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

a method for constructing nontraveling wave solutions for 1 1 dimensional evolution equations a method is proposed to construct nontraveling wave solution for 1 1 dimensional evolution equations by extending the linear traveling wave transformation of the g g expansion method into a nonlinear transformation owing to a built in arbitrary function included in the solution abundant solutions can be excited taking the vakhnenko equation as an example a series of nontraveling wave solutions with variable separation is obtained some new solutions are excited and the known solitary wave solutions are special cases of the nontraveling wave solutions

new variable separation solutions of two dimensional burgers system abstract in this paper with the help of g g expansion method we obtain some exact solutions of the riccati equation based on the riccati equation and its exact solutions we find some variable separation solutions with two arbitrary functions of two dimensional burgers system as some special examples these exact solutions can be reduced to variable separation solutions in kink solution soliton solution periodic solution and rational function solution forms

closed form discrete fractional and affine fourier transforms the discrete fractional fourier transform dfrft is the generalization of discrete fourier transform many types of dfrft have been derived and are useful for signal processing applications we introduce a new type of dfrft which are unitary reversible and flexible in addition the closed form analytic expression can be obtained it works in performance similar to the continuous fractional fourier transform frft and can be efficiently calculated by the fft since the continuous frft can be generalized into the continuous affine fourier transform aft the so called canonical transform we also extend the dfrft into the discrete affine fourier transform daft we derive two types of the dfrft and daft type 1 is similar to the continuous frft and aft and can be used for computing the continuous frft and aft type 2 is the improved form of type 1 and can be used for other applications of digital signal processing meanwhile many important properties continuous frft and aft are kept in the closed form dfrft and daft and some applications such as filter design and pattern recognition are also discussed the closed form dfrft we introduce has the lowest complexity among all current dfrfts that is still similar to the continuous frft

digital computation of linear canonical transforms we deal with the problem of efficient and accurate digital computation of the samples of the linear canonical transform lct of a function from the samples of the original function two approaches are presented and compared the first is based on decomposition of the lct into chirp multiplication fourier transformation and scaling operations the second is based on decomposition of the lct into a fractional fourier transform followed by scaling and chirp multiplication both algorithms take n log n time where n is the time bandwidth product of the signals the only essential deviation from exactness arises from the approximation of a continuous fourier transform with the discrete fourier transform thus the algorithms compute lcts with a performance similar to that of the fast fourier transform algorithm in computing the fourier transform both in terms of speed and accuracy

analytic solutions to nonlinear differential difference equations by means of the extended g g expansion method in this paper a discrete extension of the g g expansion method is applied to a relativistic toda lattice system and a discrete nonlinear schrodinger equation in order to obtain discrete traveling wave solutions closed form solutions with more arbitrary parameters which reduce to solitary and periodic waves are exhibited new rational solutions are also obtained the method is straightforward and concise and its applications in physical sciences are promising

a generalized auxiliary equation method and its application to 2 1 dimensional asymmetric nizhnik novikov vesselov equations a generalized auxiliary equation method is proposed to construct more general exact solutions of nonlinear partial differential equations with the aid of symbolic computation we choose the 2 1 dimensional asymmetric nizhnik novikov vesselov equations to illustrate the validity and advantages of the method as a result many new and more general exact non travelling wave and coefficient function solutions are obtained including soliton like solutions triangular like solitions single and combined non degenerate jacobi elliptic doubly like periodic solutions and weierstrass elliptic doubly like periodic solutions

discrete exact solutions to some nonlinear differential difference equations via the g g expansion method we extended the g g expansion method to two well known nonlinear differential difference equations the discrete nonlinear schrodinger equation and the toda lattice equation for constructing traveling wave solutions discrete soliton and periodic wave solutions with more arbitrary parameters as well as discrete rational wave solutions are revealed it seems that the utilized method can provide highly accurate discrete exact solutions to nddes arising in applied mathematical and physical sciences

sliding discrete linear canonical transform the linear canonical transform lct has been shown to be one of the most powerful tools in signal processing and in this paper we propose an adaptive approach for the computation of the discrete lct dlct termed the sliding discrete linear canonical transform sdlct first we introduce a scheme for the single point dlct which can effectively calculate a single or a few linear canonical spectra second the sdlct is proposed based on an iterative algorithm to meet the requirements of online spectral analysis when only a subset of n frequencies are required from an tilde n hbox point discrete lct n leq tilde n the additivity and reversibility properties of the proposed algorithms are also discussed in detail third the dlct convolution operation is obtained to reduce the spectral leakage of the proposed algorithm and time domain windowing is implemented via frequency domain convolution finally we present two methods to assess performance with regard to computational complexity and precision and to show the correctness of the derived results

application of the extended g g expansion method to solve the pochhammer chree equations abstract in the present paper the extended g g expansion method is used for a reliable treatment of the pochhammer chree equations which include variety of models travelling wave solutions of distinct structures namely bell shaped solitons kink shaped solitons and periodic solutions are formally derived the method demonstrates power reliability and efficiency

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

new solitary wave solutions to the kuramoto sivashinsky and the kawahara equations abstract in this work the kuramoto sivashinsky equation and the kawahara equation are studied the tanh method and the extended tanh method are used for analytic treatment for these two equations by means of these methods new solitary wave solutions are determined for each equation the two approaches are reliable and manageable

nonconforming finite element method for a generalized nonlinear schrodinger equation abstract in this paper an efficient nonconforming finite element method fem is studied with e q 1 r o t element for a generalized nonlinear schrodinger equation first we prove a novel result of the consistency error estimate with order o h2 for e q 1 r o t element which leads to the superconvergent error estimate in broken h1 norm for semi discrete scheme while previous work only derive convergent results for this element second a linearized backward euler scheme is established and a time discrete system is introduced to split the error into two parts the temporal error and the spatial error by using a rigorous analysis for the regularity of the time discrete system and the proved characters of e q 1 r o t element the optimal l2 error estimate is obtained without any time step restrictions which leads to the numerical solution can be bounded in l norm by an inverse inequality unconditionally then the supercloseness estimate is arrived at with the above achievements third global superconvergence results are deduced through interpolated postprocessing technique at last numerical examples are provided to confirm the theoretical analysis here h is the subdivision parameter and is the time step

error analysis of reconstruction from linear canonical transform based sampling in this paper we consider the performance of sampling associated with the linear canonical transform lct which generalizes a large number of classical integral transforms and fundamental operations linked to signal processing and optics first we revisit sampling approximation in the lct domain to introduce a generalized approximation operator then we derive an exact closed form expression for the integrated squared error that occurs when a signal is approximated by a basis of shifted scaled and chirp modulated versions of a generating function in the lct domain several basic properties of the approximation error are presented the derived results can be applied to a wide variety of sampling approximation schemes in the lct domain finally experimental examples are given to illustrate the theoretical derivations

a direct method for constructing the traveling wave solutions of a modified generalized vakhnenko equation abstract proposing a direct method namely the traveling wave reduction combined with a smart variable transformation a series of traveling wave solutions are constructed for a modified generalized vakhnenko equation these solutions include the jacobi elliptic function solution the weierstrass elliptic function solution and the trigonometric function solution

a homotopy for solving general polynomial systems that respects m homogeneous structures methods for finding all solutions to a polynomial system f z 0 of n equations in n unknowns using a generic homotopy have been presented by several researchers over the last ten years in this paper we describe a new method for defining a homotophy to find all solutions to a polynomial system a major feature of this new approach is that the homotopy need not to be generic or random the practical consequences are that the homotopy may be chosen to more closely resemble the system being solved with a consequent potential for reducing the arc length of paths and exploiting special structures in the original system in particular if the system is naturally m homogeneous then the number of homotopy paths that need to be tracked is reduced since many systems that arise in applications have such a structure this represents a significant advance over previous homotopies we also provide a proof that the homotopy parameter is strictly increasing on homotopy paths as a function of arc length a new result which has implications for the choice of numerical path tracking algorithms

analytic investigation of the 2 1 dimensional schwarzian korteweg de vries equation for traveling wave solutions abstract by means of the two distinct methods the exp function method and the extended g g expansion method we successfully performed an analytic study on the 2 1 dimensional schwarzian korteweg de vries equation we exhibited its further closed form traveling wave solutions which reduce to solitary and periodic waves new rational solutions are also revealed

some applications of the g g expansion method to non linear partial differential equations in the present paper we construct the traveling wave solutions involving parameters of the 2 1 dimensional higher order broer kaup equations the 2 1 dimensional breaking soliton equations the 2 1 dimensional asymmetric nizhnik novikov vesselov equations and the 2 1 dimensional bkp equations in terms of the hyperbolic functions trigonometric functions and the rational functions by using a new approach namely the g g expansion method where g g x satisfies a second order linear ordinary differential equation when the parameters are taken special values the solitary waves are derived from the traveling waves the traveling wave solutions are expressed by the hyperbolic functions the trigonometric functions and the rational functions

application of the g g expansion to travelling wave solutions of the broer kaup and the approximate long water wave equations by using the g g expansion proposed recently the travelling wave solutions involving parameters of the broer kaup equations and the approximate long water wave equations are found out the travelling wave solutions are expressed by three types of functions which are the hyperbolic functions the trigonometric functions and the rational functions when the parameters are taken as special values the solitary wave solutions are obtained c 2008 elsevier inc all rights reserved

the g g expansion method for some nonlinear evolution equations abstract in this paper the g g expansion method is applied to the liouville sine gordon and new coupled mkdv equations to obtain their some generalized exact travelling wave solutions

exact and explicit solutions to some nonlinear evolution equations by utilizing the g g expansion method in this paper we demonstrate the effectiveness of the so called g g expansion method by examining some nonlinear evolution equations with physical interest our work is motivated by the fact that the g g expansion method provides not only more general forms of solutions but also periodic and solitary waves if we set the parameters in the obtained wider set of solutions as special values then some previously known solutions can be recovered the method appears to be easier and faster by means of a symbolic computation system

fast discrete linear canonical transform based on cm cc cm decomposition and fft in this paper a discrete lct dlct irrelevant to the sampling periods and without oversampling operation is developed this dlct is based on the well known cm cc cm decomposition that is implemented by two discrete chirp multiplications cms and one discrete chirp convolution cc this decomposition doesn t use any scaling operation which will change the sampling period or cause the interpolation error compared with previous works dlct calculated by direct summation and dlct based on center discrete dilated hermite functions cddhfs the proposed method implemented by ffts has much lower computational complexity the relation between the proposed dlct and the continuous lct is also derived to approximate the samples of the continuous lct simulation results show that the proposed method somewhat outperforms the cddhfs based method in the approximation accuracy besides the proposed method has approximate additivity property with error as small as the cddhfs based method most importantly the proposed method has perfect reversibility which doesn t hold in many existing dlcts with this property it is unnecessary to develop the inverse dlct additionally because it can be replaced by the forward dlct

symbolic computation and new families of exact travelling solutions for the kawahara and modified kawahara equations in this paper we consider the nonlinear evolution equations such as kawahara equation and modified kawahara equation by using the tanh method and an exp function method the travelling wave solutions for the these equations are presented new exact travelling solutions are explicitly obtained with the aid of symbolic computation the obtained solutions include compactons solitons solitary patterns and periodic solutions

application of the g g expansion method to solve coupled mkdv equations and coupled hirota satsuma coupled kdv equations abstract in the present paper we construct the travelling wave solutions involving parameters of the coupled mkdv equations and the coupled hirota satsuma coupled kdv equations by the g g expansion method the study confirms the power of the method

compact and noncompact physical structures for the zk bbm equation a variety of exact solutions for the 2 1 dimensional zk bbm equation are developed by means of the tanh method and the sine cosine methods generalized forms of the zk bbm equation are studied the tanh and the sine cosine methods are reliable to derive solutions of distinct physical structures compactons solitons solitary patterns and periodic solutions

application of the g g expansion method to kawahara type equations using symbolic computation abstract in this paper kawahara type equations are selected to illustrate the effectiveness and simplicity of the g g expansion method with the aid of a symbolic computation system three types of more general traveling wave solutions including hyperbolic functions trigonometric functions and rational functions with free parameters are constructed solutions concerning solitary and periodic waves are also given by setting the two arbitrary parameters involved in the traveling waves as special values

symbolic computation of jacobi elliptic function solutions to nonlinear differential difference equations in this paper an algorithm is presented to find exact polynomial solutions of nonlinear differential difference equations ddes in terms of the jacobi elliptic functions the key steps of the algorithm are illustrated by the discretized mkdv lattice a maple package jacobi is developed based on the algorithm to automatically compute special solutions of nonlinear ddes the effectiveness of the package is demonstrated by applying it to a variety of equations

the ablowitz ladik lattice system by means of the extended g g expansion method abstract we analyzed the ablowitz ladik lattice system by using the extended g g expansion method further discrete soliton and periodic wave solutions with more arbitrary parameters are obtained we observed that some previously known results can be recovered by assigning special values to the arbitrary parameters

homotopy polynomial equations gross boundary data and small helmholtz systems inverse problems of the boundary measurement type appear in several geophysical contexts including dc resistivity electromagnetic induction and groundwater flow the objective is to determine a spatially varying coefficient in a partial differential equation from incomplete knowledge of the dependent variable and its normal gradient at the boundary equivalent 2d discrete inverse problems based on the helmholtz or modified helmholtz equation reduce to systems of polynomial equations indicating that there are only a finite number of exact solutions excluding certain pathological cases a homotopy procedure decides whether real positive solutions exist and if so generates the entire list the computational complexity of the algorithm scales asmm 2 wheremis the number of model parameters to be found measurement errors are accommodated by oversampling the boundary data at additional frequencies for test helmholtz and modified helmholtz inverse problems based on i perfect and ii noisy data i generate the full list of exact solutions the homotopy approach applies to large scale multidimensional geophysical inverse problems but at present is practical only for small systems up tom 9 recent advances in homotopy theory should however reduce the complexity making larger problems tractable in the future

a unified rational expansion method to construct a series of explicit exact solutions to nonlinear evolution equations a unified basic frame of rational expansion methods is presented which leads to closed form solutions of nonlinear evolution equations nlees the new unified algorithms are given to find exact rational formal polynomial solutions of nlees in terms of jacobi elliptic functions solutions of the riccati equation and solutions of the generalized riccati equation they can be implemented in symbolic computation system maple as applications of the methods we choose some physical significance nlees to illustrate the methods as a consequence we not only can successfully obtain the solutions found by most existing jacobi elliptic function methods and tanh methods but also find other new and more general solutions at the same time

application of g g expansion method to two nonlinear evolution equations abstract two nonlinear evolution equations are separately reduced to nonlinear ordinary differential equations ode by using a simple transformation when the parameters are taken as special values the traveling wave solutions are expressed by the hyperbolic functions the trigonometric functions and the rational functions the proposed method can be used for many other nonlinear evolution equations

linear canonical wigner distribution based noisy lfm signals detection through the output snr improvement analysis in our previous work we addressed a unified representation problem on wigner distribution wd in linear canonical transform lct domains by introducing a kind of closed form instantaneous cross correlation function type of wigner distribution cicfwd that can be regarded as the wd s closed form representation in the linear canonical domain we then discussed the application of cicfwd in the detection of linear frequency modulated lfm signals through the numerical simulation analysis approach and it turns out that there is a causation between lct free parameters and the detection performance the main contribution of this paper is to provide a mathematical model for this causality in order to disclose the intrinsic mechanism of non stationary signal detection performance improvement triggered by lct free parameters we first revisit the definition of cicfwd and establish some related theories including its essential properties and discretization we then formulate a mathematical theory foundation of the output signal to noise ratio snr improvement an output snr inequality relation between the traditional wd and the cicfwd for a general noisy signal and discuss how to solve this inequality to come out lct free parameters within the well established output snr improvement analysis framework we also explore lct free parameters selection results on lfm signals added with white noise finally numerical experiments are carried out to validate the correctness of strategies on determination lct free parameters and the feasibility of output snr improvement analysis approach

a generalized auxiliary equation method and its application to 2 1 dimensional korteweg de vries equations a generalized auxiliary equation method is proposed for constructing more general exact solutions of nonlinear partial differential equations with the aid of symbolic computation we choose the 2 1 dimensional korteweg de vries equations to illustrate the validity and advantages of this method as a result many new and more general exact non travelling wave and coefficient function solutions are obtained which include soliton like solutions triangular like solutions single and combined non degenerate jacobi elliptic wave function like solutions and weierstrass elliptic doubly like periodic solutions

on the validity and reliability of the g g expansion method by using higher order nonlinear equations in this study we demonstrate the validity and reliability of the so called g g expansion method via symbolic computation for illustrative examples we choose the sixth order boussinesq equation and the ninth order korteweg de vries equation as a result the power of the employed method is confirmed

the g g expansion method for finding traveling wave solutions of nonlinear partial differential equations in mathematical physics i the present paper we construct the traveling wave solutions involving parameters of the combined korteweg de vries modified korteweg de vries equation the reaction diffusion equation the compound kdv burgers equation and the generalized shallow water wave equation by using a new approach namely the g g expansion method where g g satisfies a second order linear ordinary differential equation when the parameters take special values the solitary waves are derived from the traveling waves the traveling wave solutions are expressed by the hyperbolic functions the trigonometric functions and the rational functions

existence of perturbed solitary wave solutions to a model equation for water waves abstract we prove the existence of travelling wave solutions to a fifth order partial differential equation which is a formal asymptotic approximation for water waves with surface tension these travelling waves are arbitrarily small perturbations of solitary waves but are not solitary waves themselves because they approach small amplitude oscillations for large values of the independent variable this result suggests that for bond numbers less than one third there are branches of travelling wave solutions to the water wave equations which are perturbations of supercritical elevation solitary waves and which bifurcate from froude number one and bond number one third

the g g expansion method and travelling wave solutions for a higher order nonlinear schrodinger equation abstract the higher order nonlinear schrodinger equation is reduced to a nonlinear ordinary differential equation ode by using a simple transformation various solutions of the nonlinear ode are obtained by the g g expansion method proposed recently with the aid of solutions of the nonlinear ode more explicit travelling wave solutions involving two arbitrary parameters of the higher order nonlinear schrodinger equation are found out the travelling waves are expressed by the hyperbolic functions trigonometric functions and rational functions when the parameters are taken as special values the solitary waves are also derived from the travelling waves

a generalized g g expansion method and its application to the 2 1 dimensional broer kaup equations abstract in this paper a generalized g g expansion method is used to seek more general exact solutions of the 2 1 dimensional broer kaup equations as a result non travelling wave solutions with three arbitrary functions are obtained including hyperbolic function solutions trigonometric function solutions and rational solutions the proposed method is more powerful than wang et al s method in m l wang x z li j l zhang the g g expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics phys lett a 372 2008 417 423 and it can be used for many other nonlinear evolution equations in mathematical physics

constructing of exact solutions to the kdv and burgers equations with power law nonlinearity by the extended g g expansion method abstract the extended g g expansion method is introduced to construct more general exact traveling wave solutions to the kdv and burgers equations with any order nonlinear terms the traveling wave solutions are expressed by the hyperbolic functions the trigonometric functions and the rational functions when the parameters take up special values new solitary periodic and complex solutions are also derived from the traveling wave solutions

the correct traveling wave solutions for the high order dispersive nonlinear schrodinger equation the high order dispersive nonlinear schrodinger equation is considered the exact solutions were obtained by zhang et al j l zhang m l wang x z li phys lett a 357 2006 188 195 are analyzed we can demonstrate that some solutions do not satisfy this equation to obtain the correct solutions the f expansion method is applied to solve it

weierstrass semi rational expansion method and new doubly periodic solutions of the generalized hirota satsuma coupled kdv system in the paper with the aid of symbolic computation we investigate the generalized hirota satsuma coupled kdv system via our weierstrass semi rational expansion method presented recently using the rational expansion of weierstrass elliptic function and its first order derivative as a consequence three families of new weierstrass elliptic function solutions via weierstrass elliptic function p g2 g3 and its first order derivative p g2 g3 moreover the corresponding new jacobi elliptic function solutions and solitary wave solutions are also presented and when these solitary wave solutions approach to some constants

polynomial programming using multi homogeneous polynomial continuation abstract a polynomial programming problem is a nonlinear programming problem where the objective function inequality constraints and equality constraints all consist of polynomial functions the necessary optimality conditions for such a problem can be formulated as a polynomial system of equations among whose zeros the global optimum must lie this note applies the theory of multi homogeneous also called m homogeneous polynomials and recent homotopy algorithms to the polynomial system formulation of the necessary optimality conditions significantly reducing the work of a naive homotopy approach the m homogeneous homotopy approach providing the global optimum is practical for small problems for example the geometric modeling problem of finding the distance between two polynomial surfaces is a polynomial programming problem the theory is illustrated in the last section by this geometric modeling problem and a prototype structural design problem

a tighter uncertainty principle for linear canonical transform in terms of phase derivative this study devotes to uncertainty principles under the linear canonical transform lct of a complex signal a lower bound for the uncertainty product of a signal in the two lct domains is proposed that is sharper than those in the existing literature we also deduce the conditions that give rise to the equal relation of the new uncertainty principle the uncertainty principle for the fractional fourier transform is a particular case of the general result for lct examples including simulations are provided to show that the new uncertainty principle is truly sharper than the latest one in the literature and illustrate when the new and old lower bounds are the same and when different

a generalized new auxiliary equation method and its application to the 2 1 dimensional breaking soliton equations abstract in this paper the new auxiliary equation method sirendaoreji a new auxiliary equation and exact travelling wave solutions of non linear equations phys lett a 356 2006 124 130 is improved and a generalized new auxiliary equation method is proposed to construct more general exact solutions of non linear partial differential equations with the aid of symbolic computation we choose the 2 1 dimensional breaking soliton equations to illustrate the validity and advantages of the method as a result many new and more general non travelling wave and coefficient function solutions are obtained including soliton like solutions trigonometric function solutions exponential solutions

exact traveling solutions of some nonlinear evolution equation by g g expansion method in this paper we establish exact solutions for some nonlinear evolution equations the g g expansion method was used to construct traveling wave solutions of ramani s equations in this method we take the advantages of the general solutions of the second order lode to solve effectively many nonlinear evolution equations the g g expansion method presents a wider applicability for handling nonlinear wave equations

new exact solutions of space and time fractional modified kawahara equation abstract one of the important nonlinear evolution equations in mathematical physics is the modified kawahara equation in this work by utilizing an analytic method based on the jacobi elliptic functions a group of exact solutions of the fractional modified kawahara equation had been found the method can be applied to all of the time space and space time fractional equations four problems had also been given to demonstrate the application of the method and some solutions had been illustrated by the graphics

sampling and reconstruction in arbitrary measurement and approximation spaces associated with linear canonical transform the linear canonical transform lct which generalizes many classical transforms has been shown to be a powerful tool for signal processing and optics sampling theory of the lct for bandlimited signals has blossomed in recent years however in practice signals are never perfectly bandlimited and in many cases measurement devices are nonideal the objective of this paper is to develop a sampling theorem for the lct from general measurements which can provide a suitable and realistic model of sampling and approximation for real world applications we first describe a general class of approximation spaces for the lct and provide a full characterization of their basis functions then we propose a generalized sampling theorem for arbitrary measurement and approximation spaces associated with the lct several properties of the proposed sampling theorem are also discussed furthermore the approximation error is estimated finally numerical results and several applications of the derived results are presented

analytic study on two nonlinear evolution equations by using the g g expansion method abstract the validity and reliability of the so called g g expansion method is tested by applying it to two nonlinear evolutionary equations solutions in more general forms are obtained when the parameters are taken as special values it is observed that the previously known solutions can be recovered new rational function solutions are also presented being concise and less restrictive the method can be applied to many nonlinear partial differential equations