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

law of large numbers for the sir model with random vertex weights on erdős renyi graph abstract in this paper we are concerned with the sir model with random vertex weights on erd s renyi graph g n p the erd s renyi graph g n p is generated from the complete graph c n with n vertices through independently deleting each edge with probability 1 p we assign i i d copies of a positive r v on each vertex as the vertex weights for the sir model each vertex is in one of the three states susceptible infective and removed an infective vertex infects a given susceptible neighbor at rate proportional to the production of the weights of these two vertices an infective vertex becomes removed at a constant rate a removed vertex will never be infected again we assume that at t 0 there is no removed vertex and the number of infective vertices follows a bernoulli distribution b n our main result is a law of large numbers of the model we give two deterministic functions h s t h v t for t 0 and show that for any t 0 h s t is the limit proportion of susceptible vertices and h v t is the limit of the mean capability of an infective vertex to infect a given susceptible neighbor at moment t as n grows to infinity

a theory of exact and approximate configurational invariants abstract a theory of integrable hamiltonian systems in two dimensions is formulated and applied the four dimensional phase space problem that is the vanishing of the poisson bracket between another invariant and the hamiltonian is here reduced to the solution of a series of two dimensional configuration space equations the theory is also applicable to more than two dimensions the constraints are found which admit to integrability of the orbits for magnetic or coriolis forces as well as for forces derivable from a potential when a system admits a given invariant the invariant is found for example by quadrature a number of examples including known and apparently previously unknown invariants are given the theory of exact integrals of the motion also can be extended to the derivation of approximate invariants the orbital structure of integrable or approximately integrable systems correlates with the degree maximum power of the velocity of a standardized invariant the theory admits a variational principle among other approximation techniques for the computation of a best approximate invariant the problem of the general cubic potential with one symmetric coordinate v 1 2 ax 2 1 2 by 2 cx 2 y 1 3 dy 3 of which the well studied henon heiles potential is the special case for a b and c d is examined in detail

the painleve test backlund transformation and solutions of the reduced maxwell bloch equations the painleve property for partial differential equations pdes proposed by weiss et al 1983 is studied for a system of pdes namely the reduced maxwell bloch rmb equations the rmb equations describe the propagation of short optical pulses through dielectric materials with a resonant non degenerate transition the author demonstrates that the rmb system passes the painleve test and constructs a backlund transformation and solutions of the rmb equations

the survival of one dimensional contact processes in random environments consider the inhomogeneous contact process on z1 with recovery rate 8 k at site k and infection rates a k and p k at site k due to the presence of infected neighbors at k 1 and k 1 respectively a special case of the main result in this paper is the following suppose that the environment is chosen in such a way that the 8 k s a k s and p k s are all mutually independent with the 8 k s having a common distribution and the a k s and p k s having a common distribution then the process survives if

investigation of painleve property under time singularities transformations in this paper the relation between the painleve property for ordinary differential equations and some coordinate transformations in the complex plane is investigated two sets of transformations are introduced for which the transformations of singularities structure in the complex plane are explicitly computed the first set acts only on the dependent variables while the second one acts also on the independent variables it allows for defining an extended painleve test which is coordinates invariant our results are applied on various problems arising in the singularity analysis such as the problem of negative resonances and the weak painleve conjecture the method for finding new first integrals for dynamical systems is also shown

darboux integrability of 2 dimensional hamiltonian systems with homogenous potentials of degree 3 we provide a characterization of all hamiltonian systems of the form h p12 p22 2 v q1 q2 where v is a homogenous polynomial of degree 3 which are completely integrable with darboux first integrals

motion of four dimensional rigid body around a fixed point an elementary approach i the goal of this letter is to give an elementary approach to the solution of euler frahm equations for the manakov four dimensional case for this we use the kotter approach and some results from the original papers by schottky weber and caspary we hope that such an approach will be useful for the solution of the problem of an n dimensional top

nonintegrability of some hamiltonian systems scattering and analytic continuation we consider canonical two degrees of freedom analytic hamiltonian systems with hamiltonian functionh 1 2 p12 p22 u q1 q2 where u q1 q2 1 2 v2q12 2q22 o q12 q22 3 2 and q2 u q1 0 0 under some additional not so restrictive hypothesis we present explicit conditions for the exisstence of transversal homoclinic orbits to some periodic orbits of these systems we use a theorem of lerman 1991 and an analogy between one of its conditions with the usual one dimensional quantum scattering problem the study of the scattering equation leads us to an analytic continuation problem for the solutions of a linear second order differential equation we apply our results to some classical problems

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

the complete convergence theorem holds for contact processes on open clusters of ℤd ℤ we study contact processes on open clusters of half space our result shows that the complete convergence theorem holds

painleve property and geometry abstract the painleve property for discrete hamiltonian systems implies the existence of a symplectic manifold which augments the original phase space and on which the flows exist and are analytic for all times the augmented manifold is constructed by expanding the hamilton jacobi equation a complete classification of the types of poles allowed in complex time is given for hamiltonians which separate into the direct product of hyperelliptic curves for such systems bounds on the degrees of the polynomial separating variable change and the other integrals in involution can be found from the pole series and the hamilton jacobi equation it is shown how branching can arise naturally in a painleve system

a criterion for non integrability based on poincare s theorem abstract the non integrability of a two degrees of freedom hamiltonian is investigated by a method based on a well known theorem of poincare it is proved that the perturbed system is non integrable for values of the perturbative parameter in an open interval around zero if on a dense set of resonant tori of the unperturbed system the average value of the perturbative function evaluated along the periodic orbits on each torus depends on the particular orbit an application to the separable quartic oscillator with a quadratic perturbation is made and it is shown that if the perturbation is non separable in the same coordinates the system is non integrable

non integrable systems with algebraic singularities in complex time dynamical arguments are presented which suggest that there are non integrable systems without clustering of singularities without infinite singularities or singularities with an infinite number of branches in the complex t plane several examples with only algebraic singularities are studied for which strong numerical evidence is presented for non integrability and infinitely sheeted solutions weak painleve potentials are also analysed from this point of view and all integrable cases are found to possess only finitely sheeted solutions

global analytic solutions of the radial nonlinear wave equation abstract the initial valued problem f z z 1 f z g z f z f 0 f 0 0 under suitable assumptions on the function g z f z is reduced by a well known method 2 to an operator equation of the form f n 1 f gle 1 in an abstract banach space h1 the operator n1 is a nonlinear compact and holomorphic operator defined in an open sphere centered at the origin of h1 the bar in the number means complex conjugation of and e1 is a known element in h1 the existence of a solution to the above operator equation means that there exists an analytic function f z n 1x bn zn f 0 0 which together with the first two derivatives converges absolutely for z 1 it is well known that such solutions exist under suitable assumptions on g z f z for sufficiently small here the existence of such analytic solutions is proved under more general assumptions on g z f z moreover for a given sufficiently small the theorems predict an upper bound of the solution f z for all z 1 the most important is that the existence theorems have a constructive character and can provide an answer to the question how small is the initial point f 0 as examples some equations of particular interest are studied

contact processes with random recovery rates and edge weights on complete graphs in this paper we are concerned with the contact process with random recovery rates and edge weights on the complete graph with n vertices we show that the model has a critical value which is inversely proportional to the product of the mean of the edge weight and the mean of the inverse of the recovery rate in the subcritical case the process dies out before an amount of time with order o log n with high probability as n rightarrow infty in the supercritical case the process survives for an amount of time with order exp o n with high probability as n rightarrow infty

on the one and two dimensional toda lattices and the painleve property the toda lattice and the two dimensional toda lattice 2 dtl are shown to possess a type of painleve property that is based on the use of separate singular manifolds for each dependent variable the isospectral problem for the 2 dtl found by both mikhailov and by fordy and gibbons can be simply and logically derived from this analysis some remarks are made about the connection between our work and independent work of kametaka and airhault on the relationship between the toda lattice and the second painleve transcendent

adaptive walks and extreme value theory we study biological evolution in a high dimensional genotype space in the regime of rare mutations and strong selection the population performs an uphill walk which terminates at local fitness maxima assigning fitness randomly to genotypes we show that the mean walk length is logarithmic in the number of initially available beneficial mutations with a prefactor determined by the tail of the fitness distribution this result is derived analytically in a simplified setting where the mutational neighborhood is fixed during the adaptive process and confirmed by numerical simulations

on the integrability of nonlinear dirac equations the integrability of nonlinear dirac equations is discussed applying recent results in soliton theory using the lie point transformation groups of the nonlinear dirac equations we reduce these partial differential equations to systems of ordinary differential equations and study whether these systems are integrable we also discuss whether lie backlund vector fields exist we conclude that the nonlinear dirac equations are not integrable

the algebraic complete integrability of geodesic flow on so n we study for which left invariant diagonal metrics onso n the euler arnold equations dot x x lambda x x x ij in so n lambda ij x ij lambda ij lambda ji can be linearized on an abelian variety i e are solvable by quadratures we show that merely by requiring that the solutions of the differential equations be single valued functions of complex timet suffices to prove that under a non degeneracy assumption on the metric the only such metrics are those which satisfy manakov s conditions ij bi bj ai aj 1 the case of degenerate metrics is also analyzed forn 4 this provides a new and simpler proof of a result of adler and van moerbeke 3

universality classes of interaction structures for nk fitness landscapes kauffman s nk model is a paradigmatic example of a class of stochastic models of genotypic fitness landscapes that aim to capture generic features of epistatic interactions in multilocus systems genotypes are represented as sequences of l binary loci the fitness assigned to a genotype is a sum of contributions each of which is a random function defined on a subset of k le l loci these subsets or neighborhoods determine the genetic interactions of the model whereas earlier work on the nk model suggested that most of its properties are robust with regard to the choice of neighborhoods recent work has revealed an important and sometimes counter intuitive influence of the interaction structure on the properties of nk fitness landscapes here we review these developments and present new results concerning the number of local fitness maxima and the statistics of selectively accessible that is fitness monotonic mutational pathways in particular we develop a unified framework for computing the exponential growth rate of the expected number of local fitness maxima as a function of l and identify two different universality classes of interaction structures that display different asymptotics of this quantity for large k moreover we show that the probability that the fitness landscape can be traversed along an accessible path decreases exponentially in l for a large class of interaction structures that we characterize as locally bounded finally we discuss the impact of the nk interaction structures on the dynamics of evolution using adaptive walk models

mirror transformations of hamiltonian systems we demonstrate through various examples of hamiltonian systems that symplectic structures have been encoded into the painleve test each principal balance in the painleve test induces a mirror transformation that regularizes movable singularities moreover for finite dimensional hamiltonian systems the mirror transformations are canonical

a new class of exact solutions for coupled scalar field equations the authors report a class of exact solutions valid in new regimes of parameter values for a generic system of nonlinear coupled scalar field equations which is the stationary reduction of the schrodinger boussinesq or korteweg de vries system and which extends the henon heiles system to hamiltonians with indefinite kinetic energy the results are applied to co propagating coupled nonlinear waves in magnetized plasmas

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

complex hill s equation and the complex periodic korteweg de vries equations on etudie la classe spectrale des operateurs differentiels l d 2 dx 2 q avec un potentiel q de periode un on traite des potentiels complexes et des operateurs auto adjoints

nonlinear evolution equations of hamiltonian form and the painleve analysis a number of necessary conditions for a class of nonlinear partial differential equations to pass the painleve test with the kruskal ansatz is given a theory for the admissible resonance patterns of equations in hamiltonian form is developed and discussed for some important special cases based on these results on algorithm can be described which constructs all nonlinear evolution equations of normal type and a certain hamiltonian form

non integrability of the truncated toda lattice hamiltonian at any order two sufficient conditions for the non existence of an additional analytic integral are given for hamiltonian systems with non homogeneous polynomial potential of an arbitrary degree an application is made to the truncated three particle toda lattice which is proved to be non integrable at any order

punctuated evolution for the quasispecies model abstract biological evolution in a sequence space with random fitnesses is studied within eigen s quasispecies model a strong selection limit is employed in which the population resides at a single sequence at all times evolutionary trajectories start at a randomly chosen sequence and proceed to the global fitness maximum through a small number of intermittent jumps the distribution of the total evolution time displays a universal power law tail with exponent 2 simulations show that the evolutionary dynamics is very well represented by a simplified shell model in which the sub populations at local fitness maxima grow independently the shell model allows for highly efficient simulations and provides a simple geometric picture of the evolutionary trajectories

singularity clustering in the duffing oscillator an asymptotic and numerical study is made of the singularity structure in the complex t plane of the duffing oscillator the presence of logarithmic terms in the local psi series expansion of the form t4 ln t leads to a multisheeted singularity structure of great complexity this structure is built recursively from an elemental pattern which takes the form of four armed stars of singularities this construction is deduced analytically from the properties of the mapping z t4 ln t and is confirmed quite accurately numerically a systematic resummation of the psi series in terms of lame functions is developed this series exhibits the same analytic structure at all orders and provides a semi local analytical representation of the solution which is apparently valid even in the chaotic regime

elliptic solutions and blow up in an integrable henon heiles system we consider an integrable case of the henon heiles system and use an isomorphism with the two gap kdv flow to construct families of real elliptic trajectories which are associated with two gap elliptic solitons of the kdv equation some of these solutions exhibit blow up in finite time

the euler equations on lie algebra so 4 an elementary approach to integrability condition we give a new derivation of the manakov and the product case integrability conditions of the euler equations on lie algebra so 4 fourth first integral functionally independent of the three already known integrals is obtained explicitly for all values of the parameters by an algorithmic approach

the he non heiles system revisited abstract the known integrable cases of the henon heiles system are shown to be closely related to the stationary flows of the known and only integrable fifth order single component and polynomial nonlinear evolution equations this is further evidence that these are the only integrable cases of the henon heiles system lax pairs are deduced for each of the integrable cases and used to construct the constants of motion a curious lax operator recently found by the painleve method is explained

generalized potlatch and smoothing processes we consider simple generalizations of the potlatch and smoothing processes which were introduced in 8 and studied in 5 these generalizations provide relatively simple examples of infinite interacting systems which exhibit phase transition the original potlatch and smoothing processes do not exhibit phase transition our results show that for the generalized processes phase transition does not usually occur in one or two dimensions but usually does occur in higher dimensions upper and lower bounds for the relevant critical values are obtained as one application of our results we obtain the limiting behavior of the critical values for the linear contact process in d dimensions as d thus answering a question we raised in 3 this is done by comparing the contact process with an appropriate generalized smoothing process

generalizations of the neumann system a curve theoretical approach part iii order n systems the neumann system is a well known algebraically completely integrable hamiltonian system its geometry has roots in hyperelliptic curve theory and the isospectral deformation theory of hill s operator in this paper generalizations of the neumann system are found for n sheeted riemann surfaces and the isospectral deformation theory of operators of order n trace formulas lax pairs and constants of motion are found the new systems are shown to be algebraically completely integrable

prym varieties and the geodesic flow onso n on etudie la relation entre l espace ou le flot apparait et la variete abelienne sur laquelle le flot se linearise

on the non integrability of some generalized toda lattices abstract we consider a one parameter family of two degrees of freedom generalized toda hamiltonians using ziglin s theorem with some extensions we prove the non existence of an additional single valued analytic integral except for special values of the parameter

non holonomic geodesic flows on lie groups and the integrable suslov problem on so 4 we study geodesic flows on lie groups with the left invariant non holonomic constraint in the case of the existence of an invariant measure we find new integrable non hamiltonian systems on so 4 and other six dimensional lie groups

evolutionary accessibility of modular fitness landscapes a fitness landscape is a mapping from the space of genetic sequences which is modeled here as a binary hypercube of dimension l to the real numbers we consider random models of fitness landscapes where fitness values are assigned according to some probabilistic rule and study the statistical properties of pathways to the global fitness maximum along which fitness increases monotonically such paths are important for evolution because they are the only ones that are accessible to an adapting population when mutations occur at a low rate the focus of this work is on the block model introduced by a s perelson and c a macken proc natl acad sci usa 92 9657 1995 where the genome is decomposed into disjoint sets of loci modules that contribute independently to fitness and fitness values within blocks are assigned at random we show that the number of accessible paths can be written as a product of the path numbers within the blocks which provides a detailed analytic description of the path statistics the block model can be viewed as a special case of kauffman s nk model and we compare the analytic results to simulations of the nk model with different genetic architectures we find that the mean number of accessible paths in the different versions of the model are quite similar but the distribution of the path number is qualitatively different in the block model due to its multiplicative structure a similar statement applies to the number of local fitness maxima in the nk models which has been studied extensively in previous works the overall evolutionary accessibility of the landscape as quantified by the probability to find at least one accessible path to the global maximum is dramatically lowered by the modular structure

on the singularity analysis of intersecting separatrices in near integrable dynamical systems abstract it has been recently proved by s l ziglin that transversal intersections of separatrices invariant manifolds in near integrable hamiltonian systems of two degrees of freedom imply the existence of multi valued solutions with infinitely many riemann sheets ziglin s theorem is illustrated here on a simple example and then extended and applied to non hamiltonian analytic flows x f x eg x t with x x y and g x t g x t 2 which for e 0 possess the painleve property on the other hand the theoretically expected logarithmic singularities for e 0 are obtained explicitly in solutions near the intersecting separatrices thus we conjecture that dynamical systems with the painleve property can have no separatrix intersections and hence no strange attractors etc these singularities are then numerically located and found to form certain very interesting chimney patterns in the complex t plane on which they accumulate densely the upper part of the chimneys away from the re t axis is asymptotically quite insensitive to changes in parameter values or initial conditions the singularity pattern itself however becomes denser and each chimney is seen to gradually collapse towards the re t axis as the amplitude of the driving term increases and the motion becomes more chaotic

existence of exponentially unstable periodic solutions and the non integrability of homogeneous hamiltonian systems abstract in hamiltonian systems with polynomial potential of homogeneous degree characteristic multipliers floquet multipliers permit an explicit expression for straight line periodic solutions by use of ziglin s theorem it is shown that the existence of an exponentially unstable straight line periodic solution implies the non integrability of the system

integration of a generalized henon heiles hamiltonian the generalized henon heiles hamiltonian h 1 2 px2 py2 c1x2 c2y2 axy2 bx3 3 with an additional nonpolynomial term y 2 is known to be liouville integrable for three sets of values of b a c1 c2 it has been previously integrated by genus two theta functions only in one of these cases defining the separating variables of the hamilton jacobi equations we succeed here in the two other cases to integrate the equations of motion with hyperelliptic functions

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

analytic structure of the henon heiles hamiltonian in integrable and nonintegrable regimes the solutions of the henon heiles hamiltonian are investigated in the complex time plane the use of the painleve property i e the property that the only movable singularities exhibited by the solution are poles enables successful prediction of the values of the nonlinear coupling parameter for which the system is integrable special attention is paid to the structure of the natural boundaries that are found in some of the nonintegrable regimes these boundaries have a remarkable self similar structure whose form changes as a function of the nonlinear coupling

critical value for the contact process with random recovery rates and edge weights on regular tree abstract in this paper we are concerned with contact processes with random recovery rates and edge weights on rooted regular trees t n let and be two nonnegative random variables such that p m 1 for some m 0 for each vertex x on t n x is an independent copy of while for each edge e on t n e is an independent copy of an infected vertex x becomes healthy at rate x while an infected vertex y infects an healthy neighbor z at rate proportional to y z for this model we prove that the critical value under the annealed measure approximately equals n e e 1 1 as n grows to infinity furthermore we show that the critical value under the quenched measure equals that under the annealed measure when the cluster containing the root formed with edges with positive weights is infinite

non perturbative non integrability of non homogeneous nonlinear lattices induced by non resonance hypothesis abstract we succeeded in applying ziglin s test on non integribility to non homogeneous nonlinear lattices fermi pasta ulam fpu lattices by explicit calculations of the eigenvalues of the monodromy matrices concerning the normal variational equations of lame type and with the use of the non resonance hypothesis about the eigenvalues we obtained a theorem proving the non existence of additional analytic conserved quantities other than the hamiltonian itself for the fpu lattices in the low energy limit furthermore after introducing a concept of degree of non integrability we investigated the classification of non homogeneous nonlinear lattices using a transformation r from a non homogeneous nonlinear lattice to another non homogeneous nonlinear lattice which preserves their degree of non integrability

integrating the nonintegrable analytic structure of the lorenz system revisited abstract a study of the complex time analytic structure of the lorenz system in nonintegrable parameter regimes reveals the special sets of parameter values for which one time dependent integral of motion exists furthermore the analysis yields the exact form of the part of the integral with the highest homogeneous weight and a method to construct the rest of the integral recursive clustering of singularities in the chaotic regimes of the system is observed in computer studies and explained by a simple analytic argument the analytic techniques used in these studies a systematic resummation of a logarithmic psi series appears to be quite general and can provide explicit representations of a solution even in the chaotic regimes in the neighborhood of a given movable singularity furthermore we suggest that this technique provides a type of renormalization program to study a wide class of nonintegrable systems

hamiltonian maps in the complex plane abstract the standard map is a nonintegrable discrete time analog of the vertical pendulum detailed calculations are presented and illustrated graphically for the standard map at the golden mean frequency the functional dependence of the coordinate q on the canonical angle variable is analtyically continued into the complex plane where natural boundaries are found at constant absolute values of im the boundaries represent the appearance of chaotic motion in the complex plane when the domain of analyticity shrinks to zero the kam invariant curve is destroyed two independent numerical methods with fourier analysis in the angle variable were used one based on a variation annihilation method and the other on a double expansion the results were further checked by direct solution of the complex equations of motion the numerically simpler but intrinsically complex semipendulum and semistandard map are also studied we conjecture that natural boundaries appear in the analogous analytic continuation of the invariant tori or kam surfaces of general nonintegrable systems with analytic hamiltonians

an exact solution for travelling waves of ut duxx u uk abstract an exact analytical solution for travelling waves of the fisher equation with a general nonlinearity is found the boundary values the boundedness and the stability of the solution are discussed the reduced ordinary differential equation of the equation of our study is investigated for the painleve property

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

painleve analysis and integrability of coupled non linear schrodinger equations considering a system of coupled non linear schrodinger nls equations the authors discuss the integrability properties through painleve p analysis for the two coupled nls equations they show that there exists a pair of parametric values possessing the p property for which the associated backlund transformation bt and the hirota bilinearisation are constructed these parametric choices are identical to those of zakharov and schulman 1982 who established the integrability in terms of motion invariants finally they extend the p analysis to the n coupled nls system and identify two parametric choices possessing the p property which are natural generalisations of the two coupled nls cases

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

homoclinic and heteroclinic orbits of reversible vectorfields under perturbation on demontre l existence de liens homocline heterocline entre les orbites periodiques hyperboliques d une classe de systemes hamiltoniens admettant un nombre suffisant de symetries sous une reduction isoenergetique le probleme se reduit a un probleme comprenant des champs vectoriels reversibles avec perturbations dependantes du temps admettant les memes symetries de renversement

geometrical aspects of ziglin s non integrability theorem for complex hamiltonian systems on place la methode de yoshida dans un cadre geometrique conceptuel et on etend son applicabilite

a criterion for the non existence of an additional integral in hamiltonian systems with a homogeneous potential abstract a non integrability criterion on the basis of ziglin s theorem is given for two degrees of freedom hamiltonians with a homogeneous potential of integer degree as simple examples the anisotropic kepler problem and the one dimensional newtonian three body problem are proved to be non integrable a self contained proof of ziglin s theorem n dimensional homogeneous potential version is also given

critical value for contact processes on clusters of oriented bond percolation in this paper we are concerned with contact processes on open clusters of oriented bond percolation in zd where the disease spreads along the direction of open edges we show that the two critical values in the quenched and annealed cases are equal with probability one and are asymptotically equal to dp 1 as the dimension d grows to infinity where p is the probability that the edge is open

δ exceedance records and random adaptive walks we study a modified record process where the kth record in a series of independent and identically distributed random variables is defined recursively through the condition with a deterministic sequence called the handicap for constant and exponentially distributed random variables it has been shown in previous work that the process displays a phase transition as a function of between a normal phase where the mean record value increases indefinitely and a stationary phase where the mean record value remains bounded and a finite fraction of all entries are records park et al 2015 phys rev e 91 042707 here we explore the behavior for general probability distributions and decreasing and increasing sequences focusing in particular on the case when matches the typical spacing between subsequent records in the underlying simple record process without handicap we find that a continuous phase transition occurs only in the exponential case but a novel kind of first order transition emerges when is increasing the problem is partly motivated by the dynamics of evolutionary adaptation in biological fitness landscapes where corresponds to the change of the deterministic fitness component after k mutational steps the results for the record process are used to compute the mean number of steps that a population performs in such a landscape before being trapped at a local fitness maximum

lie symmetries of the lorenz model abstract we study the generalized symmetries of the lorenz model to find the parameter values at which one or more time dependent integrals of motion exist in these cases the integrals are found trivially from the symmetries themselves a complete study of the one completely algebraically integrable case shows a the dynamics can be integrated exactly by reducing it first to a lower dimensional system b the symmetry vector field is hamiltonian these facts hold for other dissipative completely integrable dynamical systems as well the analytic study of a natural two form reveals that it is an entire function of time the foliation of phase space induced by the two form for the partially integrable cases has a simple description in terms of the coefficients occurring in the laurent series expansions of the dependent variables

on the complete and partial integrability of non hamiltonian systems abstract the methods of singularity analysis are applied to several third order non hamiltonian systems of physical significance including the lotka volterra equations the three wave interaction and the rikitake dynamo model complete integrability is defined and new completely integrable systems are discovered by means of the painleve property in all these cases we obtain integrals which reduce the equations either to a final quadrature or to an irreducible second order ordinary differential equation ode solved by painleve transcendents relaxing the painleve property we find many partially integrable cases whose movable singularities are poles at leading order with in t t 0 terms entering at higher orders in an n th order generalized rossler model a precise relation is established between the partial fulfillment of the painleve conditions and the existence of n 2 integrals of the motion

integrability partial integrability and nonintegrability for systems of ordinary differential equations the integrability of systems of ordinary differential equations with polynomial vector fields is investigated by using the singularity analysis methods three types of results are obtained first a general relationship between the degrees of first integrals and the so called kowalevskaya exponents is derived second it is shown that all solutions of algebraically integrable systems can be expanded in puiseux series third a new method to study partially integrable systems is studied these different aspects allow us to study algorithmically the integrability partial integrability and nonintegrability of differential systems

a multiple scales approach to maximal superintegrability in this paper we present a simple algorithmic test to establish if a hamiltonian system is maximally superintegrable or not this test is based on a very simple corollary of a theorem due to nekhoroshev and on a perturbative technique called multiple scales method if the outcome is positive this test can be used to suggest maximal superintegrability whereas when the outcome is negative it can be used to disprove it this method can be regarded as a finite dimensional analog of the multiple scales method as a way to produce soliton equations we use this technique to show that the real counterpart of a mechanical system found by jules drach in 1935 is in general not maximally superintegrable we give some hints on how this approach could be applied to classify maximally superintegrable systems by presenting a direct proof of the well known bertrand s theorem

the discrete self trapping equation and the painleve property from the discrete self trapping dst equation for three degrees of freedom rewritten in terms of density matrix one can derive an autonomous system of real first order ordinary differential equations by using su 3 notation performing a painleve analysis it is found that depending on the parameters of the system integrable and non integrable cases can be distinguished

trouble with gegenbauer reconstruction for defeating gibbs phenomenon runge phenomenon in the diagonal limit of gegenbauer polynomial approximations to defeat gibbs phenomenon in fourier and chebyshev series gottlieb et al d gottlieb c w shu a solomonoff h vandeven on the gibbs phenomenon i recovering exponential accuracy from the fourier partial sum of a nonperiodic analytic function j comput appl math 43 1992 81 98 developed a gegenbauer reconstruction the partial sums of the fourier or other spectral series are reexpanded as a series of gegenbauer polynomials c n m x recovering spectral accuracy even in the presence of shock waves or other discontinuities to achieve a rate of convergence which is exponential in n however gegenbauer reconstruction requires increasing the order m of the polynomials linearly with the truncation n of the series m bn for some constant b 0 when the order m is fixed it is well known that the gegenbauer series converges as n everywhere on x 1 1 if f x the function being expanded is analytic on the interval but what happens in the diagonal limit in which m n tend to infinity simultaneously we show that singularities of f x off the real axis can destroy convergence of this diagonal approximation process in the sense that the error diverges for subintervals of x 1 1 gegenbauer reconstruction must therefore be constrained to use a sufficiently small ratio of order m to truncation n this off axis singularity constraint is likely to impair the effectiveness of the reconstruction in some applications

phase transition for the large dimensional contact process with random recovery rates on open clusters in this paper we are concerned with the contact process with random recovery rates on open clusters of bond percolation on mathbb z d let xi be a random variable such that p xi ge 1 1 which ensures mathrm e frac 1 xi infty then we assign i i d copies of xi on the vertices as the random recovery rates assuming that each edge is open with probability p and the infection can only spread through the open edges then we obtain that begin aligned limsup d rightarrow infty lambda d le lambda c frac 1 p mathrm e frac 1 xi end aligned where lambda d is the critical value of the process on mathbb z d i e the maximum of the infection rates with which the infection dies out with probability one when only the origin is infected at t 0 to prove the above main result we show that the following phase transition occurs assuming that lceil log d rceil vertices are infected at t 0 where these vertices can be located anywhere then when the infection rate lambda lambda c the process survives with high probability as d rightarrow infty while when lambda lambda c the process dies out at time o log d with high probability

the speed of evolution in large asexual populations we consider an asexual biological population of constant size n evolving in discrete time under the influence of selection and mutation beneficial mutations appear at rate u and their selective effects s are drawn from a distribution g s after introducing the required models and concepts of mathematical population genetics we review different approaches to computing the speed of logarithmic fitness increase as a function of n u and g s we present an exact solution of the infinite population size limit and provide an estimate of the population size beyond which it is valid we then discuss approximate approaches to the finite population problem distinguishing between the case of a single selection coefficient g s s s b and a continuous distribution of selection coefficients analytic estimates for the speed are compared to numerical simulations up to population sizes of order 10300

singularities of the complex korteweg de vries flows on etudie le probleme isospectral de l equation de hill complexe etant donne 2g 1 valeurs propres simples dans le plan complexe trouver la classe spectrale correspondante q on considere la caracterisation de la classe isospectrale et ses sections singulieres on construit q a l aide des flots non lineaires de korteweg de vries dans c 1 les fonctions regulieres de periode 1

separability and lax pairs for henon heiles system the hamiltonian system corresponding to the generalized henon heiles hamiltonian h 1 2 px2 py2 1 2ax2 1 2by2 x2y ey3 is known to be integrable in the following three cases a b e 1 3 e 2 b 16a e 16 3 in the first two the system has been integrated by making use of genus one and genus two theta functions we show that in the third case the system can also be integrated by making use of elliptic functions finally using the fairbanks theorem we find lax pairs for each of the three integrable systems under consideration

painleve conjecture revisited the discovery of new integrable two dimensional hamiltonian systems is reported the analytic structure of the solutions makes necessary the generalization of the painlev e conjecture a widely used integrability criterion such a generalization is presented which the authors believe should replace the usual conjecture for two dimensional hamiltonian systems it is indeed compatible with all the systems already found and in addition leads to still new integrable cases

on the structure of picard fuchs type equations for liouville arnold integrable hamiltonian system on cotangent phase spaces there are studied in detail the structure properties of integral submanifold imbedding mapping for a class of algebraically liouville integrable hamiltonian systems on cotangent phase spaces and related with it so called picard fuchs type equations it is shown that these equations can be in general regularly constructed making use of a given a priori system of involutive invariants and proved that their solutions in the hamolton jacobi separable variable case give rise exactly to the integral submanifold imbedding mapping being as known a main ingredient for liouville arnold integrability by quadratures of the hamiltonian system under regard

bifurcations of invariant manifolds in the generalized henon heiles system abstract the generalized henon heiles hamiltonian h 1 2 q dot 1 2 q dot 2 2 aq 1 2 bq 2 2 q 1 2 q 2 e 3 q 2 3 in the integrable case e 6 is studied from topological point of view the topology of the real level sets for all generic values of the constants of the motion and all generic bifurcations of the liouville tori and cylinders are described by making use of the reach algebraic structure of the problem

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