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

real root classification of parametric spline functions abstract the real root classification of a given parametric spline function is a collection of possible cases of its real root distribution on every interval together with the conditions of its coefficients must be satisfied for each case this paper presents an algorithm to deal with the real root classification of a given parametric spline function two examples are provided to illustrate the proposed algorithm is flexible

an upper bound of the bezout number for piecewise algebraic curves a piecewise algebraic curve is a curve determined by the zero set of a bivariate spline function based on the discussion of the number of the zeros of homogeneous trigonometric splines with different smoothness and the common points of two piecewise algebraic curves over a star partition a better upper bound of bezout number of two piecewise algebraic curves over arbitrary triangulation is found moreover upper bounds of the bezout number b n m r n r for piecewise algebraic curves over several special partitions such as rectangular partition type 1 triangulation and type 2 triangulation are obtained

random catalytic reaction networks abstract we study networks that are a generalization of replicator or lotka volterra equations they model the dynamics of a population of object types whose binary interactions determine the specific type of interaction product such a system always reduces its dimension to a subset that contains production pathways for all of its members the network equation can be rewritten at a level of collectives in terms of two basic interaction patterns replicator sets and cyclic transformation pathways among sets although the system contains well known cases that exhibit very complicated dynamics the generic behavior of randomly generated systems is found numerically to be extremely robust convergence to a globally stable rest point it is easy to tailor networks that display replicator interactions where the replicators are entire self sustaining subsystems rather than structureless units a numerical scan of random systems highlights the special properties of elementary replicators they reduce the effective interconnectedness of the system resulting in enhanced competition and strong correlations between the concentrations

error propagation when approximating multi solitons the kdv equation as a case study abstract the paper studies the influence of the time discretizations when simulating some phenomena involving more than one solitary wave taking the kdv equation as a case study we obtain some conditions on the numerical method in order to get a more correct simulation of multi soliton solutions they are related to the evolution of the conserved quantities of the problem through the numerical integration it is shown that when approximating n solitons a method that preserves n invariants of the problem shows a better time propagation of the error than that of a general scheme as a consequence of this the simulation of some physical parameters that characterize the waves is more suitable when using conservative integrators we also show how these results can be extended to the approximation to multi solitons of any equation of the kdv hierarchy and more generally other integrable equations

stability and instability of solitary waves of korteweg de vries type considered herein are the stability and instability properties of solitary wave solutions of a general class of equations that arise as mathematical models for the unidirectional propagation of weakly nonlinear dispersive long waves special cases for which our analysis is decisive include equations of the korteweg de vries and benjamin ono type necessary and sufficient conditions are formulated in terms of the linearized dispersion relation and the nonlinearity for the solitary waves to be stable

new types of soliton solutions we announce a detailed investigation of limits of n soliton solutions of the korteweg devries kdv equation as n tends to infinity our main results provide new classes of kdv solutions including in particular new types of soliton like reflectionless solutions as a byproduct we solve an inverse spectral problem for one dimensional schrodinger operators and explicitly construct smooth and real valued potentials that yield a purely absolutely continuous spectrum on the nonnegative real axis and give rise to an eigenvalue spectrum that includes any prescribed countable and bounded subset of the negative real axis

the structure of wtc expansions and applications we construct generalized painleve expansions with logarithmic terms for a general class of non integrable scalar equations and describe their structure in detail these expansions were introduced without logarithms by weiss tabor carnevale wtc 1983 the construction of the formal solutions is shown to involve semi invariants of binary forms and tools from invariant theory are applied to the determination of the type of logarithmic terms that are required for the most general singular series the structure of the series depends strongly on whether 1 is or is not a resonance the convergence of these series is obtained as a consequence of the general results of littman and kichenassamy 1993 the results are illustrated on a family of fifth order models for water waves and other examples we also give necessary and sufficient conditions for 1 to be a resonance

solving parametric piecewise polynomial systems we deal with c r smooth continuity conditions for piecewise polynomial functions on d where d is an algebraic hypersurface partition of a domain w in r n piecewise polynomial functions of degree at most k on d that are continuously differentiable of order r form a spline space c k r we present a method for solving parametric systems of piecewise polynomial equations of the form z f 1 f n email protected email protected f 1 v x 0 f n v x 0 where f email protected c k w r w d and f w s email protected q v x for each n cell s i in d v u 1 u 2 u t is the set of parameters and x x 1 x 2 x n is the set of variables s 1 s 2 s m are all the n dimensional cells in d and email protected i 1 email protected i based on the discriminant variety method presented by lazard and rouillier we show that solving a parametric piecewise polynomial system z f 1 f n is reduced to the computation of discriminant variety of z the variety can then be used to solve the parametric piecewise polynomial system we also propose a general method to classify the parameters of z f 1 f n this method allows us to say that if there exist an open set of the parameters space where the system admits exactly a given number of distinct torsion free real zeros in every n cells in d

error propagation in numerical approximations near relative equilibria we study the propagation of errors in the numerical integration of perturbations of relative equilibrium solutions of hamiltonian differential equations with symmetries first it is shown that taking an initial perturbation of a relative equilibrium the corresponding solution is related in a first approximation to another relative equilibrium with the parameters perturbed from the original then this is used to prove that for stable relative equilibria error growth with respect to the perturbed solution is in general quadratic but only linear for schemes that preserve the invariant quantities of the problem in this sense the conclusion is similar to the one obtained when integrating unperturbed relative equilibria numerical experiments illustrate the results

asymptotic stability of solitary waves we show that the family of solitary waves 1 solitons of the korteweg de vries equation partial t u u partial x u partial x 3 u 0 is asymptotically stable our methods also apply for the solitary waves of a class of generalized korteweg de vries equations partial t u partial x f u partial x 3 u 0 in particular we study the case wheref u up 1 p 1 p 1 2 3 and 3 0 withf c4 the same asymptotic stability result for kdv is also proved for the casep 2 the modified korteweg de vries equation we also prove asymptotic stability for the family of solitary waves for all but a finite number of values ofp between 3 and 4 the solitary waves are known to undergo a transition from stability to instability as the parameterp increases beyond the critical valuep 4 the solution is decomposed into a modulating solitary wave with time varying speedc t and phase t bound state part and an infinite dimensional perturbation radiating part the perturbation is shown to decay exponentially in time in a local sense relative to a frame moving with the solitary wave asp 4 the local decay or radiation rate decreases due to the presence of aresonance pole associated with the linearized evolution equation for solitary wave perturbations

convergence of difference approximations of quasilinear evolution equations abstract the very successful theory of quasilinear evolution equations which applies to many problems of mathematical physics has been developed by t kato the theory obtains solutions of quasilinear problems via contraction mappings which are defined by means of a theory of linear evolution equations also developed by kato in the current work we show how the existence and continuous dependence theorems obtained by kato can be proved by discretization in time as opposed to earlier work in this direction the current results are much sharper concerning the continuity properties of the solutions of the discretized problem and the strength of the norms in which they converge

cayley bacharach theorem of piecewise algebraic curves the piecewise algebraic curve determined by a bivariate spline function is a generalization of the classical algebraic curve in this paper by using bezout s theorem and nother type theorem of piecewise algebraic curves the cayley bacharach theorem and hilbert function of c0 piecewise algebraic curves are presented

application of spectral deformation to the vlasov poisson system ii mathematical results this paper presents a mathematical description of the linearized vlasov poisson operator lk acting on a family of banach spaces xp related to l p r and the application of the method of spectral deformation to this model it is shown that a type a analytic family of operators lk c lk 0 lk can be associated with lk by means of this family the landau damped modes of the plasma are identified as the spectral resonances of lk existence and uniqueness of solutions to the initial value problem for the evolution equation g lk g is proven an expansion of any solution to the initial value problem with sufficiently smooth initial data is obtained in terms of the eigenfunctions of lk and a spectral integral over the essential spectrum this is applied to derive an expansion for solutions to the vlasov equation in which the landau damped portions of the distribution function are manifestly exhibited a self contained discussion of the spectral deformation method and an extension of it to ce

kowalewski s asymptotic method kac moody lie algebras and regularization we use an effective criterion based on the asymptotic analysis of a class of hamiltonian equations to determine whether they are linearizable on an abelian variety i e solvable by quadrature the criterion is applied to a system with hamiltonian h 1 mathord left vphantom 1 2 right kern nulldelimiterspace 2 sum limits i 1 l p i 2 sum limits i 1 l 1 exp left sum limits j 1 l n ij x j right parametrized by a real matrixn n ij of full rank it will be solvable by quadrature if and only if for alli j 2 n nt ij n n t jj 1 is a nonpositive integer i e the interactions correspond to the toda systems for the kac moody lie algebras the criterion is also applied to a system of gross neveu

multichannel nonlinear scattering for nonintegrable equations ii the case of anisotropic potentials and data abstract the nonlinear scattering and stability results of soffer and weinstein comm math phys 1990 are extended to the case of anisotropic potentials and data the range of nonlinearities for which the theory is shown to be valid is also extended considerably

local stability theory of solitary pulses in an active medium abstract if the spatio temporal chaos of an extended active medium is driven by repulsive or attractive interaction among localized coherent structures such as kinks and pulses its dynamics can often be described by a low dimensional dynamical system this welcomed reduction in the degrees of freedom only occurs when the localized structures remain stable during their mutual interaction it is shown that the structures can be destroyed by two types of local disturbance an expanding radiation packet and a localized disturbance which travels with the structure these two instabilities and the bifurcations they induce select the proper coherent structure and specify the dynamics of the generalized kuramoto sivashinsky gks equation in an extended domain the possibility of describing pulse radiation interaction by resonance poles also suggests that a more general coherent structure theory can be developed

phase transition in random catalytic networks the notion of auto catalytic networks has become a cornerstone in understanding the possibility of a sudden dramatic increase of diversity in biological evolution as well as in the evolution of social and economical systems here we study catalytic random networks with respect to the final outcome diversity of products we show that an analytical treatment of this long standing problem is possible by mapping the problem onto a set of nonlinear recurrence equations the solution of these equations shows a crucial dependence of the final number of products on the initial number of products and the density of catalytic production rules for a fixed density of rules we can demonstrate the existence of a phase transition from a practically unpopulated regime to a fully populated and diverse one the order parameter is the number of final products we are able to fully understand the origin of this phase transition as a crossover from one set of solutions from a quadratic equation to the other we observe a remarkable similarity of the solution of the system and the pvt diagrams in standard thermodynamics

bounds on the number of solutions of polynomial systems and the betti numbers of real piecewise algebraic hypersurfaces abstract this paper based on bihan and sottile s method which reduces a polynomial system to its gale dual system and then bounds the number of solutions of this gale system proves that a real coefficient polynomial system with n equations and with n variables involving n k 1 monomials has fewer than 27 e 5 3 88 90 i 0 k 1 2 i n 1 1 positive solutions and 27 e 10 3 88 90 i 0 k 1 2 i n 1 1 non degenerate non zero real solutions this dramatically improves f bihan and f sottile s bounds of e 2 3 4 2 k 2 n k and e 4 3 4 2 k 2 n k respectively using the new upper bound for positive solutions we establish restrictions to the sum of the betti numbers of real piecewise algebraic hypersurfaces and real piecewise algebraic curves a new bound on the number of compact components of algebraic hypersurfaces in r n is also given

the maximum number and its distribution of singular points for parametric piecewise algebraic curves abstract the piecewise algebraic curve as the zero set of a bivariate spline function is a generalization of the classical algebraic curve based on the previous method presented by lai et al 2009 we show that computing singular points of parametric piecewise algebraic curves amounts to solving parametric piecewise polynomial systems in this article we give a method to compute the maximum number and its distribution of singular points for a given parametric piecewise algebraic curve this method also produces necessary and sufficient conditions of its parameters must be satisfied an illustrated example shows that our method is flexible

symmetry and the chazy equation there are three different actions of the unimodular lie group sl 2 c on a two dimensional space in every case we show how an ordinary differential equation admitting sl 2 as a symmetry group can be reduced in order by three and the solution recovered from that of the reduced equation via a pair of quadratures and the solution to a linear second order equation a particular example is the chazy equation whose general solution can be expressed as a ratio of two solutions to a hypergeometric equation the reduction method leads to an alternative formula in terms of solutions to the lame equation resulting in a surprising transformation between the lame and hypergeometric equations finally we discuss the painleve analysis of the singularities of solutions to the chazy equation

towards a physics of evolution critical diversity dynamics at the edges of collapse and bursts of diversification systems governed by the standard mechanisms of biological or technological evolution are often described by catalytic evolution equations we study the structure of these equations and find an analogy with classical thermodynamic systems in particular we can demonstrate the existence of several distinct phases of evolutionary dynamics a phase of fast growing diversity one of stationary finite diversity and one of rapidly decaying diversity while the first two phases have been subject to previous work here we focus on the destructive aspects in particular the phase diagram of evolutionary dynamics the main message is that within a critical region massive loss of diversity can be triggered by very small external fluctuations we further propose a dynamical model of diversity which captures spontaneous creation and destruction processes fully respecting the phase diagrams of evolutionary systems the emergent time series show rich diversity dynamics including power laws as observed in actual economical data e g firm bankruptcy data we believe the present model presents a possibility to cast the famous qualitative picture of schumpeterian economic evolution into a quantifiable and testable framework

weighted energy decay for 1d wave equation we obtain a dispersive long time decay in weighted energy norms for solutions to the 1d wave equation with generic potential the decay extends the results obtained by murata for the 1d schrodinger equation

piecewise algebraic curve a piecewise algebraic curve is defined by a bivariate spline function using the techniques of the b net form of bivariate splines function discriminant sequence of polynomial cf yang lu et al sci china ser e 39 6 1996 628 and yang lu et al nonlinear algebraic equation system and automated theorem proving shanghai scientific and technological education publishing house shanghai 1996 and the number of sign changes in the sequence of coefficients of the highest degree terms of sturm sequence we determine the number of real intersection points of two piecewise algebraic curves whose common points are finite a lower bound of the number of real intersection points is given in terms of the method of rotation degree of vector field

recent researches on multivariate spline and piecewise algebraic variety the multivariate splines as piecewise polynomials have become useful tools for dealing with computational geometry computer graphics computer aided geometrical design and image processing it is well known that the classical algebraic variety in algebraic geometry is to study geometrical properties of the common intersection of surfaces represented by multivariate polynomials recently the surfaces are mainly represented by multivariate piecewise polynomials i e multivariate splines so the piecewise algebraic variety defined as the common intersection of surfaces represented by multivariate splines is a new topic in algebraic geometry moreover the piecewise algebraic variety will be also important in computational geometry computer graphics computer aided geometrical design and image processing the purpose of this paper is to introduce some recent researches on multivariate spline piecewise algebraic variety curve and their applications

a unified approach to painleve expansions abstract the painleve test for partial differential equations developed by weiss tabor and carnevale wtc is examined in detail and shown to provide a unified approach to the integrable properties of both ordinary and partial differential equations a simple modification of the wtc procedure used for partial differential equations enables us to determine the lax pairs hirota equations and auto backlund transformations for ordinary differential equations including a new lax pair for an integrable case of the henon heiles system a detailed study of the kdv hierarchy is made and a complete picture of the pattern of resonances for all solution branches is obtained the role of the singular branches is examined in detail and important new insights obtained in particular we find that each singular branch is simply a re expansion of the principal branch about a point on the pole manifold at which several isolated poles coalesce a parallel analysis is carried out for the akns hierarchy

on the asymptotic stability of solitary waves in the discrete schrodinger equation coupled to a nonlinear oscillator abstract the long time asymptotics is analyzed for finite energy solutions of the 1d discrete schrodinger equation coupled to a nonlinear oscillator the coupled system is invariant with respect to the phase rotation group u 1 for initial states close to a solitary wave the solution converges to a sum of another solitary wave and a dispersive wave which is a solution to the free schrodinger equation the proofs use the strategy of buslaev perelman v s buslaev g s perelman on the stability of solitary waves for nonlinear schrodinger equations amer math soc trans 2 164 1995 75 98 the linearization of the dynamics on the solitary manifold the symplectic orthogonal projection method of majorants etc

instability of pole singularities for the chazy equation we prove that the negative resonances of the chazy equation in the sense of painleve analysis can be related directly to its group invariance properties these resonances indicate in this case the instability of pole singularities depending on the value of a parameter in the equation an unstable isolated pole may turn into the familiar natural boundary or split into several isolated singularities in the first case a convergent series representation involving exponentially small corrections can be given this reconciles several earlier approaches to the interpretation of negative resonances on the other hand we also prove that pole singularities with the maximum number of positive resonances are stable the proofs rely on general properties of nonlinear fuchsian equations

physics of evolution selection without fitness abstract traditionally evolution is seen as a process where from a pool of possible variations of a population e g biological species or industrial goods a few variations get selected which survive and proliferate whereas the others vanish survival probabilities and proliferation rates are typically associated with the fitness of particular variations in this paper we argue that the notion of fitness is an a posteriori concept in the sense that one can assign higher fitness to species that survive but one can generally not derive or predict fitness per se proliferation rates can be measured whereas fitness landscapes i e the inter dependence of proliferation rates cannot for this reason we think that in a physical theory of evolution such notions should be avoided in this spirit here we propose a random matrix model of evolution where selection mechanisms are encoded in interaction matrices of species thereby extending the previous work of ours by a control parameter describing suppressors in the system we are able to recover some key facts of evolution dynamics endogenously such as punctuated equilibrium i e the existence of intrinsic large extinction events and at the same time periods of dramatic diversification as known e g from the fossil record further we comment on two fundamental technical problems of a physics of evolution the non closedness of its phase space and the problem of co evolving boundary conditions apparent in all systems subject to evolution

exponential stability of large amplitude traveling fronts for quasi linear relaxation systems with diffusion a b s t r a c t this paper is concerned with the stability of traveling front solutions for 2 2 quasi linear relaxation systems with small diffusion rate by applying geometric singular perturbation method special evans function estimates detailed spectral analysis and c0 semigroup theories we prove that all the non degenerate waves for semi linear relaxation systems are locally exponentially stable in some exponentially weighted spaces we also obtain the linear exponential stability of the non degenerate waves for quasi linear relaxation systems where the wave strengths can be large

fuchsian equations in sobolev spaces and blow up abstract we construct solutions of u e u which blow up precisely on a given space like hypersurface of class h s for this purpose we prove a general existence theorem for fuchsian pde in sobolev spaces the precise relation between the regularity of the data and that of the solution is shown to involve logarithmic symbols in a model situation a few further results on power nonlinearities are also included

multivariate spline and algebraic geometry abstract the purpose of this survey is to emphasize the special relationship between multivariate spline and algebraic geometry we will not only point out the algebraic geometric method of multivariate spline but also the algebraic geometric background and essence of multivariate spline especially we have made an introduction to the so called piecewise algebraic curve piecewise algebraic variety and some of their properties

lyapunov stability of ground states of nonlinear dispersive evolution equations on presente une nouvelle demonstration de la stabilite orbitale des solitons d etat fondamental de l equation de schrodinger non lineaire pour une grande classe de non linearites on demontre la stabilite de l onde solitaire pour l equation de korteweg de vries generalisee

evolutionary dynamics from a variational principle we demonstrate with a thought experiment that fitness based population dynamical approaches to evolution are not able to make quantitative falsifiable predictions about the long term behavior of some evolutionary systems a key characteristic of evolutionary systems is the ongoing endogenous production of new species these novel entities change the conditions for already existing species even darwin s demon a hypothetical entity with exact knowledge of the abundance of all species and their fitness functions at a given time could not prestate the impact of these novelties on established populations we argue that fitness is always a posteriori knowledge it measures but does not explain why a species has reproductive success or not to overcome these conceptual limitations a variational principle is proposed in a spin model like setup of evolutionary systems we derive a functional which is minimized under the most general evolutionary formulation of a dynamical system i e evolutionary trajectories causally emerge as a minimization of a functional this functional allows the derivation of analytic solutions of the asymptotic diversity for stochastic evolutionary systems within a mean field approximation we test these approximations by numerical simulations of the corresponding model and find good agreement in the position of phase transitions in diversity curves the model is further able to reproduce stylized facts of timeseries from several man made and natural evolutionary systems light will be thrown on how species and their fitness landscapes dynamically coevolve

well posedness of the initial value problem for the korteweg de vries equation 1 1 itu axu u1xu o x t e r u x 0 uo x the kdv equation which was first derived as a model for unidirectional propagation of nonlinear dispersive long waves 21 has been considered in different contexts namely in its relation with the inverse scattering method in plasma physics and in algebraic geometry see 24 and references therein our purpose is to study local and global well posedness of the ivp 1 1 in classical sobolev spaces hs r we shall say that the ivp 1 1 is locally resp globally well posed in the function space x if it induces a dynamical system on x by generating a continuous local resp global flow it was established in the works of bona and smith 3 bona and scott 2 saut and temam 30 and kato 1 5 that the ivp 1 1 is locally resp globally well posed in hs with s 3 2 resp s 2 roughly speaking global well posedness in hs depends on the available local theory and on the conservation laws satisfied by solutions of 1 1 namely

a perturbative painleve approach to nonlinear differential equations abstract we further improve the painleve test so that negative indices resonances can be treated we demand single valuedness not only for any pole like expansion as in the usual painleve test but also for every solution close to it represented as a perturbation series in a small parameter e order zero is the usual test order one already treated in a preliminary paper reduces to a linear fuchs analysis near a regular singularity and allows the introduction of all missing arbitrary coefficients higher orders lead to the analysis of a linear fuchsian type inhomogeneous equation we obtain an infinite sequence of necessary conditions for the absence of movable logarithmic branch points arising at every integer index whether positive or negative and at every order those arising at negative indices including 1 are new while some conditions may not arise before some high perturbation order we present several illustrative examples we discuss the understanding of negative indices and conclude that they are indistinguishable from positive indices just as in the fuchs theory in particular negative indices give rise to doubly infinite laurent series