isospectral problem for schrodinger operator evolutional viewpoint an isospectral transform of the schrodinger operator is considered as an evolutional problem for a transform defined by the mckean trubowitz flows associated evolutional equations are derived it is shown that for one level and two level flows these equations can be split into integrable liouville equations a relationship between the liouville equations and the darboux transforms is discussed this analysis suggests that the evolutional equations can be split into the liouville equations in the general case a hamiltonian formulation of the isospectral transform defined by the mckean trubowitz flows is presented it is shown that this transform is performed by a canonical change of variables which is related to the darboux transform

the poincare bendixson theorem for monotone cyclic feedback systems with delay abstract we consider cyclic nearest neighbor systems of differential delay equations in which the coupling between neighbors possesses a monotonicity property using a discrete integer valued lyapunov function we prove that the poincare bendixson theorem holds for such systems we also obtain results on piecewise monotonicity and stability of periodic solutions of such systems

multiplicity results of periodic solutions for a class of second order delay differential systems abstract this paper is concerned with the periodic boundary value problem 1 u t u t f t u t a e t 0 2 u 0 u 2 u 0 u 2 0 where 0 is a given constant r is a parameter the variational principle is given and some multiplicity results of periodic solutions of 1 are obtained directly via variational methods

an overlapping schwarz method for spectral element solution of the incompressible navier stokes equations efficient solution of the navier stokes equations in complex domains is dependent upon the availability of fast solvers for sparse linear systems for unsteady incompressible flows the pressure operator is the leading contributor to stiffness as the characteristic propagation speed is infinite in the context of operator splitting formulations it is the pressure solve which is the most computationally challenging despite its elliptic origins we examine several preconditioners for the consistent l2poisson operator arising in the pn pn 2spectral element formulation of the incompressible navier stokes equations we develop a finite element based additive schwarz preconditioner using overlapping subdomains plus a coarse grid projection operator which is applied directly to the pressure on the interior gauss points for large two dimensional problems this approach can yield as much as a fivefold reduction in simulation time over previously employed methods based upon deflation

factorization method and new potentials with the oscillator spectrum a one parameter family of potentials in one dimension is constructed with the energy spectrum coinciding with that of the harmonic oscillator this is a new derivation of a class of potentials previously obtained by abraham and moses with the help of the gelfand levitan formalism

existence of periodic solutions for a class of first order delay differential equations dealing with vectors abstract by the weak linking theorem and the local linking theorem we study the existence of periodic solutions for the following delay non autonomous systems 1 u t a t u t f t u t r 0 where r 0 0 r f c r r n r n satisfies f t r x f t x for all x r n and a t a i j t n n is an anti symmetric matrix valued function with a i j t c r r which is r periodic in t for all i j 1 2 n the variational principle is given some existence theorems of 4 r periodic solutions of 1 are obtained

on critical point theory for indefinite functionals in the presence of symmetries we consider functionals which are not bounded from above or from below even modulo compact perturbations and which exhibit certain symmetries with respect to the action of a compact lie group we develop a method which permits us to prove the existence of multiple critical points for such functionals the proofs are carried out directly in an infinite dimensional hilbert space and they are based on minimax arguments the applications given here are to hamiltonian systems of ordinary differential equations where the existence of multiple time periodic solutions is established for several classes of hamiltonians symmetry properties of these hamiltonians such as time translation invariancy or evenness are exploited introduction many variational problems which arise from physics or mathematical physics are indefinite in the sense that the functionals involved are not bounded from below or from above however some of these functionals defined in an appropriate function space are semidefinite in the sense that they are bounded from below or from above if perturbed with a weakly continuous functional this paper deals with functionals which are not semidefinite usually problems involving indefinite functionals are more difficult to handle and only very recently a method has been developed which permits us to treat such problems directly in an infinite dimensional function space 7 in 7 some theorems have been proved which establish the existence of at least one nontrivial critical point for such functionals in many physical situations there are problems which have symmetries with respect to the action of some lie group in this case we expect the existence of many critical points this has been established for semidefinite functionals cf e g 8 and 3 for even functionals 9 for a zp action with p prime numbers 15 16 10 and 6 for an s action in this paper we develop a method which allows us to estimate the number of critical points of indefinite functionals in the presence of symmetries more precisely received by the editors july 16 1980 and in revised form june 26 1981 1980 mathematics subject classification primary 58e05 70h05 secondary 34c25

applications of a zp index theory to periodic solutions for a class of functional differential equations abstract by means of a new geometrical index with z p group actions multiplicity results for a certain class of nonautonomous time periodic functional differential systems are obtained

a spectrally accurate method for overlapping grid solution of incompressible navier stokes equations an overlapping mesh methodology that is spectrally accurate in space and up to third order accurate in time is developed for solution of unsteady incompressible flow equations in three dimensional domains the ability to decompose a global domain into separate but overlapping subdomains eases mesh generation procedures and increases flexibility of modeling flows with complex geometries the methodology employs implicit spectral element discretization of equations in each subdomain and explicit treatment of subdomain interfaces with spectrally accurate spatial interpolation and high order accurate temporal extrapolation and requires few if any iterations yet maintains the global accuracy and stability of the underlying flow solver the overlapping mesh methodology is thoroughly validated using two dimensional and three dimensional benchmark problems in laminar and turbulent flows the spatial and temporal convergence is documented and is in agreement with the nominal order of accuracy of the solver the influence of long integration times as well as inflow outflow global boundary conditions on the performance of the overlapping grid solver is assessed in a turbulent benchmark of fully developed turbulent pipe flow the turbulent statistics with the overlapping grids is validated against published available experimental and other computation data scaling tests are presented that show near linear strong scaling even for moderately large processor counts a novel overlapping grid method for incompressible flow equations is developed the overlapping grid method is based on spectral element discretization the method maintains global spectral accuracy with polynomial refinement an explicit interface extrapolation scheme for temporal coupling is proposed the scheme allows for high order temporal accuracy with only a few iterations

adaptive local overlapping grid methods for parabolic systems in two space dimensions abstract adaptive mesh refinement techniques are described for two dimensional systems of parabolic partial differential equations solutions are calculated using galerkin s method with a piecewise bilinear basis in space and backward euler integration in time a posteriori estimates of the local discretization error of piecewise bilinear finite element solutions are obtained by a p refinement technique these error estimates are used to control a local h refinement strategy where finer grids are recursively introduced in regions where a prescribed tolerance is exceeded fine grids at a given level of refinement may overlap each other and independent solutions are generated on each of them a version of the schwarz alternating principle is used to coordinate solutions between overlapping fine grids computational results demonstrating the performance of the adaptive procedure on linear and nonlinear problems and apparent convergence of the error estimate for linear heat conduction problem and uniform global refinement is presented

multiplicity results for periodic solutions to delay differential equations via critical point theory abstract by the critical point theory we study the existence and multiplicity of periodic solutions to the following system of delay differential equations x t f x t r where x r n f c r n r n and r 0 is a given constant a sufficient condition on the existence and multiplicity of periodic solutions for eq is obtained to the authors knowledge this is the first time the existence of periodic solutions to systems of delay differential equations is dealt with by using variational approaches directly

infinite periodic solutions to a class of second order sturm liouville neutral differential equations abstract by means of variational structure and z 2 group index theory we obtain infinite periodic solutions to a class second order sturm liouville neutral delay equations p t x t s q t x t s f t x t x t x t 2 x t 2 s 0

a domain decomposition method for incompressible viscous flow a method for using domain decomposition to solve the equations of incompressible viscous flow is presented the method is described in detail and test results are given for two test problems a notable feature of the method is that the incompressibility constraint is never explicitly imposed the domain decomposition uses finite difference and spectral methods on overlapping domains with second order accurate interpolation of the velocity relating the solutions on the different domains the method is shown to be globally second order accurate by the test results

the spectral class of the quantum mechanical harmonic oscillator the purpose of this paper is to study the so calledspectral classq of anharmonic oscillatorsq d2 q having the same spectrum n 2n n 0 as the harmonic oscillatorq0 d2 x2 1 thenorming constants t n mathop lim limits x uparrow infty ell g 1 n e n x mathord left vphantom e n x e n right kern nulldelimiterspace e n x of the eigenfunctions ofq form a complete set of coordinates inq in terms of which the potential may be expressed asq x2 1 2d2 lg with theta det left delta ij e ti 1 int limits x infty e i 0 e j 0 0 leqq i j infty right e n 0 being thenth eigenfunctionq0 the spectrum and norming constants are canonically conjugate relative to the bracket f g fd gdx to wit i j 0 ti 2 j 1 or 0 according to whetheri j or not and ti tj 0 this prompts an investigation of the symplectic geometry ofq the function is related to the theta function of a singular algebraic curve numerical results are also presented

existence and multiplicity of solutions for a non periodic schrodinger equation abstract this paper is concerned with the non periodic schrodinger equation u v x u f x u x r n u x 0 as x under a general spectral assumption the existence and multiplicity of solutions are obtained for asymptotically linear nonlinearity via variational methods

circulant matrices and differential delay equations on etudie les solutions periodiques non triviales de familles parametrisees d equations differentielles a retard

stability analysis of interface temporal discretization in grid overlapping methods we investigate the stability of a temporal discretization of interface terms in grid overlapping methods a matrix stability analysis is performed on a model problem of the one dimensional diffusion equation on overlapping grids the scheme stability is first analyzed theoretically and a proof of the unconditional stability of the first order interface extrapolation scheme with the first and second order time integration for any overlap size is presented for the higher order schemes we obtain explicit estimates of the spectral radius of the corresponding discrete matrix operator and document the values of the stability threshold depending on the number of grid points and the size of overlap the influence of iterations on stability properties is also investigated numerical experiments are then presented relating the obtained stability bounds to the observed numerical values semidiscrete analysis confirms the derived scaling for the stability bounds

on multigrid for overlapping grids the solution of elliptic partial differential equations on composite overlapping grids using multigrid is discussed an approach is described that provides a fast and memory efficient scheme for the solution of boundary value problems in complex geometries the key aspects of the new scheme are an automatic coarse grid generation algorithm an adaptive smoothing technique for adjusting residuals on different component grids and the use of local smoothing near interpolation boundaries other important features include optimizations for cartesian component grids the use of over relaxed red black smoothers and the generation of coarse grid operators through galerkin averaging numerical results in two and three dimensions show that very good multigrid convergence rates can be obtained for dirichlet neumann and mixed boundary conditions a comparison to krylov based solvers shows that the multigrid solver can be much faster and require significantly less memory

systems of differential delay equations floquet multipliers and discrete lyapunov functions abstract we define a discrete integer valued lyapunov function v for cyclic nearest neighbor systems of differential delay equations possessing a feedback condition this extends analogous definitions for cyclic systems of ode s and for scalar differential delay equations we relate the values of v to the real parts of the floquet multipliers for such linear periodic systems and thereby prove all floquet subspaces are at most two dimensional

on the nonlinear fock description of quantum systems with quadratic spectra the nonlinear version of the fock approach developed previously for systems with extended equidistant spectra eleonsky v m and korolev v g 1995 j phys a math gen 28 4973 is also shown to describe systems with quadratic spectra some general properties of shift operators for such spectra are studied it is found in particular that families of shift operators related to the generators of the algebra su 1 1 as well as supersymmetry ladder operators arise naturally in that approach in a unified way