nonlocal symmetries related to backlund transformation and their applications starting from nonlocal symmetries related to backlund transformation bt many interesting results can be obtained taking the well known potential kdv pkdv equation as an example a new type of nonlocal symmetry in an elegant and compact form which comes from bt is presented and used to perform research works in two main subjects the nonlocal symmetry is localized by introducing suitable and simple auxiliary dependent variables to generate new solutions from old ones and to consider some novel group invariant solutions some other models both in finite and infinite dimensions are generated under new nonlocal symmetry the finite dimensional models are completely integrable in liouville sense which are shown equivalent to the results given through the nonlinearization method for lax pair

conformal invariance and integrable models usually an integrable nonlinear partial differential equation can be transformed to its conformal invariant form schwartz form using the conformal invariance of the integrable models we can obtain many interesting results in this paper we will focus mainly in obtaining new symmetries and new integrable models starting from the conformal invariance of an integrable model one can obtain infinitely many non local symmetries many types of 1 1 and 2 1 dimensional new sine gordon or sinh gordon extensions are obtained from the conformal flow equations of the koerteweg de vries type equations many other kinds of integrable models can be obtained from the conformal constraints of the known integrable models

explicit solutions from eigenfunction symmetry of the korteweg de vries equation in nonlinear science it is very difficult to find exact interaction solutions among solitons and other kinds of complicated waves such as cnoidal waves and painlevwaves actually even if for the most well known prototypical models such as the kortewet de vries kdv equation and the kadomtsev petviashvili kp equation this kind of problem has not yet been solved in this paper the explicit analytic interaction solutions betweensolitarywavesandcnoidalwavesareobtainedthroughthelocalizationprocedureofnonlocalsymmetries which are related to darboux transformation for the well known kdv equation the same approach also yields some other types of interaction solutions among different types of solutions such as solitary waves rational solutions bessel function solutions and or general painlev e ii solutions

infinitely many lax pairs and symmetry constraints of the kp equation starting from a known lax pair one can get some infinitely many coupled lax pairs infinitely many nonlocal symmetries and infinitely many new integrable models in some different ways in this paper taking the well known kadomtsev petviashvili kp equation as a special example we show that infinitely many nonhomogeneous linear lax pairs can be obtained by using infinitely many symmetries differentiating the spectral functions with respect to the inner parameters using a known lax pair and the darboux transformations dt infinitely many nonhomogeneous nonlinear lax pairs can also be obtained by means of the infinitely many lax pairs dt and the conformal invariance of the schwartz form of the kp equation infinitely many new nonlocal symmetries can be obtained naturally infinitely many integrable models in 1 1 dimensions 2 1 dimensions 3 1 dimensions and even in higher dimensions can be obtained by virtue of symmetry constraints of the kp equation related to the infinitely many lax pairs

non local symmetries via darboux transformations darboux transformations are used to find non local symmetries of integrable nonlinear models which include the kdv kp cdgks and 2 1 dimensional cdgks equations

backlund transformations for the isospectral and nonisospectral akns hierarchies by converting the usual lax pairs for the isospectral and nonisospectral akns hierarchies into lax pairs in riccati forms a unified explicit form of backlund transformations for these hierarchies of nonlinear evolution equations can be obtained in the isospectral case it is an auto backlund transformation however in the nonisospectral case it is not an auto backlund transformation

the lie algebraic structure of symmetries generated by hereditary symmetries based on the works of li zhu and co workers 1985 a general lie algebraic structure is given for symmetries of integrable systems which are generated by hereditary symmetries

the short pulse hierarchy we study a new hierarchy of equations containing the short pulse equation which describes the evolution of very short pulses in nonlinear media and the elastic beam equation which describes nonlinear transverse oscillations of elastic beams under tension we show that the hierarchy of equations is integrable we obtain the two compatible hamiltonian structures we construct an infinite series of both local and nonlocal conserved charges a lax description is presented for both systems for the elastic beam equations we also obtain a nonstandard lax representation

highly accurate approximations of green s and neumann functions on rectangular domains green9s and neumann functions of where is the laplacian operator on a rectangular domain are approximated to any desired degree of accuracy by finite series many applications require only a modest number of terms upper bounds for the errors in these approximations are also derived the approximating functions reveal the structural similarities and differences in green9s and neumann functions

symmetries of the kdv equation and four hierarchies of the integrodifferential kdv equations using the inverse strong symmetry of the korteweg de vries kdv equation on the trivial symmetry and 0 symmetry one gets four new sets of symmetries of the kdv equation these symmetries are expressed explicitly by the multi integrations of the jost function of the kdv equation and constitute an infinite dimensional lie algebra together with two hierarchies of the known symmetries contrary to the general belief the time independent symmetry groups of the kdv and mkdv equations are non abelian and the infinite dimensional lie algebras of the kdv and mkdv equations are nonisomorphic though two equations are related by the miura transformation starting from these sets of symmetries four hierarchies of the integrodifferential kdv equations which can be solved by the schrodinger inverse scattering transformation method are obtained some of these hierarchies enjoy a common strong symmetry and or same local conserved densities

a rapidly convergent modified green function for laplace equation in a rectangular region a computationally useful expression for the modified green function in a rectangular region is derived this can be used to obtain solutions of laplace equation or poisson equation for the electrical potential associated with any configuration of current sources that satisfies the condition of charge balance and represents a generalization of a well known result due to newman applicable to an infinite channel the starting point for the analysis is the very slowly converging fourier series expansion of the modified green function this fourier series can be separated into a fourier series in one variable that can be summed in closed form and a double fourier series that can be partly summed with respect to one index and transformed by application of the poisson summation formula and integration by residues the result of applying the poisson formula can be expressed either as a rapidly converging series of logarithmic functions or as a rapidly converging fourier series which can be integrated analytically with respect to the coordinates of the source point the use of the modified green function is demonstrated first by determining potential distributions that arise from specification of uniform current density on two or more of the boundary surfaces the potential in a cell with two electrodes embedded in opposite sides of the rectangle is then calculated and circumstances in which this potential approaches the potential in an infinite channel are determined

on the nonlocal equations and nonlocal charges associated with the harry dym hierarchy a large class of nonlocal equations and nonlocal charges for the harry dym hierarchy is exhibited they are obtained from nonlocal casimirs associated with its bi hamiltonian structure the lax representation for some of these equations is also given

abundant symmetries for the 1 1 dimensional classical liouville field theory explicit calculation shows us that there exists a rich associated symmetry structure and abundant strong symmetry configuration for the 1 1 dimensional classical liouville field theory starting from these symmetries and strong symmetries various integrable hierarchies can be obtained the liouville i iii hierarchies the modified korteweg de vries kdv hierarchy the fordy gibbons i and fordy gibbons ii hierarchies are just special cases while the kdv hierarchy caudry dodd gibbon sawada kotera hierarchy and kaup kupershmidt hierarchy can be obtained by using a same miura transformation from the modified kdv the fordy gibbons i and fordy gibbons ii hierarchies respectively two types of the generalized riccati hierarchies are also presented

hierarchy of negative order equation and its lax pair a simple method is presented to construct the hierarchy of a negative order equations from a given eigenvalue problem the hierarchy of the negative order equation we obtained is the natural continuation of a corresponding hierarchy of a higher order equation although most of the negative order equations are complicated we give their lax representations three examples are discussed to illustrate the effectiveness of our method

formal variable separation approach for nonintegrable models using a formal variable separation approach a nonlinear partial differential equation can be solved by solving ordinary different equations or even algebraic equations taking the kdv burgers and modified kdv burgers equations with background interaction as simple examples some explicit solitary wave solutions which are induced by background source and nonlinearity or dispersion are obtained

two chain rules for divided differences and faa di bruno s formula in this paper we derive two formulas for divided differences of a function of a function both formulas lead to other divided difference formulas such as reciprocal and quotient rules the two formulas can also be used to derive faa di bruno s formula and other formulas for higher derivatives of composite functions we also derive a divided difference version of faa di bruno s determinant formula

soliton solutions for a negative order non isospectral akns equation abstract in this paper the bilinear form of a negative order non isospectral akns equation is given the soliton solutions are obtained through hirota s direct method and the wronskian technique respectively

negative semi discrete kp and bkp hierarchies via nonlocal symmetries in this paper nonlocal symmetries for the bilinear semi discrete kadomtsev petviashvili kp and bilinear semi discrete b type kp bkp equations are derived by expanding these nonlocal symmetries in power series of each of two parameters we have derived bilinear negative semi discrete kp and bkp hierarchies an interesting observation is that bilinear positive and negative semi discrete kp hierarchies may be derived from the same nonlocal symmetry for the semi discrete kp equation

a generalized f expansion method with symbolic computation exactly solving broer kaup equations abstract in this paper a generalized f expansion method is applied to construct exact solutions of the 2 1 dimensional broer kaup equations as a result many general exact non travelling wave and coefficient function solutions are obtained including single and combined non degenerate jacobi elliptic function solutions soliton like solutions and trigonometric function solutions each of which contains two arbitrary functions compared with most existing f expansion methods the proposed method gives new and more general exact solutions more importantly with the aid of symbolic computation the method provides a powerful mathematical tool to solve a large many nonlinear partial differential equations

negative order korteweg de vries equations in this paper based on the regular korteweg de vries kdv system we study negative order kdv nkdv equations particularly their hamiltonian structures lax pairs conservation laws and explicit multisoliton and multikink wave solutions thorough bilinear backlund transformations the nkdv equations studied in our paper are differential and actually derived from the first member in the negative order kdv hierarchy the nkdv equations are not only gauge equivalent to the camassa holm equation through reciprocal transformations but also closely related to the ermakov pinney systems and the kupershmidt deformation the bi hamiltonian structures and a darboux transformation of the nkdv equations are constructed with the aid of trace identity and their lax pairs respectively the single and double kink wave and bell soliton solutions are given in an explicit formula through the darboux transformation the one kink wave solution is expressed in the form of tanh while the one bell soliton is in the form of sech and both forms are very standard the collisions of two kink wave and two bell soliton solutions are analyzed in detail and this singular interaction differs from the regular kdv equation multidimensional binary bell polynomials are employed to find bilinear formulation and backlund transformations which produce n soliton solutions a direct and unifying scheme is proposed for explicitly building up quasiperiodic wave solutions of the nkdv equations furthermore the relations between quasiperiodic wave solutions and soliton solutions are clearly described finally we show the quasiperiodic wave solution convergent to the soliton solution under some limit conditions

negative powers of olver recursion operators olver s concept of a recursion operator for symmetries of an evolution equation is extended to include negative powers of the operator some negative order kdv equations are derived in this way in particular the inverse image of the trivial zero symmetry generator has an elegant formulation in terms of an eigenfunction of the associated schrodinger equation used in the inverse scattering solution of the kdv equation and this formulation is used to show that this new equation is the miura transform of a sinh gordon equation

on modified green functions in exterior problems for the helmholtz equation the question of non uniqueness in boundary integral equation formulations of exterior problems for the helmholtz equation has recently been resolved with the use of additional radiating multipoles in the definition of the green function the present note shows how this modification may be included in a rigorous formalism and presents an explicit choice of coefficients of the added terms that is optimal in the sense of minimizing the least squares difference between the modified and exact green functions

nonlocal symmetries of the hirota satsuma coupled korteweg de vries system and their applications exact interaction solutions and integrable hierarchy the nonlocal symmetry is derived from the known darboux transformation dt of the hirota satsuma coupled korteweg de vries hs ckdv system and infinitely many nonlocal symmetries are given by introducing the internal parameters by extending the hs ckdv system to an auxiliary system with five dependent variables the prolongation is found to localize the so called seed nonlocal symmetry related to the dt by applying the general lie point symmetry method to this enlarged system we obtain two main results a new type of finite symmetry transformation is derived which is different from the initial dt and can generate new solutions from old ones some novel exact interaction solutions among solitons and other complicated waves including periodic cnoidal waves and painleve waves are computed through similarity reductions in addition two kinds of new integrable models are proposed from the obtained nonlocal symmetry the negative hs ckdv hierarchy by introducing the internal parameters the integrable models both in lower and higher dimensions by restricting the symmetry constraints

the approximation order of four point interpolatory curve subdivision in this paper we derive an approximation property of four point interpolatory curve subdivision based on local cubic polynomial fitting we show that when the scheme is used to generate a limit curve that interpolates given irregularly spaced points sampled from a curve in any space dimension with a bounded fourth derivative and the chosen parameterization is chordal the accuracy is fourth order as the mesh size goes to zero in contrast uniform and centripetal parameterizations yield only second order

nonlocal symmetry darboux transformation and soliton cnoidal wave interaction solution for the shallow water wave equation abstract in classical shallow water wave sww theory there exist two integrable one dimensional sww equation hirota satsuma hs type and ablowitz kaup newell segur akns type in the boussinesq approximation in this paper we mainly focus on the integrable sww equation of akns type the nonlocal symmetry in form of square spectral function is derived starting from its lax pair infinitely many nonlocal symmetries are presented by introducing the arbitrary spectrum parameter these nonlocal symmetries can be localized and the sww equation is extended to enlarged system with auxiliary dependent variables then darboux transformation for the prolonged system is found by solving the initial value problem similarity reductions related to the nonlocal symmetry and explicit group invariant solutions are obtained it is shown that the soliton cnoidal wave interaction solution which represents soliton lying on a cnoidal periodic wave background can be obtained analytically interesting characteristics of the interaction solution between soliton and cnoidal periodic wave are displayed graphically

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

numerical complexiton solutions for the complex kdv equation by the homotopy perturbation method in this paper the homotopy perturbation method is extended to investigate the numerical complexiton solutions of the complex kdv equation by constructing special forms of initial conditions three new types of realistic numerical solutions are obtained numerical positon solution expressed by the trigonometric functions numerical negaton solution expressed by the hyperbolic functions and particularly the numerical analytical complexiton solutions expressed by combinations of the two kinds of functions all these numerical solutions obtained can rapidly converge to the exact solutions obtained by lou et al illustrative numerical figures are exhibited the efficiency of the proposed method

the abundant symmetry structure of hierarchies of nonlinear equations obtained by reciprocal links explicit computation for a kawamoto type equation shows that there is a rich associated symmetry structure for four separate hierarchies of nonlinear integrodifferential equations contrary to the general belief that symmetry groups for nonlinear evolution equations in 1 1 dimensions have to be abelian it is shown that in this case the symmetry group is noncommutative its semisimple part is isomorphic to the affine lie algebra a 1 1 associated to sl 2 c in two of the additional hierarchies that were found an explicit dependence of the independent variable occurs surprisingly the generic invariance for the kawamoto type equation obtained in rogers and carillo phys scr 36 865 1987 via a reciprocal link to the mobius invariance of the singularity equation of the kaup kupershmidt kk equation only holds for one of the additional hierarchies of symmetry groups thus the generic invariance is not a universal property for the complete symmetry group of equations obtained by reciprocal links in

nonlocal topological solitons of the sine gordon equation a special nonlocal symmetry related to the b cklund transformation is obtained for the 1 1 dimensional sine gordon sg equation in its polynomial form which can further generate infinitely many nonlocal symmetries of the sg equation due to the inclusion of an arbitrary spectrum parameter this nonlocal symmetry can be localized and hence turn into a point symmetry for the augmented system through the introduction of an auxiliary function consequently finite transformation similarity reductions and explicit invariant solutions related to the nonlocal symmetry are obtained it is known that the sg equation has topological solitons or kinks or antikinks based on the group invariant solutions a different kind of topological soliton which is a topological soliton situated on a cnoidal wave background is obtained analytically and graphically displayed

the bilinear negative kadomtsev petviashvili system and its kac moody virasoro symmetry group the symmetry transformation group of the bilinear negative kadomtsev petviashvili system is studied by means of a direct method the kac moody virasoro type lie point symmetry algebra is found to be a special infinitesimal form of the symmetry group

infinitely many nonlocal symmetries and conservation laws for the 1 1 dimensional sine gordon equation abstract infinitely many nonlocal symmetries and infinitely many local and nonlocal conservation laws of the 1 1 dimensional sine gordon sg equation are derived in terms of its backlund transformation bt some special nonlocal symmetries and nonlocal conservation laws are obtained from the linearized equations of the sg equation and its bt furthermore one can derive infinitely many nonlocal symmetries from a known nonlocal symmetry but also infinitely many nonlocal conservation laws from a known nonlocal conservation law in addition infinitely many local and nonlocal conservation laws can be directly generated from the bt through the parameter expansion procedure

k symmetries and tau symmetries of evolution equations and their lie algebras on the basis of tu s work 1951 the author establishes a more general skeleton on k symmetries and tau symmetries of evolution equations and their lie algebraic structures and discuss carefully the corresponding symmetries and lie algebras for kdv and jaulent miodek hierarchies

sine gordon and modified korteweg de vries charges the sine gordon and the integrated modified korteweg de vries equations are shown to conserve the same infinite set of charges the charges are determined by a recursion relation as a consequence the solutions of all the equations generated by the charges have in common all time independent properties

a 2 1 dimensional extension for the sine gordon equation starting from the breaking soliton equation the author obtains a new integrable equation in 2 1 dimensions though the equation has no exchange symmetry of the space variables x and y the model reduces back to the known 1 1 dimensional sine gordon or liouville equation

interpolation through an iterative scheme abstract a new method of interpolation is generated through an iterative scheme on dyadic rationals the process is linear local and produces almost twice differentiable functions a fundamental interpolating function is analyzed error formulas are derived the method is applied for curve fitting