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

radii of starlikeness associated with the lemniscate of bernoulli and the left half plane abstract a normalized analytic function f defined on the open unit disk in the complex plane is in the class sl if zf z f z lies in the region bounded by the right half of the lemniscate of bernoulli given by w 2 1 sl radii for certain well known classes of functions are obtained radius problems associated with the left half plane are also investigated for these classes

an hr adaptive method for the cubic nonlinear schrodinger equation abstract the nonlinear schrodinger equation nlse is one of the most important equations in quantum mechanics and appears in a wide range of applications including optical fibre communications plasma physics and biomolecule dynamics it is a notoriously difficult problem to solve numerically as solutions have very steep temporal and spatial gradients adaptive moving mesh methods r adaptive attempt to optimise the accuracy obtained using a fixed number of nodes by moving them to regions of steep solution features this approach on its own is however limited if the solution becomes more or less difficult to resolve over the period of interest adaptive mesh refinement h adaptive where the mesh is locally coarsened or refined is an alternative adaptive strategy which is popular for time independent problems in this paper we consider the effectiveness of a combined method h r adaptive to solve the nlse in one space dimension simulations are presented indicating excellent solution accuracy compared to other moving mesh approaches the method is also shown to control the spatial error based on the user s input error tolerance evidence is also presented indicating second order spatial convergence using a novel monitor function to generate the adaptive moving mesh

a quantum cup length estimate for symplectic fixed points a new lower bound for the number of fixed points of hamiltonian automorphisms of closed symplectic manifolds m is established the new estimate extends the previously known estimates to the class of weakly monotone symplectic manifolds we prove for arbitrary closed symplectic manifolds with rational symplectic class that the cup length estimate holds true if the hofer energy of the hamiltonian automorphism is sufficiently small for arbitrary energy and on weakly monotone symplectic manifolds we define an analogon to the cup length based on the quantum cohomology ring of m providing a quantum cup length estimate

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

an envelope soliton problem the direct scattering problem of zakharov and shabat for the nonlinear schrodinger equation iu t u xx kappa u 2 u 0 is solved for the initial condition u text sech x the asymptotic solution t uparrow infty is a bound state of n centered solitons all moving at the same speed where n is the integral part of kappa 2 1 2

split step methods for the solution of the nonlinear schro dinger equation a split step method is used to discretize the time variable for the numerical solution of the nonlinear schrodinger equation the space variable is discretized by means of a finite difference and a fourier method these methods are analyzed with respect to various physical properties represented in the nls in particular it is shown how a conservation law dispersion and instability are reflected in the numerical schemes analytical and numerical instabilities of wave train solutions are identified and conditions which avoid the latter are derived these results are demonstrated by numerical examples

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

a quadratic b spline finite element method for solving nonlinear schrodinger equation a numerical solution of the nonlinear schrodinger nls equation is obtained by using the quadratic b spline finite element method test problems concerning motion birth and bound states of solitons of the nlg equation are studied to evaluate the method results are compared with various numerical methods

stability satisfied numerical approximates to the non analytical solutions of the cubic schrodinger equation abstract the time dependent complex schrodinger equation with cubic non linearity is solved by constructing differential quadrature algorithm based on sinc functions reduction the equation to a coupled system of real equations enables to approximate the space derivative terms by the proposed method the resultant ordinary differential equation system is integrated with respect to the time variable by using various explicit methods of lower and higher orders some initial boundary value problems containing some analytical and non analytical initial data are solved for experimental illustrations the computational errors between the analytical and numerical solutions are measured by the discrete maximum error norm in case the analytical solutions exist the two conserved quantities are calculated by using the numerical results in all cases the matrix stability analysis is implemented to control the time step size

on the lusternik schnirelmann category of symplectic manifolds and the arnold conjecture in arn appendix 9 arnold proposed a beautiful conjecture concerning the relation between the number of fixed points of certain i e exact or hamiltonian selfdiffeomorphisms of a closed symplectic manifold m and the minimum number of critical points of any smooth c function on m the first author succeeded in proving this form of the arnold conjecture r2 under the hypothesis that and c1 vanish on all spherical homology classes and that there is equality between the lusternik schnirelmann category of m and the dimension of m in this paper we use a fundamental property of category weight to show that for any closed symplectic manifold whose symplectic form vanishes on the image of the hurewicz map the required equality holds thus we show that the original form of the arnold conjecture holds for all symplectic manifolds having 2 m 0 c1 2 m

numerical experience with the nonlinear schrodinger equation abstract increasing the magnitude of the parameter multiplying the nonlinear term of the nonlinear schrodinger equation without changing the initial condition u x 0 sech x leads to bound states of an increasing number of solitons this results in very steep gradients in space and time and so provides a more severe test of numerical methods than before in particular we find that methods which satisfy various conservation laws theoretically may now fail to do so in practice various analytical and numerical results relevant to this situation are discussed and illustrated by numerical examples

conic regions and k uniform convexity abstract let k c denote a domain such that 1 k and k is a conic section with eccentricity equal to 1 k in this paper authors introduce the class of k uniformly convex functions k ucv with the property that the values of the expression 1 zf z f z lie inside the domain k necessary and sufficient conditions for membership in k ucv as well as sharp growth and distortion theorems for k uniformly convex functions are given the obtained results generalize the concept of uniform convexity due to a w goodman ann polon math 56 1991 87 92

radius problems in the class sl let sl denote the class of all analytic functions f in the unit disc u with the normalization f 0 f 0 1 0 and satisfying the conditionzf z f z 2 1 1 z u thus zf z f z is in the interior of the right half of the lemniscate of bernoulli x 2 y 2 2 2 x 2 y 2 0 the relations between sl and others classes geometrically defined are considered the radii of a convexity of a starlikeness and some of others of f sl are determined

classes of multivalent functions associated with a convolution operator we investigate several properties of the linear aouf silverman srivastava operator and associated classes of multivalent analytic functions which were introduced and studied by aouf et al m k aouf h silverman h m srivastava some families of linear operators associated with certain subclasses of multivalent functions comp math appl 55 2008 535 549 several theorems are extensions of earlier results of the above paper

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

symplectic methods for the ablowitz ladik discrete nonlinear schrodinger equation using several kinds of coordinate transformations we standardize the noncanonical symplectic structure of the ablowitz ladik model a l model of nonlinear schrodinger equation nlse then we employ some symplectic scheme to simulate the solitons motion and test the evolution of the discrete invariants of the a l model and also the conserved quantities of the original nlse in comparison with a higher order non symplectic scheme applied directly to the a l model we show the overwhelming superiorities of the symplectic method we also compare the implementation of the same symplectic scheme to different standardized hamiltonian systems resulting from different coordinate transformations and show that the symmetric coordinate transformation improves the numerical results obtained via the asymmetric one in preserving the invariants of the a l model and the original nlse

a differential quadrature algorithm for simulations of nonlinear schrodinger equation numerical simulations of nonlinear schrodinger equation are studied using differential quadrature method based on cosine expansion propogation of a soliton interaction of two solitons birth of standing and mobile solitons and bound state solutions are simulated the accuracy of the method dq is measured using maximum error norm the results are compared with some earlier works the lowest two conserved quantities are computed numerically for all cases

some observations on a static spatial remeshing method based on equidistribution principles this paper proposes a method of lines solution procedure with time and space adaptation for one dimensional systems of partial differential equations whose solutions display steep moving fronts the spatial remeshing algorithm which is a variation of the method published by sanz serna and christie and an extension suggested by revilla is a static remeshing method based on equidistribution principles the selection of the several algorithm components i e grid placement criterion spatial discretization scheme time integrator adaptation frequency and parameter tuning are investigated and illustrated with several test examples i e the cubic schrodinger equation a model of a single step reaction with diffusion and a model of flame propagation

the numerical treatment of the nonlinear schrodinger equation abstract the implicit midpoint rule for the integration in time of a fourth order accurate semidiscretization in space of the nonlinear schrodinger equation is studied the stability analysis for the present method is carried out by the conservation of the energy in time three increasingly difficult problems are tested by the method mentioned comparison with other methods is presented the results obtained here seem to be in better agreement with the exact solutions

a simple adaptive technique for nonlinear wave problems abstract a method for the numerical integration of the nonlinear schrodinger equation is derived which uses variable time steps and a moving spatial grid the benefits of adaptation are clearly demonstrated in the numerical experiments reported the simple technique employed to move the nodes can be applied with little coding effort to general one dimensional systems of pdes

local discontinuous galerkin methods for nonlinear schrodinger equations in this paper we develop a local discontinuous galerkin method to solve the generalized nonlinear schrodinger equation and the coupled nonlinear schrodinger equation l 2 stability of the schemes are obtained for both of these nonlinear equations numerical examples are shown to demonstrate the accuracy and capability of these methods

on analytical applications of stable homotopy the arnold conjecture critical points we prove the arnold conjecture for closed symplectic manifolds with 2 m 0 and catm dimm furthermore we prove an analog of the lusternik schnirelmann theorem for functions with generalized hyperbolicity property

orthogonal spline collocation methods for schr o dinger type equations in one space variable we examine the use of orthogonal spline collocation for the semi discreti za tion of the cubic schr o dinger equation and the two dimensional parabolic equation of tappert in each case an optimal order l 2 estimate of the error in the semidiscrete approximation is derived for the cubic schr o dinger equation we present the results of numerical experiments in which the integration in time is performed using a routine from a software library