knot concordance whitney towers and l 2 signatures we construct many examples of nonslice knots in 3 space that cannot be distinguished from slice knots by previously known invariants using whitney towers in place of embedded disks we define a geometric filtration of the 3 dimensional topological knot concordance group the bottom part of the filtration exhibits all classical concordance invariants including the cassongordon invariants as a first step we construct an infinite sequence of new obstructions that vanish on slice knots these take values in the l theory of skew fields associated to certain universal groups finally we use the dimension theory of von neumann algebras to define an l 2 signature and use this to detect the first unknown step in our obstruction theory

backward error analysis for multi symplectic integration methods a useful method for understanding discretization error in the numerical solution of odes is to compare the system of odes with the modified equations obtained through backward error analysis and using symplectic integration for hamiltonian odes provides more incite into the modified equations in this paper the ideas of symplectic integration are extended to hamiltonian pdes and this paves the way for the development of a local modified equation analysis solely as a useful diagnostic tool for the study of these types of discretizations in particular local conservation laws of energy and momentum are not preserved exactly when symplectic integrators are used to discretize but the modified equations are used to derive modified conservation laws that are preserved to higher order along the numerical solution these results are also applied to the nonlinear wave equation

whitney towers and abelian invariants of knots we relate certain abelian invariants of a knot namely the alexander polynomial the blanchfield form and the arf invariant to intersection data of a whitney tower in the 4 ball bounded by the knot we also give a new 3 dimensional algorithm for computing these invariants

a split step numerical method for the time dependent dirac equation in 3 d axisymmetric geometry abstract a numerical method is developed to solve the time dependent dirac equation in cylindrical coordinates for 3 d axisymmetric systems the time evolution is treated by a splitting scheme in coordinate space using alternate direction iteration while the wave function is discretized spatially on a uniform grid the longitudinal coordinate evolution is performed exactly by the method of characteristics while the radial coordinates evolution uses poisson s integral solution which allows to implement the radial symmetry of the wave function the latter is evaluated on a time staggered mesh by using hermite polynomial interpolation and by performing the integration analytically the cylindrical coordinate singularity problem at r 0 is circumvented by this method as the integral is well defined at the origin the resulting scheme is reminiscent of non standard finite differences in the last step of the splitting the remaining equation has a solution in terms of a time ordered exponential which is approximated to a higher order than the time evolution scheme we study the time evolution of gaussian wave packets and we evaluate the eigenstates of hydrogen like systems by using a spectral method we compare the numerical results to analytical solutions to validate the method in addition we present three dimensional simulations of relativistic laser matter interactions using the dirac equation

symplectic and multisymplectic lobatto methods for the good boussinesq equation in this paper we construct second order symplectic and multisymplectic integrators for the good boussineq equation using the two stage lobatto iiia iiib partitioned runge kutta method which yield an explicit scheme and is equivalent to the classical central difference approximation to the second order spatial derivative numerical dispersion properties and the stability of both integrators are investigated numerical results for different solitary wave solutions confirm the excellent long time behavior of symplectic and multisymplectic integrators by preserving local and global energy and momentum

symplectic integrator for nonlinear high order schrodinger equation with a trapped term in the manuscript we discuss the symplectic integrator for the numerical solution of a kind of high order schrodinger equation with trapped terms the hamiltonian formulism is discovered for it we first discretize the hamiltonian system in space to reduce it to a finite dimensional one then the symplectic midpoint scheme is applied to the temporal discretization the symplectic scheme we devise is of second order accuracy in time and 2lth order accuracy in space it is proved that it preserves the charge of the original equation veraciously the energy is not preserved explicitly for the exception of the linear case however after some computation the energy transit formula in the temporal direction is obtained in the numerical part we compare our scheme with some existing schemes including the leap frog scheme and the energy preserving scheme from the numerical evidence we find that our numerical schemes are efficient and available the numerical results are consistent with the theoretical analysis

solvability of concatenated runge kutta equations for second order nonlinear pdes in this paper we consider the solvability of concatenated runge kutta crk equations which are the nonlinear algebraic equations arising from crk methods applied to some kind of second order nonlinear partial differential equations to guarantee the existence and uniqueness of the solution of crk equations temporal and spatial step sizes of crk methods should be chosen appropriately under different assumptions with the homotopy continuation method we choose different step sizes to ensure the solvability of crk equations

difference schemes for solving the generalized nonlinear schrodinger equation this paper studies finite difference schemes for solving the generalized nonlinear schrodinger gnls equationiut uxx q u 2 u f x t u a new linearlized crank nicolson type scheme is presented by applying an extrapolation technique to the real coefficient of the nonlinear term in the gnls equation several schemes including crank nicolson type schemes hopscotch type schemes split step fourier scheme and pseudospectral scheme are adopted for solving three model problems of gnls equation which arise from many physical problems withq s s2 q s ln 1 s andq s 4s 1 s respectively the numerical results demonstrate that the linearized crank nicolson scheme is efficient and robust

single cone real space finite difference scheme for the time dependent dirac equation a finite difference scheme for the numerical treatment of the 3 1 d dirac equation is presented its staggered grid intertwined discretization treats space and time coordinates on equal footing thereby avoiding the notorious fermion doubling problem this explicit scheme operates entirely in real space and leads to optimal linear scaling behavior for the computational effort per space time grid point it allows for an easy and efficient parallelization a functional for a norm on the grid is identified it can be interpreted as probability density and is proved to be conserved by the scheme the single cone dispersion relation is shown and exact stability conditions are derived finally a single cone scheme for the two component 2 1 d dirac equation its properties and a simulation of scattering at a klein step are presented

algebraically computable piecewise linear nodal oscillators a family of piecewise linear oscillators whose oscillation can be completely characterized by algebraic methods is studied it represents up to the best of authors s knowledge the first planar example where all the oscillation properties can be determined for all the values of the bifurcation parameter in fact algebraic expressions for coordinates of representative points period and characteristic multiplier of the corresponding periodic orbit are provided thus the studied family of oscillators deserves to be considered a good benchmark for testing approximate methods of analysis in nonlinear oscillation theory the piecewise linear oscillators studied are called nodal oscillators since their relevant linear parts are of node type and they are not perturbations of the harmonic oscillator they represent real models in practice as it is shown for an electronic circuit modeling a piecewise linear version of the classical van der pol oscillator

high order time splitting hermite and fourier spectral methods in this paper we are concerned with the numerical solution of the time dependent gross pitaevskii equation gpe involving a quasi harmonic potential primarily we consider discretisations that are based on spectral methods in space and higher order exponential operator splitting methods in time the resulting methods are favourable in view of accuracy and efficiency moreover geometric properties of the equation such as particle number and energy conservation are well captured regarding the spatial discretisation of the gpe we consider two approaches in the unbounded domain we employ a spectral decomposition of the solution into hermite basis functions on the other hand restricting the equation to a sufficiently large bounded domain fourier techniques are applicable for the time integration of the gpe we study various exponential operator splitting methods of convergence orders two four and six our main objective is to provide accuracy and efficiency comparisons of exponential operator splitting fourier and hermite pseudospectral methods for the time evolution of the gpe furthermore we illustrate the effectiveness of higher order time splitting methods compared to standard integrators in a long term integration

planar linear systems with single saturated feedback abstract we study planar linear systems with a linear or non linear feedback global stabilizer subjected to saturation the number of open loop unstable eigenvalues characterizes the set of the equilibrium points and the shape of the asymptotic stability region asr some topological bifurcations on the shape of the asr are described

geometric integrators for the nonlinear schrodinger equation abstract recently an interesting new class of pde integrators multisymplectic schemes has been introduced for solving systems possessing a certain multisymplectic structure some of the characteristic features of the method are its local nature independent of boundary conditions and an equal treatment of spatial and temporal variables the nonlinear schrodinger equation nls has a multisymplectic formulation and in this paper we discuss the performance of both symplectic and multisymplectic integrators for the nls in the numerical experiments the multisymplectic concatenated midpoint scheme a centered cell discretization is shown to preserve the local conservation laws extremely well over long times and to preserve global invariants such as the norm and momentum within roundoff on the other hand an integrable hamiltonian semi discretization of nls from ablowitz and ladik al possesses a full set of global conservation laws and a noncanonical symplectic structure we generalize the generating function technique to develop symplectic integrators of arbitrary order for a general class of noncanonical systems carrying a symplectic structure of the al type another approach examined in the paper is the introduction of transformations to reduce the al system to either 1 separable form or 2 canonical form and then apply standard schemes in the new coordinates all of the discretizations are tested numerically using initial data for spatially periodic multiphase solutions the performance of the schemes as well as interrelations among various geometric features are discussed

numerical simulation of interaction between schrodinger field and klein gordon field by multisymplectic method abstract we propose a multisymplectic scheme to solve the coupled klein gordon schrodinger system the scheme preserves the multisymplectic geometry structure exactly by satisfying the discrete multisymplectic conservation law and can simulate the original waves well in a long time this scheme also has discrete quasi norm conservation law numerical experiments demonstrate the consistency between the theoretical analysis and the numerical results

multi symplectic runge kutta collocation methods for hamiltonian wave equations a number of conservative pdes like various wave equations allow for a multi symplectic formulation which can be viewed as a generalization of the symplectic structure of hamiltonian odes we show that gauss legendre collocation in space and time leads to multi symplectic integrators i e to numerical methods that preserve a symplectic conservation law similar to the conservation of symplecticity under a symplectic method for hamiltonian odes we also discuss the issue of conservation of energy and momentum since time discretization by a gauss legendre method is computational rather expensive we suggest several semi explicit multi symplectic methods based on gauss legendre collocation in space and explicit or linearly implicit symplectic discretizations in time

asymptotic stability of small gap solitons in nonlinear dirac equations we prove dispersive decay estimates for the one dimensional dirac operator and use them to prove asymptotic stability of small gap solitons in the nonlinear dirac equations with quintic and higher order nonlinear terms

symplectic and multisymplectic numerical methods for maxwell s equations in this paper we compare the behaviour of one symplectic and three multisymplectic methods for maxwell s equations in a simple medium this is a system of pdes with symplectic and multisymplectic structures we give a theoretical discussion of how some numerical methods preserve the discrete versions of the local and global conservation laws and verify this behaviour in numerical experiments we also show that these numerical methods preserve the divergence furthermore we extend the discussion on dispersion for multi symplectic methods applied to pdes with one spatial dimension to include anisotropy when applying multi symplectic methods to maxwell s equations in two spatial dimensions lastly we demonstrate how varying the courant friedrichs lewy cfl number can cause the multi symplectic methods in our comparison to behave differently which can be explained by the study of backward error analysis for pdes

multisymplectic geometry variational integrators and nonlinear pdes this paper presents a geometric variational approach to continuous and discrete mechanics and field theories using multisymplectic geometry we show that the existence of the fundamental geometric structures as well as their preservation along solutions can be obtained directly from the variational principle in particular we prove that a unique multisymplectic structure is obtained by taking the derivative of an action function and use this structure to prove covariant generalizations of conservation of symplecticity and noether s theorem natural discretization schemes for pdes which have these important preservation properties then follow by choosing a discrete action functional in the case of mechanics we recover the variational symplectic integrators of veselov type while for pdes we obtain covariant spacetime integrators which conserve the corresponding discrete multisymplectic form as well as the discrete momentum mappings corresponding to symmetries we show that the usual notion of symplecticity along an infinite dimensional space of fields can be naturally obtained by making a spacetime split all of the aspects of our method are demonstrated with a nonlinear sine gordon equation including computational results and a comparison with other discretization schemes

existence of excited states for a nonlinear dirac field we prove the existence of infinitely many stationary states for the following nonlinear dirac equation i gamma mu partial mu psi m psi bar psi psi psi 0 seeking for eigenfunctions splitted in spherical coordinates leads us to analyze a nonautonomous dynamical system inr2 the number of eigenfunctions is given by the number of intersections of the stable manifold of the origin with the curve of admissible datum this proves the existence of infinitely many stationary states ordered by the number of nodes of each component

an efficient adaptive mesh redistribution method for a non linear dirac equation this paper presents an efficient adaptive mesh redistribution method to solve a non linear dirac nld equation our algorithm is formed by three parts the nld evolution the iterative mesh redistribution of the coarse mesh and the local uniform refinement of the final coarse mesh at each time level the equidistribution principle is first employed to iteratively redistribute coarse mesh points and the scalar monitor function is subsequently interpolated on the coarse mesh in order to do one new iteration and improve the grid adaptivity after an adaptive coarse mesh is generated ideally and finally each coarse mesh interval is equally divided into some fine cells to give an adaptive fine mesh of the physical domain and then the solution vector is remapped on the resulting new fine mesh by an affine method the nld equation is finally solved by using a high resolution shock capturing method on the fixed non uniform fine mesh extensive numerical experiments demonstrate that the proposed adaptive mesh method gives the third order rate of convergence and yields an efficient and fast nld solver that tracks and resolves both small local and large solution gradients automatically

accuracy of classical conservation laws for hamiltonian pdes under runge kutta discretizations we investigate conservative properties of runge kutta methods for hamiltonian partial differential equations it is shown that multi symplecitic runge kutta methods preserve precisely the norm square conservation law based on the study of accuracy of runge kutta methods applied to ordinary and partial differential equations we present some results on the numerical accuracy of conservation laws of energy and momentum for hamiltonian pdes under runge kutta discretizations

the numerical study of a nonlinear one dimensional dirac equation we study the convergence of a difference scheme for a nonlinear one dimensional dirac equation also we show some numerical experiments by using this scheme

exact localized solutions of a family of two dimensional nonlinear spinor fields the classical equations of motion for two dimensional nonlinear spinor fields are investigated explicit solutions for monomial and logarithmic self interactions are presented furthermore properties of these classical solutions are discussed

homotopy invariants of links this paper answers a question originally asked by john milnor in 1957 it offers significant progress toward another question from that paper in m 1 an m 2 john milnor defined a collection of isotopy invariants of links stallings showed these were actually concordance i variants of the ink stallings work suggested that milnor s invariants were related to other algebraically defined invariants the massey products in the exterior of the link a connection made precise in tu and p thus the results in this paper can be reinterpreted as results about massey products in the link exterior recently mike freedman has suggested that some of these invariants may be the obstructions to the topological surgery theorem in dimension four f i if2 fl in this paper the approach to studying higher dimensional link concordance presented in o 1 is extended to the study of links in the 3 sphere a collection of new link invariants is defined and related to milnor s invariants one fundamental advantage of this approach is that these new invariants make explicit use of the three sphere containing the link the link exterior contains all the relations among milnor s invariants by accounting for the ambient sphere our invariants eliminate redundancy see theorem 15 we answer this question originally asked by milnor 1 to what extent are milnor s invariants independent of each other theorem 15 milnor also asked this question 2 can one extract invariants from the group rc n where 7r is the link group and n is the intersection of the lower central series of the link group theorem 15 answers question 1 partial results can be found in c 1 ma i on and t the only previously known results for two component links were k 2 8 where the number of linearly independent invariants are 0 1 0 5 0 6 and 4 respectively for links with three or more components tess was known

energy and quadratic invariants preserving equip multi symplectic methods for hamiltonian wave equations abstract it is well known that a numerical method which is at the same time geometric structure preserving and physical property preserving cannot exist in general for hamiltonian partial differential equations motivated by equip methods proposed in brugnano et al 2012 13 in this paper we present a novel class of parametric multi symplectic runge kutta methods called equip multi symplectic methods for hamiltonian wave equations which can also conserve energy simultaneously in a weaker sense with a suitable parameter the existence of such a parameter which enforces the energy preserving property is proved under certain assumptions on the fixed step sizes and the fixed initial condition we compare the proposed method with the classical multi symplectic runge kutta method in numerical experiments which shows the remarkable energy preserving property of the proposed method and illustrates the validity of theoretical results these theoretical and numerical results show that equip methods can be well adapted to handle hamiltonian partial differential equations

on the preservation of phase space structure under multisymplectic discretization in this paper we explore the local and global properties of multisymplectic discretizations based on finite differences and fourier spectral approximations multisymplectic ms schemes are developed for two benchmark nonlinear wave equations the sine gordon and nonlinear schrodinger equations we examine the implications of preserving the ms structure under discretization on the numerical scheme s ability to preserve phase space structure as measured by the nonlinear spectrum of the governing equation we find that the benefits of multisymplectic integrators include improved resolution of the local conservation laws dynamical invariants and complicated phase space structures

multi symplectic runge kutta nystrom methods for nonlinear schrodinger equations with variable coefficients in this paper we consider runge kutta nystrom rkn methods applied to nonlinear schrodinger equations with variable coefficients nlsevc concatenating symplectic nystrom methods in spatial direction and symplectic runge kutta methods in temporal direction for nlsevc leads to multi symplectic integrators i e to numerical methods which preserve the multi symplectic conservation law mscl we present the corresponding discrete version of mscl it is shown that the multi symplectic rkn methods preserve not only the global symplectic structure in time but also local and global discrete charge conservation laws under periodic boundary conditions we present a 4 order multi symplectic rkn method and use it in numerical simulation of quasi periodically solitary waves for nlsevc and we compare the multi symplectic rkn method with a non multi symplectic rkn method on the errors of numerical solutions the numerical errors of discrete energy discrete momentum and discrete charge the precise conservation of discrete charge under the multi symplectic rkn discretizations is attested numerically some numerical superiorities of the multi symplectic rkn methods are revealed

a real time algorithm for the location search of discontinuous conductivities with one measurement we consider an inverse problem for finding the anomaly of discontinuous electrical conductivity by one current voltage observation we develop a real time algorithm for determining the location of the anomaly this new idea is based on the observation of the pattern of a simple weighted combination of the input current and the output voltage combined with the size estimation result this algorithm gives a good initial guess for newton type schemes we give the rigorous proof for the location search algorithm both the mathematical analysis and its numerical implementation indicate our location search algorithm is very fast stable and efficient 2001 john wiley sons inc

energy dissipation preserving birkhoffian multi symplectic methods for maxwell s equations with dissipation terms in this paper we propose two new energy dissipation preserving birkhoffian multi symplectic methods birkhoffian and birkhoffian box for maxwell s equations with dissipation terms after investigating the non autonomous and autonomous birkhoffian formalism for maxwell s equations with dissipation terms we first apply a novel generating functional theory to the non autonomous birkhoffian formalism to propose our birkhoffian scheme and then implement a central box method to the autonomous birkhoffian formalism to derive the birkhoffian box scheme we have obtained four formal local conservation laws and three formal energy global conservation laws we have also proved that both of our derived schemes preserve the discrete version of the global local conservation laws furthermore the stability dissipation and dispersion relations are also investigated for the schemes theoretical analysis shows that the schemes are unconditionally stable dissipation preserving for maxwell s equations in a perfectly matched layer pml medium and have second order accuracy in both time and space numerical experiments for problems with exact theoretical results are given to demonstrate that the birkhoffian multi symplectic schemes are much more accurate in preserving energy than both the exponential finite difference time domain fdtd method and traditional hamiltonian scheme we also solve the electromagnetic pulse emp propagation problem and the numerical results show that the birkhoffian scheme recovers the magnitude of the current source and reaction history very well even after long time propagation

approximate momentum conservation for spatial semidiscretizations of semilinear wave equations we prove that a standard second order finite difference uniform space discretization of the semilinear wave equation with periodic boundary conditions analytic nonlinearity and analytic initial data conserves momentum up to an error which is exponentially small in the stepsize our estimates are valid for as long as the trajectories of the full semilinear wave equation remain real analytic the method of proof is that of backward error analysis whereby we construct a modified equation which is itself lagrangian and translation invariant and therefore also conserves momentum this modified equation interpolates the semidiscrete system for all time and we prove that it remains exponentially close to the trigonometric interpolation of the semidiscrete system these properties directly imply approximate momentum conservation for the semidiscrete system we also consider discretizations that are not variational as well as discretizations on non uniform grids through numerical example as well as arguments from geometric mechanics and perturbation theory we show that such methods generically do not approximately preserve momentum

the multisymplectic diamond scheme we introduce a class of general purpose linear multisymplectic integrators for hamiltonian wave equations based on a diamond shaped mesh on each diamond the pde is discretized by a symplectic runge kutta method the scheme advances in time by filling in each diamond locally leading to greater efficiency and parallelization and easier treatment of boundary conditions compared to methods based on rectangular meshes

numerical analysis of a multi symplectic scheme for a strongly coupled schrodinger system abstract in this paper we analyze a multi symplectic scheme for a strongly coupled nonlinear schrodinger equations we derive a general box scheme which is equivalent to the multi symplectic scheme by reduction method based on the general box scheme we prove that the multi symplectic scheme preserves not only the multi symplectic structure of the equation but also conservation law of mass in general the multi symplectic schemes are not conservative to energy in the nonlinear case so it is difficult to obtain the estimates of numerical solutions in norm hence proofs of convergence and stability are difficult for multi symplectic schemes of nonlinear equations a deduction argument and the energy analysis method are used to prove that the numerical solution is stable for initial values and second order convergent to the exact solutions in 2 norm a fixed point theorem is introduced and used to prove the unique solvability of the numerical solutions

numerical methods for hamiltonian pdes the paper provides an introduction and survey of conservative discretization methods for hamiltonian partial differential equations the emphasis is on variational symplectic and multi symplectic methods the derivation of methods as well as some of their fundamental geometric properties are discussed basic principles are illustrated by means of examples from wave and fluid dynamics

4 manifold topology ii dwyer s filtration and surgery kernels even when the fundamental group is intractable i e not good many interesting 4 dimensional surgery problems have topological solutions we unify and extend the known examples and show how they compare to the presumed counterexamples by reference to dwyer s filtration on second homology the development brings together many basic results on the nilpotent theory of links as a special case a class of links only slightly smaller than homotopically trivial links is shown to have free slices on their whitehead doubles

on multisymplecticity of sinc gauss legendre collocation discretizations for some hamiltonian pdes abstract in this paper a new multisymplectic scheme based on sinc collocation discretization in space and gauss legendre collocation discretization in time for some hamiltonian pdes is developed the scheme preserves the multisymplectic geometry structure exactly by satisfying the discrete multisymplectic conservation law moreover the discrete energy and momentum conservative properties of the multisymplectic integrator are also discussed in order to testify the superiority of the multisymplectic method it is applied to nonlinear dirac equation numerical experiments are given to illustrate the accuracy of the approximation and the conserved quantities

multi symplectic runge kutta methods for nonlinear dirac equations in this paper we consider the multi symplectic runge kutta msrk methods applied to the nonlinear dirac equation in relativistic quantum physics based on a discovery of the multi symplecticity of the equation in particular the conservation of energy momentum and charge under msrk discretizations is investigated by means of numerical experiments and numerical comparisons with non msrk methods numerical experiments presented reveal that msrk methods applied to the nonlinear dirac equation preserve exactly conservation laws of charge and momentum and conserve the energy conservation in the corresponding numerical accuracy to the method utilized it is verified numerically that msrk methods are stable and convergent with respect to the conservation laws of energy momentum and charge and msrk methods preserve not only the inner geometric structure of the equation but also some crucial conservative properties in quantum physics a remarkable advantage of msrk methods applied to the nonlinear dirac equation is the precise preservation of charge conservation law

high order compact splitting multisymplectic method for the coupled nonlinear schrodinger equations in this paper we develop a new kind of multisymplectic integrator for the coupled nonlinear schrodinger cnls equations the cnls equations are cast into multisymplectic formulation then it is split into a linear multisymplectic formulation and a nonlinear hamiltonian system the space of the linear subproblem is approximated by a high order compact hoc method which is new in multisymplectic context the nonlinear subproblem is integrated exactly for splitting and approximation we utilize an hoc sms integrator its stability and conservation laws are investigated in theory numerical results are presented to demonstrate the accuracy conservation laws and to simulate various solitons as well for the hoc sms integrator they are consistent with our theoretical analysis

existence of stationary states for nonlinear dirac equations abstract we generalize the previous result of cazenave and vasquez on the existence of stationary states for nonlinear dirac equations of the form i 3 0 m l 0 we seek solutions which are separable in spherical coordinates and we then make use of a shooting method to solve the associated problem for ordinary differential equations

on spectral stability of the nonlinear dirac equation abstract we study the point spectrum of the nonlinear dirac equation in any spatial dimension linearized at one of the solitary wave solutions we prove that in any dimension the linearized equation has no embedded eigenvalues in the part of the essential spectrum beyond the embedded thresholds we then prove that the birth of point eigenvalues with nonzero real part the ones which lead to linear instability from the essential spectrum is only possible from the embedded eigenvalues or thresholds and therefore can not take place beyond the embedded thresholds we also prove that in the nonrelativistic limit m the point eigenvalues can only accumulate to 0 and 2 m i

derivation of the discrete conservation laws for a family of finite difference schemes abstract we present conservative schemes in finite differences applied to nonlinear odes and nonlinear klein gordon schrodinger and dirac field equations a systematic approach is used and an explicit derivation of the corresponding conservation laws is given

the qualitative analysis of a class of planar filippov systems abstract we use the theory of differential inclusions filippov transformations and some appropriate poincare maps to discuss the special case of two dimensional discontinuous piecewise linear differential systems with two zones this analysis applies to uniqueness and non uniqueness for the initial value problem stability of stationary points sliding motion solutions number of closed trajectories existence of heteroclinic trajectories connecting two saddle points forming a heteroclinic cycle and existence of the homoclinic trajectory

a uniformly accurate ua multiscale time integrator pseudospectral method for the dirac equation in the nonrelativistic limit regime we propose and rigourously analyze a multiscale time integrator fourier pseudospectral mti fp method for the dirac equation with a dimensionless parameter varepsilon in 0 1 which is inversely proportional to the speed of light in the nonrelativistic limit regime i e 0 varepsilon ll 1 the solution exhibits highly oscillatory propagating waves with wavelength o varepsilon 2 and o 1 in time and space respectively due to the rapid temporal oscillation it is quite challenging in designing and analyzing numerical methods with uniform error bounds in varepsilon in 0 1 we present the mti fp method based on properly adopting a multiscale decomposition of the solution of the dirac equation and applying the exponential wave integrator with appropriate numerical quadratures by a careful study of the error propagation and using the energy method we establish two independent error estimates via two different mathematical approaches as h m 0 frac tau 2 varepsilon 2 and h m 0 tau 2 varepsilon 2 where h is the mesh size tau is the time step and m 0 depends on the regularity of the solution these two error bounds immediately imply that the mti fp method converges uniformly and optimally in space with exponential convergence rate if the solution is smooth and uniformly in time with linear convergence rate at o tau for all varepsilon in 0 1 and optimally with quadratic convergence rate at o tau 2 in the regimes when either varepsilon o 1 or 0 varepsilon lesssim tau numerical results are reported to demonstrate that our error estimates are optimal and sharp finally the mti fp method is applied to study numerically the convergence rates of the solution of the dirac equation to those of its limiting models when varepsilon to0

structure of positive decompositions of exponential operators the solution of many physical evolution equations can be expressed as an exponential of two or more operators acting on initial data accurate solutions can be systematically derived by decomposing the exponential in a product form for time reversible equations such as the hamilton or the schr odinger equation it is immaterial whether or not the decomposition coefficients are positive in fact most symplectic algorithms for solving classical dynamics contain some negative coefficients for time irreversible systems such as the fokker planck equation or the quantum statistical propagator only positive coefficient decompositions which respect the time irreversibility of the diffusion kernel can yield practical algorithms these positive time steps only forward decompositions are a highly effective class of factorization algorithms this work presents a framework for understanding the structure of these algorithms by a suitable representation of the factorization coefficients we show that specific error terms and order conditions can be solved analytically using this framework we can go beyond the sheng suzuki theorem and derive a lower bound for the error coefficient e vtv by generalizing the framework perturbatively we can further prove that it is not possible to have a sixth order forward algorithm by including only the commutator vtv ensuremath equiv mathbf v t v mathbf the pattern of these higher order forward algorithms is that in going from the 2n mathrm th to the 2n 2 mathrm th order one must include a different commutator v t 2n ensuremath 1 v in the decomposition process

numerical methods for nonlinear dirac equation abstract this paper presents a review of the current state of the art of numerical methods for nonlinear dirac nld equation several methods are extendedly proposed for the 1 1 dimensional nld equation with the scalar and vector self interaction and analyzed in the way of the accuracy and the time reversibility as well as the conservation of the discrete charge energy and linear momentum those methods are the crank nicolson cn schemes the linearized cn schemes the odd even hopscotch scheme the leapfrog scheme a semi implicit finite difference scheme and the exponential operator splitting os schemes the nonlinear subproblems resulted from the os schemes are analytically solved by fully exploiting the local conservation laws of the nld equation the effectiveness of the various numerical methods with special focus on the error growth and the computational cost is illustrated on two numerical experiments compared to two high order accurate runge kutta discontinuous galerkin methods theoretical and numerical comparisons show that the high order accurate os schemes may compete well with other numerical schemes discussed here in terms of the accuracy and the efficiency a fourth order accurate os scheme is further applied to investigating the interaction dynamics of the nld solitary waves under the scalar and vector self interaction the results show that the interaction dynamics of two nld solitary waves depend on the exponent power of the self interaction in the nld equation collapse happens after collision of two equal one humped nld solitary waves under the cubic vector self interaction in contrast to no collapse scattering for corresponding quadric case

higher order intersections in low dimensional topology we show how to measure the failure of the whitney move in dimension 4 by constructing higher order intersection invariants of whitney towers built from iterated whitney disks on immersed surfaces in 4 manifolds for whitney towers on immersed disks in the 4 ball we identify some of these new invariants with previously known link invariants such as milnor sato levine and arf invariants we also define higher order sato levine and arf invariants and show that these invariants detect the obstructions to framing a twisted whitney tower together with milnor invariants these higher order invariants are shown to classify the existence of twisted whitney towers of increasing order in the 4 ball a conjecture regarding the nontriviality of the higher order arf invariants is formulated and related implications for filtrations of string links and 3 dimensional homology cylinders are described

whitney towers and gropes in 4 manifolds many open problems and important theorems in low dimensional topology have been formulated as statements about certain 2 complexes called gropes this paper describes a precise correspondence between embedded gropes in 4 manifolds and the failure of the whitney move in terms of iterated towers of whitney disks the flexibility of these whitney towers is used to demonstrate some geometric consequences for knot and link concordance connected to n solvability k cobordism and grope concordance the key observation is that the essential structure of gropes and whitney towers can be described by embedded unitrivalent trees which can be controlled during surgeries and whitney moves it is shown that a whitney move in a whitney tower induces an ihx jacobi relation on the embedded trees

a survey on symplectic and multi symplectic algorithms we simply review symplectic and multi symplectic algorithms in the paper these algorithms have many superiorities to general non symplectic algorithms such as they have long time behavior they can preserve the symplectic geometry structure of the hamiltonian system exactly and they can simulate original physical phenomena well some commonly used methods to construct symplectic and multi symplectic schemes are simply surveyed numerical examples are also enumerated to illustrate their performances

cobordism for links in we give an explicit finite set of based links which generates under connected sum the k cobordism classes of links we show that the union of these generating sets 2 1 and in this case c m does not appear to have a natural group structure moreover we consider only linking properties as opposed to knotting of the individual components of the link the invariants we consider are independent of the concordance class of the component knots our links are oriented ordered and tame weaker equivalence relations have recently been considered by n sato k orr and the author sa 02 c2 c3 let us immediately generalize these in order to state our main results definition 1 1 a link l l1 l 1 in 53 is b1 bm cobordant to l l1 l4 where bi are ordinal numbers if there are disjointly embedded compact connected oriented surfaces vi in 53 x 0 1 such that i vi n s3 x o 1 a vi a vi ii a vil where l vio l and d vil received by the editors august 10 1989 1980 mathematics subject classification 1985 revision primary 57m25 secondary 55s30

a note on multisymplectic fourier pseudospectral discretization for the nonlinear schrodinger equation based on the multisymplectic fourier pseudospectral scheme for the nonlinear schrodinger nls equation we investigate some discrete properties corresponding to local conservation laws of the original equation the discrete normal conservation law is proved and the error estimation of local and global energy conservation laws are also obtained numerical experiments for cubic nls equation are provided to demonstrate the consistency between the theoretical analysis and the numerical results

explicit multi symplectic extended leap frog methods for hamiltonian wave equations in this paper we study the integration of hamiltonian wave equations whose solutions have oscillatory behaviors in time and or space we are mainly concerned with the research for multi symplectic extended runge kutta nystrom erkn discretizations and the corresponding discrete conservation laws we first show that the discretizations to the hamiltonian wave equations using two symplectic erkn methods in space and time respectively lead to an explicit multi symplectic integrator eleap frogi then we derive another multi symplectic discretization using a symplectic erkn method in time and a symplectic partitioned runge kutta method which is equivalent to the well known stormer verlet method in space eleap frogii these two new multi symplectic schemes are extensions of the leap frog method the numerical stability and dispersive properties of the new schemes are analyzed numerical experiments with comparisons are presented where the two new explicit multi symplectic methods and the leap frog method are applied to the linear wave equation and the sine gordon equation the numerical results confirm the superior performance and some significant advantages of our new integrators in the sense of structure preservation

multi symplectic quasi interpolation method for hamiltonian partial differential equations abstract in this paper we propose a multi symplectic quasi interpolation method for solving multi symplectic hamiltonian partial differential equations based on the method of lines we first discretize the multi symplectic pdes using quasi interpolation method and then employ appropriate time integrators to obtain the full discrete system the local conservation properties including multi symplectic conservation laws energy conservation laws and momentum conservation laws are discussed in detail for illustration we provide two concrete examples the nonlinear wave equation and the nonlinear schrodinger equation the salient feature of our multi symplectic quasi interpolation method is that it is valid both on uniform grids and nonuniform grids the numerical results show the good accuracy and excellent conservation properties of the proposed method

multi symplectic preissmann scheme for a high order wave equation of kdv type the high order wave equation of kdv type which describes many important physical phenomena has been investigated widely in last several decades here the new multi symplectic formulations for the high order wave equation of kdv type are presented and the associated local conservation laws are shown the multi symplectic discretization of each formulation is exemplified by the multi symplectic preissmann scheme the numerical experiments are given and the results verify the efficiency of the multi symplectic preissmann scheme

explicit multi symplectic methods for klein gordon schrodinger equations in this paper we propose explicit multi symplectic schemes for klein gordon schrodinger equation by concatenating suitable symplectic runge kutta type methods and symplectic runge kutta nystrom type methods for discretizing every partial derivative in each sub equation it is further shown that methods constructed in this way are multi symplectic and preserve exactly the discrete charge conservation law provided appropriate boundary conditions in the aim of the commonly practical applications a novel 2 order one parameter family of explicit multi symplectic schemes through such concatenation is constructed and the numerous numerical experiments and comparisons are presented to show the efficiency and some advantages of the our newly derived methods furthermore some high order explicit multi symplectic schemes of such category are given as well good performances and efficiencies and some significant advantages for preserving the important invariants are investigated by means of numerical experiments

splitting multisymplectic integrators for maxwell s equations in the paper we describe a novel kind of multisymplectic method for three dimensional 3 d maxwell s equations splitting the 3 d maxwell s equations into three local one dimensional lod equations then applying a pair of symplectic runge kutta methods to discretize each resulting lod equation it leads to splitting multisymplectic integrators we say this kind of schemes to be lod multisymplectic scheme lod ms the discrete conservation laws convergence dispersive relation dissipation and stability are investigated for the schemes theoretical analysis shows that the schemes are unconditionally stable non dissipative and of first order accuracy in time and second order accuracy in space as a reduction we also consider the application of lod ms to 2 d maxwell s equations numerical experiments match the theoretical results well they illustrate that lod ms is not only efficient and simple in coding but also has almost all the nature of multisymplectic integrators

rational whitney tower filtration of links we present complete classifications of links in the 3 sphere modulo framed and twisted whitney towers in a rational homology 4 ball this provides a geometric characterization of the vanishing of the milnor invariants of links in terms of whitney towers our result also says that the higher order arf invariants which are conjectured to be nontrivial measure the potential difference between the whitney tower theory in rational homology 4 balls and that in the 4 ball extensively developed by conant schneiderman and teichner

existence of standing waves for dirac fields with singular nonlinearities we prove the existence of stationary states for nonlinear dirac equations of the form i sum limits mu 0 3 gamma mu partial mu psi m psi f left bar psi psi right psi 0 e wherem 0 andf is a singular self interaction in particular in the model case wheref s s for some 0 m there exists a solution of e of the form t x ei t x wherex0 t andx x1 x2 x3 such that has compact support if 0 1 3 then is of classc1 if 1 3 1 then is continuously differentiable except on some sphere x r where is infinite

conservation of wave action under multisymplectic discretizations in this paper we discuss the conservation of wave action under numerical discretization by variational and multisymplectic methods both the abstract wave action conservation defined with respect to a smooth periodic one parameter ensemble of flow realizations and the specific wave action based on an approximated and averaged lagrangian are addressed in the numerical context it is found that the discrete variational formulation gives rise in a natural way not only to the discrete wave action conservation law but also to a generalization of the numerical dispersion relation to the case of variable coefficients indeed a fully discrete analogue of the modulation equations arises on the other hand the multisymplectic framework gives easy access to the conservation law for the general class of multisymplectic runge kutta methods a numerical experiment confirms conservation of wave action to machine precision and suggests that the solution of the discrete modulation equations approximates the numerical solution to order cal o varepsilon on intervals of cal o varepsilon 1

local structure preserving algorithms for the good boussinesq equation in this paper we derive a series of local structure preserving algorithms for the good boussinesq equation including multisymplectic geometric structure preserving algorithms local energy preserving algorithms and local momentum preserving algorithms the outstanding advantage of the proposed algorithms is that they conserve these local structures in any time space region exactly for example the proposed local energy preserving algorithms preserve the local energy conservation law in any local domain therefore the local structure preserving algorithms overcome the shortage of global structure preserving algorithms on the boundary conditions especially with suitable boundary conditions such as periodic or homogeneous boundary conditions the local structure preserving algorithms will be global structure preserving algorithms numerical results verify the theoretical analysis

lyapunov stability of ground states of nonlinear dispersive evolution equations on presente une nouvelle demonstration de la stabilite orbitale des solitons d etat fondamental de l equation de schrodinger non lineaire pour une grande classe de non linearites on demontre la stabilite de l onde solitaire pour l equation de korteweg de vries generalisee

the multi symplecticity of partitioned runge kutta methods for hamiltonian pdes in this article we consider partitioned runge kutta prk methods for hamiltonian partial differential equations pdes and present some sufficient conditions for multi symplecticity of prk methods of hamiltonian pdes

numerical simulations of time dependent partial differential equations when a time dependent partial differential equation pde is discretized in space with a spectral approximation the result is a coupled system of ordinary differential equations odes in time this is the notion of the method of lines mol and the resulting set of odes is stiff the stiffness may be even exacerbated sometimes the linear terms are the primarily responsible for the stiffness with a rapid exponential decay of some modes as in a dissipative pde or a rapid oscillation of some modes as in a dispersive pde therefore for a time dependent pde which combines low order nonlinear terms with higher order linear terms it is desirable to use a higher order approximation both in space and in time along our research we have focused on a particular case of spectral methods the so called pseudo spectral methods to solve numerically time dependent pdes using different techniques an integrating factor in de la hoz and vadillo 2010 an exponential time differencing method in de la hoz and vadillo 2008 and differentiation matrices in the theoretical frame of matrix differential equations in de la hoz and vadillo 2012 2013a b this paper which is a unified review of those contributions aims at providing a better understanding of those methods by illustrating their variety and more importantly their power furthermore we also give emphasis to choosing adequate schemes to advance in time

on structure preserving discontinuous galerkin methods for hamiltonian partial differential equations energy conservation and multi symplecticity abstract in this paper we present and study discontinuous galerkin dg methods for one dimensional multi symplectic hamiltonian partial differential equations we particularly focus on semi discrete schemes with spatial discretization only and show that the proposed dg methods can simultaneously preserve the multi symplectic structure and energy conservation with a general class of numerical fluxes which includes the well known central and alternating fluxes applications to the wave equation the benjamin bona mahony equation the camassa holm equation the korteweg de vries equation and the nonlinear schrodinger equation are discussed some numerical results are provided to demonstrate the accuracy and long time behavior of the proposed methods numerically we observe that certain choices of numerical fluxes in the discussed class may help achieve better accuracy compared with the commonly used ones including the central fluxes

existence of periodic orbits of stable saturated systems the saturation of linear controllers produces the undesirable existence of equilibrium points or periodic orbits of the closed loop system this typical nonlinear behavior has been observed in real systems or by means of simulation of certain examples however there are only a few studies in which the properties of saturated systems have been examined rigorously and a proof of the existence of periodic orbits created by the saturation of the controller is lacking in this paper we choose an example of an open loop stable linear control system with an stabilizing saturated linear feedback to prove rigorously the existence of a periodic orbit

grope cobordism and feynman diagrams we explain how the usual algebras of feynman diagrams behave under the grope degree introduced in ct we show that the kontsevich integral rationally classifies grope cobordisms of knots in 3 space when the class is used to organize gropes this implies that the grope cobordism equivalence relations are highly nontrivial in dimension 3 we also show that the class is not a useful organizing complexity in 4 dimensions since only the arf invari ant survives in contrast measuring gropes according to height does lead to very interesting 4 dimensional information cot finally several low degree calculations are explained in particular we show that s equivalence is the same relation as grope cobordism based on the smallest tree with an internal vertex

energy evolution of multi symplectic methods for maxwell equations with perfectly matched layer boundary abstract in this paper we consider the energy evolution of multi symplectic methods for three dimensional maxwell equations with perfectly matched layer boundary and present the energy evolution laws of maxwell equations under the discretization of multi symplectic yee method and general multi symplectic runge kutta methods

split step spectral schemes for nonlinear dirac systems abstract the paper considers split step spectral schemes for the numerical integration of nonlinear dirac systems in 1 1 dimensions proofs of stability and convergence are given along with numerical experiments which clearly show the superiority of the suggested methods over standard and split step finite difference algorithms

quadratic invariants and multi symplecticity of partitioned runge kutta methods for hamiltonian pdes in this paper we study the preservation of quadratic conservation laws of runge kutta methods and partitioned runge kutta methods for hamiltonian pdes and establish the relation between multi symplecticity of runge kutta method and its quadratic conservation laws for schrodinger equations and dirac equations it reveals that multi symplectic runge kutta methods applied to equations with appropriate boundary conditions can preserve the global norm conservation and the global charge conservation respectively

multi symplectic runge kutta nystrom methods for nonsmooth nonlinear schrodinger equations abstract in this paper we discuss multi symplectic ms runge kutta nystrom rkn methods applied to a kind of nonsmooth nonlinear schrodinger equations nlses i e the nlses with delta potentials based on some theoretical suggestions proposed in 1 e g the weak reformulation of multi symplectic hamiltonian system mshs for the nlses and some newly addressed local and global conservation laws concatenating rkn algorithms for this weak mshs are novelly constructed and the multi symplecticity of them is revealed under the msrkn discretizations we show that some intrinsic characters of this weak mshs including the global symplectic structure in time and the normalization conservation laws are preserved analytically just like that for the smooth nlses 11 numerical experiments validate our methods and the errors of numerical solutions discrete energies and discrete parity symmetry are also investigated numerically comparisons with non ms schemes our ones exhibit some advantages especially for the precise preservation of normalization conservation laws

error analysis of discrete conservation laws for hamiltonian pdes under the central box discretizations the central box scheme has been the most successful of the multisymplectic integrators for hamiltonian pdes in this paper we investigate conservative properties of the central box scheme for hamiltonian pdes and derive the error formulas of discrete local and global conservation laws of energy and momentum we apply these results to the nonlinear schrodinger equation and klein gordon equation numerical experiments are presented to verify the theoretical predications

existence of localized solutions for a classical nonlinear dirac field we prove the existence of stationary states for nonlinear dirac equations of the form i gamma mu partial mu psi m psi f bar psi psi psi 0 we seek solutions which are separable in spherical coordinates and we use a shooting method for solving the associated problem of ordinary differential equations