the painleve property for partial differential equations in this paper we define the painleve property for partial differential equations and show how it determines in a remarkably simple manner the integrability the backlund transforms the linearizing transforms and the lax pairs of three well known partial differential equations burgers equation kdv equation and the modified kdv equation this indicates that the painleve property may provide a unified description of integrable behavior in dynamical systems ordinary and partial differential equations while at the same time providing an efficient method for determining the integrability of particular systems

spread of information and infection on finite random networks the modeling of epidemic like processes on random networks has received considerable attention in recent years while these processes are inherently stochastic most previous work has been focused on deterministic models that ignore important fluctuations that may persist even in the infinite network size limit in a previous paper for a class of epidemic and rumor processes we derived approximate models for the full probability distribution of the final size of the epidemic as opposed to only mean values in this paper we examine via direct simulations the adequacy of the approximate model to describe stochastic epidemics and rumors on several random network topologies homogeneous networks erd os r enyi er random graphs barabasi albert scale free networks and random geometric graphs we find that the approximate model is reasonably accurate in predicting the probability of spread however the position of the threshold and the conditional mean of the final size for processes near the threshold are not well described by the approximate model even in the case of homogeneous networks we attribute this failure to the presence of other structural properties beyond degree degree correlations and in particular clustering which are present in any finite network but are not incorporated in the approximate model in order to test this hypothesis we perform additional simulations on a set of er random graphs where degree degree correlations and clustering are separately and independently introduced using recently proposed algorithms from the literature our results show that even strong degree degree correlations have only weak effects on the position of the threshold and the conditional mean of the final size on the other hand the introduction of clustering greatly affects both the position of the threshold and the conditional mean similar analysis for the barabasi albert scale free network confirms the significance of clustering on the dynamics of rumor spread for this network though with its highly skewed degree distribution the addition of positive correlation had a much stronger effect on the final size distribution than was found for the simple random graph

consensus dynamics on random rectangular graphs abstract a random rectangular graph rrg is a generalization of the random geometric graph rgg in which the nodes are embedded into a rectangle with side lengths a and b 1 a instead of on a unit square 0 1 2 two nodes are then connected if and only if they are separated at a euclidean distance smaller than or equal to a certain threshold radius r when a 1 the rrg is identical to the rgg here we apply the consensus dynamics model to the rrg our main result is a lower bound for the time of consensus i e the time at which the network reaches a global consensus state to prove this result we need first to find an upper bound for the algebraic connectivity of the rrg i e the second smallest eigenvalue of the combinatorial laplacian of the graph this bound is based on a tight lower bound found for the graph diameter our results prove that as the rectangle in which the nodes are embedded becomes more elongated the rrg becomes a large world i e the diameter grows to infinity and a poorly connected graph i e the algebraic connectivity decays to zero the main consequence of these findings is the proof that the time of consensus in rrgs grows to infinity as the rectangle becomes more elongated in closing consensus dynamics in rrgs strongly depend on the geometric characteristics of the embedding space and reaching the consensus state becomes more difficult as the rectangle is more elongated

grammian solutions and pfaffianization of a non isospectral and variable coefficient kadomtsev petviashvili equation abstract in this paper we first present the grammian determinant solutions to the non isospectral and variable coefficient kadomtsev petviashvili vckp equation then by using the pfaffianization procedure of hirota and ohta a new non isospectral and variable coefficient integrable coupled system is generated moreover gramm type pfaffian solutions of the pfaffianized system are proposed

exact solutions to a mixed toda lattice hierarchy through the inverse scattering transform starting from the toda spectral problem a mixed hierarchy of the isospectral equations and the nonisospectral equations is constructed through discrete zero curvature equations soliton solutions of the hierarchy are obtained by the inverse scattering transform meanwhile a new kind of time evolution of scattering data for the toda spectral problem is presented

darboux transformations for the isospectral and nonisospectral mkp equation abstract darboux transformations for the isospectral and nonisospectral modified kadomtsev petviashvili mkp equations are investigated in the isospectral case it is an auto darboux transformation however in the nonisospectral case it is not auto darboux tranformation

on recursion operators abstract we observe that application of a recursion operator of the burgers equation does not produce the expected symmetries this is explained by the incorrect assumption that d x 1 d x 1 we then proceed to give a method to compute the symmetries using the recursion operator as a first order approximation

random geometric graphs we analyze graphs in which each vertex is assigned random coordinates in a geometric space of arbitrary dimensionality and only edges between adjacent points are present the critical connectivity is found numerically by examining the size of the largest cluster we derive an analytical expression for the cluster coefficient which shows that the graphs are distinctly different from standard random graphs even for infinite dimensionality insights relevant for graph bipartitioning are included

cylindrical korteweg de vries equation and painleve property the authors demonstrate that the lax representation for the cylindrical korteweg de vries equation can be obtained with the help of the painleve property of this equation

random rectangular graphs a generalization of the random geometric graph rgg model is proposed by considering a set of points uniformly and independently distributed on a rectangle of unit area instead of on a unit square 0 1 2 the topological properties of the random rectangular graphs rrgs generated by this model are then studied as a function of the rectangle sides lengths a and b 1 a and the radius r used to connect the nodes when a 1 we recover the rgg and when a the very elongated rectangle generated resembles a one dimensional rgg we obtain here analytical expressions for the average degree degree distribution connectivity average path length and clustering coefficient for rrg these results provide evidence that show that most of these properties depend on the connection radius and the side length of the rectangle usually in a monotonic way the clustering coefficient however increases when the square is transformed into a slightly elongated rectangle and after this maximum it decays with the increase of the elongation of the rectangle we support all our findings by computational simulations that show the goodness of the theoretical models proposed for rrgs

soliton interactions for the korteweg devries equation a new perspective it can be shown that the n soliton solution of the korteweg devries equation can be decomposed into n separate solitons cf gardner et al 1974 calogero and degasperis and yoneyama 1982 however it is not immediately clear from the form of their solutions how the separate solitons relate directly to the single soliton solution here the two soliton case is considered and a decomposition is sought which can be clearly related to the single soliton solution although it appears that there is a family of such decompositions it is shown that only one of these is correct although this decomposition is equivalent to the decomposition given previously by gardner et al calogero and degasperis and yoneyama the form given here is different it is suggested that the form of solution produced here is a more appropriate representation of the solution since it is clear how it relates directly to the single soliton solution and it is easy through this form to analyse the interaction of the two solutions

exact solutions to kdv equations with variable coefficients and or nonuniformities abstract a class of kdv equations with variable coefficients and or nonuniformities is discussed we derive easily the constraint conditions which must apply to this class in order to obtain exact solutions by a direct method we obtain solutions in a number of generic as well as specific cases

the curve cone of almost complex 4 manifolds in this paper we study the curve cone of an almost complex 4 manifold which is tamed by a symplectic form in particular we prove the cone theorem as in mori theory for all such manifolds using the seiberg witten theory for small rational surfaces and minimal ruled surfaces we study the configuration of negative curves we define abstract configuration of negative curves which records the homological and intersection information of curves combinatorial blowdown is the main tool to study these configurations as an application of our investigation of the curve cone we prove the nakai moishezon type duality for all almost kahler structures on cp2 kcp2 with k 9 and minimal ruled surfaces with a negative curve this is proved using a version of gram schmidt orthogonalization process for the j tamed symplectic inflation

geometrical and spectral study of beta skeleton graphs we perform an extensive numerical analysis of skeleton graphs a particular type of proximity graphs in a skeleton graph bsg two vertices are connected if a proximity rule that depends of the parameter 0 is satisfied moreover for 1 there exist two different proximity rules leading to lune based and circle based bsgs first by computing the average degree of large ensembles of bsgs we detect differences which increase with the increase of between lune based and circle based bsgs then within a random matrix theory rmt approach we explore spectral and eigenvector properties of random bsgs by the use of the nearest neighbor energy level spacing distribution and the entropic eigenvector localization length respectively the rmt analysis allows us to conclude that a localization transition occurs at 1

random spherical graphs this work addresses a modification of the random geometric graph rgg model by considering a set of points uniformly and independently distributed on the surface of a d 1 sphere with radius r in a d dimensional euclidean space instead of on an unit hypercube 0 1 d then two vertices are connected by a link if their great circle distance is at most s in the case of d 3 the topological properties of the random spherical graphs rsgs generated by this model are studied as a function of s we obtain analytical expressions for the average degree degree distribution connectivity average path length diameter and clustering coefficient for rsgs by setting r sqrt pi 2 pi we also show the differences between the topological properties of rsgs and those of two dimensional rggs and random rectangular graphs rrgs surprisingly in terms of the average clustering coefficient rsgs look more similar to the analytical estimation for rggs than rggs themselves when their boundary effects are considered we support all our findings by computer simulations that corroborate the accuracy of the theoretical models proposed for rsgs

line soliton interactions of a nonisospectral and variable coefficient kadomtsev petviashvili equation an extension of the dressing method is developed to solve a nonisospectral and variable coefficient generalization of the kadomtsev petviashvili equation nonstandard dynamics of the one soliton solutions are presented decomposition of the two solitons and their interactions are studied in detail

new exact solutions for a generalized variable coefficient kdv equation new exact solutions for a generalized variable coefficient kdv equation were obtained using the generalized expansion method r sabry m a zahran e g fan phys lett a 326 2004 93 the obtained solutions include solitary wave solutions besides jacobi and weierstrass doubly periodic wave solutions

connectivity distribution of spatial networks we study spatial networks constructed by randomly placing nodes on a manifold and joining two nodes with an edge whenever their distance is less than a certain cutoff we derive the general expression for the connectivity distribution of such networks as a functional of the distribution of the nodes we show that for regular spatial densities the corresponding spatial network has a connectivity distribution decreasing faster than an exponential in contrast we also show that scale free networks with a power law decreasing connectivity distribution are obtained when a certain information measure of the node distribution integral of higher powers of the distribution diverges we illustrate our results on a simple example for which we present simulation results finally we speculate on the role played by the limiting case p k proportional variant k 1 which appears empirically to be relevant to spatial networks of biological origin such as the ones constructed from gene expression data

a new method for finding traveling wave solutions of the generalized kdv equation with variable coefficient in this paper a transformation between two ansatz equations is presented and applied to discuss the generalized kdv equation with variable coefficient a lot of exact traveling wave solutions are obtained in terms of elliptic functions the method can also be used to study other nonlinear partial differential equations

symmetries and a hierarchy of the general kdv equation two groups old and new of symmetries and their lie algebra properties for the kdv and cylindrical kdv equations are unified and extended to the general kdv equation

the exact solutions for the nonisospectral akns hierarchy through the inverse scattering transform abstract exact solutions for the nonisospectral akns hierarchy are obtained through the inverse scattering transform some explicit forms of these solutions are further given reductions are also considered exact solutions of some nonisospectral hierarchies such as the nonisospectral mkdv hierarchy the nonisospectral kdv hierarchy the nonisospectral nonlinear schrodinger hierarchy and the nonisospectral sine gordon hierarchy are obtained

nonpropagating solitons of the variable coefficient and nonisospectral korteweg de vries equation the method of inverse scattering is extended to solve the initial value problem of a variable coefficient and nonisospectral korteweg de vries equation with time varying boundary condition one and two soliton solutions are examined in detail by an appropriate decomposition soliton interactions and asymptotic behaviors are investigated oscillating and asymptotically standing two soliton solutions are discussed

non propagating solitons of the non isospectral and variable coefficient modified kdv equation absbact the akns system associated with a non isospectral and variable coefficient mkdv equation is presented the method of inverse scattering is adapted to the non isospectral situation to determine the time evolution of the scattering data n soliton olutions are obtained examples of oscillating or standing one solitons with unusual dynamics are given an in depth study of the two soliton case is carried out by appropriately decomposing the solution into individual soliton elements in order to examine their interactions breathers are also constructed

random geometric graphs with general connection functions in the original 1961 gilbert model of random geometric graphs nodes are placed according to a poisson point process and links formed between those within a fixed range motivated by wireless ad hoc networks soft or probabilistic connection models have recently been introduced involving a connection function h r that gives the probability that two nodes at distance r are linked directly connect in many applications not only wireless networks it is desirable that the graph is connected that is every node is linked to every other node in a multihop fashion here the connection probability of a dense network in a convex domain in two or three dimensions is expressed in terms of contributions from boundary components for a very general class of connection functions it turns out that only a few quantities such as moments of the connection function appear good agreement is found with special cases from previous studies and with numerical simulations

investigation on a nonisospectral fifth order korteweg de vries equation generalized from fluids in this paper a nonisospectral fifth order korteweg de vries equation generalized from fluids is investigated with symbolic computation such equation is transformed into its bilinear form through a proposed dependent variable transformation with one more parameter than those in the existing literature n soliton solutions backlund transformation and lax pair in the explicit forms are constructed based on the above results the characteristic line method is applied to discuss the features of the solitons for the nonisospectral problem i e the controllable solitonic velocities and widths four types of solitonic structures with the different solitonic velocities widths amplitudes and backgrounds are also illustrated

auto backlund transformation lax pairs and painleve property of a variable coefficient korteweg de vries equation i using the painleve property of partial differential equations the auto backlund transformation and lax pairs for a korteweg de vries kdv equation with time dependent coefficients are obtained the lax pair criterion also makes it possible for some new models of the variable coefficient kdv equation to be found that can represent nonsoliton dynamical systems this can explain the wave breaking phenomenon in variable depth shallow water

the bi hamiltonian structure of some nonlinear fifth and seventh order differential equations and recursion formulas for their symmetries and conserved covariants using a bi hamiltonian formulation we give explicit formulas for the conserved quantities and infinitesimal generators of symmetries for some nonlinear fifth and seventh order nonlinear partial differential equations among them the caudrey dodd gibbon sawada kotera equation and the kupershmidt equation we show that the lie algebras of the symmetry groups of these equations are of a very special form among the c vector fields they are generated from two given commuting vector fields by a recursive application of a single operator furthermore for some higher order equations those multisoliton solutions which for t asymptotically decompose into traveling wave solutions are characterized as eigenvector decompositions of certain operators

integrable nonlinear evolution equations with time dependent coefficients a simple and straightforward method for generating completely integrable nonlinear evolution equations with time dependent coefficients is presented herein for the equations under consideration the solutions to those given by vector fields which are independent of time are given thus explicit links between equations are obtained as an application of the proposed method it is shown that the linear superposition with arbitrary time dependent coefficients of different members of an integrable hierarchy is again integrable furthermore it turns out that for some integrable equations like the korteweg de vries kdv the benjamin ono bo or the kadomtsev petviashvili kp the resolvent operator of lower order flows can be explicitly obtained from that of any higher order flow those flows demonstrated for the kdv which can be generated by lie homomorphisms coming from first order problems are completely classified many well known equations which can be found in the literature are of that type as an application of such a first order link a direct link is given from kdv to the cylindrical kdv and from there to the kp with nontrivial dependence on the second spatial variable

pseudopotentials lax pairs and backlund transformations for some variable coefficient nonlinear equations using the method developed by nucci 1989 the author has obtained the pseudopotentials lax pairs backlund transformations and the singularity manifold equations of some variable coefficient nonlinear equations these equations are generalized korteweg de vries generalized modified korteweg de vries and generalized boussinesq equations

degree correlations in random geometric graphs spatially embedded networks are important in several disciplines the prototypical spatial network we assume is the random geometric graph of which many properties are known here we present new results for the two point degree correlation function in terms of the clustering coefficient of the graphs for two dimensional space in particular with extensions to arbitrary finite dimensions

painleve property auto backlund transformation lax pairs and reduction to the standard form for the korteweg de vries equation with nonuniformities it is demonstrated that the kdv equation with nonuniformities ut a t u b x t u x c t uux d t uxxx e x t 0 has the painleve property if the compatibility condition among the coefficients of it holds bt a lc b bbx dbxxx 2ah hl d c2 dh dt ce x 2a2 al d3 c4 da dt l d c l d c2 d dt l d c where l d dt lg and h t is an arbitrary function of t the auto backlund transformation and lax pairs for this equation are obtained by truncating the laurent expansion furthermore assuming the compatibility condition then the kdv equation with nonuniformities is transformable via suitable variable transformations to the standard kdv

symmetry constraint of mkdv equations by binary nonlinearization abstract a symmetry constraint for mkdv integrable hierarchy is presented by binary nonlinearization the spatial part and the temporal parts of the lax pairs and the adjoint lax pairs of mkdv equations are all constrained as finite dimensional liouville integrable hamiltonian systems whose integrals of motion are explicitly given in terms of the proposed symmetry constraint mkdv equations are decomposed into two finite dimensional liouville integrable constrained systems and thus a kind of separation of variables for mkdv equations is established

negatons positons rational like solutions and conservation laws of the korteweg de vries equation with loss and non uniformity terms solitons negatons positons rational like solutions and mixed solutions of a non isospectral equation the korteweg de vries equation with loss and non uniformity terms are obtained through the wronskian technique the non isospectral characteristics of the motion behaviours of some solutions are described with some figures made by using mathematica we also derive an infinite number of conservation laws of the equation

painleve analysis and reducibility to the canonical form for the generalized kadomtsev petviashvili equation the most general kadomtsev petviashvili kp type equation ut a t x y u b t x y ux c t x y uux d t x y uxxx x k t x y uyy e t x y is studied and the conditions for the coefficients in order that it owns complete integrability are determined via a painleve test finally it is proved that the above conditions are the same as those requested for reducing the equation to the canonical form via suitable transformations

topological versus spectral properties of random geometric graphs in this work we perform a detailed statistical analysis of topological and spectral properties of random geometric graphs rggs a graph model used to study the structure and dynamics of complex systems embedded in a two dimensional space rggs g n l consist of n vertices uniformly and independently distributed on the unit square where two vertices are connected by an edge if their euclidian distance is less than or equal to the connection radius l 0 sqrt 2 to evaluate the topological properties of rggs we chose two well known topological indices the randic index r g and the harmonic index h g we characterize the spectral and eigenvector properties of the corresponding randomly weighted adjacency matrices by the use of random matrix theory measures the ratio between consecutive eigenvalue spacings the inverse participation ratios and the information or shannon entropies s g first we review the scaling properties of the averaged measures topological and spectral on rggs then we show that i the averaged scaled indices r g and h g are highly correlated with the average number of nonisolated vertices v g and ii surprisingly the averaged scaled shannon entropy s g is also highly correlated with v g therefore we suggest that very reliable predictions of eigenvector properties of rggs could be made by computing topological indices

equivalence groupoid of a class of variable coefficient korteweg de vries equations we classify the admissible transformations in a class of variable coefficient korteweg de vries equations as a result a full description of the structure of the equivalence groupoid of the class is given the class under study is partitioned into six disjoint normalized subclasses the widest possible equivalence group for each subclass is found which appears to be generalized extended in five cases ways for improvement of transformational properties of the subclasses are proposed using gaugings of arbitrary elements and mapping between classes the group classification of one of the subclasses is carried out as an illustrative example

enumerative geometry of calabi yau 4 folds gromov witten theory is used to define an enumerative geometry of curves in calabi yau 4 folds the main technique is to find exact solutions to moving multiple cover integrals the resulting invariants are analogous to the bps counts of gopakumar and vafa for calabi yau 3 folds we conjecture the 4 fold invariants to be integers and expect a sheaf theoretic explanation

random geometric graphs in high dimension many machine learning algorithms used for dimensional reduction and manifold learning leverage on the computation of the nearest neighbors to each point of a data set to perform their tasks these proximity relations define a so called geometric graph where two nodes are linked if they are sufficiently close to each other random geometric graphs where the positions of nodes are randomly generated in a subset of r d offer a null model to study typical properties of data sets and of machine learning algorithms up to now most of the literature focused on the characterization of low dimensional random geometric graphs whereas typical data sets of interest in machine learning live in high dimensional spaces d 10 2 in this work we consider the infinite dimensions limit of hard and soft random geometric graphs and we show how to compute the average number of subgraphs of given finite size k e g the average number of k cliques this analysis highlights that local observables display different behaviors depending on the chosen ensemble soft random geometric graphs with continuous activation functions converge to the naive infinite dimensional limit provided by erdos renyi graphs whereas hard random geometric graphs can show systematic deviations from it we present numerical evidence that our analytical results exact in infinite dimensions provide a good approximation also for dimension d 10

on the structure during interaction of the two soliton solution of the korteweg de vries equation by analysing the structure during interaction of the soliton elements in the two soliton solution of the korteweg de vries equation it is shown how to calculate the duration of the interaction for two soliton elements with any amplitude ratio it is also clarified that the soliton elements as defined here always pass through one another irrespective of the magnitude of the amplitude ratio of the soliton elements this conclusion confirms most previous reports on the interaction of the soliton elements but contradicts another report which states that when amplitudes of the soliton elements are nearly equal the soliton elements do not pass through each other but rather advance toward each other bounce off one another and then recede from each other however it should be noted that in the last case the definition of the soliton elements is different from that given here

analytical solutions of kdv equation with relaxation effect of inhomogeneous medium kdv korteweg de vries equation with relaxation effect of inhomogeneous medium with time changing can be employed in many different physical fields in this paper some new analytical solutions of the equation are obtained which may be very useful in numerical simulation by using of the truncated expansion and jacobi elliptic function expansion methods

painleve analysis rational and special solutions of variable coefficient korteweg de vries equations painleve analysis of two variable coefficient korteweg de vries equations are considered the painleve test for these equations is applied the condition for the painleve property of the equations are found some rational and special solutions are presented

exact solutions for kdv system equations hierarchy abstract exact solutions for kdv system equations hierarchy are obtained by using the inverse scattering transform exact solutions of isospectral kdv hierarchy nonisospectral kdv hierarchies and equations related to the kdv spectral problem are obtained by reduction the interaction of two solitons is investigated

soliton solutions to the 3rd nonisospectral akns system abstract bilinear form of the nonisospectral akns equation is given the n soliton solutions and double wronskian solution are obtained through hirota s direct method and wronskian technique respectively the nonisospectral mkdv equation and its multi soliton solutions are presented by reducing

symmetries conservation laws and hamiltonian structures of the non isospectral and variable coefficient kdv and mkdv equations an infinite number of form invariant symmetries is obtained and a one to one correspondence between symmetries and conservation laws is established for the non isospectral and variable coefficient generalizations of both the kdv and the mkdv equations two families of symmetries and their lie algebraic structures are constructed some interesting facts about their hamiltonian structures are presented

on the kdv type equation with variable coefficients in this paper the kdv type equations with variable coefficients appearing a lot in the literature are discussed generally the explicit transformations which transform various kdv mkdv and kp type nonlinear evolution equations into their canonical forms with constant coefficients are given the results presented here tie up many of the investigations scattered in the literature

n soliton solutions to the bogoyavlenskii schiff equation and a quest for the soliton solution in 3 1 dimensions we study the integrable systems in higher dimensions which can be written by the trilinear form instead of by the hirota s bilinear form we explicitly discuss the bogoyavlenskii schiff equation in 2 1 dimensions its analytical proof of multisoliton solution and a new feature are given being guided by the strong symmetry we also propose a new equation in 3 1 dimensions

impact of boundaries on fully connected random geometric networks many complex networks exhibit a percolation transition involving a macroscopic connected component with universal features largely independent of the microscopic model and the macroscopic domain geometry in contrast we show that the transition to full connectivity is strongly influenced by details of the boundary but observe an alternative form of universality our approach correctly distinguishes connectivity properties of networks in domains with equal bulk contributions it also facilitates system design to promote or avoid full connectivity for diverse geometries in arbitrary dimension

symmetries and a hierarchy of the general modified kdv equation two groups of symmetries and their lie algebra properties for the modified kdv equation are extended to the general modified kdv equation and the miura transformation between the general kdv equation and the general modified equation is also established

nonintegrable reductions of the self dual yang mills equations in a metric of plane wave type symmetry reductions of the self dual yang mills equations for sl 2 c bundles with the background metric ds2 2 du dv dx2 f2 u dy2 are considered one of the field components in the reduced equations can be cast into jordan normal form after gauge transformations the reduced equations for the two possible normal forms are equivalent respectively to certain generalizations of the korteweg de vries kdv equation and the nonlinear schrodinger nls equation it is shown that the generalized kdv and nls equations fail the painleve test except when the metric is flat the generalized kdv equation is transformed to a simple form in the case when f u ua and it is shown that one may obtain either the kdv equation or the cylindrical kdv equation by this method only when the metric is flat

infinitely many commuting symmetries and constants of motion in involution for explicitly time dependent evolution equations the existing symmetry approach to exactly solvable time evolution equations in one spatial dimension is extended to the case that the equation depends explicitly on the time and space coordinates in this context commuting symmetries constants of motion in involution strong and hereditary symmetries squared eigenfunction operators and hamiltonian structures are discussed the cylindrical korteweg de vries equation is used as an illustrative example

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