symmetries and algebras of the integrable dispersive long wave equations in 2 1 dimensional spaces similarly to the kadomtsev petviashvili kp equation a set of generalized symmetries with arbitrary functions of t is given by a simple constructable formula for the integrable dispersive long wave equations in 2 1 space dimensions these symmetries constitute an infinite dimensional lie algebra which is a generalization to the known winfinity algebra

on the symmetry groups of the intrinsic generalized wave and sine gordon equations the intrinsic generalized sine gordon and wave equations are integrable n dimensional generalizations of the sine gordon and wave equations respectively it is shown that their lie point symmetry groups are finite dimensional they consist only of translations in the first case and translations and dilations in the second the symmetries are used to obtain classes of exact invariant solutions

symmetries of the kadomtsev petviashvili equation generalized symmetries with arbitrary functions of time t for the well known 2 1 dimensional integrable model kadomtsev petviashvili kp equation are found by means of the extended mastersymmetry approach then an explicit and simple constructive formula for the symmetries of the kp equation is derived directly from the symmetry definition equation without using complicated recursion operators all the known symmetries appear as special cases of those obtained in this paper the general infinite dimensional lie algebra constituted by these symmetries is also given

symmetry reduction for nonlinear relativistically invariant equations symmetry reduction is studied for the relativistically invariant scalar partial differential equation h u u 2 u 0 in n 1 dimensional minkowski space m n 1 the introduction of k symmetry variables 1 k as invariants of a subgroup g of the poincare group p n 1 having generic orbits of codimension k n in m n 1 reduces the equation to a pde in k variables all codimension 1 symmetry variables in m n 1 n arbitrary reducing the equation studied to an ode are found as well as all codimension 2 and 3 variables for the low dimensional cases n 2 3 the type of equation studied includes many cases of physical interest in particular nonlinear klein gordon equations such as the sine gordon equation and hamilton jacobi equations

lie point symmetry and some new soliton like solutions of the konopelchenko dubrovsky equations abstract in this paper the infinite dimensional symmetry groups of the 2 1 dimensional konopelchenko dubrovsky kd equation is found by the classical lie group method the symmetry groups are used to perform symmetry reduction and together with tanh function method we obtain soliton like solutions moreover a characterization of the corresponding lie algebra is given

conformal invariant expansion and high dimensional painleve integrable models a conformal invariant asymptotic expansion approach to solving real nonlinear physical systems is proposed in particular the 3 1 dimensional kadomtsev petviashvili 3dkp equation is solved approximately by using the conformal invariant asymptotic expansion in some special cases the approximate solutions of the 3dkp become exact many types of 3 1 dimensional painleve integrable models are obtained at the same time

reduction of the n dimensional static vacuum einstein equation and generalized schwarzschild solutions abstract we consider the static vacuum einstein spacetime when the spatial factor is conformal to a n dimensional pseudo euclidean space the most general ansatz that reduces the resulting system of partial differential equations to a system of ordinary differential equations is completely described we obtain the entire set of solutions of the reduced system where the classical schwarzschild solution arises as a particular solution in addition we show that the riemannian spatial factors associated to these solutions are foliated by parallel hypersurfaces of constant mean curvature

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

exact solutions of the multidimensional classical φ6 field equations obtained by symmetry reduction the often used 6 model of classical critical phenomena is studied in 3 1 dimensional minkowski and euclidean spaces the euler lagrange equations describing the kinetics of the scalar order parameter are in this case nonlinear klein gordon equations the method of symmetry reduction is systematically applied to derive all the solutions invariant under subgroups with generic orbits of codimension 1 whenever the obtained ordinary differential equations have the painleve property they can be transformed to one of two standard forms these are then solved in terms of elliptic functions or elementary ones this results in a large number of new exact solutions particularly interesting solutions are found in the immediate vicinity of the tricritical point our treatment of the 6 theory is complete only for the four dimensional spaces m 3 1 and e 4 but many of the results are given for the more general cases of m n 1 and e n 1

lie symmetry analysis and some new exact solutions of the wu zhang equation the lie symmetry analysis and the basic similarity reductions are performed for the wu zhang equation a 2 1 dimensional nonlinear dispersive wave equation some new exact solutions generated from the similarity transformation are provided they demonstrate some new three dimensional features of a single solitary wave and two interacting solitary waves

lie symmetries of soliton equations with self consistent sources via source generation procedure abstract in this paper the source generation procedure sgp proposed by hu and wang x b hu h y wang construction of dkp and bkp equation with self consistent sources inverse problems 22 2006 1903 1920 is utilized to derive lie symmetries of bilinear soliton equations with self consistent sources sescs such as kpescs bkpescs and differential difference kpescs furthermore it is shown that these lie symmetries constitute generators of the corresponding lie symmetry algebras

pfaffianization of the two dimensional toda lattice pfaffianization procedure due to hirota and ohta is applied to the two dimensional toda lattice as a result a pfaffianized version of the two dimensional toda lattice is found

nonclassical analysis and painleve property for the kupershmidt equations by means of the nonclassical lie approach and the direct method all the similarity reductions of the kupershmidt equations are found two reduction equations which can also be obtained by the classical lie approach are the painleve type ii and iv respectively the other three reduction equations which cannot be obtained by the classical lie approach are only linear or riccati type

full symmetry groups painleve integrability and exact solutions of the nonisospectral bkp equation abstract based on the generalized symmetry group method presented by lou and ma lou and ma non lie symmetry groups of 2 1 dimensional nonlinear systems obtained from a simple direct method j phys a math gen 38 2005 l129 firstly both the lie point groups and the full symmetry group of the nonisospectral bkp equation are obtained at the same time a relationship is constructed between the new solutions and the old ones of equation secondly the nonisospectral bkp can be proved to be painleve integrability by combining the standard wtc approach with the kruskal s simplification some solutions are obtained by using the standard truncated painleve expansion finally based on the relationship by the generalized symmetry group method and some solutions by using the standard truncated painleve expansion some interesting solution are constructed

similarity solutions of the kadomtsev petviashvili equation using the direct method introduced by clarkson and kruskal ck 1989 recently the author obtains the similarity reductions of the kadomtsev petviashvili equation kpe some of them possess the same painleve properties as that of the boussinesq equation be obtained by ck however the similarity solutions of kpe may include some more than three arbitrary functions of time there exist some other types of similarity reductions of the kpe different from the similarity reductions of the be obtained by ck

infinite dimensional symmetries of two dimensional generalized burgers equations the conditions for a class of generalized burgers equations which a priori involve nine arbitrary functions of one or two variables to allow an infinite dimensional symmetry algebra are determined although this algebra can involve up to two arbitrary functions of time it does not allow a virasoro subalgebra this result reconfirms a long standing fact that variable coefficient generalizations of a nonintegrable equation should be expected to remain as such

self similar solutions for a coupled system of nonlinear schrodinger equations this work is devoted to the study of self similar solutions of the 2 1 dimensional coupled nonlinear schrodinger equations which occur in nonlinear optics 1 psi 1 z psi 1 xx psi 1 yy eta mod psi 1 to 2 1 h mod psi 2 mod 2 psi 1 0 and square root 1 psi 2 psi 2 xx psi 2 yy eta mod psi 2 mod 2 1 h mod psi 1 psi 2 0 where psi 1 and psi 2 are complex functions h is a non vanishing real parameter eta or 1 and in or 1 the authors give the point symmetry properties of the model and calculate generic 2 1 dimensional symmetry reductions some exact and approximate solutions are obtained in particular they use a variational approach to determine and classify a set of physically relevant localized nonsingular self similar solutions

lie symmetries of a generalised nonlinear schrodinger equation i the symmetry group and its subgroups the symmetry group of the generalised non linear schrodinger equation i psi t delta psi a0 psi a1 mod psi mod 2 psi a2 mod psi mod 4 psi in three space dimensions is shown to be the extended galilei group g 3 for a1a2 not 0 and the galilei similitude group gd 3 including a dilation for a1 oor a2 0 all lie subgroups of g 3 and gd 3 are found they will be used in a subsequent paper to obtain group invariant solutions of the equation

symmetries and invariant solutions of the two dimensional variable coefficient burgers equation we discuss symmetries and reductions of the two dimensional burgers equation with variable coefficient we classify one dimensional and two dimensional subalgebras of the burgers symmetry algebra which is infinite dimensional into conjugacy classes under the adjoint action of the symmetry group invariance under one dimensional subalgebras provides reductions to lower dimensional partial differential equations further reductions of these equations to second order ordinary differential equations are obtained through invariance under two dimensional subalgebras the reduced odes are then analysed and shown that they belong to the polynomial class of second order equations which can be linearized only for particular values of parameters figuring in the coefficient

reductions by isometries of the self dual yang mills equations in four dimensional euclidean space a classification of the subgroups of the four dimensional euclidean group is given the anti self dual yang mills equations in four dimensional euclidean space are then reduced for each class representative and for any gauge group when a strict invariance of the gauge fields is required or for the gauge group so 3 if the gauge fields remain invariant up to gauge transformations amongst the residual systems many constitute sets of first order ordinary differential equations sharing the painleve property

maximal abelian subalgebras of pseudounitary lie algebras abstract the task of classifying and constructing all maximal abelian subalgebras of su p q p q 1 is reduced to that of classifying orthogonally indecomposable masas these are either maximal abelian nilpotent subalgebras represented by nilpotent matrices in any finite dimensional representation or for p q they can be nonorthogonally decomposable and their study can be reduced to a construction of manss of sl p c two types of manss of su p q are shown to exist one rowed and non one rowed numerous classification theorems are proven and applied to obtain all masas of su p 1 su p 2 and su p q with p q 6 physical applications are discussed

on asymptotic stability of delay difference systems in this paper we obtain some criteria for determining the asymptotic stability of the zero solution for some classes of delay difference systems by using a discrete version of the second lyapunov method our results can be well suited for computational purposes

some constraints and solutions of the kadomtsev petviashvili equation we illustrate the relations among the symmetry invariant group and constraint for a differential equation applied to the kadomtsev petviashvili equation some constraints and solutions are given in particular the equation associated with symmetry sigma of the kp equation is introduced and discussed

are all the equations of the kadomtsev petviashvili hierarchy integrable the kadomtsev petviashvili kp hierarchy is an infinite set of nonlinear partial differential equations in which the number of independent variables increases indefinitely as one proceeds down the hierarchy since these equations were obtained as part of a group theoretical approach to soliton equations it would appear that the kp hierarchy provides integrable scalar equations with an arbitrary number of independent variables it is shown by investigating a specific equation in 3 1 dimensions that the higher equations in the kp hierarchy are only integrable in a conditional sense the equation under study taken in isolation does not pass certain well known and reliable integrability tests thus applying painleve analysis we find that solutions exist allowing movable critical points furthermore solitary wave solutions are shown to exist that do not behave like solitons in multiple collisions on the other hand if the dependence of a solution on the first 2 1 variables is restricted by the fact that it should also satisfy the kp equation itself then the integrability conditions in the other dimensions are satisfied conditional integrability thus means that linear techniques will provide only those solutions of equations in the hierarchy that simultaneously satisfy lower equations in the same hierarchy

w gravity n 2 strings and 2 2 su yang mills instantons it is conjectured that w gravity can be interpreted as the gauge theory of diffeomorphisms in the space of dimensionally reduced d 2 2 su yang mills instantons these diffeomorphisms preserve a volume three form and are those that furnish the correspondence between the dimensionally reduced plebanski equation and the kp equation in 1 2 dimensions a supersymmetric extension furnishes super w gravity the super plebanski equation generates self dual complexified super gravitational backgrounds sdsg in terms of the super plebanski second heavenly form since the latter equation yields n 1 d 4 sdsg complexified backgrounds associated with the complexified cotangent space of the riemannian surface t c required in the formulation of su complexified self dual yang mills theory sdym it naturally follows that the recently constructed d 2 2 n 4 sdsym theory as the consistent background of the open n 2 superstring can be embedded into the n 1 su complexified self dual super yang mill

negative flows of the potential kp hierarchy we construct a grassmannian like formulation for the potential kp hierarchy including additional negative flows our approach will generalize the notion of a r function to include negative flows we compare the resulting hierarchy with results by hirota satsuma and bogoyavlenskii

symmetry reduction for the kadomtsev petviashvili equation using a loop algebra the kadomtsev petviashvili kp equation ut 3uux 2 1 4 uxxx x 3 uyy 4 0 allows an infinite dimensional lie group of symmetries i e a group transforming solutions amongst each other the lie algebra of this symmetry group depends on three arbitrary functions of time t and is shown to be related to a subalgebra of the loop algebra a 1 4 low dimensional subalgebras of the symmetry algebra are identified specifically all those of dimension n 3 and also a physically important six dimensional lie algebra containing translations dilations galilei transformations and quasirotations new solutions of the kp equation are obtained by symmetry reduction using the one dimensional subalgebras of the symmetry algebra these solutions contain up to three arbitrary functions of t

point symmetries of conditionally integrable nonlinear evolution equations the lie point symmetries of the first two equations in the kadomtsev petviashvili kp hierarchy introduced by jimbo and miwa are investigated the first is the potential kp equation the second involves four independent variables and is called the jimbo miwa jm equation the joint symmetry algebra for the two equations is shown to have a kac moody virasoro structure whereas the symmetry algebra of the jm equation alone does not subgroups of the joint symmetry group are used to perform symmetry reduction and to obtain invariant solutions

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

pancyclicity and bipancyclicity of conditional faulty folded hypercubes a graph is said to be pancyclic if it contains cycles of every length from its girth to its order inclusive and a bipartite graph is said to be bipancyclic if it contains cycles of every even length from its girth to its order the pancyclicity or the bipancyclicity of a given network is an important factor in determining whether the network s topology can simulate rings of various lengths an n dimensional folded hypercube fq n is an attractive variant of an n dimensional hypercube q n that is obtained by establishing some extra edges between the vertices of q n fq n for any odd n is known to be bipartite in this paper we explore the pancyclicity and bipancyclicity of fq n for any fq n n 2 with at most 2n 3 faulty edges where each vertex is incident to at least two fault free edges we prove that there exists a fault free cycle of every even length from 4 to 2 n and when n 2 is even we prove there also exists a fault free cycle of every odd length from n 1 to 2 n 1 the result is optimal with respect to the number of faulty edges tolerated

lie symmetries and infinite dimensional lie algebras of certain 1 1 dimensional nonlinear evolution equations in this paper we discuss the lie symmetries symmetry algebra and similarity reductions of two different equations introduced in the recent literature namely i a new coupled integrable dispersionless equation and ii a new coupled hyperbolic variational equation we point out that both the systems admit in contradistinction to conventional lie algebras in 1 1 dimensional systems infinite dimensional lie algebras furthermore we find physically interesting solutions for special choices of the symmetry parameters

maximal abelian subalgebras of real and complex symplectic lie algebras we provide guidelines for classifying maximal abelian subalgebras masa s of the symplectic lie algebras sp 2n r and sp 2n c into conjugacy classes under the lie groups sp 2n r and sp 2n c respectively the task of classifying all masa s is reduced to the classification of orthogonally indecomposable oid masa s two types of orthogonally indecomposable masa s of sp 2n c exist 1 indecomposable maximal abelian nilpotent subalgebras mans s 2 decomposable masa s their classification reduces to a classification of mans s of sl n c four types of orthogonally indecomposable masa s of sp 2n r exist 1 absolutely indecomposable masa s mans s 2 relatively indecomposable masa s their classification reduces to a classification of mans s of su p q for p q n 3 decomposable absolutely oid masa s involving mans s of sl n r 4 decomposable relatively oid masa s involving mans s of sl n 2 c for n even low dimensional cases of sp 2n f n 1 2 3 f r or c are treated exhau

non lie symmetry groups of 2 1 dimensional nonlinear systems obtained from a simple direct method a new direct method is developed to find finite symmetry groups of nonlinear mathematical physics systems using the direct method for the well known 2 1 dimensional kadomtsev petviashvili equation and the ablowitz kaup newell segur system both the lie point symmetry groups and the non lie symmetry groups are obtained the lie symmetry groups obtained via traditional lie approaches are only special cases furthermore the expressions of the exact finite transformations of the lie groups are much simpler than those obtained via the standard approaches

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

group theoretical analysis of dispersive long wave equations in two space dimensions abstract the symmetry algebra of an integrable dispersive long wave equation in two space dimensions is shown to be infinite dimensional and to have a kac moody virasoro structure the corresponding symmetry group is used to generate a large number of new solutions in particular solitons kinks and periodic waves these waves have wave crests of quite general shapes the results are compared to those for a nonintegrable two space dimensional dispersive long wave equation for which the symmetry group is finite dimensional and wave crests are always straight lines

symmetry algebras of the potential nizhnik novikov veselov model two sets of generalized symmetries of the potential nizhnik novikov veselov equation are given in a simple formula by developing a generalized formal series method for searching symmetries of one type of generalized partial differential equations these two sets of symmetries constitute an infinite dimensional lie algebra which contains three types of virasoro algebras and three w type algebras as its special cases

lie symmetries of a generalized kuznetsov zabolotskaya khokhlov equation abstract we consider a class of generalized kuznetsov zabolotskaya khokhlov gkzk equations and determine its equivalence group which is then used to give a complete symmetry classification of this class the infinite dimensional symmetry is used to reduce such equations to 1 1 dimensional pdes special attention is paid to group theoretical properties of a class of generalized dispersionless kp gdkp or zabolotskaya khokhlov equations as a subclass of gkzk equations the conditions are determined under which a gdkp equation is invariant under a lie algebra containing the virasoro algebra as a subalgebra this occurs if and only if this equation is completely integrable a similar connection is shown to hold for generalized kp equations

symmetry reduction of the wess zumino model and supersymmetric reduced systems an investigation of the supersymmetric reduction of the wess zumino model is carried out for the subgroups of the poincare group the necessary conditions for residual supersymmetry derived from super poincare transformations and a classification of the subalgebras of the poincare algebra which might give rise to supersymmetric residual systems are obtained two supersymmetric reductions of the off shell model with interactions by bidimensional subgroups are presented

on the calogero degasperis fokas equation in 2 1 dimensions abstract in this paper we study a 2 1 dimensional integrable calogero degasperis fokas equation derivable by using a method proposed by calogero a catalogue of classical symmetry reductions are given these reductions to partial differential equations in 1 1 admit symmetries which lead to further reductions i e to second order ordinary differential equations these odes provide several classes of solutions all of them are expressible in terms of known functions some of them are expressible in terms of the second and third painleve trascendents the corresponding solutions of the 2 1 dimensional equation involve up to three arbitrary smooth functions consequently the solutions exhibit a rich variety of qualitative behaviour indeed by making appropriate choices for the arbitrary functions we exhibit solitary waves coherent structures and bound states

edge pancyclicity and edge bipancyclicity of faulty folded hypercubes let f v and f e be sets of faulty vertices and faulty edges respectively in the folded hypercube fq n so that f v f e n 2 for n 2 choose any fault free edge e if n 3 then there is a fault free cycle of length l in fq n containing e for every even l ranging from 4 to 2 n 2 f v if n 2 is even then there is a fault free cycle of length l in fq n containing e for every odd l ranging from n 1 to 2 n 2 f v 1

continuous symmetries of difference equations lie group theory was originally created more than 100 years ago as a tool for solving ordinary and partial differential equations in this article we review the results of a much more recent program the use of lie groups to study difference equations we show that the mismatch between continuous symmetries and discrete equations can be resolved in at least two manners one is to use generalized symmetries acting on solutions of difference equations but leaving the lattice invariant the other is to restrict them to point symmetries but to allow them to also transform the lattice

uniform asymptotic stability in nonlinear volterra discrete systems abstract we employ the notion of total stability to obtain new criteria for uniform asymptotic stability of the zero solution of a nonlinear volterra discrete system resolvent equation methods are employed and a summability criterion on the resolvent kernel is obtained also we obtain a new difference equation that the resolvent r n s satisfies

algebra of pseudodifferential operators and symmetries of equations in the kadomtsev petviashvili hierarchy point symmetries are obtained for all equations in the kp hierarchy the lie algebra for each equation is infinite dimensional and involves several arbitrary functions of the corresponding time tn the symmetry algebra is a semidirect sum of a virasoro algebra and a kac moody one the positive part of this algebra is embedded into the known w algebra of kp symmetries and into the free fermion algebra l the corresponding action on the tau function is presented the negative part of the point symmetries does not fit into the free fermion algebra but is embedded into a p algebra based on the algebra of pseudodifferential operators

group invariant solutions of a nonlinear acoustics model based on a recent classification of subalgebras of the symmetry algebra of the zabolotskaya khokhlov equation all similarity reductions of this equation into ordinary differential equations are obtained large classes of group invariant solutions of the equation are also determined and some properties of the reduced equations and exact solutions are discussed

the twisted n cube with application to multiprocessing it is shown that by exchanging any two independent edges in any shortest cycle of the n cube n or 3 its diameter decreases by one unit this leads to the definition of a new class of n regular graphs denoted tq sub n with 2 sup n vertices and diameter n 1 which has the n 1 cube as subgraph other properties of tq sub n such as connectivity and the lengths of the disjoints paths are also investigated moreover it is shown that the complete binary tree on 2 sup n 1 vertices which is not a subgraph of the n cube is a subgraph of tq sub n how these results can be used to enhance hypercube multiprocessors is discussed

the subalgebras of the rank two symplectic lie algebra abstract the semisimple subalgebras of the rank 2 symplectic lie algebra sp 4 c are well known and we recently classified its levi decomposable subalgebras in this article we classify the solvable subalgebras of sp 4 c up to inner automorphism this completes the classification of the subalgebras of sp 4 c more broadly speaking in completing the classification of the subalgebras of sp 4 c we have completed the classification of the subalgebras of the rank 2 semisimple lie algebras

every edge lies on cycles embedding in folded hypercubes with vertex fault tolerant the folded hypercube is a well known variation of hypercube structure and can be constructed from a hypercube by adding a link to every pair of vertices with complementary addresses an n dimensional folded hypercube fq n for short for any odd n is known to be bipartite in this paper let f be a faulty vertex in fq n it has been shown that 1 every edge of fq n f lies on a fault free cycle of every even length l with 4 l 2 n 2 where n 3 2 every edge of fq n f lies on a fault free cycle of every odd length l with n 1 l 2 n 1 where n 2 is even in terms of every edge lies on a fault free cycle of every odd length in fq n f our result improves the result of cheng et al 2013 where odd cycle length up to 2 n 3

impulsive control of discrete systems with time delay this note deals with impulsive control of discrete systems with time delay it s the first time that an impulsive controller was introduced for discrete systems with time delay firstly some stability criteria for delay discrete systems with impulse effects are obtained by using lyapunov functionals and razumikhin technique these stability criteria show that if the impulses satisfy some conditions then the original stable systems can keep their stability property even with impulse effects then by using lyapunov functionals or lyapunov functions together with razumikhin technique two stability criteria for the discrete systems with time delay under impulsive control are given these two results show that unstable systems can be stable under impulsive control some examples are also presented to illustrate the efficiency of the obtained results

maurer cartan equations for lie symmetry pseudogroups of differential equations a new method of constructing structure equations of lie symmetry pseudogroups of differential equations dispensing with explicit solutions of the infinitesimal determining systems of the pseudogroups is presented and illustrated by the examples of the kadomtsev petviashvili and korteweg de vries equations

variable coefficient davey stewartson system with a kac moody virasoro symmetry algebra we study the symmetry group properties of the variable coefficient davey stewartson vcds system the lie point symmetry algebra with a kac moody virasoro kmv structure is shown to be isomorphic to that of the usual constant coefficient ds system if and only if the coefficients satisfy some conditions these conditions turn out to coincide with those for the vcds system to be transformable to the ds system by a point transformation the equivalence group of the vcds system is applied to pick out the integrable subsystems from a class of non integrable ones additionally the full symmetry group of the ds system is derived explicitly without exponentiating its symmetry algebra lump solutions rationally localized in all directions in 2 introduced by ozawa for the ds system are shown to hold even for the vcds system precisely when the system belongs to the integrable class i e equivalent to the ds system these solutions can be used for establishing exact blow up solutions in finite time in the s

nonclassical symmetry reductions for the kadomtsev petviashvili equation abstract in this paper we discuss dimensional reductions of the physically and mathematically significant kadomtsev petviashvili equation in particular we exhibit some new reductions of the kadomtsev petviashvili equation which are obtained by a generalization of the direct method for determining similarity reductions recently developed by clarkson and kruskal j math phys 30 1989 2201 new solutions of the kadomtsev petviashvili equation are obtained in terms of weierstrass elliptic functions solutions of the lame equations and the first second and fourth painleve transcendents a group theoretical explanation of the new solutions is provided in terms of conditional symmetries leaving only specific subsets of solutions of the kadomtsev petviashvili equation invariant

symmetries of discrete dynamical systems differential difference equations of the form un fn t un 1 un un 1 are classified according to their continuous lie point symmetry groups it is shown that for nonlinear equations the symmetry group can be at most seven dimensional the integrable toda lattice is a member of this class and has a four dimensional symmetry group

analysis of the symmetry group and exact solutions of the dispersionless kp equation in n 1 dimensions the lie algebra of the symmetry group of the n 1 dimensional generalization of the dispersionless kadomtsev petviashvili equation is obtained and identified as a semi direct sum of a finite dimensional simple lie algebra and an infinite dimensional nilpotent subalgebra group transformation properties of solutions under the subalgebra sl 2 r are presented known explicit analytic solutions in the literature are shown to be actually group invariant solutions corresponding to certain specific infinitesimal generators of the symmetry group

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

periodicity and stability of linear volterra difference systems abstract we give sufficient conditions for a system of volterra difference equations to have a unique periodic solution then a criterion for uniform asymptotic stability is established

equations of arbitrary order invariant under the kadomtsev petviashvili symmetry group by means of a simple new approach a general kadomtsev petviashvili kp family with an arbitrary function of group invariants of arbitrary order is proposed it is proved that the general kp family possesses a common infinite dimensional kac moody virasoro lie point symmetry algebra the known fourth order one can be re obtained as a special example the finite transformation group is presented in a clearer form the kac moody virasoro group invariant solutions and the kac moody group invariant solutions of the kp family are determined by the boussinesq and kdv families respectively

painleve analysis lie symmetries and invariant solutions of potential kadomstev petviashvili equation with time dependent coefficients in this paper the painleve analysis of 2 1 dimensional variable coefficients potential kadomstev petviashvili vcpkp equation s x email protected t s email protected x email protected t s x x x email protected t s y y 0 has been performed to study the painleve properties the vcpkp equation is also reduced to a 1 1 dimensional partial differential equation via classical lie group method and reduced equations are further studied to obtain certain general solutions

the third painleve equation and associated special polynomials in this paper we are concerned with rational solutions algebraic solutions and associated special polynomials with these solutions for the third painleve equation piii these rational and algebraic solutions of piii are expressible in terms of special polynomials defined by second order bilinear differential difference equations which are equivalent to toda equations the structure of the roots of these special polynomials is studied and it is shown that these have an intriguing highly symmetric and regular structure using the hamiltonian theory for piii it is shown that these special polynomials satisfy pure difference equations fourth order bilinear differential equations as well as differential difference equations further representations of the associated rational solutions in the form of determinants through schur functions are given

symmetry full symmetry groups and some exact solutions to a generalized davey stewartson system the lie symmetry algebra of a generalized davey stewartson gds system is obtained the general element of this algebra depends on eight arbitrary functions of time which has a kac moody virasoro loop algebra structure and is isomorphic to that of the standard integrable davey stewartson equations under certain conditions imposed on parameters and arbitrary functions then based on the symmetry group direct method proposed by lou and ma j phys a 38 l129 2005 the full symmetry groups of the gds system are obtained from the full symmetry groups both the lie symmetry group and a group of discrete transformations can be obtained finally some exact solutions involving sech sech2 sech2 and tanh tanh2 tanh2 type solitary wave solutions are presented by a generalized subequation expansion method

general theorems for stability and boundedness for nonlinear functional discrete systems abstract we consider the nonlinear functional discrete system x n 1 g n x s 0 s n def g n x and obtain sufficient conditions for stability properties of the zero solution and for the boundedness of solutions

path embedding in faulty hypercubes abstract there are some interesting results concerning longest paths or even cycles embedding in faulty hypercubes this paper considers the embeddings of paths of all possible lengths between any two fault free vertices in faulty hypercubes let fv respectively fe denote the number of faulty vertices respectively edges in an n dimensional hypercube qn we prove that there exists a fault free path of length l between any two distinct fault free vertices u and v in qn with f v f e n 2 for each l satisfying d q n u v 2 l 2 n 2 f v 1 and 2 l d q n u v the bounds on path length l and faulty set size f v f e for a successful embedding are tight that is the result does not hold if l d q n u v 2 or l 2 n 2 f v 1 or f v f e n 2 moreover our result improves some known results

2 1 dimensional models with virasoro type symmetry algebra according to a theory proposed in a previous paper by lou and hu 1994 starting from a realization of a virasoro type symmetry algebra we can obtain various 2 1 dimensional integrable models under the condition that the models possess the infinitely dimensional virasoro type symmetry algebra an explicit realization of the generalized virasoro type symmetry algebra is used to obtain concrete invariant models a set of equations which possesses the same infinite dimensional kac moody virasoro type lie point symmetry algebra as that of the kadomtsev petviashvilli equation is given

fault tolerant path embedding in folded hypercubes with both node and edge faults the folded hypercube fq n is a well known variation of the hypercube structure fq n is superior to q n in many measurements such as diameter fault diameter connectivity and so on let email protected fq n resp email protected fq n denote the set of faulty nodes resp faulty edges in fq n in the case that all nodes in fq n are fault free it has been shown that fq n contains a fault free path of length 2 n 1 resp 2 n 2 between any two nodes of odd resp even distance if email protected fq n n 1 where n 1 is odd and fq n contains a fault free path of length 2 n 1 between any two nodes if email protected fq n n 2 where n 2 is even in this paper we extend the above result to obtain two further properties which consider both node and edge faults as follows 1 fq n contains a fault free path of length at least 2 n 2 email protected fq n 1 resp 2 n 2 email protected fq n 2 between any two fault free nodes of odd resp even distance if email protected fq n email protected fq n n 1 where n 1 is odd 2 fq n contains a fault free path of length at least 2 n 2 email protected fq n 1 between any two fault free nodes if email protected fq n email protected fq n n 2 where n 2 is even

kdv extensions with painleve property by means of the conformal invariance mobious transformation invariance the well known kdv equation is extended to i a 1 1 dimensional space time symmetric form ii two 2 1 dimensional space isotropic forms and iii general 3 1 dimensional and n dimensional forms the extensions are proven to be integrable under the meaning that they possess the painleve property