the painleve property for partial differential equations in this paper we define the painleve property for partial differential equations and show how it determines in a remarkably simple manner the integrability the backlund transforms the linearizing transforms and the lax pairs of three well known partial differential equations burgers equation kdv equation and the modified kdv equation this indicates that the painleve property may provide a unified description of integrable behavior in dynamical systems ordinary and partial differential equations while at the same time providing an efficient method for determining the integrability of particular systems

proof verification and the hardness of approximation problems we show that every language in np has a probablistic verifier that checks membership proofs for it using logarithmic number of random bits and by examining a constant number of bits in the proof if a string is in the language then there exists a proof such that the verifier accepts with probability 1 i e for every choice of its random string for strings not in the language the verifier rejects every provided proof with probability at least 1 2 our result builds upon and improves a recent result of arora and safra 1998 whose verifiers examine a nonconstant number of bits in the proof though this number is a very slowly growing function of the input length as a consequence we prove that no max snp hard problem has a polynomial time approximation scheme unless np p the class max snp was defined by papadimitriou and yannakakis 1991 and hard problems for this class include vertex cover maximum satisfiability maximum cut metric tsp steiner trees and shortest superstring we also improve upon the clique hardness results of feige et al 1996 and arora and safra 1998 and show that there exists a positive e such that approximating the maximum clique size in an n vertex graph to within a factor of n e is np hard

generalized riccati equation expansion method and its application to the 3 1 dimensional jumbo miwa equation based on the computerized symbolic system maple and a riccati equation a generalized riccati equation expansion method for constructing soliton like solutions of non linear evolution equations nees is presented by a new general ansatz the proposed method is more powerful than most of the existing tanh methods the extended tanh function method the modified extended tanh function method and generalized hyperbolic function method by use of the method we not only can successfully recover the previously known formal solutions but also construct new and more general formal solutions for some nees making use of the method we study the the 3 1 dimensional jumbo miwa equation and obtain rich new families of the exact solutions including the non travelling wave and coefficient functions soliton like solutions singular soliton like solutions periodic form solutions

solitonic excitations and interactions in an α helical protein modeled by three coupled nonlinear schrodinger equations with variable coefficients governing the molecular excitations associated with bioenergy transport in an helical protein through the soliton modes three coupled higher order nonlinear schrodinger equations with variable coefficients are investigated via symbolic computation using bell polynomials a bilinear form and backlund transformation are derived for this model furthermore explicit n soliton solutions are constructed in terms of the double wronskian determinant solitonic excitations are found to remain stable against the disorders of the parameters under certain constraints finally propagation characteristics and interactions of the solitonic excitations are discussed soliton amplitudes are related to the energy modulation coefficients and both the soliton properties and energy distribution during the interactions are affected by the inhomogeneity coefficients of the protein with time dependent disorders

a simple yet accurate finite element procedure for computing stress intensity factors a new finite element procedure for calculating stress intensity factors in elastic crack problems is developed in common with a number of other approaches in the literature the procedure combines the analytical singular fields present in a problem with a finite element treatment of the residual regular problem what distinguishes the procedure is the use of path independent integrals to balance the analytical and numerical contributions a set of test problems with exact solutions is analysed and demonstrates that the procedure is readily implemented and can accurately evaluate stress intensity factors with a modest amount of computational effort the application of two competing methods to the test problems further demonstrates that the new procedure is markedly superior in both its initial accuracy and its rate of convergence the paper concludes with two additional illustrations of the procedure as applied to the single edge crack and the centre crack these also yield accurate results for little computational effort

backlund transformation and multi soliton solutions for a 2 1 dimensional korteweg de vries system via symbolic computation abstract in this paper the 2 1 dimensional korteweg de vries system is symbolically investigated by the bilinear method the n soliton solution is presented then based on the backlund transformation in bilinear form a new backlund transformation is obtained and new representation of the n soliton solution is derived a class of novel multi soliton solutions are obtained by the new backlund transformation and the availability of symbolic computation is demonstrated

optical solitons in a monomode fiber we discuss the propagation of optical solitons in a monomode fiber as a model of long distance high bit rate transmission system we give several new results which did not appear in our previous papers on this subject such as 1 a derivation of the perturbed nonlinear schrodinger equation from the maxwell equation 2 on the integrability of the perturbed nonlinear schrodinger equation 3 a discussion of the soliton as a stable fixed point of certain infinite dimensional map generated by a transmission system with periodic excitations

on the distributional complexity of disjointness we prove that the distributional communication complexity of the predicate disjointness with respect to a very simple measure on inputs is n

some optimal inapproximability results we prove optimal up to an arbitrary e 0 inapproximability results for max e k sat for k 3 maximizing the number of satisfied linear equations in an over determined system of linear equations modulo a prime p and set splitting as a consequence of these results we get improved lower bounds for the efficient approximability of many optimization problems studied previously in particular for max e2 sat max cut max di cut and vertex cover

probabilistic checking of proofs a new characterization of np we give a new characterization of np the class np contains exactly those languages l for which membership proofs a proof that an input x is in l can be verified probabilistically in polynomial time using logarithmic number of random bits and by reading sublogarithmic number of bits from the proof we discuss implications of this characterization specifically we show that approximating clique and independent set even in a very weak sense is np hard

bell polynomial manipulations on the backlund transformations and lax pairs for some soliton equations with one tau function in the framework of bell polynomial manipulations under investigation hereby are three single field bilinearizable equations the 1 1 dimensional shallow water wave model boiti leon manna pempinelli model and 2 1 dimensional sawada kotera model based on the concept of scale invariance a direct and unifying bell polynomial scheme is employed to achieve the backlund transformations and lax pairs associated with those three soliton equations note that the bell polynomial expressions and bell polynomial typed backlund transformations for those three soliton equations can be respectively cast into the bilinear equations and bilinear backlund transformations with symbolic computation consequently it is also shown that the bell polynomial typed backlund transformations can be linearized into the corresponding lax pairs

averaged dynamics of optical pulses described by a nonlinear schrodinger equation with periodic coefficients a nonlinear schrodinger equation with periodic coefficients as it appears e g in nonlinear optics is considered the high frequency variable part of the dispersion may be even much larger than the mean value the ratio of the length of the dispersion map to the period of a solution is assumed as one small parameter the second one corresponds to the integral over the variable part of the dispersion for the averaged dynamics we propose a procedure based on the bogolyubov method as a result we obtain the asymptotic equation in the dominating order as well as with the next corrections the equation is valid for all combinations of the small parameters the explicit forms of the coefficients are presented for a two step dispersion map with an exponential loss function the forms of the bright and black soliton solutions are discussed the results are compared to those from other averaging methods namely the multiple scale method and the method based on lie transformations

types of solutions of the variable coefficient nonlinear schrodinger equation with symbolic computation by using hirota s bilinear method and symbolic computation solutions for a variable coefficient nonlinear schrodinger equation are obtained theoretically it is found that the type of the solutions changes with the different choices of the group velocity dispersion coefficient beta 2 z according to those solutions the relevant properties and features of physical and optical interest are illustrated in addition an effective technique for controlling the shape of the pulses is presented the results of this paper will be valuable to the study of the future development of ultrahigh rate and long distance optical communication systems

bell polynomial approach and n soliton solution for the extended lotka volterra equation in plasmas symbolically investigated in this paper is the extended lotka volterra elv equation which can govern the kinetics of the discrete peaks of the weak langmuir turbulence in plasmas without the linear damping and random noise binary bell polynomials are applied to the bilinearization of the discrete system bilinear backlund transformation of the elv equation is constructed n soliton solution in terms of the extended casorati determinant is also presented and verified propagation and interaction behaviors of the langmuir turbulence are analyzed it is demonstrated that the number of the interacting langmuir waves can influence the soliton velocity and amplitude as well as the collision phase shift graphic illustrations of the solitonic collisions show that the repulsion effects and nonlinear interactions are also associated with the number of the interacting langmuir waves

exact travelling wave solutions for a generalized nonlinear schrodinger equation a wide class of exact travelling wave solutions of a generalized nonlinear schrodinger equation gnls is obtained and analysed in detail this class of solutions incorporates bright and dark solitary waves periodic waves unbounded waves and other solitary waves as asymptotic limits of the periodic or unbounded modes the method of analysis adopted is based on reducing the gnls to an ordinary differential equation and studying the phase plane of the resulting dynamical system application of the obtained results to the problem of propagation of femtosecond duration pulses in nonlinear optical fibres is also discussed

symbolic computation and new families of exact soliton like solutions to the integrable broer kaup bk equations in 2 1 dimensional spaces in this paper new families of soliton like solutions are obtained for 2 1 dimensional integrable broer kaup equations by using the symbolic computation method developed by gao and tian sample solutions obtained from these methods are presented solitary wave solutions are merely a special case in one family the method can also be extended to other types of nonlinear evolution equations in mathematical physics

symbolic computation backlund transformation and analytic n soliton like solution for a nonlinear schrodinger equation with nonuniformity term from certain space laboratory plasmas abstract with symbolic computation a bilinear backlund transformation is presented for a nonlinear schrodinger equation with nonuniformity term from certain space laboratory plasmas and correspondingly the one soliton like solution is derived from the backlund transformation simultaneously the n soliton like solution in double wronskian form is also given besides the authors verify that the n 1 and n soliton like solutions satisfy the backlund transformation the results obtained in this paper might be valuable for the study of the nonuniform media

hidden variables joint probability and the bell inequalities it is shown that the following statements about a quantum correlation experiment are mutually equivalent 1 there is a deterministic hidden variables model for the experiment 2 there is a factorizable stochastic model 3 there is one joint distribution for all observables of the experiment returning the experimental probabilities 4 there are well defined compatible joint distributions for all pairs and triples of commuting and noncommuting observables 5 the bell inequalities hold

a parallel repetition theorem we show that a parallel repetition of any two prover one round proof system mip 2 1 decreases the probability of error at an exponential rate no constructive bound was previously known the constant in the exponent in our analysis depends only on the original probability of error and on the total number of possible answers of the two provers the dependency on the total number of possible answers is logarithmic which was recently proved to be almost the best possible u feige and o verbitsky proc 11th annual ieee conference on computational complexity ieee computer society press los alamitos ca 1996 pp 70 76

solitonic solutions for a variable coefficient variant boussinesq system in the long gravity waves variable coefficient variant boussinesq vcvb system is able to describe the nonlinear and dispersive long gravity waves traveling in two horizontal directions with varying depth in this paper with symbolic computation a lax pair associated with the vcvb system under some constraints for variable coefficients is derived and based on the lax pair two sorts of basic darboux transformations are presented by applying the darboux transformations some solitonic solutions are obtained with the relevant constraints given in the text in addition the vcvb system is transformed to a variable coefficient broer kaup system solitonic solutions and procedure of getting them could be helpful to solve the nonlinear and dispersive problems in fluid dynamics

the use of backlund transformations in obtaining n soliton solutions in wronskian form the wronskian formulation of the n soliton solutions of various nonlinear evolution equations modified korteweg de vries with both zero and non zero asymptotic conditions sine gordon korteweg de vries kadomtsev petviashvili and boussinesq equations are obtained by inductive use of backlund transformations in hirota s 1980 bilinear notation

soliton solutions backlund transformation and wronskian solutions for the extended 2 1 dimensional konopelchenko dubrovsky equations in fluid mechanics abstract this paper is to investigate the extended 2 1 dimensional konopelchenko dubrovsky equations which can be applied to describing some phenomena in the stratified shear flow the internal and shallow water waves and plasmas bilinear form equations are transformed from the original equations and n soliton solutions are derived via symbolic computation bilinear form backlund transformation and single soliton solution are obtained and illustrated wronskian solutions are constructed from the backlund transformation and single soliton solution

a hybrid micro macro bem with micro scale inclusion crack interactions local analysis schemes capable of detailed representations of micro features of a problem are integrated with a macro scale bem technique capable of handling finite geometries and realistic boundary conditions this paper focuses on micro scale interactions among cracks and inclusions as well as their ramifications on macro scale damage evaluations the micro scale effects are introduced into the macro scale bem computations through an augmented fundamental solution obtained from an integral equation representation of the micro scale features the proposed hybrid micro macro bem formulation allows complete decomposition of the real problem into two sub problems one residing entirely at the micro level while the other resides at the macro level this allows for investigations of the micro structural attributes while retaining the macro scale geometric features and actual boundary conditions for the structural component under consideration as a first attempt dilute inclusion densities with strong inclusion crack and crack crack interactions are considered the numerical results obtained from the hybrid bem analysis establish the accuracy and effectiveness of the proposed micro macro computational scheme for this class of problems the proposed micro macro bem formulation can be easily extended to investigate the effects of other micro features e g interfaces short or continuous fibers in the context of linear elasticity on macro scale failure modes observed in structural components

on the effect of quarter point element size on fracture criteria the effect of the quarter point element size on the solution parameters of fracture criteria is investigated in view of its effect on the computed stress intensity factors an analytical relationship between the error in calculating crack propagation increment and stress intensity for single mode problems is obtained while for mixed mode problems the investigation is based on numerical experimentation the singular element size is found to have a substantial effect on both crack propagation angle and increment for the geometries considered

numerical solutions of a variable coefficient nonlinear schrodinger equation for an inhomogeneous optical fiber abstract this paper investigates a variable coefficient nonlinear schrodinger equation for an inhomogeneous optical fiber numerical one and two solitonic envelopes of the electrical field via the fourth order split step runge kutta split step fourier and runge kutta methods with equal grids in the axis and equal grids in the axis are graphically presented respectively where and represent the retarded time and normalized distance along the fiber 2 norm of the relative errors between the analytical solutions under the painleve integrability condition and numerical solutions are given where the cpu time is also shown relative errors and cpu time of the numerical one and two soliton solutions with equal grids in the axis are bigger than those with equal grids in the axis which does not mean that the results with equal grids in the axis are infeasible compared with the numerical solutions with equal grids in the axis those with equal grids in the axis are closer to the analytical solutions without the painleve integrability condition with respect to the relative errors and cpu time one could choose the split step fourier method to derive the numerical one soliton solutions while the numerical two soliton solutions are gotten with the rk method the attenuation coefficient makes the amplitudes of the solitons decrease the group velocity dispersion coefficient leads to the periodic solitons while the effect of the attenuation coefficient is more obvious than that of the nonlinearity parameter effects of r 1 and i 1 on the solitonic weak interaction between the two solitons are investigated solitonic weak interaction between the two solitons enhances with r 1 and i 1 increasing where r 1 and i 1 are the frequencies of the two solitons

nonlinear pulse propagation in dispersion managed system abstract we study nonlinear pulse propagation in an optical transmission system with dispersion managements this is particularly important for designing an ultra fast and long haul communication system in the next generation there exists a quasi stationary pulse solution in such a system whose width and chirp are rapidly oscillating with the period of dispersion management this pulse also has several new features such as enhanced power when compared with the soliton case with a uniform dispersion and reduction of timing jitter in a wavelength division multiplexed system the method used here is so called the guiding center or normal form theory based on the lie transform and the averaging method for the case of a small dispersion management the core equation obtained as the normal form is given by the nonlinear schrodinger nls equation with a small correction while for the case with a rather large management the core equation is shown to be the nls equation having a nontrapping quadratic potential because of this potential a steady pulse propagating in such a system requires enhanced power

n fold darboux transformation and solitonic interactions of a variable coefficient generalized boussinesq system in shallow water abstract under consideration in this paper is a variable coefficient generalized boussinesq system for the long weakly nonlinear and weakly dispersive surface waves in shallow water with the aid of symbolic computation n fold darboux transformation n dt is constructed for that system analytic solutions of the system are obtained via the n dt elastic interactions of three bell shaped and periodic bell shaped solitons are obtained fusion interactions and periodic fusion fission interactions of the solitary waves are graphically analyzed which are inelastic

solitary wave solutions to the generalized coupled mkdv equation with multi component using the darboux matrix method the multi solitary wave solutions of the generalized coupled mkdv equation with multi component are obtained the obtained solution formulas provide us with a comprehensive approach to construct exact solutions for the generalized coupled mkdv equation by some basic solutions of the boiti and tu spectral problem

solitons backlund transformation and lax pair for the 2 1 dimensional boiti leon pempinelli equation for the water waves under investigation in this paper is the 2 1 dimensional boiti leon pempinelli blp equation for the water waves by virtue of the binary bell polynomials and symbolic computation the bilinear form for the blp equation is obtained furthermore soliton solutions are presented and soliton interaction properties including the elastic inelastic and elastic inelastic collisions are discussed by the graphical analysis besides the backlund transformation in the form of the binary bell polynomials is derived via the backlund transformation the shock wave solutions and lax pair are both constructed

n soliton solutions for the 2 1 dimensional hirota maccari equation in fluids plasmas and optical fibers abstract under investigation in this paper is the hirota maccari equation which is a generalized 2 1 dimensional model in fluid dynamics plasma physics and optical fiber communication with the aid of the hirota bilinear method and symbolic computation the corresponding n soliton solutions are given and illustrated the characteristic face method and asymptotic analysis are applied to discuss the solitonic propagation and collision including the bidirectional solitons elastic interactions and inelastic interactions finally a kind of special phenomenon with the parameters varying is investigated which might provide people with useful information on the dynamics of the relevant fields

free bits pcps and nonapproximability towards tight results this paper continues the investigation of the connection between probabilistically checkable proofs pcps and the approximability of np optimization problems the emphasis is on proving tight nonapproximability results via consideration of measures such as the free bit complexity and the amortized free bit complexity of proof systems the first part of the paper presents a collection of new proof systems based on a new error correcting code called the long code we provide a proof system that has amortized free bit complexity of 2 epsilon implying that approximating maxclique within n frac13 e and approximating the chromatic number within n frac15 e are hard assuming np neq corp for any e 0 we also derive the first explicit and reasonable constant hardness factors for min vertex cover msat 2 and max cut and we improve the hardness factor for msat 3 we note that our nonapproximability factors for maxsnp problems are appreciably close to the values known to be achievable by polynomial time algorithms finally we note a general approach to the derivation of strong nonapproximability results under which the problem reduces to the construction of certain gadgets the increasing strength of nonapproximability results obtained via the pcp connection motivates us to ask how far this can go and whether pcps are inherent in any way the second part of the paper addresses this the main result is a reversal of the connection due to feige et al fglss connection j acm 43 1996 pp 268 292 where the latter had shown how to translate proof systems for np into np hardness of approximation results for maxclique we show how any np hardness of approximation result for maxclique yields a proof system for np roughly our result says that for any constant f if maxclique is np hard to approximate within n 1 1 f then np subseteq overline fpcp log f the latter being the class of languages possessing proofs of logarithmic randomness and amortized free bit complexity f this suggests that pcps are inherent to obtaining nonapproximability results furthermore the tight relation suggests that reducing the amortized free bit complexity is necessary for improving the nonapproximability results for maxclique the third part of our paper initiates a systematic investigation of the properties of pcp and fpcp free pcp as a function of the following various parameters randomness query complexity free bit complexity amortized free bit complexity proof size etc we are particularly interested in triviality results which indicate which classes are not powerful enough to capture np we also distill the role of randomized reductions in this area and provide a variety of useful transformations between proof checking complexity classes

unique games with entangled provers are easy we consider one round games between a classical verifier and two provers who share entanglement we show that when the constraints enforced by the verifier are unique constraints i e permutations the value of the game can be well approximated by a semidefinite program sdp essentially the only algorithm known previously was for the special case of binary answers as follows from the work of tsirelson in 1980 among other things our result implies that the variant of the unique games conjecture where we allow the provers to share entanglement is false our proof is based on a novel quantum rounding technique showing how to take a solution to an sdp and transform it into a strategy for entangled provers using our approximation by an sdp we also show a parallel repetition theorem for unique entangled games

solitonic propagation and interaction for a generalized variable coefficient forced korteweg de vries equation in fluids under investigation is a generalized variable coefficient forced korteweg de vries equation in fluids and other fields from the bilinear form of such equation the n soliton solution and a type of analytic solution are constructed with symbolic computation analytic analysis indicates that 1 dispersive and dissipative coefficients affect the solitonic velocity 2 external force term affects the solitonic velocity and background 3 line damping coefficient and some parameters affect the solitonic velocity background and amplitude solitonic propagation and interaction can be regarded as the combination of the effects of various variable coefficients according to a constraint among the nonlinear dispersive and line damping coefficients in this paper the possible applications of our results in the real world are also discussed in three aspects i e solution with the constraint solution without the constraint and approximate solution

generalized xor non locality games with graph description on a square lattice we propose a family of non locality unique games for 2 parties based on a square lattice on an arbitrary surface we show that due to structural similarities with error correction codes of kitaev for fault tolerant quantum computation the games have classical values computable in polynomial time for d 2 measurement outcomes by representing games in their graph form for arbitrary d and underlying surface we provide their classification into equivalence classes with respect to relabeling of measurement outcomes for a selected set of permutations which define the winning conditions a case study of games with periodic boundary conditions is presented in order to verify their impact on classical and quantum values of the family of games it suggests that quantum values suffer independently from presence of different winning conditions that can be imposed due to periodicity as long as no local restrictions are in place

soliton solutions and a backlund transformation for a generalized nonlinear schrodinger equation with variable coefficients from optical fiber communications under investigation in this paper is a generalized nonlinear schrodinger model with variable dispersion nonlinearity and gain loss which could describe the propagation of optical pulse in inhomogeneous fiber systems by employing the hirota method one and two soliton solutions are obtained with the aid of symbolic computation furthermore a general formula which denotes multi soliton solutions is given some main properties of the solutions are discussed simultaneously as one important property of nonlinear evolution equation the backlund transformation in bilinear form is also constructed which is helpful on future research and as far as we know is firstly proposed in this paper

the complexity of approximating a nonlinear program we consider the problem of finding the maximum of a multivariate polynomial inside a convex polytope we show that there is no polynomial time approximation algorithm for this problem even one with a very poor guarantee unless p np we show that even when the polynomial is quadratic i e quadratic programming there is no polynomial time approximation unless np is contained in quasi polynomial time

painleve property soliton like solutions and complexitons for a coupled variable coefficient modified korteweg de vries system in a two layer fluid model abstract as a model derived from a two layer fluid system which describes the atmospheric and oceanic phenomena a coupled variable coefficient modified korteweg de vries system is concerned in this paper with the help of symbolic computation its integrability in the painleve sense is investigated furthermore hirota s bilinear method is employed to construct the bilinear forms through the dependent variable transformations and soliton like solutions and complexitons are derived finally effects of variable coefficients are discussed graphically and it is concluded that the variable coefficients control the propagation trajectories of solitons and complexitons