new exact solutions of the broer kaup equations in 2 1 dimensional spaces abstract with the aid of maple several new kinds of exact solutions for the broer kaup equations in 2 1 dimensional spaces are obtained by using a new ansatz this approach can also be applied to other nonlinear evolution equations

new candidates for arbitrage free stock price models via generalized conditional symmetry method abstract in this paper a new perspective upon generating arbitrage free stock price models is proposed the generalized conditional symmetry gcs method is applied to the governing second order 1 1 partial differential equation which does contain a rational parameter p drawn from the interval 1 2 1 we investigate the conditions that yield the concerned equation admitting a special class of second order gcss the determining system is solved in several special cases and from invariance surface condition associated to each of the gcs operator for all values of p some invariant solutions are pointed out new candidate models for arbitrage free stock price are derived

generalized multiple riccati equations rational expansion method with symbolic computation to construct exact complexiton solutions of nonlinear partial differential equations in this paper based on symbolic computation and the idea of rational expansion method a generalized multiple riccati equations rational expansion method gmrere is proposed to uniformly construct a series of exact complexiton solutions for nonlinear partial differential equations compared with most existing tanh function methods and rational expansion methods the proposed method not only recovers some known solutions but also finds some new and general complexiton solutions the efficiency of the method can be demonstrated on 2 1 dimensional burgers equation

new complexiton solutions of the nonlinear evolution equations using a generalized rational expansion method with symbolic computation abstract in this paper we study complexiton solutions of nonlinear evolution equations by using a generalized ansatz with the help of symbolic computation maple we obtain new types of complexiton solutions of the 2 1 dimensional burgers equation the solutions contain the combination of hyperbolic function and elliptic function trigonometric function and elliptic function

homotopy perturbation method for fractional kdv equation in this paper the homotopy perturbation method is directly applied to derive approximate solutions of the fractional kdv equation the results reveal that the proposed method is very effective and simple for solving approximate solutions of fractional differential equations

homotopy perturbation method for a type of nonlinear coupled equations with parameters derivative abstract in this paper homotopy perturbation method is directly extended to investigate nonlinear coupled equations with parameters derivative and to derive their numerical solutions these nonlinear coupled equations with parameters derivative contain many important mathematical physics equations and reaction diffusion equations by choosing different values of the parameters in general formal numerical solutions as a result a very rapidly convergent series solution is obtained the efficiency and accuracy of the method are verified by using two famous examples coupled burgers and mkdv equations numerical solutions show that good results have been achieved

riemann roch and abel jacobi theory on a finite graph abstract it is well known that a finite graph can be viewed in many respects as a discrete analogue of a riemann surface in this paper we pursue this analogy further in the context of linear equivalence of divisors in particular we formulate and prove a graph theoretic analogue of the classical riemann roch theorem we also prove several results analogous to classical facts about riemann surfaces concerning the abel jacobi map from a graph to its jacobian as an application of our results we characterize the existence or non existence of a winning strategy for a certain chip firing game played on the vertices of a graph

chip firing on dynkin diagrams and mckay quivers this paper establishes new connections between the representation theory of finite groups and sandpile dynamics two classes of avalanche finite matrices and their critical groups integer cokernels are studied from the viewpoint of chip firing sandpile dynamics namely the cartan matrices of finite root systems and the mckay cartan matrices for finite subgroups g of general linear groups in the root system case the recurrent and superstable configurations are identified explicitly and are related to minuscule dominant weights in the mckay cartan case for finite subgroups of the special linear group the cokernel is related to the abelianization of the subgroup g in the special case of the classical mckay correspondence the critical group and the abelianization are shown to be isomorphic

special solutions from the variable separation approach the davey stewartson equation a variable separation procedure for the davey stewartson ds equation is proposed by using a prior ansatz to its bilinear form the reduced equations for two variable separated fields have the same trilinear form although they possess different independent variables the trilinear equation can be changed to a spacetime symmetric form and can be solved by means of a boussinesq type equation system whenever a pair of solutions of the reduced fields are obtained a corresponding solution of the ds equation can be obtained algebraically the single dromion solution and some kinds of positon solutions are obtained explicitly

painleve integrability of a 2 1 dimensional reaction diffusion equation exact solutions and their interactions we investigate the singularity structure analysis of a 2 1 dimensional coupled nonlinear extension of the reaction diffusion nlerd equation by means of the painleve p test following the weiss et al s formalism j math phys 24 522 1983 we prove the arbitrariness of the expansion coefficients of the observables thus without the use of the kruskal s simplification we obtain a backlund transformation of the coupled nlerd equation via a consistent truncation procedure stemming from the weiss s methodology j weiss m tabor and g carnevale j math phys 25 13 1984 in the wake of such results we unveil a typical spectrum of localized and periodic coherent patterns we also investigate the scattering properties of such structures and we unearth two peculiar soliton phenomena namely the fusion and the fission

quasiperiodic tilings with tenfold symmetry and equivalence with respect to local derivability two 2d quasiperiodic tilings with generalized tenfold symmetry are derived from the lattice a4r the reciprocal of the root lattice a4 both tilings are built from four tiles triangles in one case rhombi and hexagons in the other after a brief description of the tilings and their structures the authors introduce the equivalence concept of mutual local derivability they present its key properties and its application to several tenfold tilings and discuss some implications on a future classification of tilings in position space

stochastic exact solutions of the wick type stochastic nls equation based upon the hermite transformation and the extended sub equation expansion method soliton like solutions of the stochastic schrodinger equation in the white noise environment are obtained via maple

non classical symmetry reduction example of the boussinesq equation a symmetry of an equation will leave the set of all solutions invariant a conditional symmetry will leave only a subset of solutions defined by some differential condition invariant the authors show how a specific class of conditional symmetries can be used to reduce a partial differential equation to an ordinary one in particular for the boussinesq equation these conditional symmetries together with the ordinary ones provide all possible reductions to ordinary differential equations a group theoretical explanation of the recently obtained new reductions is provided

trees parking functions syzygies and deformations of monomial ideals for a graph g we construct two algebras whose dimensions are both equal to the number of spanning trees of g one of these algebras is the quotient of the polynomial ring modulo certain monomial ideal while the other is the quotient of the polynomial ring modulo certain powers of linear forms we describe the set of monomials that forms a linear basis in each of these two algebras the basis elements correspond to g parking functions that naturally came up in the abelian sandpile model these ideals are instances of the general class of monotone monomial ideals and their deformations we show that the hilbert series of a monotone monomial ideal is always bounded by the hilbert series of its deformation then we define an even more general class of monomial ideals associated with posets and construct free resolutions for these ideals in some cases these resolutions coincide with scarf resolutions we prove several formulas for hilbert series of monotone monomial ideals and investigate when they are equal to hilbert series of deformations in the appendix we discuss the abelian sandpile model

generalized method and new exact wave solutions for 2 1 dimensional broer kaup kupershmidt system in this paper a new algebra method i e generalized elliptic equation rational expansion method is devised to uniformly construct a series of exact solutions for nonlinear partial differential equations compared with most existing tanh methods the proposed method not only recover some known solutions but also find some new and general solutions the efficiency of the method can be demonstrated on the 2 1 dimensional broer kaup kupershmidt system as a result we obtain many new types of solutions include rational formal solitary wave solutions rational formal triangular periodic wave solutions rational formal jacobi and weierstrass double period solutions and rational wave solutions

application of rational expansion method for stochastic differential equations abstract by means of the hermite transformation a new general ansatz and the symbolic computation system maple we apply the riccati equation rational expansion method 24 to uniformly construct a series of stochastic non traveling wave solutions for stochastic differential equations to illustrate the effectiveness of our method we take the stochastic mkdv equation as an example and successfully construct some new and more general solutions the method can also be applied to solve other nonlinear stochastic differential equation or equations

enumerating linear systems on graphs the divisor theory of graphs views a finite connected graph g as a discrete version of a riemann surface divisors on g are formal integral combinations of the vertices of g and linear equivalence of divisors is determined by the discrete laplacian operator for g as in the case of riemann surfaces we are interested in the complete linear system d of a divisor d the collection of nonnegative divisors linearly equivalent to d unlike the case of riemann surfaces the complete linear system of a divisor on a graph is always finite we compute generating functions encoding the sizes of all complete linear systems on g and interpret our results in terms of polyhedra associated with divisors and in terms of the invariant theory of the dual of the jacobian group of g if g is a cycle graph our results lead to a bijection between complete linear systems and binary necklaces our results also apply to a model in which the laplacian is replaced by an invertible integral m matrix

backlund transformation and wronskian solitons for the 2 1 dimensional nizhnik novikov veselov equations abstract korteweg de vries type equations are seen to describe the shallow water waves stratified internal waves ion acoustic waves plasma physics and lattice dynamics an isotropic extension of which are the 2 1 dimensional nizhnik novikov veselov equations hereby based on the hirota bilinear method and symbolic computation we derive the bilinear form and backlund transformation for such an extension n soliton solutions in the wronskian form are given and it can be verified that the backlund transformation can connect the n 1 and n soliton solutions solitonic propagation and collision are discussed the larger amplitude soliton moves faster and then overtakes the smaller one after the collisions the solitons keep their original shapes and velocities invariant except for the phase shift collisions among the two and three solitons are all elastic

new similarity reductions of the boussinesq equation some new similarity reductions of the boussinesq equation which arises in several physical applications including shallow water waves and also is of considerable mathematical interest because it is a soliton equation solvable by inverse scattering are presented these new similarity reductions including some new reductions to the first second and fourth painleve equations cannot be obtained using the standard lie group method for finding group invariant solutions of partial differential equations they are determined using a new and direct method that involves no group theoretical techniques

approximate solution for system of differential difference equations by means of the homotopy analysis method abstract in this paper the homotopy analysis method is successfully applied to solve approximate solutions of the differential difference system the solution of another relativistic toda lattice system is considered comparisons made between the results of the proposed method and exact solutions reveal that the homotopy analysis method is very effective in solving differential difference system

self organized critical state of sandpile automaton models we study a general bak tang wiesenfeld type automaton model of self organized criticality in which the toppling conditions depend on local height but not on its gradient we characterize the critical state and determine its entropy for an arbitrary finite lattice in any dimension the two point correlation function is shown to satisfy a linear equation the spectrum of relaxation times describing the approach to the critical state is also determined exactly

new applications of the singular manifold method to the 2 1 dimensional burgers equations new applications of the singular manifold method are presented to a 2 1 dimensional burgers equation and a higher order burgers equation by choosing different seed solutions auto backlund transformations cole hopf transformations and general functional separation solutions containing three arbitrary functions are obtained for the equations in question localized coherent structures are for the first time expressed by jacobi elliptic functions not infinite periodic limit and their interaction properties are numerically studied

a new coupled sub equations expansion method and novel complexiton solutions of 2 1 dimensional burgers equation in this paper based on symbolic computation and the idea of rational expansion method a new coupled sub equations method is devised to uniformly construct a series of exact complexiton solutions for nonlinear evolution equations compared with most existing tanh function methods and other sophisticated methods the proposed method not only recover some known solutions but also find some new and general solutions which include many new types of complexiton solutions the efficiency of the method can be demonstrated on 2 1 dimensional burgers equation

modified method of simplest equation and its applications to the bogoyavlenskii equation in this paper we construct the exact traveling wave solutions of the bogoyavlenskii equation using modified method of simplest equation the simplest equation herewith is a special riccati equation in the meantime the proposed method in this work is proved to be a powerful mathematical tool for obtaining exact solutions of other nonlinear partial differential equations in mathematical physics and other research fields

lie transformations similarity reduction and solutions for the nonlinear madelung fluid equations with external potential the application of lie group methods to a system of coupled nonlinear partial differential equations representing what is usually called a madelung fluid is shown the generating operators of the transformation group that depends on five arbitrary group constants will be constructed and all subclasses of systems of ordinary differential equations derived by similarity reduction will be presented in tabular form two subclasses of physical interest are investigated in detail and the similarity solutions are compared with solutions found earlier by the application of inverse scattering transform techniques to the cubic nonlinear schrodinger equation similarity solutions for the madelung equations with linear external potential x f0x are presented

2 1 dimensional compacton solutions with and without completely elastic interaction properties taking the 2 1 dimensional broer kaup kupershmidt system as a simple example some special types of 2 1 dimensional compacton solutions are constructed it is shown that there is quite rich interaction behaviour between two travelling compactons for some types of compactons the interactions among them may not be completely elastic for some others the interactions are completely elastic there is no phase shift for the interactions of the 2 1 dimensional compactons discussed in this paper

the symmetry reductions of a turbulence model in this paper we obtain symmetry reductions of the system of two coupled parabolic partial differential equations which model the evolution of turbulent bursts using the classical lie method of infinitesimals the reduction to systems of ordinary differential equations odes are obtained from the optimal system of subalgebras these systems admit symmetries which lead to further reductions an algorithm presented by bluman for reducing the order of odes allows us to reduce some of these systems invariant under a two parameter group directly to first order odes systems the hidden symmetries of some of these systems are obtained and some new exact solutions have been derived

a new korteweg de vries equation based sub equation method and its application to the 2 1 dimensional korteweg de vries equation with the help of the symbolic computation system maple we present korteweg de vries equation based sub equation method being concise and straightforward it is applied to the 2 1 dimensional korteweg de vries equation it is shown that n soliton solution of the 2 1 dimensional korteweg de vries equation can be found by this new method the method can be applied to other nonlinear partial differential equations in mathematical physics

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

linear b spline finite element method for the improved boussinesq equation in this paper we develop and validate a numerical procedure for solving a class of initial boundary value problems for the improved boussinesq equation the finite element method with linear b spline basis functions is used to discretize the nonlinear partial differential equation in space and derive a second order system involving only ordinary derivatives it is shown that the coefficient matrix for the second order term in this system is invertible consequently for the first time the initial boundary value problem can be reduced to an explicit initial value problem to which many accurate numerical methods are readily applicable various examples are presented to validate this technique and demonstrate its capacity to simulate wave splitting wave interaction and blow up behavior

similarity solutions of the deformed maxwell bloch system the authors have found similarity reductions for the deformed maxwell bloch system to the fifth and second painleve equations asymptotics of the solutions of these equations which are relevant to two level atomic systems with pumping have also been derived

some novel nonlinear coherent excitations of the davey stewartson system exact solutions of many integrable 2 1 2 spatial and 1 temporal dimensional systems of nonlinear evolution equations e g the davey stewartson model can be obtained by a special separation of variables procedure by choosing the jacobi elliptic functions as the building blocks exact doubly periodic solutions are obtained analytically here two sets of elliptic functions with two different independent moduli are employed and the resulting wave packets are expressed as rational functions of elliptic functions by taking the long wave limit in one spatial variable peculiar wave patterns localized in one direction but periodic in the other direction will arise by taking the long wave limit in both spatial variables exponentially localized wave patterns which differ from the known dromions will result the boundary conditions relating to these localized structures are studied

generalized riccati equation rational expansion method and its application by directly extending the riccati equation rational expansion method a new method is developed for constructing a series of exact analytical solutions of an averaged dispersion managed fiber system equation adme including some new rational formal soliton like solutions triangle like solutions and rational solutions

infinite dimensional symmetry algebras and an infinite number of conserved quantities of the 2 1 dimensional davey stewartson equation an infinite dimensional symmetry algebra of the davey stewartson equation is explicitly presented it is shown that this algebra is a gauge generalization of the symmetry transformation for the schrodinger equation and that the virasoro algebra appears as the subalgebra an infinite number of conserved quantities associated with the transformations are also obtained

similarity reductions of the kp equation by a direct method basing on a direct method developed by clarkson and kruskal 1989 the kadomtsev petviashvili kp equation has been reduced to three types of 1 1 dimensional partial differential equations which are equivalent to the three types of the similarity reduction equations obtained by the classical lie approach but with different independent variables more arbitrary functions which have been missed by the classical lie approach have been included in the solutions of the kp equation for instance the third type of reduction obtained by the direct method can be divided into three subcases and the third type of solution of the kp equation obtained by the classical lie approach is only a special case of one subcase of the authors results

singular reduction operators in two dimensions the notion of singular reduction operators i e of singular operators of nonclassical conditional symmetry of partial differential equations in two independent variables is introduced all possible reductions of these equations to first order odes are exhaustively described as examples properties of singular reduction operators of 1 1 dimensional evolution and wave equations are studied it is shown how to favourably enhance the derivation of nonclassical symmetries for this class by an in depth prior study of the corresponding singular vector fields

on the infinite dimensional symmetry group of the davey stewartson equations the lie algebra of the group of point transformations leaving the davey stewartson equations dse s invariant is obtained the general element of this algebra depends on four arbitrary functions of time the algebra is shown to have a loop structure a property shared by the symmetry algebras of all known 2 1 dimensional integrable nonlinear equations subalgebras of the symmetry algebra are classified and used to reduce the dse s to various equations involving only two independent variables

hodograph transformations of linearizable partial differential equations in this paper an algorithmic method is developed for transforming quasilinear partial differential equations of the form u t g u u nx f u u x cdots u n 1 x u mx equiv partial m u partial x m where dg du not equiv 0 into semilinear equations i e equations of the above form with g u 1 this crucially involves the use of hodograph transformations i e transformations involving the interchange of dependent and independent variables furthermore the most general quasilinear equation of the above form is found that can be mapped via a hodograph transformation to a semilinear form this algorithm provides a method for establishing whether a given quasilinear equation is linearizable i e is solvable in terms of either a linear partial differential equation or of a linear integral equation in particular this method is used to show how the painleve tests may be applied to quasilinear equations this appears to resolve the problem that solutions of lineariz

blowup of solutions for improved boussinesq type equation the paper studies the existence and uniqueness of local solutions and the blowup of solutions to the initial boundary value problem for improved boussinesq type equation utt uxx uxxtt u xx by a galerkin approximation scheme combined with the continuation of solutions step by step and the fourier transform method it proves that under rather mild conditions on initial data the above mentioned problem admits a unique generalized solution u w2 0 t h2 0 1 as long as c2 r in particular when s asp where a 0 is a real number and p 1 is an integer specially a 0 if p is an odd number the solution blows up in finite time moreover two examples of blowup are obtained numerically

new exact solutions for the konopelchenko dubrovsky equation using an extended riccati equation rational expansion method and symbolic computation in this paper based on computerized symbolic computation and a new general ansatz an extended riccati equation rational expansion method is presented to construct multiple exact solutions for nonlinear evolution equations and implemented in a computer algebraic system the validity and reliability of the method are tested by its application to the konopelchenko dubrovsky equation the method can be applied to other nonlinear evolution equations in mathematical physics

symmetries of differential equations from sophus lie to computer algebra the topic of this article is the symmetry analysis of differential equations and the applications of computer algebra to the extensive analytical calculations which are usually involved in it the whole area naturally decomposes into two parts depending on whether ordinary or partial differential equations are considered we show how a symmetry may be applied to lower the order of an ordinary differential equation or to obtain similarity solutions of partial differential equations the computer algebra packages sods and spde respectively which have been developed to perform almost all algebraic manipulations necessary to determine the symmetry group of a given differential equation are presented furthermore it is argued that the application of computer algebra systems has qualitatively changed this area of applied mathematics

linear systems on tropical curves a tropical curve gamma is a metric graph with possibly unbounded edges and tropical rational functions are continuous piecewise linear functions with integer slopes we define the complete linear system d of a divisor d on a tropical curve gamma analogously to the classical counterpart we investigate the structure of d as a cell complex and show that linear systems are quotients of tropical modules finitely generated by vertices of the cell complex using a finite set of generators d defines a map from gamma to a tropical projective space and the image can be modified to a tropical curve of degree equal to mathrm deg d the tropical convex hull of the image realizes the linear system d as a polyhedral complex

revisitation of the localized excitations of the 2 1 dimensional kdv equation in a previous paper lou s y 1995 j phys a math gen 28 7227 a generalized dromion structure was revealed for the 2 1 dimensional kdv equation which was first derived by boiti et al boiti m leon j j p manna m and pempinelli f 1986 inverse problems 2 271 using the idea of the weak lax pair in this paper using a backlund transformation and the variable separation approach we find there exist much more abundant localized structures for the 2 1 dimensional kdv equation the abundance of the localized structures of the model is introduced by the entrance of an arbitrary function of the seed solution some special types of dromion solution lumps breathers instantons and the ring type of soliton are discussed by selecting the arbitrary functions appropriately the dromion solutions can be driven by sets of straight line and curved line ghost solitons the dromion solutions may be located not only at the cross points of the lines but also at the closed points of the curves the breathers may breathe both in amplitude and in shape

root lattices and quasicrystals it is shown that root lattices and their reciprocals might serve as the right pool for the construction of quasicrystalline structure models all noncrystallographic symmetries observed so far are covered in minimal embedding with maximal symmetry

new classes of symmetries for partial differential equations new classes of symmetries for partial differential equations are introduced by writing a given partial differential equation s in a conserved form a related system t with potentials as additional dependent variables is obtained the lie group of point transformations admitted by t induces a symmetry group of s new symmetries may be obtained for s that are neither point nor lie backlund symmetries they are determined by a completely algorithmic procedure significant new symmetries are found for the wave equation with a variable wave speed and the nonlinear diffusion equation

infinitely many symmetries of the davey stewartson equation using a formal series symmetry method for the davey stewartson equation dse we find that there exist two sets of infinitely many formal series symmetries of the dse however different from the kadomtsev petviashvili equation kpe we failed to get infinitely many truncated symmetries only six truncated symmetries can be obtained simply from the formal series symmetries

symmetries and exact solutions of the sharma tass olver equation abstract the simple symmetry reduction procedure is repeatedly used to obtain infinitely many symmetries as well as exact solutions of the sharma tass olver sto equation some sets of exact solutions such as singular rational complexitons and both analytical and singular rational negaton positons with new soliton fission and fusion phenomena are obtained

a new variable coefficient korteweg de vries equation based sub equation method and its application to the 3 1 dimensional potential ytsf equation with the aid of maple we present variable coefficient korteweg de vries equation based sub equation method the key idea of our method is to take advantage of the variable coefficient korteweg de vries equation and its various solutions to generate various solutions of nonlinear evolution equations the efficiency of the method can be demonstrated on the 3 1 dimensional potential ytsf equation and we construct successfully its new styles of solutions

critical groups of group representations abstract this paper investigates the critical group of a faithful representation of a finite group it computes the order of the critical group in terms of the character values and gives some restrictions on its subgroup structure it also computes the precise structure of the critical group both for the regular representation of any finite group and for the reflection representation of s n

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

application of the extended homotopy perturbation method to a kind of nonlinear evolution equations based on the homotopy perturbation method we propose a new extended homotopy perturbation method we apply the method to solve a kind of nonlinear evolution equations with the help of the symbolic computation system maple the success of the paper lies on the fact that we do not conform to the traditional homotopy mode but construct a new homotopy of the given equations the results reveal that we can obtain the approximation solution of a kind of nonlinear evolution equations

on a unified approach to transformations and elementary solutions of painleve equations an algorithmic method is developed for investigating the transformation properties of second order equations of painleve type this method which utilizes the singularity structure of these equations yields explicit transformations which relate solutions of the painleve equations ii vi with different parameters these transformations easily generate rational and other elementary solutions of the equations the relationship between painleve equations and certain new equations quadratic in the second derivative of painleve type is also discussed

dynamical survey of a generalized zakharov equation and its exact travelling wave solutions we investigate the dynamical behavior of a generalized zakharov equation for the complex envelope of the high frequency wave and the real low frequency field by analyzing its phase portraits following a dynamical system approach in different parameter regions we depict phase portraits of a travelling wave system through discussing the bifurcation of phase portraits we unwrap explicit miscellaneous travelling waves including localized and periodic ones

new exact travelling wave solutions for the shallow long wave approximate equations based upon a generally projective riccati equation method which is a direct and unified algebraic method for constructing more general form travelling wave solutions of nonlinear partial differential equations and implemented in a computer algebraic system we consider the shallow long wave approximate equations new and more general form solutions are obtained including kink shaped solitons bell shaped solitons singular solitons and periodic solutions the properties of the new formal solitary wave solutions are shown by some figures

numerical solutions of coupled burgers equations with time and space fractional derivatives in this paper by introducing the fractional derivative in the sense of caputo the adomian decomposition method is directly extended to study the coupled burgers equations with time and space fractional derivatives as a result the realistic numerical solutions are obtained in a form of rapidly convergent series with easily computable components the figures show the effectiveness and good accuracy of the proposed method

nonclassical symmetries of a porous medium equation with absorption in this paper new symmetry reductions and exact solutions are presented for the porous medium equation with absorption those spatial forms for which the equation can be reduced to an ordinary differential equation are studied the symmetry reductions and exact solutions presented are derived by using the nonclassical method developed by bluman and cole and are unobtainable by lie classical method

random tilings of high symmetry i mean field theory we study random tiling models in the limit of high rotational symmetry in this limit a mean field theory yields reasonable predictions for the configurational entropy of free boundary rhombus tilings in two dimensions we base our mean field theory on an iterative tiling construction inspired by the work of de bruijn in addition to the entropy we consider correlation functions phason elasticity and the thermodynamic limit tilings of dimension other than two are considered briefly

the anisotropic heisenberg spin chain and the nonlinear schrodinger equation abstract it is shown that the solutions of the continuous anisotropic heisenberg spin chain ahsc can be obtained from the linear integral equation which was proposed in a previous paper for the solutions of the isotropic heisenberg spin chain ihsc and the nonlinear schrodinger equation nls an explicit expression is obtained for the miura transformation which maps the solutions of the ahsc on solutions of the nls in the second part of the paper we investigate the similarity solutions of these partial differential equations which leads to ordinary differential equations of painleve type as an application we discuss some new solutions of painleve iv

a riemann roch theorem in tropical geometry recently baker and norine have proven a riemann roch theorem for finite graphs we extend their results to metric graphs and thus establish a riemann roch theorem for divisors on abstract tropical curves

on two approaches of finding exact solutions to nonlinear evolution equations in this work we give some discussions on the sec q tanh q method of finding exact solutions to the nonlinear evolution equations presented by liu and jiang we conclude that the method is just one special case of the generalized projective riccati equation method therefore these deformed hyperbolic function solutions do not pertain to a new class then based on the generalized projective riccati equation method more exact solutions of the hirota satsuma coupled kdv systems are obtained

the fourth painleve equation and associated special polynomials in this article rational solutions and associated polynomials for the fourth painleve equation are studied these rational solutions of the fourth painleve equation are expressible as the logarithmic derivative of special polynomials the okamoto polynomials the structure of the roots of these okamoto polynomials is studied and it is shown that these have a highly regular structure the properties of the okamoto polynomials are compared and contrasted with those of classical orthogonal polynomials further representations are given of the associated rational solutions in the form of determinants through schur functions

matching rules and growth rules for pentagonal quasicrystal tilings preuve de l existence de ces regles pour un ensemble infini de pavages pentagonaux 2d autres que le pavage de peurose

numerical solutions for fractional kdv burgers equation by adomian decomposition method abstract based on the symbolic computation system maple adomian decomposition method developed for differential equations of integer order is directly extended to derive explicit and numerical solutions of the fractional kdv burgers equation the fractional derivatives are described in the caputo sense according to my knowledge this paper represents the first available numerical solutions of the nonlinear fractional kdv burgers equation with time and space fractional derivatives finally the solutions of our model equation are calculated in the form of convergent series with easily computable components

lie transformations nonlinear evolution equations and painleve forms we present the results of a systematic investigation of invariance properties of a large class of nonlinear evolution equations under a one parameter continuous lie group of transformations it is shown that in general the corresponding invariant variables the subclass of which is the usual similarity variables lead to ordinary differential equations of painleve type in the case of inverse scattering transform solvable equations as conjectured by ablowitz ramani and segur this is found to be also true for certain higher spatial dimensional versions such as the kadomtsev petviashivilli two dimensional sine gordon and ernst equations for the nonsolvable equations considered here this invariance study leads to ordinary differential equations with movable critical points

group invariant solutions of differential equations we introduce the concept of a weak symmetry group of a system of partial differential equations that generalizes the nonclassical method introduced by bluman and cole for finding group invariant