a non existence result for low energy sign changing solutions of the brezis nirenberg problem in dimensions 4 5 and 6 abstract we consider the brezis niremberg problem p e u u p 1 u e u in u 0 on where is a smooth bounded domain in r n n 4 5 6 p 1 2 n n 2 is the critical sobolev exponent and e is a positive parameter the main result of the paper generalizes the result of a iacopetti and f pacella 10 precisely we show that there are no low energy sign changing solutions u e with max u e min u e 0 or as e goes to zero

understanding complex dynamics by means of an associated riemann surface we provide an example of how the complex dynamics of a recently introduced model can be understood via a detailed analysis of its associated riemann surface thanks to this geometric description an explicit formula for the period of the orbits can be derived which is shown to depend on the initial data and the continued fraction expansion of a simple ratio of the coupling constants of the problem for rational values of this ratio and generic values of the initial data all orbits are periodic and the system is isochronous for irrational values of the ratio there exist periodic and quasi periodic orbits for different initial data moreover the dependence of the period on the initial data shows a rich behavior and initial data can always be found with arbitrarily large periods

a direct approach to finding exact invariants for one dimensional time dependent classical hamiltonians for a classical hamiltonian h 1 2 p2 v q t with an arbitrary time dependent potential v q t exact invariants that can be expressed as series in positive powers of p i q p t n 0pnfn q t are examined the method is based on direct use of the equation di dt i t i h 0 a recursion relation for the coefficients fn q t is obtained all potentials that admit an invariant quadratic in p are found and for those potentials all invariants quadratic in p are determined the feasibility of extending the analysis to find invariants that are polynomials in p of higher degree than quadratic is discussed the systems for which invariants quadratic in p have been found are transformed to autonomous systems by a canonical transformation

dynamical singularities for complex initial conditions and the motion at a real separatrix this work investigates singularities occurring at finite real times in the classical dynamics of one dimensional double well systems with complex initial conditions the objective is to understand the relationship between these singularities and the behavior of the systems for real initial conditions an analytical treatment establishes that the dynamics of a quartic double well system possesses a doubly infinite sequence of singularities these are associated with initial conditions that converge to those for the real separatrix as the singularity time becomes infinite this confluence of singularities is shown to lead to the unstable behavior that characterizes the real motion at the separatrix numerical calculations confirm the existence of a large number of singularities converging to the separatrix for this and two additional double well systems the approach of singularities to the real axis is of particular interest since such behavior has been related to the formation of chaos in nonintegrable systems the properties of the singular trajectories which cause this convergence to the separatrix are identified the hyperbolic fixed point corresponding to the potential energy maximum responsible for the characteristic motion at a separatrix also plays a critical role in the formation of the complex singularities by delaying trajectories and then deflecting them into asymptotic regions of space from where they are directly repelled to infinity in a finite time

similarity reductions from extended painleve expansions for nonintegrable evolution equations abstract the use of singular manifold expansions to find exact solutions to nonintegrable evolution equations is extended to include arbitrary resonance coefficients in such a way as to make the resulting infinite series exactly resummable the technique involves the use of a rescaling ansatz analogous to that used to analyze the psi series of nonintegrable ordinary differential equations the result is a similarity reduction of the equation in which the constrained singular manifold plays the role of a similarity variable the method is capable of yielding new solutions corresponding to either classical or nonclassical conditional lie symmetries

dynamical systems on infinitely sheeted riemann surfaces this paper is part of a program that aims to understand the connection between the emergence of chaotic behaviour in dynamical systems in relation with the multi valuedness of the solutions as functions of complex time in this work we consider a family of systems whose solutions can be expressed as the inversion of a single hyperelliptic integral the associated riemann surface r c is known to be an infinitely sheeted covering of the complex time plane ramified at an infinite set of points whose projection in the plane is dense the main novelty of this paper is that the geometrical structure of these infinitely sheeted riemann surfaces is described in great detail which allows to study global properties of the flow such as asymptotic behaviour of the solutions periodic orbits and their stability or sensitive dependence on initial conditions the results are then compared with a numerical integration of the equations of motion following the recent approach of calogero the real time trajectories of the system are given by paths on r that are projected to a circle on the complex plane due to the branching of r the solutions may have different periods or may not be periodic at all

li yau type estimates for a nonlinear parabolic equation on complete manifolds abstract let m g be a complete noncompact riemannian manifold with the m dimensional bakry emery ricci curvature bounded below in this paper we give a local li yau type gradient estimate for the positive solutions to a general nonlinear parabolic equation u t u u a u log u q u in m 0 where a r is a c 2 smooth function and q q x t is a function which generalizes many previous well known gradient estimate results

on the connection between hyperelliptic separability and painleve integrability we consider systems of odes which are associated with some physically significant examples shallow water equilibrium solutions travelling waves of the harry dym equation a lotka volterra system of competing species and the geodesic flow on the triaxial ellipsoid the first three are shown to share the following properties i they are hyperelliptically separable systems hss and after a suitable nonlinear time transformation become algebraically completely integrable aci and ii they are of the weak painleve type and become full painleve after the application of this transformation the geodesic flow on the other hand although it passes the usual painleve test does not possess a full set of free constants and thus one may not conclude whether it has the painleve property or not this system is also hss and becomes aci after the application of a suitable nonlinear time transformation we also combine our geometric analytical investigation with a numerical analysis of the system in the complex plane and show that there is perfect correspondence between the results of the two approaches this correspondence strengthens the reliability of such numerical studies and helps us better understand their implication in cases where such nonlinear transformations to complete integrability are not available

on the non integrability of a family of duffing van der pol oscillators we investigate the non integrability of a family of duffing van der pol oscillators x alpha x x2 1 x beta x3 gamma cos omega t by studying the analytic properties of the dynamics in complex time we find that the solutions of have no worse than algebraic singularities at t with only t t 1 2 terms present in their series expansions unlike for example the alpha 0 duffing case where typically log t t terms arise still when integrating around long enough contours a remarkably intricate pattern of square root singularities emerges on different sheets which appears to prevent solutions from ever returning to the original sheet such evidence of infinitely sheeted solutions termed the iss property has also been observed in a number of hamiltonian systems and is illustrated here on a simple example of a single first order differential equation we suggest that the iss property is a necessary condition for non integrability i e non existence of a complete set of analytic single valued constants of the motion which would permit the complete integration of a dynamical system in terms of quadratures

integrability and the painleve property for low dimensional systems we examine a system described by two first order nonlinear differential equations from the point of view of integrability the singularity analysis in the complex time plane is used to investigate the painleve property which according to the ablowitz ramani segur conjecture is a prerequisite for integrability for infinite dimensional systems we show that for such low dimensional systems the painleve analysis is still a most useful guide but integrable cases also exist which do not possess the painleve property

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 new class of integrable systems we present a family of dynamical systems associated with the motion of a particle in two space dimensions these systems possess a second integral of motion quadratic in velocities apart from the hamiltonian and are thus completely integrable they were found through the derivation and subsequent resolution of the integrability condition in the form of a partial differential equation pde for the potential a most important point is that the same pde was derived through considerations on the analytic structure of the singularities of the solutions weak painleve property

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

normal form solutions of dynamical systems in the basin of attraction of their fixed points abstract the normal form theory of poincare siegel and arnol d is applied to an analytically solvable lotka volterra system in the plane and a periodically forced dissipative duffing s equation with chaotic orbits in its 3 dimensional phase space for the planar model we determine exactly how the convergence region of normal forms about a nodal fixed point is limited by the presence of singularities of the solutions in the complex t plane despite such limitations however we show in the case of a periodically driven system that normal forms can be used to obtain useful estimates of the basin of attraction of a stable fixed point of the poincare map whose boundary is formed by the intersecting invariant manifolds of a second hyperbolic fixed point nearby

exact solutions of the multidimensional derivative nonlinear schrodinger equation for many body systems of criticality the authors investigate the problem of strongly interacting many body systems near criticality as recently described in terms of nonlinear dynamics by dixon and tuszynski 1989 they show that in the first order of approximation the equation of motion for the order parameter can be mapped on the derivative nonlinear schrodinger equation and thus can be solved exactly they solve the equations both in the absence and presence of current densities

period doubling bifurcations chaos phase locking and devil s staircase in a bonhoeffer van der pol oscillator abstract the effect of a periodic input current a 1 cos t in the bonhoeffer van der pol oscillator along with a bias a 0 is investigated numerically as the parameter a 1 is varied in the absence of bias by holding the other parameters at constant values typical period doubling bifurcation sequence is found to occur leading to chaotic motion in agreement with the feigenbaum scenario when the bias is switched on at the transition to chaos frequency locking is observed in the system the frequency locked intervals exhibit complete devil s staircase similar to the one observed in belousov zhabotinsky reaction

on the parabolic kernel of the schrodinger operator etude des equations paraboliques du type q x t t u x t 0 sur une variete riemannienne generale introduction estimations de gradients inegalites de harnack majorations et minorations des solutions fondamentales equation de la chaleur et noyau de green operateur de schrodinger

on the analytic structure of the driven pendulum the analytic structure of the solution of the driven pendulum is investigated through painleve analysis in the complex time plane the existence is pointed out of a two armed infinite sheeted riemann structure of the singularities after an exponential transformation

investigating non integrability and chaos in complex time the question of integrability of systems of ordinary differential equations odes has been studied extensively for centuries recently however owing to an upsurge of interest in chaos a lot of studies have focused on the investigation of non integrable systems where chaos often manifests itself as extremely sensitive dependence on initial conditions of a dense set of real solutions in real time here we argue that non integrability can be efficiently investigated by solving odes in complex time and review recent results which strongly suggest that for a system of odes to be non integrable it is necessary that it possess a dense set of infinitely sheeted solutions in the complex domain

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

galoisian obstruction to the integrability of general dynamical systems abstract the galoisian approach to study the integrability of classical hamiltonian systems the so called morales ramis theory has been proved to be useful and powerful by many applications here two analogous forms of the morales ramis theory for general dynamical systems both in vector field and mapping forms are given galois groups of the corresponding variational equations are studied and some necessary conditions of the system to possess a certain number of integrals are presented several applications are given at last to illustrate our results

the role of the green s function in a non linear elliptic equation involving the critical sobolev exponent this paper is concerned with non linear elliptic problems of the type pe u u n 2 n 2 eu u 0 on u 0 on where is a smooth and bounded domain in rn n 4 and e 0 we show that if the ue are solutions of pe which concentrate around a point as e 0 then this point cannot be on the boundary of and is a critical point of the regular part of the green s function conversely we show that for n 5 and any non degenerate critical point x0 of the regular part of the green s function there exist solutions of pe concentrating around x0 as e 0

necessary and sufficient conditions for finite time singularities in ordinary differential equations abstract this paper gives a theorem which provides necessary and sufficient conditions for the existence of a real finite value of the independent variable at which the general solution of a certain class of ordinary differential equations diverges to infinity this class is a large subset of the set of all autonomous non linear polynomial ordinary differential equation these conditions involve the asymptotic form of the local series representation for the general solutions around the singularities and can be checked algorithmically

on exact solutions for damped anharmonic oscillators for arbitrary functions f1 f2 and f3 the anharmonic oscillator x f1 t x f2 t x f3 t x3 0 cannot be solved in closed form i e the general solution cannot be expressed as elliptic functions the authors apply the painleve test to obtain the constraint on the functions f1 f2 and f3 for which the equation passes the test the constraint on f1 f2 and f3 i e the differential equation which f1 f2 and f3 obey is discussed and solutions are given

integrability of hamiltonians with third and fourth degree polynomial potentials the weak painleve property as a criterion of integrability is applied to the case of simple hamiltonians describing the motion of a particle in two dimensional polynomial potentials of degree three and four this allows a complete identification of all the integrable cases of cubic potentials in the case of quartic potentials although our results are not exhaustive some new integrable cases are discovered in both cases the integrability is explicited by a direct calculation of the second integral of motion of the system

explicit integrability for hamiltonian systems and the painleve conjecture we analyze a class of hamiltonian systems in two dimensions for which we proved that a second constant of motion exists it is shown that using the two first integrals the equations of motion can be written in a form which allows their integration by quadratures an analysis of the equations of motion in this reduced form establishes the behavior of the solutions in the complex time plane it is shown explicitly that the systems belonging to this class possess the weak painleve property i e their solutions in complex time can present singularities of a specific algebraic type

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

mel nikov s function for two dimensional mappings this paper calculates explicitly the mel nikov function for general perturbations of two dimensional maps possessing a saddle connection this result may be considered as the discrete analogue of ziglin s calculation of the mel nikov function for time periodic perturbations of two dimensional continuous flows key words integrable planar maps perturbations homoclinic orbits mel nikov s function poisson summation formula

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

on the application of normal forms near attracting fixed points of dynamical systems abstract it is shown on an integrable example in the plane that normal form solutions need not converge over the full basin of attraction of fixed points of dissipative dynamical systems their convergence breaks down at a singularity in the complex time plane of the exact solutions of the problem however as is demonstrated on a nonintegrable example with 3 dimensional phase space the region of convergence of normal forms can be large enough to extend almost to a nearby hyperbolic fixed point whose invariant manifolds embrace the attracting fixed point forming a complicated basin boundary thus in such problems normal forms are shown to be useful in practice as a tool for finding large regions of initial conditions for which the solutions are necessarily attracted to the fixed point at t

new aspects of integrability of force free duffing van der pol oscillator and related nonlinear systems in this paper we show that the force free duffing van der pol oscillator is completely integrable for a specific parametric choice we derive a general solution for this parametric choice further we describe a procedure to construct the transformation which removes the time dependent part from the first integral and provide the general solution by quadrature the procedure is shown to have a wider applicability through additional examples we also show that through our method one can deduce linearizing transformations in a simple and straightforward way and illustrate it with a specific example

new similarity reductions and painleve analysis for the symmetric regularised long wave and modified benjamin bona mahoney equations the author discusses similarity reductions and painleve analysis for the symmetric regularised long wave and modified benjamin bona mahoney equations 1972 both of which arise in several physical applications including shallow water waves both equations are thought to non integrable i e not solvable by inverse scattering since numerical studies show that the interaction of solitary waves in inelastic in particular we determine some new similarity reductions of the symmetric regularised long wave equation these new similarity reductions are not obtainable using the classical lie group method for finding group invariant solutions of partial differential equations they are determined using a new and direct method which involves no group theoretical techniques it is shown that every similarity reduction of both the symmetric regularised long wave and modified benjamin bona mahoney equations obtained using the classical lie group method reduces the associated partial differential equation to an ordinary differential equation of painleve type whereas the new similarity solution of the symmetric regularised long wave equation reduces it to an ordinary differential equation which is not of painleve type it is also shown that neither the symmetric regularised long wave equation nor the modified benjamin bona mahoney equation possesses the painleve property for partial differential equations as defined by weiss et al 1983

the generalized damped cubic equation integrability and general solution the nonlinear ordinary differential equation of the form u f1u f2u f3u3 o representing a generalized damped cubic equation is fully analysed the author finds the general expression for f2 in terms of f1 and f3 that renders the equation integrable also he shows that if f2 satisfies the above mentioned condition the former nonlinear differential equation can always be reduced through adequate changes of variable to the second painleve transcendent

the analytic structure of dynamical systems and self similar natural boundaries abstract in this paper we investigate the analytic complex time structure of the movable singularities for several dynamical systems in general it is found that there exists a direct connection between the occurencce of a certain type of multiple valuedness of the singularities and the existence of a class of remarkable self similar natural boundaries for these systems an asymptotic description of the distribution of singularities in the natural boundary is developed this provides a description of the fine scale structure of these natural boundaries that agrees closely with the numerical calculations

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

a matrix li yau hamilton estimate for kahler ricci flow in this paper we prove a new matrix li yau hamilton lyh estimate for kahler ricci flow on manifolds with nonnegative bisectional curvature the form of this new lyh estimate is obtained by the interpolation consideration originated in ch by chow this new inequality is shown to be connected with perelman s entropy formula through a family of differential equalities in the rest of the paper we show several applications of this new estimate and its corresponding estimate for linear heat equation these include a sharp heat kernel comparison theorem generalizing the earlier result of li and tian lt a manifold version of stoll s theorem st on the characterization of algebraic divisors and a localized monotonicity formula for analytic subvarieties which sharpens the bishop volume comparison theorem motivated by the connection between the heat kernel estimate and the reduced volume monotonicity of perelman p we prove a sharp lower bound heat kernel estimate for the time dependent heat equation which is in a certain sense dual to perelman s monotonicity of the reduced volume as an application of this new monotonicity formula we show that the blow down limit of a certain type long time solution is a gradient expanding soliton we also illustrate the connection between the new lyh estimate and the hessian comparison theorem of fin on the forward reduced distance local monotonicity formulae on entropy and forward reduced volume are also derived

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

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

gradient estimates for the heat equation under the ricci flow the paper considers a manifold m evolving under the ricci flow and establishes a series of gradient estimates for positive solutions of the heat equation on m

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

interpolating between constrained li yau and chow hamilton harnack inequalities for a nonlinear parabolic equation abstract we establish a one parameter family of harnack inequalities connecting the constrained trace li yau differential harnack inequality for a nonlinear parabolic equation to the constrained trace chow hamilton harnack inequality for this nonlinear equation with respect to evolving metrics related to the ricci flow on a 2 dimensional closed manifold this result can be regarded as a nonlinear version of the previous work of y zheng and the author j y wu y zheng interpolating between constrained li yau and chow hamilton harnack inequalities on a surface arch math 94 2010 591 600

the singularity analysis for nearly integrable systems homoclinic intersections and local multivaluedness abstract in this study a new perturbative scheme for nonintegrable ordinary differential equations is proposed these perturbative expansions are based on the singularity analysis of the unperturbed system and is performed in the neighborhood of its singularities under suitable conditions on the homoclinic structure of the unperturbed system the melnikov vector can be computed based on the knowledge of the laurent expansions of the solutions the existence of transverse homoclinic intersections is therefore explicitly related to the existence of critical points for the solutions in the complex plane of the independent variable

gradient estimates for a simple elliptic equation on complete non compact riemannian manifolds in this paper we study the local gradient estimate for the positive solution to the following equation u aulog u bu 0i nm where a 0 b are real constants m is a complete non compact riemannian manifold our result is optimal in the sense when m g is a complete non compact expanding gradient ricci soliton by definition m g is called an expanding gradient ricci soliton if for some constant c 0 it satisfies that rc cg d 2 f

order and the ubiquitous occurrence of chaos abstract for a large class of ode s which includes the van der pol equation we determine analytically the asymptotic location of the singularities in the complex t plane by integrating these ode s numerically we show that if the singularities are dense which is the generic case the solution is chaotic in the sense of sensitive dependence on initial conditions in the exceptional case that the singularities are not dense the solution exhibits order taxis chaos is ubiquitous even for first order ode s in complex t

blow up and symmetry of sign changing solutions to some critical elliptic equations abstract in this paper we continue the analysis of the blow up of low energy sign changing solutions of semi linear elliptic equations with critical sobolev exponent started in m ben ayed k el mehdi f pacella blow up and nonexistence of sign changing solutions to the brezis nirenberg problem in dimension three ann inst h poincare anal non lineaire in press in addition we prove axial symmetry results for the same kind of solutions in a ball

a nonexistence result for sign changing solutions of the brezis nirenberg problem in low dimensions abstract we consider the brezis nirenberg problem u u u 2 2 u in u 0 on where is a smooth bounded domain in r n n 3 2 2 n n 2 is the critical sobolev exponent and 0 is a positive parameter the main result of the paper shows that if n 4 5 6 and is close to zero there are no sign changing solutions of the form u p u 1 p u 2 w where p u i is the projection on h 0 1 of the regular positive solution of the critical problem in r n centered at a point and w is a remainder term some additional results on norm estimates of w and about the concentrations speeds of tower of bubbles in higher dimensions are also presented

oscillatory finite time singularities in finance population and rupture abstract we present a simple two dimensional dynamical system where two nonlinear terms exerting respectively positive feedback and reversal compete to create a singularity in finite time decorated by accelerating oscillations the power law singularity results from the increasing growth rate the oscillations result from the restoring mechanism as a function of the order of the nonlinearity of the growth rate and of the restoring term a rich variety of behavior is documented analytically and numerically the dynamical behavior is traced back fundamentally to the self similar spiral structure of trajectories in phase space unfolding around an unstable spiral point at the origin the interplay between the restoring mechanism and the nonlinear growth rate leads to approximately log periodic oscillations with remarkable scaling properties three domains of applications are discussed 1 the stock market with a competition between nonlinear trend followers and nonlinear value investors 2 the world human population with a competition between a population dependent growth rate and a nonlinear dependence on a finite carrying capacity 3 the failure of a material subjected to a time varying stress with a competition between positive geometrical feedback on the damage variable and nonlinear healing

differential harnack estimates for backward heat equations with potentials under the ricci flow abstract in this paper we derive a general evolution formula for possible harnack quantities as a consequence we prove several differential harnack inequalities for positive solutions of backward heat type equations with potentials including the conjugate heat equation under the ricci flow we shall also derive perelman s harnack inequality for the fundamental solution of the conjugate heat equation under the ricci flow

the transition from regular to irregular motions explained as travel on riemann surfaces we introduce and discuss a simple hamiltonian dynamical system interpretable as a three body problem in the complex plane and providing the prototype of a mechanism explaining the transition from regular to irregular motions as travel on riemann surfaces the interest of this phenomenology illustrating the onset in a deterministic context of irregular motions is underlined by its generality suggesting its eventual relevance to understand natural phenomena and experimental investigations here only some of our main findings are reported without detailing their proofs a more complete presentation will be published elsewhere

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

global stability of a class of discontinuous dual billiards an infinite parameter family of discontinuous area preserving maps is studied using geometrical methods necessary and sufficient conditions are determined for the existence of some bounding invariant sets which guarantee global stability it is shown that under some additional constraints all orbits become periodic most of them lyapounov stable and with a maximal period in any bounded domain of phase space this yields a class of maps acting on a discrete phase space

invertible point transformations painleve analysis and anharmonic oscillators the techniques of an invertible point transformation and the painleve analysis can be used to construct integrable ordinary differential equations the authors compare both techniques for anharmonic oscillators

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

symplectic structures their backlund transformations and hereditary symmetries abstract it is shown that compatible symplectic structures lead in a natural way to hereditary symmetries we recall that a hereditary symmetry is an operator valued function which immediately yields a hierarchy of evolution equations each having infinitely many commuting symmetries all generated by this hereditary symmetry furthermore this hereditary symmetry usually describes completely the soliton structure and the conservation laws of these equations this result then provide us with a method for constructing hereditary symmetries and hence exactly solvable evolution equations in addition we show how symplectic structures transform under backlund transformations this leads to a method for generating a whole class of symplectic structures from a given one several examples and applications are given illustrating the above results also the connection of our results with those of gelfand and dikii and of magri is briefly pointed out

on the solvability of painleve ii and iv we introduce a rigorous methodology for studying the riemann hilbert problems associated with certain integrable nonlinear ordinary differential equations for concreteness we investigate the painleve ii and painleve iv equations we show that the cauchy problems for these equations admit in general global meromorphic in t solutions furthermore for special relations among the monodromy data and for t on stokes lines these solutions are bounded for finite t

newtonian dynamics in the plane corresponding to straight and cyclic motions on the hyperelliptic curve μ2 νn 1 n z ergodicity isochrony and fractals we study the complexification of the one dimensional newtonian particle in a monomial potential we discuss two classes of motions on the associated riemann surface the rectilinear and the cyclic motions corresponding to two different classes of real and autonomous newtonian dynamics in the plane the rectilinear motion has been studied in a number of papers while the cyclic motion is much less understood for small data the cyclic time trajectories lead to isochronous dynamics for bigger data the situation is quite complicated computer experiments show that for sufficiently small degree of the monomial the motion is generically isochronous with integer period which depends in a quite sensitive way on the initial data if the degree of the monomial is sufficiently high computer experiments show essentially chaotic behavior we suggest a possible theoretical explanation of these different behaviors we also introduce a two parameter family of two dimensional mappings describing the motion of the center of the circle as a convenient representation of the cyclic dynamics we call such a mapping the center map computer experiments for the center map show a typical multifractal behavior with periodicity islands therefore the above complexification procedure generates dynamics amenable to analytic treatment and possessing a high degree of complexity

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

a gradient estimate for all positive solutions of the conjugate heat equation under ricci flow abstract we establish a point wise gradient estimate for all positive solutions of the conjugate heat equation this contrasts to perelman s point wise gradient estimate which works mainly for the fundamental solution rather than all solutions like perelman s estimate the most general form of our gradient estimate does not require any curvature assumption moreover assuming only lower bound on the ricci curvature we also prove a localized gradient estimate similar to the li yau estimate for the linear schrodinger heat equation the main difference with the linear case is that no assumptions on the derivatives of the potential scalar curvature are needed a classical harnack inequality follows

asymptotic analysis of time singularities for a class of time dependent hamiltonians we analyse the local and global structure of time singularities for a class of quasi integrable hamiltonian systems in the arnold liouville sense we show that there is good agreement between the numerically observed local behaviour of the solutions and the perturbative scheme we produce using asymptotic approximations of the solution around the singularities we also prove the convergence of the psi series associated to the movable singularities of the systems considered we also propose a simple model in order to analyse the global structure of the singularities in the directions of exponential growth of the potential in time

a simple and unified approach to identify integrable nonlinear oscillators and systems in this paper we consider a generalized second order nonlinear ordinary differential equation ode of the form x k1xq k2 x k3x2q 1 k4xq 1 1x 0 where ki s i 1 2 3 4 1 and q are arbitrary parameters which includes several physically important nonlinear oscillators such as the simple harmonic oscillator anharmonic oscillator force free helmholtz oscillator force free duffing and duffing van der pol oscillators modified emden type equation and its hierarchy generalized duffing van der pol oscillator equation hierarchy and so on and investigate the integrability properties of this rather general equation we identify several new integrable cases for arbitrary value of the exponent q q r the q 1 and q 2 cases are analyzed in detail and the results are generalized to arbitrary q our results show that many classical integrable nonlinear oscillators can be derived as subcases of our results and significantly enlarge the list of integrable equations that exists in the contemporary literature t

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

lie symmetries and infinite dimensional lie algebras of certain nonlinear dissipative systems we point out that the dissipative force free duffing oscillator and holmes rand nonlinear oscillator admit infinite dimensional symmetry algebras for certain parametric choices for which the systems become integrable from the nature of the symmetry vector fields one can also write down the integrals of motion for the above systems in addition we point out that the other dissipative systems discussed in the literature also admit infinite dimensional lie algebras

the kowalewski top 99 years later a lax pair generalizations and explicit solutions a natural lax pair for the kowalewski top is derived by using a general group theoretic approach this gives a new insight into the algebraic geometry of the top and leads to its complete solution via finite band integration theory