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

the oscillation and asymptotically monotone solutions of second order quasilinear differential equations abstract we offer sufficient conditions for the oscillation of all solutions of the perturbed quasilinear differential equation a t y 1 y q t y p t y y as well as for the existence of a positive monotone solution of the damped differential equation a t y 1 y b t y 1 y h t y 0 where 0 examples that dwell upon the importance of our results are also included

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

painleve expansions for nonintegrable evolution equations abstract the wtc method is applied to a variety of nonintegrable nonlinear evolution equations and shown to provide a systematic procedure for obtaining special solutions some of the special techniques developed for integrable systems are also shown to be applicable to nonintegrable equations

oscillation and nonoscillation theorems for second order nonlinear difference equations abstract some new oscillation and nonoscillation theorems are obtained for second order nonlinear difference equations a p n a x n q n a x n f n x n b x n which generalize and improve all results proved by popenda in j math anal appl 123 1987 34 48

oscillation theorems for higher order nonlinear delay difference equations abstract some new oscillation results for difference equations of the form 1 1 and forced difference equations of the form 1 2 are established the results obtained are discrete analogues of some known results for delay differential equations

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

oscillation and nonoscillation theorems for second order difference equations in this paper we are concerned with the oscillatory and nonoscillatory behavior of the solutions of some second order difference equations let r be the set of real numbers n n n l where n is given nonnegative integer for the function x n t r the difference operator d l1 a is fixed real constant will be defined as follows d x x ax n e n instead of a we shall write a we define inductively a x a atix for k 1 we shall study difference equations of the form

hamiltonian structure and integrability of the stationary kuramoto sivashinsky equation in this article the stationary kuramoto sivashinsky ks equation is studied it is shown first that this equation can be transformed into a one dimensional hamiltonian system describing the motion of a particle in a time dependent potential second working with this formalism and coming back then to the initial ks equation a subset of initial conditions is derived for which the corresponding solutions u x will behave like the dominant pole 120 x x0 3 deduced from the painleve analysis and will explode in a finite time finally special attention is given to the well known kuramoto tsuzuki solution and it is shown that provided the independent variable x is initialized in a specified domain this solution explodes also like 120 x x0 3

subtree prune and regraft a reversible real tree valued markov process we use dirichlet form methods to construct and analyze a reversible markov process the stationary distribution of which is the brownian continuum random tree this process is inspired by the subtree prune and regraft spr markov chains that appear in phylogenetic analysis a key technical ingredient in this work is the use of a novel gromov hausdorff type distance to metrize the space whose elements are compact real trees equipped with a probability measure also the investigation of the dirichlet form hinges on a new path decomposition of the brownian excursion

the earth mover s distance as a metric for image retrieval we investigate the properties of a metric between two distributions the earth mover s distance emd for content based image retrieval the emd is based on the minimal cost that must be paid to transform one distribution into the other in a precise sense and was first proposed for certain vision problems by peleg werman and rom for image retrieval we combine this idea with a representation scheme for distributions that is based on vector quantization this combination leads to an image comparison framework that often accounts for perceptual similarity better than other previously proposed methods the emd is based on a solution to the transportation problem from linear optimization for which efficient algorithms are available and also allows naturally for partial matching it is more robust than histogram matching techniques in that it can operate on variable length representations of the distributions that avoid quantization and other binning problems typical of histograms when used to compare distributions with the same overall mass the emd is a true metric in this paper we focus on applications to color and texture and we compare the retrieval performance of the emd with that of other distances

linearity inside nonlinearity exact solutions to the complex ginzburg landau equation abstract systems of two linear partial differential equations pdes with constant coefficients define a natural basis of elementary functions to build some exact solutions to nonlinear pdes for the one dimensional complex ginzburg landau equation the usual representations of the complex field re a or a are multivalued which makes difficult the search for exact solutions in this manner fortunately the painleve analysis naturally introduces an elementary single valued like representation of a by two fields z grad respectively complex abd real uniquely defined by an explicit expression for and a ze i another important feature is the invariance by parity on a which increases the class of expected solutions this allows to retrieve quite easily the four famous solutions of nozaki and bekki represented by the constant coefficients of two linear partial differential equations and a finite set of constants

travelling wave solutions for the generalized zakharov kuznetsov equation with higher order nonlinear terms abstract with the aid of symbolic computation and two auxiliary ordinary differential equations the new generalized algebraic method is extended to the generalized zakharov kuznetsov gzk equation with higher order nonlinear terms for constructing a series of new and more general travelling wave solutions because of the higher order nonlinear terms the equation can not be directly dealt with by the method and require some kinds of techniques by means of two proper transformations we transform the gzk equation to an ordinary differential equation that is easy to solve and find a rich variety of new exact travelling wave solutions for the gzk equation which include soliton solutions combined soliton solutions triangular periodic solutions jacobi elliptic function solutions combined jacobi elliptic function solutions and rational function solutions the method used here can be also extended to other nonlinear partial differential equations with higher order nonlinear terms

link between solitary waves and projective riccati equations many solitary wave solutions of nonlinear partial differential equations can be written as a polynomial in two elementary functions which satisfy a projective hence linearizable riccati system from that property the authors deduce a method for building these solutions by determining only a finite number of coefficients this method is much shorter and obtains more solutions than the one which consists of summing a perturbation series built from exponential solutions of the linearized equation they handle several examples for the henon heiles hamiltonian system they obtain several exact solutions one of them defines a new solitary wave solution for a coupled system of boussinesq and nonlinear schrodinger equations for a third order dispersive equation with two monomial nonlinearities they isolate all cases where the general solution is single valued

solutions of a 2 1 dimensional dispersive long wave equation a special type of multisoliton solution with a particular dispersion relation is obtained for wu zhang equation which describes 2 1 dimensional dispersive long waves by the standard weiss tabor carnvale painleve truncation expansion using a nonstandard truncation of a modified conte s invariant painleve expansion two different types of soliton solutions without any dispersive relation is found two types of periodic wave solutions expressed by jacobi elliptic functions are found by the truncations of a special extended painleve expansion the soliton solutions are special cases of the corresponding periodic solutions

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

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

ricci ollivier curvature of the rooted phylogenetic subtree prune regraft graph abstract statistical phylogenetic inference methods use tree rearrangement operations such as subtree prune regraft spr to perform markov chain monte carlo mcmc across tree topologies the structure of the graph induced by tree rearrangement operations is an important determinant of the mixing properties of mcmc motivating the study of the underlying spr graph in greater detail in this paper we investigate the spr graph of rooted trees rspr graph in a new way by calculating the ricci ollivier curvature with respect to uniform and metropolis hastings random walks this value quantifies the degree to which a pair of random walkers from specified points move towards each other negative curvature means that they move away from one another on average while positive curvature means that they move towards each other in order to calculate this curvature we develop fast new algorithms for rspr graph computation we then develop formulas characterizing how the number of rspr neighbors of a tree changes after an rspr operation is applied to that tree these give bounds on the curvature as well as a flatness in the limit theorem indicating that paths of small topology changes are easy to traverse however we find that large topology changes i e moving a large subtree give pairs of trees with negative curvature we show using simulation that mean access time distributions depend on distance degree and curvature demonstrating the relevance of these results to stochastic tree search

a new truncation in painleve analysis the author observes that when the expansion variable in painleve analysis satisfies a system of riccati equations truncation at a level higher than constant level is allowed this extends the range of exact solutions to nonlinear partial differential equations that one is able to obtain using truncated painleve expansions

traveling wave solutions of the generalized nonlinear evolution equations solitary wave solutions for a family of nonlinear evolution equations with an arbitrary parameter in the exponents are constructed some of the obtained solutions seem to be new

asymptotic behavior of solutions of second order nonlinear difference equations 0 2 j 1 b n u n g n y y h 1 1 or equivalently e y a0 by g n y ey where e is the operator defined by ey n y n 1 2 let z z2 be a fundamental system of solutions of the linear difference equation lz e2z f aez bz 0 3 satisfying z o 1 z l 0 zz o 0 22 l 1 4 we will prove that there exists a ball b 0 p c r2 such that if y o y 1 e b 0 p then the solution y of equation 1 is defined on all n and for n co it satisfies

transcendents defined by nonlinear fourth order ordinary differential equations three ordinary differential equations are considered the general solutions of these equations are shown to be the essentially transcendental functions with respect to their initial conditions irreducibility of these equations is discussed

curvature concentration and error estimates for markov chain monte carlo we provide explicit nonasymptotic estimates for the rate of convergence of empirical means of markov chains together with a gaussian or exponential control on the deviations of empirical means these estimates hold under a positive curvature assumption expressing a kind of metric ergodicity which generalizes the ricci curvature from differential geometry and on finite graphs amounts to contraction under path coupling

quasi exact solutions of nonlinear differential equations abstract the concept of quasi exact solutions of nonlinear differential equations is introduced quasi exact solution expands the idea of exact solution for additional values of parameters of differential equation these solutions are approximate ones of nonlinear differential equations but they are close to exact solutions quasi exact solutions of the the kuramoto sivashinsky the korteweg de vries burgers and the kawahara equations are founded

asymptotic and oscillatory behavior of solutions of general nonlinear difference equations of second order abstract the asymptotic and oscillatory behavior of solutions of some general second order nonlinear difference equations of the form a n h y n 1 y n p n y n q n 1 f y n 1 0 n z is studied oscillation criteria for their solutions when pn is of constant sign are established results are also presented for the damped forced equation a n h y n 1 y n p n y n q n 1 f y n 1 e n n z examples are inserted in the text for illustrative purposes

method for deriving rational solutions of some nonlinear evolution equations maps for nonlinear evolution equations are discussed an algorithmic method for deriving rational solutions is presented this approach is illustrated by examples which have solutions in the form of two truncated expansions

the oscillation of solutions of difference equations abstract using the solutions of a suitable inequality we shall discuss the asymptotic properties of the solutions of a generalized emden fowler difference equation in the absence of specified constant sign solutions of certain inequalities oscillatory behavior of all solutions of this equation are also deduced

exact solutions of the complex modified korteweg de vries equation firstly some similarity reductions of the complex modified korteweg de vries equation cmkdv which arises in the asymptotic interpretation of one dimensional plane wave propagation in a quadratic micropolar medium are discussed although it is not a soliton equation solvable by inverse scattering transformation its similarity reductions obtained by the use of lie group methods are of mathematical interest secondly the painleve analysis developed by weiss et al 1983 for nonlinear partial differential equations is applied to the cmkdv equation and the data obtained by the truncation technique yield some analytical solutions of the ordinary modified korteweg de vries equation and travelling wave solutions of the cmkdv equation which are also solutions of the similarity reduction obtained by classical lie group analysis

the painleve property for partial differential equations ii backlund transformation lax pairs and the schwarzian derivative in this paper we investigate the painleve property for partial differential equations by application to several well known partial differential equations burgers kdv mkdv bousinesq higher order kdv and kp equations it is shown that consideration of the singular manifold leads to a formulation of these equations in terms of the schwarzian derivative this formulation is invariant under the moebius group acting on dependent variables and is shown to obtain the appropriate lax pair linearization for the underlying nonlinear pde

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

linear hamiltonian difference systems disconjugacy and jacobi type conditions abstract we consider a linear hamiltonian difference system for the so called singular case so that discrete sturm liouville equations of higher order are included in our theory we introduce the concepts of focal points for matrix valued and generalized zeros for vector valued solutions of the system and define disconjugacy for linear hamiltonian difference systems we prove a reid roundabout theorem which gives conditions equivalent to positive definiteness of a certain discrete quadratic functional among them the strengthened jacobi s condition and a condition on a certain riccati difference equation the key to this theorem is a discrete version of picone s identity furthermore for the sake of generalization of our theorem we introduce controllability for linear hamiltonian difference systems and prove a reid roundabout theorem for a more general functional and more general boundary conditions

oscillation and non oscillation of solutions of second order difference equations involving generalized difference abstract in this paper sufficient conditions are obtained for oscillation non oscillation of solutions of second order difference equations involving generalized difference of the form a p n 1 a y n 1 q n y n 0 n 1 and a p n 1 a y n 1 q n y n f n n 1 where a is defined by a y n y n 1 ay n a 0 necessary and sufficient conditions are obtained for oscillation of the first equation with a 0 and q n q a non zero constant the signs of q n and a play an important role for oscillation non oscillation of these two equations

ricci curvature of markov chains on metric spaces abstract we define the coarse ricci curvature of metric spaces in terms of how much small balls are closer in wasserstein transportation distance than their centers are this definition naturally extends to any markov chain on a metric space for a riemannian manifold this gives back after scaling the value of ricci curvature of a tangent vector examples of positively curved spaces for this definition include the discrete cube and discrete versions of the ornstein uhlenbeck process moreover this generalization is consistent with the bakry emery ricci curvature for brownian motion with a drift on a riemannian manifold positive ricci curvature is shown to imply a spectral gap a levy gromov like gaussian concentration theorem and a kind of modified logarithmic sobolev inequality the bounds obtained are sharp in a variety of examples

analytic solutions of a microstructure pde and the kdv and kadomtsev petviashvili equations by invariant painleve analysis and generalized hirota techniques abstract truncated painleve expansions invariant painleve analysis and generalized hirota expansions are employed in combination to solve partially reduce to quadrature the integrable kdv and kp equations and a nonintegrable generalized microstructure gms equation although the multisolitons of the kdv and kp equations are very well known the solutions obtained here for all the three nlpdes are novel and non trivial the solutions obtained via invariant painleve analysis are all complicated rational functions with arguments which themselves are confluent hypergeometric kdv or trigonometric gms functions of various distinct non traveling kdv and traveling wave variables in some cases this is slightly reminiscent of doubly periodic elliptic function solutions when nonlinear ode systems are reduced to quadratures the solutions obtained by the use of recently generalized hirota type expansions in the truncated painleve expansions are closer in functional form to conventional hyperbolic secant solutions although with non trivial traveling wave arguments which are distinct for the three nlpdes considered here

on the behavior of solutions of a class of nonlinear partial differential equations the behavior of the steady state or the traveling wave solutions for a class of nonlinear partial differential equations is studied the nonlinearity in these equations is expressed by the presence of the convective term it is shown that the steady state or the traveling wave solution may explode at a finite value of the spatial or the characteristic variable this holds whatever the order of the spatial derivative term in these equations furthermore new special solutions of a set of equations in this class are also found

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

reduction technique for matrix nonlinear evolution equations solvable by the spectral transform the main purpose of this paper is to describe a technique of reduction whereby from the class of evolution equations for matrices of order n solvable via the spectral transform associated to the matrix linear schrodinger eigenvalue problem one derives subclasses of nonlinear evolution equations involving less than n2 fields to illustrate the method from the equations for matrices of order 2 two subclasses of equations for 2 fields rather than 4 are obtained the first class coincides or rather includes that solvable via the spectral transform associated to the generalized zakharov shabat spectral problem further reduction to nonlinear evolution equations for a single field reproduces a number of well known equations but also yields a novel one highly nonlinear the second class also yields highly nonlinear equations some examples are given including another novel evolution equation for a single field

invariance principle for variable speed random walks on trees we consider stochastic processes on complete locally compact tree like metric spaces t r t r on their natural scale with boundedly finite speed measure given a triple t r t r such a speed motion on t r t r can be characterized as the unique strong markov process which if restricted to compact subtrees satisfies for all x y tx y t and all positive bounded measurable ff ex y0dsf xs 2 t dz r y c x y z f z ex 0 ydsf xs 2 t dz r y c x y z f z where c x y z c x y z denotes the branch point generated by x y zx y z if t r t r is a discrete tree xx is a continuous time nearest neighbor random walk which jumps from vv to v vv v at rate 12 v r v v 112 v r v v 1 if t r t r is path connected xx has continuous paths and equals the brownian motion which was recently constructed in trans amer math soc 365 2013 3115 3150 in this paper we show that speed n n motions on tn rn tn rn converge weakly in path space to the speed motion on t r t r provided that the underlying triples of metric measure spaces converge in the gromov hausdorff vague topology introduced in stochastic process appl 126 2016 2527 2553

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

from singular manifold equations to integrable evolution equations the singular manifold equations are used to construct nonlinear integrable partial differential equations a number of well known integrable equations hierarchies of korteveg de vries caudrey dodd gibbon kaup kuperschmidt and harry dym are obtained some new integrable equations are presented a possible approach to classification of integrable evolution equations is discussed

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 of the equation for surface waves in a convecting fluid abstract a method for finding exact solutions and the first integrals is presented the basic idea of the method is to use the value of the fuchs index that appears in the painleve test to construct the auxiliary equation for finding the first integrals and exact solutions of nonlinear differential equations it allows us to obtain the first integrals and new exact solutions of some nonlinear ordinary differential equations the main feature of the method is that we do not assign a solution function at the beginning we find this function during calculations this approach is conceptually equivalent to the third step of the painleve test and sometimes allows us to change this step our approach generalizes a number of other methods for finding exact solutions of nonlinear differential equations we demonstrate a method for finding the traveling wave solutions and the first integrals of the well known nonlinear evolution equation for description of surface waves in a convecting liquid the general solution of this equation at some conditions on parameters and new traveling wave solutions of the fourth order equation are found

oscillation theorems for perturbed nonlinear second order difference equations abstract sufficient conditions for the oscillation of all solutions of perturbed difference equations of the form a n 1 y n 1 q n y n p n y n y n n n are obtained examples are inserted to illustrate the results

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

fixed parameter algorithms for maximum agreement forests we present new and improved fixed parameter algorithms for computing maximum agreement forests of pairs of rooted binary phylogenetic trees the size of such a forest for two trees corresponds to their subtree prune and regraft distance and if the agreement forest is acyclic to their hybridization number these distance measures are essential tools for understanding reticulate evolution our algorithm for computing maximum acyclic agreement forests is the first depth bounded search algorithm for this problem our algorithms substantially outperform the best previous algorithms for these problems

exact solitary waves in a convecting fluid the perturbed korteweg de vries equation ut lambda 1uux lambda 2uxxx lambda 3uxxxx lambda 4 uux x lambda 5uxx 0 which describes the evolution of long shallow waves in a convecting fluid when the critical rayleigh number slightly exceeds its critical value admits two types of exact solitary wave solution

nonoscillation and oscillation of solutions of a class of third order difference equations in this paper sufficient conditions are obtained for nonoscillation oscillation of all solutions of a class of nonlinear third order difference equations of the form a pn a2yn qn a2yn f n yn ayn a2yn involving the generalized difference operator a which is defined as ayn yn 1 ayn a 0

on the exactly solvable equation s_t beta s gamma 2 s_x _x alpha beta s gamma 2 s_x occurring in two phase flow in porous media the symmetry approach of fokas j math phys 21 1980 pp 1318 1325 is used to show that the most general equation of the form s t g s s xx f s s x dg ds ne 0 which is exactly solvable is equivalent to the equation s t beta s gamma 2 s x x alpha beta s gamma 2 s x this equation is then mapped to the linear heat equation through a series of backlund transformations the above result is used to solve an initial boundary value problem modeling the process of waterflooding i e the displacement of oil initially in a porous reservoir by continuously injected water

painleve analysis and backlund transformation in the kuramoto sivashinsky equation the authors apply the painleve test to the kuramoto sivashinsky nonlinear partial differential equation which describes many interesting fluid motions the equation passes the test in a weak sense although the painleve analysis does not yield the general solution they perform the backlund transformation in order to obtain new particular solutions the authors find explicitly all the solutions of the set of equations which define the painleve backlund transformation they reduce to only one type namely the already known kink shaped solitary wave