two novel methods and multi mode periodic solutions for the fermi pasta ulam model we introduce two novel methods for studying periodic solutions of the fpu model both numerically and rigorously one is a variational approach based on the dual formulation of the problem and the other involves computer assisted proofs these methods are used e g to construct a new type of solutions whose energy is spread among several modes associated with closely spaced resonances

a difference scheme for a nonlinear partial integrodifferential equation a difference method for the numerical integration of a nonlinear partial integrodifferential equation is considered the integral term is treated by means of a convolution quadrature suggested by lubich some results from lubich s discretized fractional calculus play a crucial role in proving consistency the verification of stability and convergence is based on the nonnegative character of the real quadratic form associated with the convolution quadrature a stability result is derived that is applicable to equations and numerical methods far more general than those treated in this paper

integral representation for the eigenstates of the spin system with inverse square interactions we consider the calogero sutherland moser system with spin 1 2 we give the integral representation for the ground state wavefunction excited states of the calogero spin model confined in the harmonic potential are also given

an extended hydrogen atom with yangian sl 2 the hydrogen atom is extended by deforming the loop algebra which is shown to be the symmetrical algebra of the hydrogen atom to the case with yangian a realization of is proposed using this yangian the energy spectrum of the extended hydrogen atom is calculated

numerical analysis of volterra integro differential equations for viscoelastic rods and membranes abstract we consider the initial boundary value problems for a homogeneous volterra integro differential equations for viscoelastic rods and membranes in a bounded smooth domain the memory kernel of the equation is made up in a complicated way from the distinct moduli of stress relaxation for compression and shear which is challenging to approximate the literature reported on the numerical solution of this model is extremely sparse in this paper we will study the second order continuous time galerkin approximation for its space discretization and propose a fully discrete scheme employing the crank nicolson method for the time discretization then we derive the uniform long time error estimates in the norm l t 1 0 l x 2 for the finite element solutions some numerical results are presented to illustrate our theoretical error bounds

time stepping discontinuous galerkin methods for fractional diffusion problems time stepping hp hp versions discontinuous galerkin dg methods for the numerical solution of fractional subdiffusion problems of order alpha with 1 alpha 0 1 0 will be proposed and analyzed generic hp hp version error estimates are derived after proving the stability of the approximate solution for h h version dg approximations on appropriate graded meshes near t 0 t 0 we prove that the error is of order o k max 2 p frac alpha 2 o kmax 2 p 2 where k k is the maximum time step size and p ge 1 p 1 is the uniform degree of the dg solution for hp hp version dg approximations by employing geometrically refined time steps and linearly increasing approximation orders exponential rates of convergence in the number of temporal degrees of freedom are shown finally some numerical tests are given

a second order accurate numerical method for a fractional wave equation we study a generalized crank nicolson scheme for the time discretization of a fractional wave equation in combination with a space discretization by linear finite elements the scheme uses a non uniform grid in time to compensate for the singular behaviour of the exact solution at t 0 with appropriate assumptions on the data and assuming that the spatial domain is convex or smooth we show that the error is of order k2 h2 where k and h are the parameters for the time and space meshes respectively

discontinuous galerkin method for an evolution equation with a memory term of positive type we consider an initial value problem for a class of evolution equations incorporating a memory term with a weakly singular kernel bounded by c t s 1 where 0 1 for the time discretization we apply the discontinuous galerkin method using piecewise polynomials of degree at most q 1 for q 1 or 2 for the space discretization we use continuous piecewise linear finite elements the discrete solution satisfies an error bound of order k 9 h 2 l k where k and h are the mesh sizes in time and space respectively and l k max 1 log k 1 in the case q 2 we prove a higher convergence rate of order k 3 h 2 l k at the nodes of the time mesh typically the partial derivatives of the exact solution are singular at t 0 necessitating the use of non uniform time steps we compare our theoretical error bounds with the results of numerical computations

symmetric invariant manifolds in the fermi pasta ulam lattice abstract the fermi pasta ulam fpu lattice with periodic boundary conditions and n particles admits a large group of discrete symmetries the fixed point sets of these symmetries naturally form invariant symplectic manifolds that are investigated in this short note for each k dividing n we find k degree of freedom invariant manifolds they represent short wavelength solutions composed of k fourier modes and can be interpreted as embedded lattices with periodic boundary conditions and only k particles inside these invariant manifolds other invariant structures and exact solutions are found which represent for instance periodic and quasi periodic solutions and standing and travelling waves some of these results have been found previously by other authors via a study of mode coupling coefficients and recently also by investigating bushes of normal modes the method of this paper is similar to the latter method and much more systematic than the former we arrive at previously unknown results without any difficult computations it is shown moreover that similar invariant manifolds exist also in the klein gordon kg lattice and in the thermodynamic and continuum limits

asymptotic resonance interaction of modes and subharmonic bifurcation we study the existence of small amplitude oscillations near elliptic equilibria of autonomous systems which mix different normal modes the reference problem is the fermi pasta ulam model a chain of nonlinear oscillators with nearest neighborhood interaction we develop a new bifurcation approach that locates secondary bifurcations from the unimodal primary branches two sufficient conditions for bifurcation are given one involves only the arithmetic properties of the eigenvalues of the linearized system asymptotic resonance while the other takes into account the nonlinear character of the interaction between normal modes nonlinear coupling both conditions are checked for the fermi pasta ulam problem

numerical solution for a sub diffusion equation with a smooth kernel in this paper we study the numerical solution of an initial value problem of a sub diffusion type for the time discretization we apply the discontinuous galerkin method and we use continuous piecewise finite elements for the space discretization optimal order convergence rates of our numerical solution have been shown we compare our theoretical error bounds with the results of numerical computations we also present some numerical results showing the super convergence rates of the proposed method

numerical solution of parabolic integro differential equations by the discontinuous galerkin method the numerical solution of a parabolic equation with memory is considered the equation is first discretized in time by means of the discontinuous galerkin method with piecewise constant or piecewise linear approximating functions the analysis presented allows variable time steps which as will be shown call then efficiently be selected to match singularities in the solution induced by singularities in the kernel of the memory term or by nonsmooth initial data the combination with finite element discretizat in space is also studied

proof of nishida s conjecture on anharmonic lattices we prove nishida s 1971 conjecture stating that almost all low energetic motions of the anharmonic fermi pasta ulam lattice with fixed endpoints are quasi periodic the proof is based on the formal computations of nishida the kam theorem discrete symmetry considerations and an algebraic trick that considerably simplifies earlier results

discretization with variable time steps of an evolution equation with a positive type memory term we consider the linearized form of an evolution equation used to model viscoelasticity and heat conduction in materials with memory the equation is discretized in time using several schemes based on the backward euler and crank nicolson methods and incorporating appropriate quadratures for the memory term by permitting variable time steps we remove a restriction present in earlier related work by the first two authors

a numerical method for a partial integro differential equation a method is considered for the integration in time of a partial integro differential equation the discretization technique employed is patterned after an idea of ch lubich error bounds are derived for both smooth and nonsmooth initial data

a spectral order method for inverting sectorial laplace transforms laplace transforms which admit a holomorphic extension to some sector strictly containing the right half plane and exhibiting a potential behavior are considered a spectral order parallelizable method for their numerical inversion is proposed the method takes into account the available information about the errors arising in the evaluations several numerical illustrations are provided

adaptive finite element methods for parabolic problems i a linear model problem this paper is the first part in a series of papers on adaptive finite element methods for parabolic problems in this paper an adaptive algorithm is presented and analyzed for choosing the space a

adaptive discretization of an integro differential equation with a weakly singular convolution kernel an integro differential equation involving a convolution integral with a weakly singular kernel is considered the kernel can be that of a fractional integral the integro differential equation is discretized using the discontinuous galerkin method with piecewise constant basis functions sparse quadrature is introduced for the convolution term to overcome the problem with the growing amount of data that has to be stored and used in each time step a priori and a posteriori error estimates are proved an adaptive strategy based on the a posteriori error estimate is developed finally the precision and effectiveness of the algorithm are demonstrated in the case that the convolution is a fractional integral this is done by comparing the numerical solutions with analytical solutions

on metastability in fpu we present an analytical study of the fermi pasta ulam fpu model with periodic boundary conditions we analyze the dynamics corresponding to initial data with one low frequency fourier mode excited we show that correspondingly a pair of kdv equations constitute the resonant normal form of the system we also use such a normal form in order to prove the existence of a metastability phenomenon more precisely we show that the time average of the modal energy spectrum rapidly attains a well defined distribution corresponding to a packet of low frequencies modes subsequently the distribution remains unchanged up to the time scales of validity of our approximation the phenomenon is controlled by the specific energy

topics in quantum integrable systems recent developments in the theory of quantum integrable particle systems in one dimension with inverse square interactions are reviewed first the yangian symmetry is introduced and the energy spectra of the related spin models are discussed the character of the su n 1 wznw theory is shown to be closely related with the rogers szego polynomial second the infinite dimensional representation for solutions of the yang baxter equation and the reflection equation is given based on the representation the dunkl operators associated with the classical root systems are constructed the macdonald polynomial and its generalization are discussed in connection with the eigenstates for the trigonometric case finally some results on short range interacting systems are mentioned

symmetry of the calogero model confined in the harmonic potential yangian and w algebra the su nu calogero model confined in the harmonic potential is considered we show that the model has yangian symmetry y su nu unifying with the creation annihilation operators which constitute the su nu loop algebra we reveal the algebraic structure of this system the relation with the energy spectrum is also discussed

energy localization on q tori long term stability and the interpretation of fermi pasta ulam recurrences we focus on two approaches that have been proposed in recent years for the explanation of the so called fermi pasta ulam fpu paradox i e the persistence of energy localization in the low q fourier modes of fermi pasta ulam nonlinear lattices preventing equipartition among all modes at low energies in the first approach a low frequency fraction of the spectrum is initially excited leading to the formation of natural packets exhibiting exponential stability while in the second emphasis is placed on the existence of q breathers i e periodic continuations of the linear modes of the lattice which are exponentially localized in fourier space following ideas of the latter we introduce in this paper the concept of q tori representing exponentially localized solutions on low dimensional tori and use their stability properties to reconcile these two approaches and provide a more complete explanation of the fpu paradox

birkhoff coordinates for the toda lattice in the limit of infinitely many particles with an application to fpu abstract in this paper we study the birkhoff coordinates cartesian action angle coordinates of the toda lattice with periodic boundary condition in the limit where the number n of the particles tends to infinity we prove that the transformation introducing such coordinates maps analytically a complex ball of radius r n in discrete sobolev analytic norms into a ball of radius r n with r r 0 independent of n if and only if 2 then we consider the problem of equipartition of energy in the spirit of fermi pasta ulam we deduce that corresponding to initial data of size r n 2 0 r 1 and with only the first fourier mode excited the energy remains forever in a packet of fourier modes exponentially decreasing with the wave number finally we consider the original fpu model and prove that energy remains localized in a similar packet of fourier modes for times one order of magnitude longer than those covered by previous results which is the time of formation of the packet the proof of the theorem on birkhoff coordinates is based on a new quantitative version of a vey type theorem by kuksin and perelman which could be interesting in itself

numerical asymptotic stability for the integro differential equations with the multi term kernels we prove that a second order backward difference semi discretization approximation for the integro differential equations with the multi term kernels preserves the weighted asymptotic integrabilities of continuous solutions the method of proof extend and simulate numerically those introduced by hannsgen and wheeler relying on deep frequency domain techniques

effective hamiltonian for travelling discrete breathers hamiltonian chains of oscillators in general probably do not sustain exact travelling discrete breathers however solutions which look like moving discrete breathers for some time are not difficult to observe in numerics in this paper we propose an abstract framework for the description of approximate travelling discrete breathers in hamiltonian chains of oscillators the method is based on the construction of an effective hamiltonian enabling one to describe the dynamics of the translation degree of freedom of moving breathers error estimate on the approximate dynamics is also studied the concept of the peierls nabarro barrier can be made clear in this framework we illustrate the method with two simple examples namely the salerno model which interpolates between the ablowitz ladik lattice and the discrete nonlinear schrodinger system and the fermi pasta ulam chain

post processing discontinuous galerkin solutions to volterra integro differential equations analysis and simulations this paper presents a superconvergence extraction technique for volterra integro differential equations with smooth and non smooth kernels specifically extracting superconvergence is done via a post processed discontinuous galerkin dg method obtained from interpolating the dg solution using lagrange polynomials at the nodal points a global superconvergence error bound in the l norm is established for a non smooth kernel a family of non uniform time meshes is used to compensate for the singular behaviour of the exact solution near t 0 the derived theoretical results are numerically validated in a sample of test problems demonstrating higher than expected convergence rates

yang baxter equation in long range interacting systems we consider the su p spin chains with long range interactions and the spin generalization of the calogero sutherland models we show that their properties derive from a transfer matrix obeying the yang baxter equation we obtain the expression of the conserved quantities of the dynamical models and we diagonalise them in the spin chain case we establish the connection between the degeneracies of the spectrum and the representation theory of the yangians we use a correspondence with the dynamical models to diagonalise the hamiltonian finally we extend the previous results to the case of a trigonometric r matrix

differential difference operators associated to reflection groups there is a theory of spherical harmonics for measures invariant under a finite reflection group the measures are products of powers of linear functions whose zero sets are the mirrors of the reflections in the group times the rotation invariant measure on the unit sphere in rn a commutative set of differential difference operators each homogeneous of degree 1 is the analogue of the set of first order partial derivatives in the ordinary theory of spherical harmonics in the case of r2 and dihedral groups there are analogues of the cauchy riemann equations which apply to gegenbauer and jacobi polynomial expansions the analysis of orthogonality structures for polynomials in several variables is a problem of vast dimensions this paper is part of an ongoing program to establish a workable theory for one particular class the underlying structure is based on finite coxeter groups these are finite groups acting on euclidean space generated by reflections in the zero sets of a collection of linear functions the roots the weight functions for the orthogonality are products of powers of these linear functions restricted to the surface of the unit sphere in addition the weight function is required to be invariant under the action of the group the resulting theory has strong similarities to the theory of spherical harmonics this was established in previous papers of the author 3 4 5 most notably a homogeneous polynomial is orthogonal to all polynomials of lower degree if and only if it is annihilated by a certain second order differential difference operator ordinary partial differentiation acts as an endomorphism on ordinary harmonic functions the use of such operators and their adjoints leads to recurrence formulas and orthogonal decompositions for harmonic polynomials in this paper we construct a commutative set of first order differentialdifference operators associated to the second order operator previously mentioned received by the editors june 1 1987 1980 mathematics subject classification 1985 revision primary 33a45 33a65 20h15 secondary 20c30 42c10 51f15

a superconvergent discontinuous galerkin method for volterra integro differential equations smooth and non smooth kernels abstract we study the numerical solution for volerra integro differential equations with smooth and non smooth kernels we use an h version discontinuous galerkin dg method and derive nodal error bounds that are explicit in the parameters of interest in the case of non smooth kernel it is justified that the start up singularities can be resolved at superconvergence rates by using non uniformly graded meshes our theoretical results are numerically validated in a sample of test problems

on averaging reduction and symmetry in hamiltonian systems abstract the existence of periodic orbits for hamiltonian systems at low positive energies can be deduced from the existence of nondegenerate critical points of an averaged hamiltonian on an associated reduced space alternatively in classical kinetic plus potential energy hamiltonians the existence of such orbits can often be established by elementary geometrical arguments the present paper unifies the two approaches by exploiting discrete symmetries including reversing diffeomorphisms that occur in a given system the symmetries are used to locate the periodic orbits in the averaged hamiltonian and thence in the original hamiltonian when the periodic orbits are continued under perturbations admitting the same symmetries in applications to the henon heiles hamiltonian it is illustrated how higher order averaging can sometimes be used to overcome degeneracies encountered at first order

an hp version discontinuous galerkin method for integro differential equations of parabolic type we study the numerical solution of a class of parabolic integro differential equations with weakly singular kernels we use an hp version discontinuous galerkin dg method for the discretization in time we derive optimal hp version error estimates and show that exponential rates of convergence can be achieved for solutions with singular temporal behavior near t 0 caused by the weakly singular kernel moreover we prove that by using nonuniformly refined time steps optimal algebraic convergence rates can be achieved for the h version dg method we then combine the dg time stepping method with a standard finite element discretization in space and present an optimal error analysis of the resulting fully discrete scheme our theoretical results are numerically validated in a series of test problems

results on normal forms for fpu chains in this paper we prove among other results that near the equilibirum position any periodic fpu chain with an odd number n of particles admits a birkhoff normal form up to order 4 whereas any periodic fpu chain with n even admits a resonant normal form up to order 4 this resonant normal form of order 4 turns out to be completely integrable further for n odd we obtain an explicit formula of the hessian of its hamiltonian at the fixed point

symmetry and resonance in periodic fpu chains the symmetry and resonance properties of the fermi pasta ulam chain with periodic boundary conditions are exploited to construct a near identity transformation bringing this hamiltonian system into a particularly simple form this birkhoff gustavson normal form retains the symmetries of the original system and we show that in most cases this allows us to view the periodic fpu hamiltonian as a perturbation of a nondegenerate liouville integrable hamiltonian according to the kam theorem this proves the existence of many invariant tori on which motion is quasiperiodic experiments confirm this qualitative behaviour we note that one can not expect this in lower order resonant hamiltonian systems so the periodic fpu chain is an exception and its special features are caused by a combination of special resonances and symmetries

nonsmooth data error estimates for approximations of an evolution equation with a positive type memory term we study the numerical approximation of an integro differential equation which is intermediate between the heat and wave equations the proposed discretization uses convolution quadrature based on the first and second order backward difference methods in time and piecewise linear finite elements in space optimal order error bounds in terms of the initial data and the inhomogeneity are shown for positive times without assumptions of spatial regularity of the data

time reversal symmetry in dynamical systems a survey abstract in this paper we survey the topic of time reversal symmetry in dynamical systems we begin with a brief discussion of the position of time reversal symmetry in physics after defining time reversal symmetry as it applies to dynamical systems we then introduce a major theme of our survey namely the relation of time reversible dynamical sytems to equivariant and hamiltonian dynamical systems we follow with a survey of the state of the art on the theory of reversible dynamical systems including results on symmetric periodic orbits local bifurcation theory homoclinic orbits and renormalization and scaling some areas of physics and mathematics in which reversible dynamical systems arise are discussed in an appendix we provide an extensive bibliography on the topic of time reversal symmetry in dynamical systems

boundary k matrix elliptic dunkl operator and quantum many body systems the infinite dimensional representation of the boundary k operator the solution of the reflection equation is considered the trigonometric k matrix is studied as a degenerate case of the elliptic operator a method to construct the elliptic dunkl operator is proposed and the relationship with the quantum many body problem is also discussed

superconvergence of a discontinuous galerkin method for fractional diffusion and wave equations we consider an initial boundary value problem for partial tu partial t alpha nabla 2u f t that is for a fractional diffusion 1 alpha 0 or wave 0 alpha 1 equation a numerical solution is found by applying a piecewise linear discontinuous galerkin dg method in time combined with a piecewise linear conforming finite element method in space the time mesh is graded appropriately near t 0 but the spatial mesh is quasi uniform previously we proved that the error measured in the spatial l 2 norm is of order k 2 alpha h 2 ell k uniformly in t where k is the maximum time step h is the maximum diameter of the spatial finite elements alpha min alpha 0 le0 and ell k max 1 log k here we prove convergence of order k 3 2 alpha ell k h 2 at each time level t n for 1 alpha 1 thus if 1 2 alpha 1 then the dg solution is superconvergent which generalizes a known result for the classical heat equation i e the case alpha 0 a

the symmetries of solitons in this article we will retrace one of the great mathematical adventures of this century the discovery of the soliton and the gradual explanation of its remarkable properties in terms of hidden symmetries we will take an historical approach starting with a famous numerical experiment carried out by fermi pasta and ulam on one of the first electronic computers and with zabusky and kruskal s insightful explanation of the surprising results of that experiment and of a follow up experiment of their own in terms of a new concept they called solitons solitons however raised even more questions than they answered in particular the evolution equations that govern solitons were found to be hamiltonian and have infinitely many conserved quantities pointing to the existence of many non obvious symmetries we will cover next the elegant approach to solitons in terms of the inverse scattering transform and lax pairs and finally explain how those ideas led step by step to the discovery that loop groups acting by dressing transformations give a conceptually satisfying explanation of the secret soliton symmetries

near integrability of periodic fpu chains abstract the fpu chain with periodic boundary conditions is studied by birkhoff gustavson normalisation in the cases of up to 6 particles and for chains with an odd number of particles the normal forms are integrable which permits us to apply kam theory this leads to the presence of many invariant tori on which the motion is quasi periodic thus we explain the recurrence phenomena and the small size of chaos observed in experiments furthermore we find a certain clustering of modes

analysis of a temporal discretization for a semilinear fractional diffusion equation abstract this paper considers a temporal discretization of a semilinear fractional diffusion equation with nonsmooth initial data with appropriately graded temporal grids first order temporal accuracy in the norm of l 2 0 t l 2 is derived by a new discrete gronwall s inequality and then first order temporal accuracy in the norm of l 0 t l 2 is established for 1 2 1 finally numerical results are provided to verify the theoretical results

integrable n particle hamiltonians with yangian or reflection algebra symmetry we use the dunkl operator approach to construct one dimensional integrable models describing n particles with internal degrees of freedom these models are described by a general hamiltonian belonging to the centre of the yangian or the reflection algebra which ensures that they admit the corresponding symmetry in particular the open problem of the symmetry is answered for the bn type sutherland model with spin and for a generalized bn type nonlinear schrodinger hamiltonian