symplectic methods for the nonlinear schrodinger equation abstract in this paper we show that the spatial discretization of the nonlinear schrodinger equation leads to a hamiltonian system which can be simulated with symplectic numerical schemes in particular we apply two symplectic integrators to the nonlinear schrodinger equation and we demonstrate that they are able to produce accurate results and to preserve very well the invariants of the original system such as the energy and charge

on numerical methods for hamiltonian pdes and a collocation method for the vlasov maxwell equations abstract hamiltonian partial differential equations often have implicit conservation laws constants of the motion embedded within them it is not in general possible to preserve these conservation laws simply by discretization in conservative form because there is frequently only one explicit conservation law however by using weighted residual methods and exploiting the hamiltonian structure of the equations it is shown that at least some of the conservation laws are preserved in a method of lines continuous in time in particular the hamiltonian can always be exactly preserved as a constant of the motion other conservation laws in particular linear and quadratic casimirs and momenta can sometimes be conserved too depending on the details of the equations under consideration and the form of discretization employed collocation methods also offer automatic conservation of linear and quadratic casimirs some standard discretization methods when applied to hamiltonian problems are shown to be derived from a numerical approximation to the exact poisson bracket of the system a method for the vlasov maxwell equations based on legendre gauss lobatto collocation is presented as an example of these ideas

dispersive properties of multisymplectic integrators multisymplectic ms integrators i e numerical schemes which exactly preserve a discrete space time symplectic structure are a new class of structure preserving algorithms for solving hamiltonian pdes in this paper we examine the dispersive properties of ms integrators for the linear wave and sine gordon equations in particular a leapfrog in space and time scheme a member of the lobatto runge kutta family of methods and the preissman box scheme are considered we find the numerical dispersion relations are monotonic and that the sign of the group velocity is preserved the group velocity dispersion gvd is found to provide significant information and succinctly explain the qualitative differences in the numerical solutions obtained with the different schemes further the numerical dispersion relations for the linearized sine gordon equation provides information on the ability of the ms integrators to capture the sine gordon dynamics we are able to link the numerical dispersion relations to the total energy of the various methods thus providing information on the coarse grid behavior of ms integrators in the nonlinear regime

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

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

chaotic and homoclinic behavior for numerical discretizations of the nonlinear schro dinger equation abstract certain conservative discretizations of the nls can produce irregular behavior we consider the diagonal discretization as a conservative perturbation of the integrable discretization and study the homoclinic crossings in its nonlinear spectrum we find that irregularity sets in when two homoclinic structures are present and in this case many and continual homoclinic crossings occur throughout the irregular time series we indicate a melnikov analysis to study the consequences of this homoclinic behavior

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

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

error propagation in numerical approximations near relative equilibria we study the propagation of errors in the numerical integration of perturbations of relative equilibrium solutions of hamiltonian differential equations with symmetries first it is shown that taking an initial perturbation of a relative equilibrium the corresponding solution is related in a first approximation to another relative equilibrium with the parameters perturbed from the original then this is used to prove that for stable relative equilibria error growth with respect to the perturbed solution is in general quadratic but only linear for schemes that preserve the invariant quantities of the problem in this sense the conclusion is similar to the one obtained when integrating unperturbed relative equilibria numerical experiments illustrate the results

finite element approximation for 2nd order elliptic eigenvalue problems with nonlocal boundary or transition conditions in this paper we use two symplectic schemes to simulate the ablowitz ladik model associated to the cubic nonlinear schrodinger equation and we compare them with nonsymplectic methods

hybrid chebyshev polynomial scheme for solving elliptic partial differential equations abstract in this paper we propose hybrid chebyshev polynomial scheme hcps which couples the chebyshev polynomial scheme and the method of fundamental solutions into a single matrix system this hybrid formulation requires solving only one system of equations and opens up the possibilities for solving a large class of partial differential equations in this paper we consider various boundary value problems and in particular the challenging cauchy navier equation the solution is approximated by the sum of the particular solution and the homogeneous solution chebyshev polynomials are used to approximate a particular solution of the given partial differential equation and the method of fundamental solutions is used to approximate the homogeneous solution numerical results show that our proposed approach is efficient accurate and stable

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

model order reduction for nonlinear schrodinger equation we apply the proper orthogonal decomposition pod to the nonlinear schrodinger nls equation to derive a reduced order model the nls equation is discretized in space by finite differences and is solved in time by structure preserving symplectic mid point rule a priori error estimates are derived for the pod reduced dynamical system numerical results for one and two dimensional nls equations coupled nls equation with soliton solutions show that the low dimensional approximations obtained by pod reproduce very well the characteristic dynamics of the system such as preservation of energy and the solutions

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

on the numerical solution of the sine gordon equation in this the first of two papers on the numerical solution of the sine gordon equation we investigate the numerical behavior of a double discrete completely integrable discretization of the sine gordon equation for certain initial values in the vicinity of homoclinic manifolds this discretization admits an instability in the form of grid scale oscillations we clarify the nature of the instability through an analytical investigation supported by numerical experiments in particular a perturbation analysis of the associated linear spectral problem shows that the initial values used for the numerical experiments lie exponentially close to a homoclinic manifold this paves the way for the second paper where we use the non linear spectrum as a basis for comparing different numerical schemes

energy conserving methods for the nonlinear schrodinger equation in this paper we further develop recent results in the numerical solution of hamiltonian partial differential equations pdes brugnano et al 2015 by means of energy conserving methods in the class of line integral methods in particular the runge kutta methods named hamiltonian boundary value methods hbvms we shall use hbvms for solving the nonlinear schrodinger equation nlse of interest in many applications we show that the use of energy conserving methods able to conserve a discrete counterpart of the hamiltonian functional confers more robustness on the numerical solution of such a problem

on the numerical integration of ordinary differential equations by symmetric composition methods differential equations of the form dot x x a b are considered where the vector fields a and b can be integrated exactly enabling numerical integration of x by composition of the flows of a and b various symmetric compositions are investigated for order complexity and reversibility free lie algebra theory gives simple formulae for the number of determining equations for a method to have a particular order a new more accurate way of applying the methods thus obtained to compositions of an arbitrary first order integrator is described and tested the determining equations are explored and new methods up to 100 times more accurate at constant work than those previously known are given

inverse buckling problem for inhomogeneous columns abstract this study is motivated by our recent study elishakoff i rollot o 1999 new closed form solutions for buckling of a variable stiffness column by mathematica journal of sound and vibration 224 172 182 which presented new closed form solutions for buckling of columns with variable stiffness the column was simply supported at both ends in the present study we rederive in other means the same solution and moreover obtain new solutions for two other sets of the boundary conditions by posing an inverse buckling problem it is shown that the buckling load is dependent upon a single stiffness coefficient by suitable choice of this parameter the buckling load can be made arbitrarily large

symplectic integration of hamiltonian wave equations the numerical integration of a wide class of hamiltonian partial differential equations by standard symplectic schemes is discussed with a consistent hamiltonian approach we discretize the hamiltonian and the poisson structure separately then form the the resulting ode s the stability accuracy and dispersion of different explicit splitting methods are analyzed and we give the circumstances under which the best results can be obtained in particular when the hamiltonian can be split into linear and nonlinear terms many different treatments and examples are compared

a new solution of schrodinger equation based on symplectic algorithm an explicit three dimensional symplectic finite difference time domain method is introduced to solve the schrodinger equation the method is obtained by using a symplectic scheme with a propagator of exponential differential operators for the time direction approximation and fourth order collocated finite differences for the space discretization firstly a high order symplectic framework for discretizing the schrodinger equation is presented secondly the comparisons on numerical stability dispersion are also provided between the new symplectic scheme and other commonly used methods finally numerical examples are given to evaluate the performance of the proposed method and the advantages on the accuracy and stability are also further demonstrated