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

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

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

flexible regression approach to propensity score analysis and its relationship with matching and weighting in propensity score analysis the frequently used regression adjustment involves regressing the outcome on the estimated propensity score and treatment indicator this approach can be highly efficient when model assumptions are valid but can lead to biased results when the assumptions are violated we extend the simple regression adjustment to a varying coefficient regression model that allows for nonlinear association between outcome and propensity score we discuss its connection with some propensity score matching and weighting methods and show that the proposed analytical framework can shed light on the intrinsic connection among some mainstream propensity score approaches stratification regression kernel matching and inverse probability weighting and handle commonly used causal estimands we derive analytic point and variance estimators that properly take into account the sampling variability in the estimated propensity score extensive simulations show that the proposed approach possesses desired finite sample properties and demonstrates competitive performance in comparison with other methods estimating the same causal estimand the proposed methodology is illustrated with a study on right heart catheterization

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

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

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

an iterative minimization formulation for saddle point search this paper proposes and analyzes an iterative minimization formulation for searching index 1 saddle points of an energy function we give a general and rigorous description of eigenvector following

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

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

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

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 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

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

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

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

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

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

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

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

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