integrability and soliton solutions for an n coupled nonlinear schrodinger system in optical fibers abstract in this paper an n coupled nonlinear schrodinger system in optical fibers with energy exchanging and different self phase modulation and cross phase modulation terms is investigated the system is found to be integrable in the sense of passing the painleve analysis the lax pair for the system is given in the form of the block matrices through generalizing that for a 2 coupled nonlinear schrodinger system with the ablowitz kaup newell segur scheme through the hirota bilinear method with an auxiliary function one and two soliton solutions are derived via symbolic computation one soliton solutions can present the profile of one peak or two peaks in different components with the asymptotic analysis two soliton solutions are found to admit the elastic collision and the propagation and collision of solitons are analyzed

maximum norm error estimates of a linear cadi method for the klein gordon schrodinger equations abstract in this paper an efficient linear compact alternating direction implicit cadi scheme with second order temporal accuracy and fourth order spatial accuracy is proposed for approximating solution to the initial boundary problem for the two dimensional klein gordon schrodinger equations furthermore maximum norm error estimates of numerical solutions are obtained by using the energy method and the mathematical induction method finally some numerical experiments are presented to support the theoretical results

twisted fourier mukai functors due to a theorem by orlov every exact fully faithful functor between the bounded derived categories of coherent sheaves on smooth projective varieties is of fourier mukai type we extend this result to the case of bounded derived categories of twisted coherent sheaves and at the same time we weaken the hypotheses on the functor as an application we get a complete description of the exact functors between the abelian categories of twisted coherent sheaves on smooth projective varieties

phase portraits of the two body problem with manev potential the manev systems are two body problems defined by a potential of the form a r b r2 where r is the distance between the two particles and a and b are arbitrary constants the hamiltonian h pr2 p 2 r2 2 a r b r2 and the angular momentum p r2 associated with manev systems are two first integrals which are independent and in involution let ih respectively ic be the set of points of the phase space on which h respectively p takes the value h respectively c since h and p are first integrals the sets ih ic and ihc ihic are invariant under the flow of the manev systems we characterize the global flow of these systems when a and b vary thus we describe the foliation of the phase space by the invariant sets ih and the foliation of ih by the invariant sets ihc

regular article a fast spectral algorithm for nonlinear wave equations with linear dispersion spectral algorithms offer very high spatial resolution for a wide range of nonlinear wave equations on periodic domains including well known cases such as the korteweg de vries and nonlinear schrodinger equations for the best computational efficiency one needs also to use high order methods in time while somehow bypassing the usual severe stability restrictions we use linearly implicit multistep methods with the innovation of choosing different methods for different ranges in fourier space high accuracy at low wavenumbers and a stability at high wavenumbers this new approach compares favorably to alternatives such as split step and integrating factor or linearly exact methods

two numerical meshless techniques based on radial basis functions rbfs and the method of generalized moving least squares gmls for simulation of coupled klein gordon schrodinger kgs equations in the present study three numerical meshless methods are being considered to solve coupled klein gordon schrodinger equations in one two and three dimensions first the time derivative of the mentioned equation will be approximated using an implicit method based on crank nicolson scheme then kansa s approach rbfs pseudo spectral ps method and generalized moving least squares gmls method will be used to approximate the spatial derivatives the proposed methods do not require any background mesh or cell structures so they are based on a meshless approach applying three techniques reduces the solution of the one two and three dimensional partial differential equations to the solution of linear system of algebraic equations as is well known the use of kansa s approach makes the coefficients matrix in the above linear system of algebraic equations to be ill conditioned and we applied lu decomposition technique but when we employ ps method fasshauer 2007 the matrix of coefficients in the obtained linear system of algebraic equations is well conditioned also the gmls technique yields a well conditioned linear system because a shifted and scaled polynomial basis will be used at the end of this paper we provide some examples on one two and three dimensions for obtaining numerical simulations also the obtained numerical results show the applicability of the proposed three methods to find the numerical solution of the kgs equations

efficient and accurate numerical methods for the klein gordon schrodinger equations in this paper we present efficient unconditionally stable and accurate numerical methods for approximations of the klein gordon schrodinger kgs equations with without damping terms the key features of our methods are based on i the application of a time splitting spectral discretization for a schrodinger type equation in kgs ii the utilization of fourier pseudospectral discretization for spatial derivatives in the klein gordon equation in kgs iii the adoption of solving the ordinary differential equations odes in phase space analytically under appropriate chosen transmission conditions between different time intervals or applying crank nicolson leap frog for linear nonlinear terms for time derivatives the numerical methods are either explicit or implicit but can be solved explicitly unconditionally stable and of spectral accuracy in space and second order accuracy in time moreover they are time reversible and time transverse invariant when there is no damping terms in kgs conserve or keep the same decay rate of the wave energy as that in kgs without or with a linear damping term keep the same dynamics of the mean value of the meson field and give exact results for the plane wave solution extensive numerical tests are presented to confirm the above properties of our numerical methods for kgs finally the methods are applied to study solitary wave collisions in one dimension 1d as well as dynamics of a 2d problem in kgs

variational iteration method for coupled nonlinear schrodinger equations in this paper we apply the variational iteration method proposed by ji huan he to simulate numerically a system of two coupled nonlinear one dimensional schrodinger equations subjected initially to a prescribed periodic wave solution test examples are given to demonstrate the accuracy and capability of the method with different wave wave interaction coefficients the accuracy of the method is verified by ensuring that the energy conservation remains almost constant the numerical results obtained with a minimum amount of computation show that the variational iteration method is much easier more convenient and efficient for solving nonlinear partial differential equations

on the l convergence of a difference scheme for coupled nonlinear schrodinger equations in this article a finite difference scheme for coupled nonlinear schrodinger equations is studied the existence of the difference solution is proved by brouwer fixed point theorem with the aid of the fact that the difference solution satisfies two conservation laws the finite difference solution is proved to be bounded in the discrete l norm then the difference solution is shown to be unique and second order convergent in the discrete l norm finally a convergent iterative algorithm is presented

a fourth order explicit schemes for the coupled nonlinear schrodinger equation abstract in this paper we derive numerical methods for solving the coupled nonlinear schrodinger equation we discretize the space derivative by central difference formulas of fourth order the resulting ordinary differential system is solved by the fourth order explicit runge kutta method neumann and periodic boundary conditions are used the method is tested for accuracy and the conserved quantities these methods conserve the three conserved quantities conserve exactly for at least five decimal places a comparison has been made with some existing methods

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

symmetric periodic solutions of the anisotropic manev problem we consider the manev potential in an anisotropic space i e such that the force acts differently in each direction using a generalization of the poincare continuation method we study the existence of periodic solutions for weak anisotropy in particular we find that the symmetric periodic orbits of the manev system are perturbed to periodic orbits in the anisotropic problem

nonconforming finite element method for a generalized nonlinear schrodinger equation abstract in this paper an efficient nonconforming finite element method fem is studied with e q 1 r o t element for a generalized nonlinear schrodinger equation first we prove a novel result of the consistency error estimate with order o h2 for e q 1 r o t element which leads to the superconvergent error estimate in broken h1 norm for semi discrete scheme while previous work only derive convergent results for this element second a linearized backward euler scheme is established and a time discrete system is introduced to split the error into two parts the temporal error and the spatial error by using a rigorous analysis for the regularity of the time discrete system and the proved characters of e q 1 r o t element the optimal l2 error estimate is obtained without any time step restrictions which leads to the numerical solution can be bounded in l norm by an inverse inequality unconditionally then the supercloseness estimate is arrived at with the above achievements third global superconvergence results are deduced through interpolated postprocessing technique at last numerical examples are provided to confirm the theoretical analysis here h is the subdivision parameter and is the time step

on convergence and stability of the explicit difference method for solution of nonlinear schrodinger equations the first boundary value problem for a nonlinear schrodinger equation is investigated the conditional convergence and stability on the initial data of the explicit three level difference scheme of dufort frankel type in c and w 2 1 norms are proved grid analogues of energy conservation laws and grid multiplicative inequalities are used

fourth order time stepping for stiff pdes a modification of the exponential time differencing fourth order runge kutta method for solving stiff nonlinear pdes is presented that solves the problem of numerical instability in the scheme as proposed by cox and matthews and generalizes the method to nondiagonal operators a comparison is made of the performance of this modified exponential time differencing etd scheme against the competing methods of implicit explicit differencing integrating factors time splitting and fornberg and driscoll s sliders for the kdv kuramoto sivashinsky burgers and allen cahn equations in one space dimension implementation of the method is illustrated by short matlab programs for two of the equations it is found that for these applications with fixed time steps the modified etd scheme is the best

on convergence and stability of a numerical scheme of coupled nonlinear schrodinger equations we consider the numerical solution of coupled nonlinear schrodinger equations we prove the stability and convergence in the l 2 space for an explicit scheme the estimations of which are used for the implicit scheme and compare both methods as a test we compare the numerical solutions of the manakov system with known analytical solitonic solutions and as an example of the general system evolution of two impulses with different group velocity model of interaction of pulses in optic fibers as a last example a rectangular pulse evolution shows asymptotic behavior typical for nonlinear schrodinger equation asymptotics with the same initial conditions

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

fourth order compact and energy conservative difference schemes for the nonlinear schrodinger equation in two dimensions abstract in this paper a fourth order compact and energy conservative difference scheme is proposed for solving the two dimensional nonlinear schrodinger equation with periodic boundary condition and initial condition and the optimal convergent rate without any restriction on the grid ratio at the order of o h 4 2 in the discrete l 2 norm with time step and mesh size h is obtained besides the standard techniques of the energy method a new technique and some important lemmas are proposed to prove the high order convergence in order to avoid the outer iteration in implementation a linearized compact and energy conservative difference scheme is derived numerical examples are given to support the theoretical analysis

on the anisotropic manev problem we consider the manev potential given by the sum between the inverse and the inverse square of the distance in an anisotropic space i e such that the force acts differently in each direction using mcgehee coordinates we blow up the collision singularity paste a collision manifold to the phase space study the flow on and near the collision manifold and find a positive measure set of collision orbits besides frontal homothetic frontal nonhomothetic and spiraling collisions and ejections we put into the evidence the surprising class of oscillatory collision and ejection orbits using the infinity manifold we further tackle capture and escape solutions in the zero energy case by finding the connection orbits between equilibria and or cycles at impact and at infinity we describe a large class of capture collision and ejection escape solutions

central configurations and total collisions for quasihomogeneous n body problems abstract we consider n body problems given by potentials of the form r a r b with a b constants 0 a b to analyze the dynamics of the problem we first prove some properties related to central configurations including a generalization of moulton s theorem then we obtain several qualitative properties for collision and near collision orbits in the manev type case a 1 at the end we point out some new relationships between central configurations relative equilibria and homothetic solutions

higher order exponential time differencing scheme for system of coupled nonlinear schrodinger equations the coupled nonlinear schrodinger equations are highly used in modeling the various phenomena in nonlinear fiber optics like propagation of pulses efficient and reliable numerical schemes are required for analysis of these models and for improvement of the fiber communication system in this paper we introduce a new version of the cox and matthews third order exponential time differencing runge kutta etd3rk scheme based on the 1 2 pade approximation to the exponential function in addition we present its local extrapolation form to improve temporal accuracy to the fourth order the developed scheme and its extrapolation are seen to be strongly stable which have ability to damp spurious oscillations caused by high frequency components in the solution a computationally efficient algorithm of the new scheme based on a partial fraction splitting technique is presented in order to investigate the performance of the novel scheme we considered the system of two and four coupled nonlinear schrodinger equations and performed several numerical experiments on them the numerical experiments showed that the developed numerical scheme provide an efficient and reliable way for computing long range solitary solutions given by coupled nonlinear schrodinger equations and conserved the conserved quantities mass and energy exactly to at least five decimal places

finite difference method for generalized zakharov equations abstract a conservative difference scheme is presented for the initial boundary value problem for generalized zakharov equations the scheme canbe implicit or semiexplicit depending on the choice of a parameter on thebasis of a priori estimates and an inequality about norms convergence of thedifference solution is proved in order 0 h2 t2 which is better than previous results introductionthe zakharov equations 20 1 1 iet exx ne 0 1 2 ntt n e 2 xx 0describe the propagation of langmuir waves in plasmas here the complexunknown function e is the slowly varying envelope of the highly oscillatoryelectric field and the unknown real function n denotes the fluctuation of the ion density about its equilibrium value the global existence of a weak solution for the zakharov equations in one dimension is proved in 19 and existence and uniqueness of a smooth solutionfor the equations are obtained provided smooth initial data are prescribed numerical methods for the zakharov equations are studied only in 5 9 10

numerical simulation of nonlinear schro dinger systems a new conservative scheme we present a new numerical scheme for nonlinear schrodinger type equations the scheme conserves the energy and charge of the systems and it is linearly implicit numerical experiments for several physical problems show that our scheme is stable accurate and efficient

a class of conservative orthogonal spline collocation schemes for solving coupled klein gordon schrodinger equations abstract a class of discrete time orthogonal spline collocation schemes for solving coupled klein gordon schrodinger equations with initial and boundary conditions are considered these schemes are constructed by using piecewise cubic hermite interpolations in space combined with finite difference methods in time it is proved that the schemes have the conservation laws of discrete energy and possess second order accuracy in maximum norm and fourth order in l 2 norm for time and space respectively the conservation laws and rate of convergence are verified in numerical experiments moreover the propagations of solitary waves and collisions of two head on solitary waves are also well simulated

numerical integration of a coupled korteweg de vries system abstract we introduce a numerical method for general coupled korteweg de vries systems the scheme is valid for solving cauchy problems for an arbitrary number of equations with arbitrary constant coefficients the numerical scheme takes its legality by proving its stability and convergence which gives the conditions and the appropriate choice of the grid sizes the method is applied to the hirota satsuma hs system and compared with its known explicit solution investigating the influence of initial conditions and grid sizes on accuracy we also illustrate the method to show the effects of constants with a transition to nonintegrable cases

linear stability of relative equilibria in the charged three body problem a relative equilibrium is a periodic orbit of the n body problem that rotates uniformly maintaining the same central configuration for all time in this paper we generalize some results of r moeckel and we apply it to study the linear stability of relative equilibria in the charged three body problem we find necessary conditions to have relative equilibria linearly stable for the collinear charged three body problem for planar relative equilibria we obtain necessary and sufficient conditions for linear stability in terms of the parameters masses and electrostatic charges in the last case we obtain a stability inequality which generalizes the routh condition of celestial mechanics we also proof the existence of spatial relative equilibria and the existence of planar relative equilibria of any triangular shape

a new class of time discretization schemes for the solution of nonlinear pdes we consider issues of stability of time discretization schemes withexacttreatment of thelinear part elp schemes for solving nonlinear pdes a distinctive feature of elp schemes is the exact evaluation of the contribution of the linear term that is if the nonlinear term of the equation is zero then the scheme reduces to the evaluation of the exponential function of the operator representing the linear term computing and applying the exponential or other functions of operators with variable coefficients in the usual manner requires evaluating dense matrices and is highly inefficient it turns out that computing the exponential of strictly elliptic operators in the wavelet system of coordinates yields sparse matrices for a finite but arbitrary accuracy this observation makes our approach practical in a number of applications in particular we consider applications of elp schemes to advection diffusion equations we study the stability of these schemes and show that both explicit and implicit elp schemes have distinctly different stability properties if compared with known implicit explicit schemes for example we describe explicit schemes with stability regions similar to those of typical implicit schemes used for solving advection diffusion equations

high order linear compact conservative method for the nonlinear schrodinger equation coupled with the nonlinear klein gordon equation abstract in this paper we design a linear compact conservative numerical scheme which preserves the original conservative properties to solve the klein gordon schrodinger equation the proposed scheme is based on using the finite difference method the scheme is three level and linear implicit priori estimate and the convergence of the finite difference approximate solutions are discussed by the discrete energy method numerical results demonstrate that the present scheme is conservative efficient and of high accuracy

unconditional stability and convergence of crank nicolson galerkin fems for a nonlinear schrodinger helmholtz system the paper is concerned with the unconditional stability and optimal l 2 error estimates of linearized crank nicolson galerkin fems for a nonlinear schrodinger helmholtz system in mathbb r d d 2 3 by introducing a corresponding time discrete system we separate the error into two parts i e the temporal error and the spatial error since the latter is tau independent the uniform boundedness of numerical solutions in l infty norm follows an inverse inequality immediately without any restrictions on time stepsize then optimal error estimates are obtained in a routine way numerical examples in both two and three dimensional spaces are given to illustrate our theoretical results

on the global dynamics of the anisotropic manev problem we study the global flow of the anisotropic manev problem which describes the planar motion of two bodies under the influence of an anisotropic newtonian potential with a relativistic correction term we first find all the heteroclinic orbits between equilibrium solutions then we generalize the poincare melnikov method and use it to prove the existence of infinitely many transversal homoclinic orbits invoking a variational principle and the symmetries of the system we finally detect infinitely many classes of periodic solutions

dufort frankel type methods for linear and nonlinear schrodinger equations in this paper we propose a so called dufort frankel type method for the linear and nonlinear schrodinger equations schemes of the dufort frankel type are explicit and have solution independent stability criteria moreover the dufort frankel type method satisfies a discrete conservation law for the schrodinger equations and this is important to get reliable numerical solutions

wavelet transformation induced multi time scaling watmus model for coupled transient electro magnetic and structural dynamics finite element analysis abstract multi functional devices that integrate electromagnetic and mechanical fields are gaining importance in a wide variety of applications the combined mechanical and electromagnetic regimes encompassed in these structures make it necessary to develop effective multi physics analysis tools at a range of temporal scales an important consideration is that the different fields governing multi physics response may have large frequency discrepancies e g the ultra high electromagnetic frequencies and moderate vibration frequencies computational analyses of these discrepant frequency problems using conventional time integration schemes can become intractable this paper develops a framework for coupling transient electromagnetic and dynamic fields to predict the evolution of electric and magnetic fields and their fluxes in a vibrating substrate undergoing finite deformation it addresses the issue of time integration with large frequency ratios by introducing a novel wavelet transformation induced multi time scaling watmus method in the finite element framework the method significantly enhances the computational efficiency in comparison with conventional single time scale integration methods an adaptive enhancement of watmus scheme allows for the optimal wavelet bases in the transformation and integration step sizes the accuracy and efficiency of the proposed watmus scheme is verified by comparing the results with the single time scale simulations of the coupled transient electromagnetic nonlinear vibration problems

approximate solutions to the zakharov equations via finite differences abstract an energy preserving linearly implicit finite difference scheme is presented for computing solutions to the periodic initial value problem for the zakharov equations solitary waves and colliding solitary waves are computed and a comparison is made with previous calculations

optimal point wise error estimate of a compact difference scheme for the klein gordon schrodinger equation abstract in this paper we propose a compact finite difference scheme for computing the klein gordon schrodinger equation kgse with homogeneous dirichlet boundary conditions the proposed scheme not only conserves the total mass and energy in the discrete level but also is linearized in practical computation except for the standard energy method a new technique is introduced to obtain the optimal convergent rate without any restriction on the grid ratios at the order of o h 4 2 in the l norm with time step and mesh size h finally numerical results are reported to test the theoretical results

exponential time differencing for stiff systems we develop a class of numerical methods for stiff systems based on the method of exponential time differencing we describe schemes with second and higher order accuracy introduce new runge kutta versions of these schemes and extend the method to show how it may be applied to systems whose linear part is nondiagonal we test the method against other common schemes including integrating factor and linearly implicit methods and show how it is more accurate in a number of applications we apply the method to both dissipative and dispersive partial differential equations after illustrating its behavior using forced ordinary differential equations with stiff linear parts

error estimate of fourth order compact scheme for linear schrodinger equations a fourth order compact difference scheme is proposed for two dimensional linear schrodinger equations with periodic boundary conditions by using the discrete energy method it is proven that the difference scheme is uniquely solvable unconditionally stable and convergent a maximum norm error estimate and thus an asymptotic expansion of the discrete solution are also obtained using the expansion of the difference solution high order approximations could be achieved by richardson extrapolations extension to three dimensional problems is also discussed numerical experiments are included to support the theoretical results and comparisons with the crank nicolson method are presented to show the effectiveness of our method

smoothing schemes for reaction diffusion systems with nonsmooth data cox and matthews s m cox p c matthews exponential time differencing for stiff systems j comput phys 176 2002 430 455 developed a class of exponential time differencing runge kutta schemes etdrk for nonlinear parabolic equations kassam and trefethen a k kassam ll n trefethen fourth order time stepping for stiff pdes siam j sci comput 26 2005 1214 1233 have shown that these schemes can suffer from numerical instability and they proposed a modified form of the fourth order etdrk4 scheme they use complex contour integration to implement these schemes in a way that avoids inaccuracies when inverting matrix polynomials but this approach creates new difficulties in choosing and evaluating the contour for larger problems neither treatment addresses problems with nonsmooth data where spurious oscillations can swamp the numerical approximations if one does not treat the problem carefully such problems with irregular initial data or mismatched initial and boundary conditions are important in various applications including computational chemistry and financial engineering we introduce a new version of the fourth order cox matthews kassam trefethen etdrk4 scheme designed to eliminate the remaining computational difficulties this new scheme utilizes an exponential time differencing runge kutta etdrk scheme using a diagonal pade approximation of matrix exponential functions while to deal with the problem of nonsmooth data we use several steps of an etdrk scheme using a sub diagonal pade formula the new algorithm improves computational efficiency with respect to evaluation of the high degree polynomial functions of matrices having an advantage of splitting the matrix polynomial inversion problem into a sum of linear problems that can be solved in parallel in this approach it is only required that several backward euler linear problems be solved in serial or parallel numerical experiments are described to support the new scheme

a gabriel theorem for coherent twisted sheaves the aim of this work is to give a generalization of gabriel s theorem on coherent sheaves to coherent twisted sheaves on noetherian schemes we start by showing that we can recover a noetherian scheme x from the category coh x of coherent twisted sheaves over x where lies in the cohomological brauer group of x this follows from the bijective correspondence between closed subsets of x and serre subcategories of finite type of coh x moreover any equivalence between coh x and coh y where x and y are noetherian schemes and alpha in br x br y induces an isomorphism between x and y

an element free based solution for nonlinear schrodinger equations using the icvmls ritz method an improved complex variable moving least squares ritz icvmls ritz method is proposed for predicting numerical solution of the two dimensional nonlinear schrodinger equation in this element free solution procedure the icvmls approximation is employed to reduce the number of unknown coefficients in the trial function it follows by the ritz procedure to derive the final algebraic equation system through discretizing the constructed energy formulation of the nonlinear schrodinger equation the central differencing scheme and newton s algorithm are adopted to solve the nonlinear equation system numerical experiments are conducted on the final form of the governing equation system to demonstrate the accuracy and efficiency of the element free icvmls ritz method by comparing the computed results with the available analytical solutions

analysis of a conservative high order compact finite difference scheme for the klein gordon schrodinger equation abstract in this paper we propose and analyze a conservative high order compact finite difference scheme for the klein gordon schrodinger kgs equation with dirichlet boundary condition by introducing the time derivative of one solution as a separate dependent variable we rewrite the kgs equation as a system of partial differential equations pdes and obtain a two level compact finite difference scheme based on the energy method and matrix knowledge on the vector form we convert the point wise form of the proposed compact scheme into an equivalent vector form and analyze its conservative and convergence properties we prove that the proposed scheme preserves the total energy and charge in the discrete forms and the convergence rate of the scheme without any restrictions on the grid ratio is at the order of o h 4 2 in l 2 norm where h and are spatial and temporal steps respectively numerical experiments are carried out to verify the performance of the scheme

the global flow of the manev problem the manev problem a two body problem given by a potential of the form a r b r2 where r is the distance between particles and a b are positive constants comprises several important physical models having its roots in research done by isaac newton we provide its analytic solution then completely describe its global flow using mcgehee coordinates and topological methods and offer the physical interpretation of all solutions we prove that if the energy constant is negative the orbits are generically precessional ellipses except for a zero measure set of initial data for which they are ellipses for zero energy the orbits are precessional parabolas and for positive energy they are precessional hyperbolas in all these cases the set of initial data leading to collisions has positive measure

finiteness of polygonal relative equilibria for generalised quasi homogeneous n body problems and n body problems in spaces of constant curvature we prove for generalisations of quasi homogeneous n body problems with centre of mass zero and n body problems in spaces of negative constant gaussian curvature that if the masses and rotation are fixed there exists for every order of the masses at most one equivalence class of relative equilibria for which the point masses lie on a circle as well as that there exists for every order of the masses at most one equivalence class of relative equilibria for which all but one of the point masses lie on a circle and rotate around the remaining point mass the method of proof is a generalised version of a proof by j m cors g r hall and g e roberts on the uniqueness of co circular central configurations for power law potentials

numerical study of the soliton waves of the coupled nonlinear schrodinger system a new six point scheme of the coupled nonlinear schrodinger cnls system is derived from the symplectic scheme to study the collision behaviors of soliton waves we prove that the new six point scheme preserves the square conservation which matches the square conservation of the cnls system numerical experiments show that the new six point scheme has excellent long time numerical behaviors from the numerical experiment results we find that the collisions of the soliton waves in the cnls system are sensitive to the collision velocity and the cross phase modulational coefficient

near collision dynamics for particle systems with quasihomogeneous potentials abstract we study collision and ejection orbits of 3 particle systems having the potential w u v where u and v are homogeneous functions of degree a and b respectively with 1 a b we show that for b 2 collision and ejection orbits tend to form asymptotically a central configuration for the case b 2 which corresponds to maneff s gravitational law we find a set of collision and ejection orbits reaching the triple collision manifold without asymptotic phase this set contains an uncountable union of manifolds and has positive measure within the set of all rectilinear solutions

a compact fourth order l stable scheme for reaction diffusion systems with nonsmooth data exponential time differencing runge kutta etdrk schemes based on diagonal pade approximations for the numerical solution of reaction diffusion systems containing nonsmooth data have the disadvantage of producing poor numerical results when the time steps are too large relative to the spatial steps on the other hand methods based on sub diagonal pade approximations do not suffer from this bottleneck in this paper we present a novel modified version of cox and matthews fourth order etdrk scheme based on sub diagonal pade approximation to matrix exponential functions in combination with compact fourth order finite difference scheme in space for direct integration of reaction diffusion problems with nonsmooth data for an efficient implementation of the scheme a partial fraction splitting technique is utilized in which it is required to solve several backward euler type linear systems at each time step moreover the design of the algorithm offers parallel implementation so we implement the proposed algorithm in parallel on two processors utilizing matlabmpi a parallel message passing version of matlab and obtain that the parallel version is computationally more efficient than the existing schemes considered in this paper we investigate the amplification factor of the scheme and plot its boundaries of stability regions which give an indication of the stability of the scheme calculation of the local truncation error and an empirical convergence analysis demonstrates the fourth order accuracy of the proposed scheme accuracy computational efficiency and reliability of the new scheme are demonstrated with numerical examples and comparing it with existing schemes

numerical comparison of five difference schemes for coupled klein gordon schrodinger equations in quantum physics in this paper we compare the classical conservative properties of five difference schemes applied to the coupled klein gordon schr odinger equations in quantum physics and investigate the numerical behaviour of the schemes in the implementation numerical results reveal merits and shortcomings of the schemes and show that all five schemes are stable in the classical conservation laws in the sense of preservation of classical conservative properties conservative schemes are better than others

optimal error analysis of cranknicolson schemes for a coupled nonlinear schrdinger system in 3d the paper is concerned with the time step condition of the commonly used semi implicit cranknicolson finite difference schemes for a coupled nonlinear schrdinger system in three dimensional space we present the optimal l2 error estimate without any restriction on time step while all previous works require certain time step conditions our approach is based on a rigorous analysis in both real and imaginary parts of the energy estimate inequality of the error function numerical examples for both two dimensional and three dimensional models are investigated and numerical results illustrate our theoretical analysis

nineteen dubious ways to compute the exponential of a matrix twenty five years later in principle the exponential of a matrix could be computed in many ways methods involving approximation theory differential equations the matrix eigenvalues and the matrix characteristic polyn

spectral method for solving the system of equations of schro dinger klein gordon field in this paper we construct the conservative spectral scheme for the periodic initial value problem for a system of equations of the complex schrodinger field interacting with the real klein gordon field and estimate the error which is xv gf nk nn xv xv nk ixnn xv 1 o k2 n 1

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

convergence of an energy preserving scheme for the zakharov equations in one space dimension an energy preserving linearly implicit finite difference scheme is presented for approximating solutions to the periodic cauchy problem for the one dimensional zakharov system of two nonlinear partial differential equations first order convergence estimates are obtained in a standard energy norm in terms of the initial errors and the usual discretization errors

compact finite difference schemes with high accuracy for one dimensional nonlinear schrodinger equation abstract in this paper two compact finite difference schemes are presented for the numerical solution of the one dimensional nonlinear schrodinger equation the discrete l 2 norm error estimates show that convergence rates of the present schemes are of order o h 4 2 numerical experiments on some model problems show that the present schemes preserve the conservation laws of charge and energy and are of high accuracy

solving the generalized nonlinear schrodinger equation via quartic spline approximation this paper is concerned with a new conservative finite difference method for solving the generalized nonlinear schrodinger gnls equation iut uxx f u 2 u 0 the numerical scheme is constructed through the semidiscretization and an application of the quartic spline approximation central difference and extrapolation formulae are used for approximating the neumann boundary conditions introduced both continuous and discrete energy conservation and the stability property are investigated the numerical method provides an efficient and reliable way for computing long time solitary solutions given by the gnls equation numerical examples are given to demonstrate our conclusions

a conservative numerical scheme for a class of nonlinear schrodinger equation with wave operator in this paper the initial boundary value problem of a class of nonlinear schrodinger equation with wave operator is considered an explicit and efficient finite difference scheme is presented this is a scheme of four levels with a discrete conservative law convergence and stability are proved

a conservative difference scheme for the zakharov equations a new conservative difference scheme is presented for the periodic initial value problem of zakharov equations the scheme can be implicit or semi explicit depending on the choice of a parameter the discretization of the initial condition is of second order accuracy which is consistent with the accuracy of the scheme on the basis of a priori estimates and an inequality about norms convergence of the difference solutions is proved in the energy norm numerical experiments with the schemes are done for several test cases computational results demonstrate that the new semi explicit scheme with a new initial approximation is more accurate and computationally efficient

unconditional optimal error estimates for bdf2 fem for a nonlinear schrodinger equation abstract this paper analyzes unconditional optimal error estimates for a 2 step backward differentiation formula bdf2 method for a nonlinear schrodinger equation in the analysis we split an error estimate into two parts one from the spatial discretization and the other from the temporal discretization we present the boundedness of the solution of the time discrete system in the certain strong norms and the error estimates for time discretization by these boundedness and temporal error estimates we obtain the l 2 error estimates without any conditions on a time step size numerical experiments are provided to validate our analysis and check the efficiency of our method

efficient semi implicit schemes for stiff systems when explicit time discretization schemes are applied to stiff reaction diffusion equations the stability constraint on the time step depends on two terms the diffusion and the reaction the part of the stability constraint due to diffusion can be totally removed if the linear diffusions are treated exactly using integration factor if or exponential time differencing etd methods for systems with severely stiff reactions those methods are not efficient because the reaction terms in if or etd are still approximated with explicit schemes in this paper we introduce a new class of semi implicit schemes which treats the linear diffusions exactly and explicitly and the nonlinear reactions implicitly a distinctive feature of the scheme is the decoupling between the exact evaluation of the diffusion terms and implicit treatment of the nonlinear reaction terms as a result the size of the nonlinear system arising from the implicit treatment of the reactions is independent of the number of spatial grid points it only depends on the number of original equations unlike the case in which standard implicit temporal schemes are directly applied to the reaction diffusion system the stability region for this class of methods is much larger than existing methods using an explicit treatment of reaction terms in particular the one with second order accuracy is unconditionally linearly stable with respect to both diffusion and reaction direct numerical simulations on test equations as well as morphogen systems from developmental biology show the new semi implicit schemes are efficient robust and accurate

compact alternating direction implicit scheme for the two dimensional fractional diffusion wave equation in this paper we consider the numerical method for solving the two dimensional fractional diffusion wave equation with a time fractional derivative of order alpha 1 alpha 2 a difference scheme combining the compact difference approach for spatial discretization and the alternating direction implicit adi method in the time stepping is proposed and analyzed the unconditional stability and h 1 norm convergence of the scheme are proved rigorously the convergence order is mathcal o tau 3 alpha h 1 4 h 4 2 where tau is the temporal grid size and h 1 h 2 are spatial grid sizes in the x and y directions respectively in addition a crank nicolson adi scheme is presented and the corresponding error estimates are also established numerical results are presented to support our theoretical analysis and indicate that the compact adi scheme reduces the storage requirement and cpu time successfully

error estimates of a conservative finite difference fourier pseudospectral method for the klein gordon schrodinger equation abstract a semi linearized time stepping method based on the finite difference in time and fourier pseudospectral discretization for spatial derivatives is constructed and analyzed for the klein gordon schrodinger equation the resulting numerical scheme is proved to conserve the total mass and energy in the discrete levels the maximum norm error estimates reflecting the second order accuracy in time and spectral accuracy in space are established by using the standard energy method coupled with the induction argument an efficient numerical algorithm is reported to speed up the evaluation of resulting algebra equation by means of the fast discrete fourier transform numerical experiments are presented to show the effectiveness of our method and to confirm our analysis

exact solutions of multi component nonlinear schrodinger and klein gordon equations applying the sec q tanh q method phys lett a 298 2002 253 we find a class of exact solution of multi component nonlinear schrodinger and klein gordon equations and generalize the correspond results in phys lett a 298 2002 253 and j phys a math gen 34 2001 4281

convergence of a conservative difference scheme for a class of klein gordon schrodinger equations in one space dimension a conservative difference scheme is presented for the initial boundary value problem of a class of klein gordon schrodinger equations the scheme can be implicit or implicit explicit depending on the choice of a parameter on the basis of the priori estimates and an inequality about norms convergence of the difference solution is proved with order o h 2 t 2 in the energy norm

existence of a lower bound for the distance between point masses of relative equilibria for generalised quasi homogeneous n body problems and the curved n body problem we prove that if for relative equilibrium solutions of a generalisation of quasi homogeneous n body problems the masses and rotation are given then the minimum distance between the point masses of such a relative equilibrium has a universal lower bound that is not equal to zero we furthermore prove that the set of such relative equilibria is compact and prove related results for n body problems in spaces of constant gaussian curvature

the global flow of the quasihomogeneous potentials of manev schwarchild type abstract in this paper we study two body problems defined by a potential of the form v r a r b r2 c r3 where r is the distance between the two particles and a b c are arbitrarily chosen constants the hamiltonian h r p r p 1 2 p r 2 p 2 r 2 a r b r 2 c r 3 and the angular momentum p r 2 are two first integrals independent and in involution let ih respectively im be the set of points on the phase space on which h respectively p takes the value h respectively m since h and p are first integrals the sets ih im and ihm ih im are invariant under the systems associated to the hamiltonian h we characterize the global flow of the systems when a b and c vary describing the foliation of the phase space by the invariant sets ih the foliation of ih by the invariant sets ihm and the movement of the flow over ihm

on tsertsvadze s difference scheme for the kuramoto tsuzuki equation in this paper we have proved the second order convergence in l norm of the tsertsvadze s difference scheme for the kuramoto tsuzuki equation the existence uniqueness and iterative algorithm are also discussed in detail furthermore a l second order convergent linearized difference scheme is given for inhomogeneous equation all results are obtained without any restrictions on the meshsizes at last a numerical example is presented

an unconditionally stable three level explicit difference scheme for the schro dinger equation with a variable coefficient a three level explicit difference scheme for the schrodinger equation with a variable coefficient is established it is proved by the discrete energy method that the scheme is unconditionally stable

stability in a rhomboidal 5 body problem with generalized central forces we consider a system of five mass points r1 r2 r3 and r4 with masses m1 m2 m and m3 m4 m moving about a single massive body r0 with mass m0 at its center which is assumed to be the origin of the coordinates system we assume that the central body r0 makes a generalized force on the four mass points and that such a force is generated by a manev s type potential i e characterized by a potential of the form 1r er2 on the other hand is assumed that the attraction between the bodies r1 r2 r3 and r4 is of the newtonian type this model represents several cases for instance when the central body is a spheroid or a radiating source first we prove the existence of three different relative rhomboidal solutions and its central configuration is as follows 1 the rhombus is a square and all primaries have equal masses 2 the rhombus is not a square but all masses are equal and 3 the rhombus is not a square and the pairs of primaries have different masses the first two cases present two pa

an efficient conservative difference scheme for fractional klein gordon schrodinger equations abstract in the paper we give an efficient conservative scheme for the fractional klein gordon schrodinger equations based on the central difference scheme the crank nicolson scheme and leap frog scheme first we use central difference scheme for discretizing the system in space direction second we use crank nicolson and leap frog scheme for discretizing the system in time direction we find that the scheme can be decoupled linearized and suitable for parallel computation to increase computing efficiency and preserve mass and energy conservation laws the convergence of the scheme is discussed and it is shown that the scheme is of the accuracy o 2 h 2 the numerical experiments are given and verify the correctness of theoretical results and the efficiency of the scheme

numerical analysis for fourth order compact conservative difference scheme to solve the 3d rosenau rlw equation in this paper a fourth order compact and energy conservative difference scheme for three dimensional rosenau rlw equation is proposed the scheme is a two level and nonlinear implicit scheme it is proved by the discrete energy method that the compact scheme is solvable the convergence and stability of the difference scheme is obtained and its numerical convergence order is o 2 h4 in the l norm we discuss an iterative algorithm for solving the nonlinear algebraical system generated by the nonlinear compact scheme and prove its convergence numerical experiment results show that the theory is accurate and the method is efficient and reliable

parametric spline schemes for the coupled nonlinear schrodinger equation abstract in this study the parametric cubic spline scheme is implemented to find the approximate solution of the coupled nonlinear schrodinger equations this scheme is based on the crank nicolson method in time and parametric cubic spline functions in space the error analysis and stability of the scheme are investigated and the numerical results show that we can get different precision schemes by choosing suitably parameter values and this scheme is unconditionally stable two problems are solved to illustrate the efficiency of the methods as well as to compare with other methods

time compact difference methods for wave propagation in discontinuous media in b gustafsson and e mossberg siam j sci comput 26 2004 pp 259 271 a fourth order one step method was constructed for the solution of wave propagation problems the method is based on the first order system form of the pde and uses a staggered grid both in space and time the method was also applied with good results to a problem with discontinuous coefficients without using any special procedure across the discontinuity in this paper we will analyze a model problem from acoustics and demonstrate the theoretical foundation for this behavior furthermore we shall present more detailed numerical experiments which confirm the theoretical results

the locally extrapolated exponential time differencing lod scheme for multidimensional reaction diffusion systems this paper introduces the local extrapolation of first order locally one dimensional exponential time differencing scheme for numerical solution of multidimensional nonlinear reaction diffusion systems this novel scheme has the benefit of solving multidimensional problems in locally one dimensional fashion by implementing sequences of tridiagonal matrix solvers instead of solving a banded system the storage size needed for solving systems in higher dimensions with this scheme is similar to that needed for one spatial dimension systems the stability monotonicity and convergence of the locally extrapolated exponential time differencing scheme have been examined stability analysis shows that the scheme is strongly stable l stable and is particularly beneficial to nonlinear partial differential equations with irregular initial data or discontinuity involving initial and boundary conditions due to its ability to damp spurious oscillations caused by high frequency components in the solution the performance of the novel scheme has been investigated by testing it on a two dimensional schnakenberg model two and three dimensional brusselator models and a three dimensional enzyme kinetics of michaelis menten type reaction diffusion model numerical experiments demonstrate the efficiency accuracy and reliability of the scheme locally extrapolated lod exponential time differencing scheme is developed multidimensional nonlinear reaction diffusion systems are considered monotonicity and stability are analyzed the order of convergence is examined

semiorthogonal decompositions and birational geometry of del pezzo surfaces over arbitrary fields we study the birational properties of geometrically rational surfaces from a derived categorical perspective in particular we give a criterion for the rationality of a del pezzo surface s over an arbitrary field namely that its derived category decomposes into zero dimensional components when s has degree at least 5 we construct explicit semiorthogonal decompositions by subcategories of modules over semisimple algebras arising as endomorphism algebras of vector bundles and we show how to retrieve information about the index of s from brauer classes and chern classes associated to these vector bundles