a note on colored homfly polynomials for hyperbolic knots from wzw models using the correspondence between chern simons theories and wess zumino witten models we present the necessary tools to calculate colored homfly polynomials for hyperbolic knots for two bridge hyperbolic knots we derive the colored homfly invariants in terms of crossing matrices of the underlying wess zumino witten model our analysis extends previous works by incorporating non trivial multiplicities for the primaries appearing in the crossing matrices so as to describe colorings of homfly invariants beyond the totally symmetric or anti symmetric representations of su n the crossing matrices directly relate to 6j symbols of the quantum group mathcal u q su n we present powerful methods to calculate such quantum 6j symbols for general n this allows us to determine previously unknown colored homfly polynomials for two bridge hyperbolic knots we give explicitly the homfly polynomials colored by the representation 2 1 for two bridge hyperbolic knots with up to eight crossings yet the scope of application of our techniques goes beyond knot theory e g our findings can be used to study correlators in wess zumino witten conformal field theories or in the limit to classical groups to determine color factors for yang mills amplitudes

higher order nonlinear degenerate parabolic equations this paper is concerned with nonlinear degenerate parabolic equations of the form ut 1 m 1 d f u d2m 1u 0 with f u u n n 1 near u 0 and d x under appropriate boundary conditions it is shown that there exists a weak solution u some of the main results of the paper are that u 0 if u0 0 and that the support of u t when u0 0 increases with t for the last property we require that n 2 and m 1

a boundedness criterion for generalized calderon zygmund operators in 1965 a p calderon showed the l2 boundedness of the so called first calderon commutator this is one of the first examples of a non convolution operator associated to a kernel which satisfies certain size and smoothness properties comparable to those of the kernel of the hilbert transform these properties together with the l2 boundedness imply the lp boundedness for all p s in 1 oo many operators in analysis such as certain classes of pseudo differential operators and the cauchy integral operator on a curve are associated with kernels having these properties for these operators one of the major questions is if they are bounded on l2 we are going to give necessary and sufficient conditions for such an operator to be bounded on l2 they are essentially that the images of the function 1 under the actions of the operator and its adjoint both lie in bmo in the case of the aforementioned first commutator this can be checked by an integration by parts in the first section we present some basic notions and state the theorem which is proved in sections 2 and 3 in section 4 we show how to recover some classical results in sections 5 and 6 we construct a functional calculus for small perturbations of a and in section 7 we show a connection between the theory of calderon zygmund operators and kato s conjecture it is a pleasure to express our thanks to r r coifman and y meyer for suggesting many elegant simplifications in our proofs and most of the applications we also wish to thank stephen semmes for several pertinent remarks

analytic study of the fifth order integrable nonlinear evolution equations by using the tanh method abstract the tanh method is used to derive explicit travelling wave solutions for two fifth order integrable nonlinear evolution equations the fifth order kdv equation fkdv and the fifth order caudrey dodd gibbon cdg equation are used as vehicles for our analysis new families of solitary wave solutions and periodic solutions are formally obtained for these equations the presented framework reveals quite a number of useful features of the method used here

variable weak hardy spaces and their applications abstract let p r n 0 be a variable exponent function satisfying the globally log holder continuous condition in this article the authors first introduce the variable weak hardy space on r n wh p r n via the radial grand maximal function and then establish its radial or non tangential maximal function characterizations moreover the authors also obtain various equivalent characterizations of wh p r n respectively by means of atoms molecules the lusin area function the littlewood paley g function or g function as an application the authors establish the boundedness of convolutional type and non convolutional order calderon zygmund operators from h p r n to wh p r n including the critical case when p n n or when p n n where p ess inf x r n p x

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

a new generalized riccati equation rational expansion method to a class of nonlinear evolution equations with nonlinear terms of any order abstract with the aid of symbolic computation a new algebraic method named generalize riccati equation rational expansion grere method is devised for constructing travelling wave solutions for nonlinear evolution equations nees with nonlinear terms of any order compared with most existing tanh methods for finding travelling wave solutions the proposed method not only recovers the results by most known algebraic methods but also provides new and more general solutions for illustration we apply the generalized method to a class of nonlinear evolution equations with nonlinear term of any order as a result we obtain several new kinds of exact solutions this approach can also be applied to other nonlinear evolution equations with nonlinear terms of any order

a new method to obtain either first or second order reductions for parametric polynomial odes abstract for a n th order ordinary differential equation that is either polynomial on the dependent variable and its derivatives or polynomial in the derivatives of the dependent variable a new method to obtain either first or second order reductions respectively is introduced the corresponding reduced equations are generalized polynomials in the dependent variable or in its derivative the method can be applied to equations that may have integer parameters in the exponents of the polynomial of the initial equation a procedure to determine in function of the parameters if the given equation may admit such reductions and to restrict the possible variations of the exponents in the reduced equations is also provided the introduced method strictly generalizes several methods that have recently appeared in the literature as applications of the method this study considers first order reductions for a generalized emden fowler equation and second order reductions for the travelling waves equation of a fifth order korteweg de vries equation and for a painleve chazy equation as far as we know some of the reported reductions are new several maple programs to determine if a given polynomial equation admits any of the considered reductions and if this is the case to determine the reduced equations are also included

novel interacting phenomena in 2 1 dimensional akns system abstract by means of an new auxiliary equation method based on mapping method and the second order linear ordinary differential equation some types of variable separation solutions with two arbitrary lower dimensional functions of the 2 1 dimensional ablowitz kaup newell segur system are derived based on the derived variable separation excitation some new special types of non localized solutions such as doubly periodic wave and line periodic wave are constructed by choosing appropriate functions in addition we also investigate the annihilation phenomenon and non elastic interaction graphically

new exact solutions for mch and mdp equations by auxiliary equation method abstract with the aid of symbolic computation auxiliary equation method is introduced to investigate modified forms of camassa holm and degasperis procesi equations a series of new exact traveling wave solutions including smooth solitary wave solution peakons singular solution periodic wave solution jacobi elliptic solution are obtained in general form these new exact solutions will enrich previous results and help us further understand the physical structures of these two nonlinear equations

racah wigner algebra for q deformed algebras the concepts of vector coupling coefficients and recoupling coefficients are generalized to q deformed algebras their properties under complex conjugation and permutation of irreps are derived relations between the coefficients for example the racah backcoupling rule are proved it is shown that these properties may be used to recursively calculate the coupling and recoupling coefficients for all q algebras the 3 jm and 6 j symbols for su 2 q are used to illustrate the building up method

a fourth order numerical scheme for solving the modified burgers equation a finite difference scheme based on fourth order rational approximants to the matrix exponential term in a two time level recurrence relation is proposed for the numerical solution of the modified burgers equation the resulting nonlinear system which is analyzed for stability is solved using an already known modified predictor corrector scheme the results arising from the experiments are compared with the corresponding ones known from the available literature

large n duality lagrangian cycles and algebraic knots we consider knot invariants in the context of large n transitions of topological strings in particular we consider aspects of lagrangian cycles associated to knots in the conifold geometry we show how these can be explicitly constructed in the case of algebraic knots we use this explicit construction to explain a recent conjecture relating study of stable pairs on algebraic curves with homfly polynomials furthermore for torus knots using the explicit construction of the lagrangian cycle we also give a direct a model computation and recover the homfly polynomial for this case

some special types of solutions of a class of the n 1 dimensional nonlinear wave equations some special types of solutions with arbitrary functions for the n 1 dimensional sine cosine gordon equation the n 1 dimensional double sinh gordon equation and the n 1 dimensional sinh cosinh gordon equations are constructed by means of the separation transformation approach as an illustrative sample the properties of some solutions are shown by their figures

exact solutions to the two dimensional korteweg de vries burgers equation two methods are described for obtaining exact travelling monotone shock wave solutions to the two dimensional korteweg de vries burgers equation they are compared with two methods that have been reported recently by other authors

long wave instabilities and saturation in thin film equations hocherman and rosenau conjectured that long wave unstable cahn hilliard type interface models develop nite time singularities when the nonlinearity in the destabilizing term grows faster at large amplitudes than the nonlinearity in the stabilizing term phys d 67 113 125 1993 we consider this conjecture for a class of equations often used to model thin lms in a lubrication context by showing that if the solutions are uniformly bounded above or below e g are nonnegative then the destabilizing term can be stronger than previously conjectured yet the solution still remains globally bounded for example they conjecture that for the long wave unstable equation ht h n hxxx x h m hx x m n leads to blow up using a conservation of volume constraint for the case m n 0 we conjecture a dierent critical exponent for possible singularities of nonnegative solutions we prove that nonlinearities with exponents below the conjectured critical exponent yield globally bounded solutions specically for the above equation solutions are bounded if m n 2 the bound is proved using energy methods and is then used to prove the existence of non negative weak solutions in the sense of distributions we present preliminary numerical evidence suggesting that m n 2 can allow blow up

solitons and discrete eigenfunctions of the recursion operator of non linear evolution equations i the caudrey dodd gibbon sawada kotera equation the solution of the third order isospectral equation of the caudrey dodd gibbon sawada kotera equation cdgske for soliton potential is obtained recursively from the riccati equation derived by iterating once the auto backlund transformation it is then shown that the discrete eigenfunctions of the sixth order recursion operator for this equation can be written in terms of the solutions of the isospectral equation the behaviour of the 1 soliton solution which has certain novel features is studied a sine gordon like equation resembling the double sine gordon equation is derived from the cdgske

generalized tanh method with symbolic computation and generalized shallow water wave equation abstract we report that symbolic computation with the generalized tanh method leads to new soliton like solutions for a 2 1 dimensional generalization of the shallow water wave equations

soliton solutions and doubly periodic wave solutions for a new generalized hirota satsuma coupled system in this paper a new algebraic method with symbolic computation is presented to uniformly construct a series of exact solutions for general nonlinear equations compared with most existing tanh methods the proposed method not only gives new and more general solutions but also provides a guideline to classify the various types of the solutions according to the values of some parameters for illustration we apply the proposed method to solve a new generalized hirota satsuma coupled system and explicitly construct a series of exact solutions which include the soliton solutions and elliptic doubly periodic solutions as special cases in addition the links among our proposed method the tanh method and the extended method are also clarified generally

a bounded linear integrator for some diffusive nonlinear time dependent partial differential equations we propose a numerical method to approximate the solutions of generalized forms of two bi dimensional models of mathematical physics namely the burgers fisher and the burgers huxley equations in one dimensional form the literature in the area gives account of the existence of analytical solutions for both models in the form of traveling wave fronts bounded within an interval i of the real numbers motivated by this fact we propose a finite difference methodology that guarantees that under certain analytical conditions on the model and computer parameters estimates within i will evolve discretely into new estimates which are likewise bounded within i additionally we establish the preservation in the discrete domain of the skew symmetry of the solutions of the models under study our computational implementation of the method confirms numerically that the properties of positivity and boundedness are preserved under the analytical constraints derived theoretically our simulations show a good agreement between the analytical solutions derived in the present work and the corresponding numerical approximations

auto backlund transformation and similarity reductions for coupled burger s equation in this work we applied backlund transformation and similarity reduction for coupled burger s equation clarkson and kruskal developed a direct and simple method to obtain more similarity solutions of nonlinear partial differential equation we received our inspiration from fan s article as far as we think that our work is an extended one from the fan s has done in his paper as a result of this study we obtained solitary wave solutions and traveling wave solutions of coupled burger s equation

new molecular characterizations of anisotropic musielak orlicz hardy spaces and their applications abstract let r n 0 0 be an anisotropic growth function and a a general expansive matrix on r n let h a r n be the anisotropic musielak orlicz hardy space associated with a defined via the non tangential grand maximal function in this article the authors establish a new molecular characterization of h a r n via its known atomic characterizations which positively answer a question mentioned by li et al 2016 34 as an application the authors obtain a criterion on the boundedness of linear operators on h a r n which further implies the boundedness of integral anisotropic calderon zygmund operators on h a r n in addition the boundedness of these operators from h a r n to the musielak orlicz space l r n is also presented

bilinear operators on herz type hardy spaces the authors prove that bilinear operators given by finite sums of products of calderon zygmund operators on rn are bounded from hk11 q1 hk22 q2 into hk p q if and only if they have vanishing moments up to a certain order dictated by the target space herehk q are homogeneous herztype hardy spaces with 1 p 1 p1 1 p2 0 pi 1 q 1 q1 1 q2 1 q1 q2 1 q 1 2 and n qi i as an application they obtain that the commutator of a calderon zygmund operator with a bmo function maps a herz space into itself

reductions of pdes to second order odes and symbolic computation a new method to obtain second order reductions for ordinary differential equations which are polynomial in the derivatives of the dependent variable is presented the method is applied to obtain reductions and new solutions to several well known equations of mathematical physics a lubrication equation a thin film equation the zoomeron equation and a family of 5 th order partial differential equations which includes the caudrey dodd gibbon sawada kotera kaup kupershmidt ito and lax equations some pieces of computer algebra code to derive the reductions are also included

persistence of travelling wave solutions of a fourth order diffusion system in this paper the extended burgers huxley equation with the fourth order derivative is considered first the convergence to the uniform steady state is proved which means the solution of the equation with positive initial data will remain positive for time t sufficiently large then the persistence of the travelling wave solution for the extended equation on the unbounded domain is investigated we have proved that this solution will persist under small perturbation of the equation

traveling wave solutions for nonlinear equations using symbolic computation abstract in this paper a direct and unified algorithm for constructing multiple travelling wave solutions of nonlinear partial differential equations pdes is presented and implemented in a computer algebraic system the key idea of this method is to take full advantage of a riccati equation involving a parameter and use its solutions to replace the tanh function in the tanh method it is quite interesting that the sign of the parameter can be used to exactly judge the number and types of such travelling wave solutions in this way we can successfully recover the previously known solitary wave solutions that had been found by the tanh method and other more sophisticated methods more importantly for some equations with no extra effect we also find other new and more general solutions at the same time by introducing appropriate transformations our method is further extended to the nonlinear pdes whose balancing numbers may be any nonzero real numbers the efficiency of the method can be demonstrated on a large variety of nonlinear pdes such as those considered in this paper burgers huxley equation coupled korteweg de vries equation caudrey dodd gibbon kawada equation active dissipative dispersive media equation generalized fisher equation and nonlinear heat conduction equation

an algebraic method for finding a series of exact solutions to integrable and nonintegrable nonlinear evolution equations an algebraic method is devised to uniformly construct a series of exact solutions for general integrable and nonintegrable nonlinear evolution equations compared with most existing tanh methods the jacobi function expansion method or other sophisticated methods the proposed method not only gives new and more general solutions but also provides a guideline to classify the various types of the solutions according to the values of some parameters the solutions obtained in this paper include a polynomial solutions b exponential solutions c rational solutions d triangular periodic wave solutions e hyperbolic and solitary wave solutions and f jacobi and weierstrass doubly periodic wave solutions the efficiency of the method can be demonstrated on a large variety of nonlinear equations such as those considered in this paper new 2 1 dimensional calogero kdv equation 3 1 dimensional jimbo miwa equation symmetric regular long wave equation drinfel d sokolov wilson equation 2 1 dimensional generalized dispersive long wave equation double sine gordon equation calogero degasperis fokas equation and coupled schrodinger boussinesq equation in addition the links among our proposed method the tanh method the extended method and the jacobi function expansion method are also clarified generally

the symmetric group algebraic formulas for some sf 6j symbols and sf sf1 sf2 3jm symbols explicit rank dependent expressions have been obtained for some symmetric group sf 6j symbols and some sf sf1 sf2 3jm symbols using butler s recursion method a key point in deriving these results is the use of the reduced notation introduced by murnaghan to label irreps various symmetries of the 6j and 3jm symbols have been imposed these include the complex conjugation permutation and transpose conjugation we incorporate a new symmetry that arises from the occurrence of the two isomorphic direct product groups sf1 sf2 and sf2 sf1 as subgroups of sf in relation to the tables of 6j and 3jm symbols presented a discussion is given of the symmetric group unitary group duality

polynomial invariants for torus knots and topological strings we make a precision test of a recently proposed conjecture relating chern simons gauge theory to topological string theory on the resolution of the conifold first we develop a systematic procedure to extract string amplitudes from vacuum expectation values vevs of wilson loops in chern simons gauge theory and then we evaluate these vevs in arbitrary irreducible representations of su n for torus knots we find complete agreement with the predictions derived from the target space interpretation of the string amplitudes we also show that the structure of the free energy of topological open string theory gives further constraints on the chern simons vevs our work provides strong evidence towards an interpretation of knot polynomial invariants as generating functions associated to enumerative problems

hardy space estimates for limited ranges of muckenhoupt weights abstract in this article we give a full necessary and sufficient set of conditions for a calderon zygmund operator to be bounded on weighted hardy spaces h w p where w is an muckenhoupt weight and 0 p in fact this result is new even when 1 p since it allows for h w p boundedness of an operator when 1 p q and w a q where it is possible that h w p l w p these singular integral results are achieved by proving littlewood paley stein square function type estimates from h w p into l w p for 0 p and a muckenhoupt weight w which are interesting results in their own right new techniques involving a weight invariant spaces are also used to prove the weighted estimates for calderon zygmund operators more precisely we prove the following bmo type weight invariance properties for a fixed s 0 the weighted sobolev bmo spaces i s b m o w coincide for all w a the weighted p type triebel lizorkin spaces f w s 2 coincide for all w a and these two classes of spaces coincide with each other as well all of which have comparable norms up to constants depending on an a p character of the weight w a

herz spaces and restricted summability of fourier transforms and fourier series a general summability method the so called summability is considered for multi dimensional fourier transforms and fourier series a new inequality for the hardy littlewood maximal function is verified it is proved that if the fourier transform of is in a herz space then the restricted maximal operator of the means of a distribution is of weak type 1 1 provided that the supremum in the maximal operator is taken over a cone like set from this it follows that t f f over a cone like set a e for all f l1 r d moreover

calderon zygmund operators on herz type hardy spaces abstract we consider the boundedness of calderon zygmund operators from h k p q r n to h k p q r n where h k p q r n is the hardy space associated with the herz space k p q r n and h k p q r n is the local version of h k p q r n we show calderon s commutator is bounded from h k p q to h k p q

on a conditionally stable nonlinear method to approximate some monotone and bounded solutions of a generalized population model in this work we design a numerical method to approximate some solutions of a generalization of a nonlinear diffusion reaction model which appears in the context of population dynamics the existence of traveling wave solutions for the equation under consideration is a well known fact some of such solutions are positive bounded from above and monotone in both space and time motivated by these facts we propose an explicit nonlinear finite difference methodology to approximate consistently the solutions of the model under investigation in the linear regime the method is consistent of first order in time and of second order in space under certain flexible parameter conditions the method is capable of preserving the positivity the boundedness and the spatial and the temporal monotonicity of the traveling wave solutions moreover we establish analytically and numerically that the nonlinear method is conditionally stable a computational implementation of our technique shows that the method preserves in practice the mathematical features of interest of the exact solutions considered

a skew symmetry preserving computational technique for obtaining the positive and the bounded solutions of a time delayed advection diffusion reaction equation abstract in this work we consider a one dimensional time delayed advective version of the well known fisher kolmogorov petrovsky piscounov equation from population dynamics which extends several models from mathematical physics including the classical wave equation the nonlinear klein gordon equation a fitzhugh nagumo equation from electrodynamics and the burgers huxley equation and the newell whitehead segel equation from fluid mechanics we propose a skew symmetry preserving finite difference scheme for approximating the solutions of the model under investigation and establish conditions on the model coefficients and the numerical parameters under which the method provides positive or bounded approximations for initial data which are likewise positive or bounded respectively the derivation of the conditions under which the positivity and the boundedness of the approximations is guaranteed is based on the properties of the inverses of m matrices in fact the conditions obtained here assure that the iterative method is described in vector form through the multiplication by a matrix of this type we provide simulations in order to show that the technique is indeed conditionally positivity preserving and boundedness preserving

multilinear theory of strongly singular calderon zygmund operators and applications abstract the main purpose of this paper is to study the multilinear strongly singular calderon zygmund operator whose kernel is more singular near the diagonal than that of the standard multilinear calderon zygmund operator the sharp maximal estimate for this class of multilinear singular integrals is established and as applications its boundedness on product of weighted lebesgue spaces and product of variable exponent lebesgue spaces is obtained respectively moreover the endpoint estimates of the types l l b m o b m o b m o b m o and l m o l m o l m o are established for the multilinear strongly singular calderon zygmund operator these results improve the corresponding known ones for the standard multilinear calderon zygmund operator furthermore we remove the size condition assumption for the kernel of the multilinear strongly singular calderon zygmund operator which has been imposed in the literature for the case of the standard multilinear calderon zygmund operator to get the sharp maximal estimates see remarks 1 1 1 4 extra care is needed to deal with the mean oscillation over balls with small radius to overcome the stronger singularity in establishing such sharp maximal estimates and endpoint estimates

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

exact travelling wave solutions of nonlinear evolution equations in 1 1 and 2 1 dimensions abstract based upon the symbolic computation system maple the projective riccati equation method is improved to seek for travelling wave solutions in 1 1 and 2 1 dimensional nonlinear evolution equations by introducing two new elementary bell shaped and kink shaped functions the validity and reliability of the method is tested by applying it to the 2 1 dimensional generalized of the shallow water wave equation potential kdv equation and burgers kdv equation

solitary solutions of some nonlinear evolution equations in this paper the traveling wave reduction combined with the homogeneous balance method hbm is used to find new exact solutions by using less studied nonlinear partial differential equations npdg of higher order the usual starting point is a special transformation converting the npdg in its two variables x and t into an nonlinear ordinary differential equation node in the single variable x using the hyperbolic tangent method new exact solutions for the three npdgs are studied the new feature of this paper is of course the fact that we are dealing with npdgs which cannot be found by studying the relevant literature therefore we believe it is time for publishing some special solutions of this interesting kind of equations clearly we do not use a new formalism but with the aid of the tanh method we are able to calculate at least exact traveling wave solutions solitary waves which can physically interpreted as an action of wave propagation simultaneous we point out the necessity of such sophisticated methods since a general theory of npdgs does not exist the existence of nontrivial noncomplex valued solutions of the ramani equation enables us to generate families of real valued solutions of a nonlinear sixth order system special differential transformations are leading to connections between traveling wave solutions of associated nonlinear evolution equations

application of he s exp function method for nonlinear evolution equations in this paper he s exp function method is used to construct solitary and soliton solutions of nonlinear evolution equations modified equal width burger huxley and nonlinear heat conduction equations are chosen to illustrate the effectiveness of the method the method is straightforward and concise and its applications are promising the exp function method presents a wider applicability for handling nonlinear wave equations

generalized algebraic method and new exact traveling wave solutions for 2 1 dimensional dispersive long wave equation abstract with the help of the symbolic computation system maple a new generalized algebraic method to uniformly construct solutions in terms of special function of nonlinear partial differential equations is presented by means of a more general ansatz as an application of the method we choose a 2 1 dimensional dispersive long wave equation to illustrate the method as a result we can successfully obtain the solutions found by the method proposed by fan e g fan phys lett a 300 2002 243 and find other new and more general solutions at the same time which include polynomial solutions exponential solutions rational solutions triangular periodic wave solutions hyperbolic and soliton solutions jacobi and weierstrass doubly periodic wave solutions

on classes of integrable systems and the painleve property the caudrey dodd gibbon equation is found to possess the painleve property investigation of the backlund transformations for this equation obtains the kuperschmidt equation a certain transformation between the kuperschmidt and caudrey dodd gibbon equation is obtained this transformation is employed to define a class of p d e s that identically possesses the painleve property for equations within this class backlund transformations and rational solutions are investigated in particular the sequences of higher order kdv caudrey dobb gibbon and kuperschmidt equations are shown to possess the painleve property

non classical symmetries and the singular manifold method the burgers and the burgers huxley equations in this paper a generalization of the direct method of clarkson and kruskal 1989 for finding similarity reductions of partial differential equations is found and discussed for the burgers and burgers huxley equations the generalization incorporates the singular manifold method largely based upon the painleve property this singular manifold can be used as a reduced variable furthermore a sort of inverse procedure is hereby developed through which we find the equations that yield the vector field components associated to the symmetries of the pde this procedure also displays the profound relationship among the symmetries and the singular manifold as a reduced variable the symmetries found in this way are shown to be those corresponding to the so called non classical symmetries by bluman and cole 1974 and olver and rosenau 1986

a rational generalization of fan s method and its application to generalized shallow water wave equations in this paper we employ a rational expansion to generalize fan s method for exact travelling wave solutions for nonlinear partial differential equations pdes to verify the reliability of the proposed method the generalized shallow water wave gsww equation has been investigated as an example kinds of new exact travelling wave solutions of a rational form have been obtained this indicates that the proposed method provides a more general result for exact solution of nonlinear equations

positivity preserving finite volume scheme for the nagumo type equations on distorted meshes abstract in this paper we present a nonlinear positivity preserving finite volume scheme for the nagumo type equations with anisotropic tensor diffusion coefficient for the diffusion term we use the positivity preserving finite volume scheme for the time direction we use the backward euler approximation we deal with nonlinear reaction term implicitly and decompose nonlinear reaction coefficient into two nonnegative functions thus we get a system of nonlinear algebraic equations the advantages of our scheme are that it can be applied to distorted meshes and has no severe constraint on the time step the numerical results verify the theoretical result

the numerical solution of a generalized burgers huxley equation through a conditionally bounded and symmetry preserving method in this article we propose a non standard finite difference scheme to approximate the solutions of a generalized burgers huxley equation from fluid dynamics our numerical method preserves the skew symmetry of the partial differential equation under study and under some analytical constraints of the model constants and the computational parameters involved it is capable of preserving the boundedness and the positivity of the solutions in the linear regime the scheme is consistent to first order in time due partially to the inclusion of a tuning parameter in the approximation of a temporal derivative and to second order in space we compare the results of our computational technique against the exact solutions of some particular initial boundary value problems our simulations indicate that the method presented in this work approximates well the theoretical solutions and moreover that the method preserves the boundedness of solutions within the analytical constraints derived here in the problem of approximating solitary wave solutions of the model under consideration we present numerical evidence on the existence of an optimum value of the tuning parameter of our technique for which a minimum relative error is achieved finally we linearly perturb a steady state solution of the partial differential equation under investigation and show that our simulations still converge to the same constant solution establishing thus robustness of our method in this sense

two new reductions methods for polynomial differential equations and applications to nonlinear pdes abstract for ordinary differential equations of the form p y v v v n 0 which are polynomial in the variables v v v n two new reduction methods to first order equations are considered the reduced equations are of the forms v q 1 y v 1 q and v q 2 y v 1 q where q 1 and q 2 are two polynomials of degree p in v 1 q and v respectively whose coefficients depend on y in contrast to most of the known reduction methods of these types which use either q 1 or q 2 in this paper the values of the positive integers p q are not predetermined a procedure to obtain the possible values of the integers for which a reduction of any of these types may exist is provided as a consequence new reductions that cannot be obtained by other known methods may be found the new methods have been applied to obtain some reductions and consequently new solutions for three polynomial ordinary differential equations related to well known equations in mathematical physics the kuramoto sivashinsky equation a generalized benney equation and a 5 th order kdv equation some pieces of computer algebra code written in maple and implementing the underlying algorithms to derive the reductions are also included

racah coefficients of quantum group uq n racah coefficients of uq n are derived from subduction coefficients of hecke algebras hf q by using the schur weyl duality relation between uq n and hf q the racah coefficients of uq n for the resulting irreps n1 n2 nk o with sigma i 1kni or 5 and some of them for the resulting irreps with multiplicity and sigma i 1kni or 6 are tabulated

commutators of hardy operators in this paper we establish hardy s integral inequalities for commutators generating hardy operators with one sided dyadic cmo functions

application of the method of simplest equation for obtaining exact traveling wave solutions for the extended korteweg de vries equation and generalized camassa holm equation a version of the method of the simplest equation called modified method of simplest equation is applied to the extended korteweg de vries equation and to generalized camassa holm equation exact traveling wave solutions of these two nonlinear partial differential equations are obtained the equations of bernoulli riccati and the extended tanh equation are used as simplest equations some of the obtained solutions correspond to surface water waves

some estimates for bochner riesz operators on the weighted herz type hardy spaces abstract in this paper by using the atomic decomposition and molecular characterization of the homogeneous and non homogeneous weighted herz type hardy spaces h k q p w 1 w 2 h k q p w 1 w 2 we obtain some weighted boundedness properties of the bochner riesz operator and the maximal bochner riesz operator on these spaces for n 1 p 1 q 0 p 1 and 1 q

knot invariants from topological recursion on augmentation varieties using the duality between wilson loop expectation values of su n chern simons theory on s 3 and topological open string amplitudes on the local mirror of the resolved conifold we study knots on s 3 and their invariants encoded in colored homfly polynomials by means of topological recursion in the context of the local mirror calabi yau threefold of the resolved conifold we generalize the topological recursion of the remodelled b model in order to study branes beyond the class of toric harvey lawson special lagrangians as required for analyzing non trivial knots on s 3 the basic ingredients for the proposed recursion are the spectral curve given by the augmentation variety of the knot and the calibrated annulus kernel encoding the topological annulus amplitudes associated to the knot we present an explicit construction of the calibrated annulus kernel for torus knots and demonstrate the validity of the topological recursion we further argue that if an explicit form of the calibrated annulus kernel is provided for any other knot the proposed topological recursion should still be applicable we study the implications of our proposal for knot theory which exhibit interesting consequences for colored homfly polynomials of mutant knots