equations the new method is numerically stable and computationally fast numerical experiments are presented to testify the robustness and efficiency of the proposed method

on the coupling of 3d bem and fem frame model applied to elastodynamic analysis this work presents the coupling of framed structures approached by finite elements with three dimensional bodies represented by the boundary element method the coupling is particularly dedicated to analyse three dimensional half space stiffened by piles and other composite domain problems for which static and dynamic cases have been taken into account some numerical examples are analysed to point out the power and the accuracy of the proposed scheme

on numerical computation of integrals with integrands of the form f x sin w xr on 0 1 with existing numerical integration methods and algorithms it is difficult in general to obtain accurate approximations to integrals of the form 0 1f x sin wx r dxor 0 1f x cos wx r dx r 0 where f is a sufficiently smooth function on 0 1 gautschi has developed software as scripts in matlab for computing these integrals for the special case r w 1 in this paper an algorithm as a mathematica program is developed for computing these integrals to arbitrary precision for any given values of the parameters in a certain range numerical examples are given of testing the performance of the algorithm program

filon clenshaw curtis rules for highly oscillatory integrals with algebraic singularities and stationary points in this paper we propose and analyze composite filon clenshaw curtis quadrature rules for integrals of the form i k a b f g int a b f x exp mathrm i kg x mathrm d x where k geq 0 f may have integrable singularities and g may have stationary points our composite rule is defined on a mesh with m subintervals and requires mn 1 evaluations of f it satisfies an error estimate of the form c n k r m n 1 r where r is determined by the strength of any singularity in f and the order of any stationary points in g and c n is a constant which is independent of k and m but depends on n the regularity requirements on f and g are explicit in the error estimates for fixed k the rate of convergence of the rule as m rightarrow infty is the same as would be obtained if f was smooth moreover the quadrature error decays at least as fast as k rightarrow infty as does the original integral i k a b f g for the case of nonlinear oscill

differential equations with general highly oscillatory forcing terms the concern of this paper is in expanding and computing initial value problems of the form y f y h t where the function h oscillates rapidly for 1 asymptotic expansions for such equations are well understood in the case of modulated fourier oscillators graphic 1 and they can be used as an organizing principle for very accurate and affordable numerical solvers however there is no similar theory for more general oscillators and there are sound reasons to believe that approximations of this kind are unsuitable in that setting we follow in this paper an alternative route demonstrating that for a much more general family of oscillators e g linear combinations of functions of the form ei g k t it is possible to expand y t in a different manner each r th term in the expansion is graphic 2 for some 0 and it can be represented as an r dimensional highly oscillatory integral because computation of multivariate highly oscillatory integrals is fairly well understood this provides a powerful method for an effective discretization of a numerical solution for a large family of highly oscillatory ordinary differential equations 1 embed inline graphic 1 gif 2 embed inline graphic 2 gif

young diagrams and n soliton solutions of the kp equation we consider n soliton solutions of the kp equation an n soliton solution is a solution u x y t which has the same set of n line soliton solutions in both asymptotics y and y the n soliton solutions include all possible resonant interactions among those line solitons we then classify those n soliton solutions by defining a pair of n numbers n n with n n 1 n n n j 1 2n which labels n line solitons in the solution the classification is related to the schubert decomposition of the grassmann manifolds gr n 2n where the solution of the kp equation is defined as a torus orbit then the interaction pattern of n soliton solution can be described by the pair of young diagrams associated with n n we also show that n soliton solutions of the kdv equation obtained by the constraint u y 0 cannot have resonant interaction

on graded meshes for weakly singular volterra integral equations with oscillatory trigonometric kernels we compare the collocation methods on graded meshes with that on uniform meshes for the solution of the weakly singular volterra integral equation of the second kind with oscillatory trigonometric kernels due to in general unbounded derivatives at the left endpoint of the interval of integration we should approximate the solution of the integral equation by collocation methods on graded meshes however we show that this non smooth behavior of the solution has little effect on the approximate solution when the kernels of integral equation are highly oscillatory numerical examples are given to confirm the proposed results

efficient quadrature of highly oscillatory integrals using derivatives in this paper we explore quadrature methods for highly oscillatory integrals generalizing the method of stationary phase we expand such integrals into asymptotic series in inverse powers of the frequency the outcome is two families of methods one based on a truncation of the asymptotic series and the other extending an approach implicit in the work of filon filon 1928 proc r soc edinb 49 38 47 1 both kinds of methods approximate the integral as a linear combination of function values and derivatives with coefficients that may depend on frequency we determine asymptotic properties of these methods proving perhaps counterintuitively that their performance drastically improves as frequency grows the paper is accompanied by numerical results that demonstrate the potential of this set of ideas 1 ref 4

modified equations rational solutions and the painleve property for the kadomtsev petviashvili and hirota satsuma equations we propose a method for finding the lax pairs and rational solutions of integrable partial differential equations that is when an equation possesses the painleve property a backlund transformation is defined in terms of an expansion about the singular manifold this backlund transformation obtains 1 a type of modified equation that is formulated in terms of schwarzian derivatives and 2 a miura transformation from the modified to the original equation by linearizing the ricati type miura transformation the lax pair is found on the other hand consideration of the distinct backlund transformations of the modified equations provides a method for the iterative construction of rational solutions this also obtains the lax pairs for the modified equations in this paper we apply this method to the kadomtsev petviashvili equation and the hirota satsuma equations

triangular rogue wave cascades by numerically applying the recursive darboux transformation technique we study high order rational solutions of the nonlinear schr odinger equation that appear spatiotemporally as triangular arrays of peregrine solitons these can be considered as rogue wave cascades and complement previously discovered circular cluster forms in this analysis we reveal a general parametric restriction for their existence and investigate the interplay between cascade and cluster forms as a result we demonstrate how to generate many more hybrid rogue wave solutions including semicircular clusters that resemble claws

nondegeneracy morse index and orbital stability of the kp i lump solution using backlund transformation we analyze the spectral property of the kp i lump solution it is proved that the lump is nondegenerate and its morse index equals one as a consequence we show that it is orbitally stable

generating mechanism for higher order rogue waves we introduce a mechanism for generating higher order rogue waves hrws of the nonlinear schrodinger nls equation the progressive fusion and fission of n degenerate breathers associated with a critical eigenvalue 0 creates an order n hrw by adjusting the relative phase of the breathers in the interacting area it is possible to obtain different types of hrws the value 0 is a zero point of an eigenfunction of the lax pair of the nls equation and it corresponds to the limit of the period of the breather tending to infinity by employing this mechanism we prove two conjectures regarding the total number of peaks as well as a decomposition rule in the circular pattern of an order n hrw

letter to the editor on quadrature of bessel transformations a method for integral transformations of highly oscillatory functions bessel functions is presented it is based on the filon type method and the decay of the error can be increased as increases the effectiveness and accuracy of the quadrature is tested for both large arguments and higher orders of bessel functions in the case where the orders are nonnegative integers

oscillation free solutions to volterra integral and integro differential equations with periodic force terms volterra equations governed by oscillatory functions arise frequently in applied fields to construct efficient algorithms for solving these equations oscillatory properties of their solutions should be studied in this paper oscillatory orders of solutions to a class of highly oscillatory volterra integral equations are presented then some modified solutions are given with the help of asymptotic expansions of highly oscillatory integrals finally theoretical analysis verifies these modified solutions enjoy less oscillation than the original one and extensions to highly oscillatory volterra integro differential equations are considered

computation of integrals with oscillatory and singular integrands using chebyshev expansions we present a general method for computing oscillatory integrals of the form 1 1f x g x e i w xdx where f is sufficiently smooth on 1 1 w is a positive parameter and g is a product of singular factors of algebraic or logarithmic type based on a chebyshev expansion of f and the properties of chebyshev polynomials the proposed method for such integrals is constructed with the help of the expansion of the oscillatory factor e i w x furthermore due to numerically stable recurrence relations for the modified moments the devised scheme can be employed to compute oscillatory integrals with algebraic or logarithmic singularities at the end or interior points of the interval of integration numerical examples are provided to confirm our analysis

solutions of the kp equation associated to rank three commuting differential operators over a singular elliptic curve abstract we give a method for finding explicit solutions of the kadomtsev petviashvili kp equation which are associated to rank three commuting ordinary differential operators explicit formulae for the coefficients of rank three commuting differential operators are used to reduce the kp equation to a nonlinear partial differential equation in only two variables this equation which is obtained in a similar way to the rank two krichever novikov kn equation is easier to solve than kp and some solutions are given

second order nonlinear schrodinger equation breather solutions in the degenerate and rogue wave limits we present an explicit analytic form for the two breather solution of the nonlinear schr odinger equation with imaginary eigenvalues it describes various nonlinear combinations of akhmediev breathers and kuznetsov ma solitons the degenerate case when the two eigenvalues coincide is quite involved the standard inverse scattering technique does not generally provide an answer to this scenario we show here that the solution can still be found as a special limit of the general second order expression and appears as a mixture of polynomials with trigonometric and hyperbolic functions a further restriction of this particular case where the two eigenvalues are equal to i produces the second order rogue wave with two free parameters considered as differential shifts the illustrations reveal a precarious dependence of wave profile on the degenerate eigenvalues and differential shifts thus we establish a hierarchy of second order solutions revealing the interrelated nature of the general case the rogue wave and the degenerate breathers

on a family of solutions of the kadomtsev petviashvili equation which also satisfy the toda lattice hierarchy we describe the interaction pattern in the x y plane for a family of soliton solutions of the kadomtsev petviashvili kp equation the solutions considered also satisfy the finite toda lattice hierarchy we determine completely their asymptotic patterns for y and we show that all the solutions except the 1 soliton solution are of resonant type consisting of arbitrary numbers of line solitons in both asymptotics that is arbitrary n incoming solitons for y interact to form arbitrary n outgoing solitons for y we also discuss the interaction process of those solitons and show that the resonant interaction creates a web like structure having n 1 n 1 holes

quadrature methods for multivariate highly oscillatory integrals using derivatives while there exist effective methods for univariate highly oscillatory quadrature this is not the case in a multivariate setting in this paper we embark on a project extending univariate theory to more variables inter alia we demonstrate that in the absence of critical points and subject to a nonresonance condition an integral over a simplex can be expanded asymptotically using only function values and derivatives at the vertices a direct counterpart of the univariate case this provides a convenient avenue towards the generalization of asymptotic and filon type methods as formerly introduced by the authors in a single dimension to simplices and more generally to polytopes the nonresonance condition is bound to be violated once the boundary of the domain of integration is smooth in effect its violation is equivalent to the presence of stationary points in a single dimension we further explore this issue and propose a technique that often can be used in this situation yet much remains to be done to understand more comprehensively the influence of resonance on the asymptotics of highly oscillatory integrals