point wise error estimate of a conservative difference scheme for the fractional schrodinger equation in this paper a new conservative difference scheme is proposed for solving the nonlinear space fractional schrodinger equation the riesz space fractional derivative is approximated by the second order accurate weighted and shifted grunwald difference operator building on a careful analysis of this operator the conservation properties including the mass and energy are analyzed and some norm equivalences are established then the numerical solution is shown to be bounded in various norms uniformly with respect to the discretization parameters and optimal order bounds on the global error of the scheme in the l 2 norm semi h 2 norm and l norm are derived the uniform bounds on the numerical solutions as well as the error bounds hold unconditionally in the sense that no restriction on the size of the time step in terms of the spatial mesh size needs to be assumed numerical tests are performed to support our theoretical results

gauss seidel type methods for energy states of a multi component bose einstein condensate in this paper we propose two iterative methods a jacobi type iteration ji and a gauss seidel type iteration gsi for the computation of energy states of the time independent vector gross pitaevskii equation vgpe which describes a multi component bose einstein condensate bec a discretization of the vgpe leads to a nonlinear algebraic eigenvalue problem naep we prove that the gsi method converges locally and linearly to a solution of the naep if and only if the associated minimized energy functional problem has a strictly local minimum the gsi method can thus be used to compute ground states and positive bound states as well as the corresponding energies of a multi component bec numerical experience shows that the gsi converges much faster than ji and converges globally within 10 20 steps

on the computation of ground state and dynamics of schrodinger poisson slater system in this paper we deal with the computation of ground state and dynamics of the schrodinger poisson slater sps system to this end backward euler and time splitting pseudospectral methods are proposed for the nonlinear schrodinger equation with the nonlocal hartree potential approximated by solving a poisson equation the approximation approaches for the hartree potential include fast convolution algorithms which are accelerated by using fft in 1d and fast multipole method fmm in 2d and 3d and sine fourier pseudospectral methods the inconsistency in 0 mode in fourier pseudospectral approach is pointed out which results in a significant loss of high order of accuracy as expected for spectral methods numerical comparisons show that in 1d the fast convolution and sine pseudospectral approaches are compatible while in 3d the fast convolution approach based on fmm is second order accurate and the fourier pseudospectral approach is better than it from both efficiency and accuracy point of view among all these approaches the sine pseudospectral one is the best candidate in the numerics of the sps system finally we apply the backward euler and time splitting sine pseudospectral methods to study the ground state and dynamics of 3d sps system in different setups

realizing the berezin integral as a superspace contour integral integration on supermanifolds using contours and the covariant differential forms of kostant b kostant graded manifolds graded lie theory and prequantisation in lecture notes in mathematics vol 570 springer berlin 1977 is described the good properties that these integrals naturally acquire are considered it is then shown that the formal process of integration over even and odd variables introduced by berezin f a berezin the method of second quantization academic new york 1966 using partly covariant and partly contravariant forms can be regarded as a special case of these contour integrals over covariant forms

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

long time asymptotic expansion for the damped semilinear wave equation the first initial boundary value problem is considered for the damped semilinear wave equation with the quadratic nonlinearity for small initial data and homogeneous boundary conditions its solution is constructed in the form of a series in the eigenfunctions of the laplace operator the long time asymptotic expansion is obtained which shows the nonlinear effects of amplitude and frequency multiplication the same results hold for the admissible initial data that are not small

analysis on generalized superspaces the analysis over commutative algebras generalized supercommutative algebras that is differentiation and integration for functions defined on superspace over a commutative algebra is studied

dimension reduction of the schrodinger equation with coulomb and anisotropic confining potentials we consider dimension reduction for the three dimensional 3d schrodinger equation with the coulomb interaction and an anisotropic confining potential to lower dimensional models in the limit of infinitely strong confinement in one or two space dimensions and obtain formally the surface adiabatic model sam or surface density model sdm in two dimensions 2d and the line adiabatic model lam in one dimension 1d efficient and accurate numerical methods for computing ground states and dynamics of the sam sdm and lam models are presented based on efficient and accurate numerical schemes for evaluating the effective potential in lower dimensional models they are applied to find numerically convergence and convergence rates for the dimension reduction from 3d to 2d and 3d to 1d in terms of ground state and dynamics which confirm some existing analytical results for the dimension reduction in the literature in particular we explain and demonstrate that the standard schrodinger poisson system in 2

global properties of supermanifolds we construct new examples of supermanifolds and determine the vector bundle structure of the supermanifolds commonly used in physics we show that any supermanifold admits a foliation whose leaves are locally tangent to the soul directions in the coordinate charts and which is one of a nested sequence of foliations we point out that the existence of these foliations implies restrictions on the possible topologies of supermanifolds for example a compact supermanifold with a single even dimension must have vanishing euler characteristic we also show that a globally defined superfield on a nice compact supermanifold must be constant along the leaves of the foliations by this mechanism the global topology of a supermanifold can be used to impose physically interesting constraints on superfields as an example we exhibit a supermanifold which has the local geometry of flat superspace but is such that all globally defined superfields are chiral

on the ground states and dynamics of space fractional nonlinear schrodinger gross pitaevskii equations with rotation term and nonlocal nonlinear interactions in this paper we propose some efficient and robust numerical methods to compute the ground states and dynamics of fractional schrodinger equation fse with a rotation term and nonlocal nonlinear interactions in particular a newly developed gaussian sum gausum solver is used for the nonlocal interaction evaluation 31 to compute the ground states we integrate the preconditioned krylov subspace pseudo spectral method 4 and the gausum solver for the dynamics simulation using the rotating lagrangian coordinates transform 14 we first reformulate the fse into a new equation without rotation then a time splitting pseudo spectral scheme incorporated with the gausum solver is proposed to simulate the new fse in parallel to the numerical schemes we also prove some existence and nonexistence results for the ground states dynamical laws of some standard quantities including the mass energy angular momentum and the center of mass are stated the ground states properties with respect to the fractional order and or rotating frequencies dynamics involving decoherence and turbulence together with some interesting phenomena are reported

finite dimensional exponential attractors for semilinear wave equations with damping abstract we consider the initial value problem for a class of second order evolution equations that includes among others the 3d sine gordon equation with damping and the 3d klein gordon type equations with damping we show the existence of a set with finite fractal dimension that contains the global attractor and attracts all smooth solutions at an exponential rate

fast calculation of energy and mass preserving solutions of schrodinger poisson systems on unbounded domains this paper deals with the numerical solution of the time dependent schrodinger poisson system in the spherically symmetric case since the problem is posed on an unbounded domain one has to introduce artificial boundary conditions to confine the computational domain the main topic of this work is the construction of a so called discrete transparent boundary condition tbc for a crank nicolson type predictor corrector scheme for solving the schrodinger poisson system this scheme has the property of mass and energy conservation exactly on the discrete level we propose different strategies for the discrete tbc and present an efficient implementation finally a numerical example illustrate the findings and shows the comparison results between the different approaches

stability of standing waves for the fractional schrodinger choquard equation abstract in this paper we consider the stability of standing waves for the fractional schrodinger choquard equation with an l 2 critical nonlinearity by using the profile decomposition of bounded sequences in h s and variational methods we prove that the standing waves are orbitally stable we extend the study of bhattarai for a single equation bhattarai 2017 to the l 2 critical case

inertial manifolds for nonlinear evolutionary equations abstract in this paper we introduce the concept of an inertial manifold for nonlinear evolutionary equations in particular for ordinary and partial differential equations these manifolds which are finite dimensional invariant lipschitz manifolds seem to be an appropriate tool for the study of questions related to the long time behavior of solutions of the evolutionary equations the inertial manifolds contain the global attractor they attract exponentially all solutions and they are stable with respect to perturbations furthermore in the infinite dimensional case they allow for the reduction of the dynamics to a finite dimensional ordinary differential equation

superconformal primary fields on a graded riemann sphere primary superfields for a two dimensional euclidean superconformal field theory are constructed as sections of a sheaf over a graded riemann sphere the transformation law is found to be the same as that of an o n extended primary field the construction is then applied to the n 3 neveu schwarz case various quantities in the n 3 theory are calculated such as elements of the super mobius group and the two point function applications of the construction to calculate three point functions and fusion rules in a manifestly supersymmetric fashion are discussed

solution of time independent schrodinger equation by the imaginary time propagation method numerical solution of eigenvalues and eigenvectors of large matrices originating from discretization of linear and non linear schrodinger equations using the imaginary time propagation itp method is described convergence properties and accuracy of 2nd and 4th order operator splitting methods for the itp method are studied using numerical examples the natural convergence of the method is further accelerated with a new dynamic time step adjustment method the results show that the itp method has better scaling with respect to matrix size as compared to the implicitly restarted lanczos method an efficient parallel implementation of the itp method for shared memory computers is also demonstrated

finite dimensionality and regularity of attractors for a 2 d semilinear wave equation with nonlinear dissipation we consider a semilinear wave equation defined on a two dimensional bounded domain with a nonlinear dissipation our main result is that the flow generated by the model is attracted by a finite dimensional global attractor in addition this attractor has additional regularity properties that depend on regularity properties of nonlinear functions in the equation to our knowledge this is a first result of this type in the context of higher dimensional wave equations

variational calculus on supermanifolds and invariance properties of superspace field theories the foundations of a variational calculus on fibered supermanifolds are outlined giving applications to the formulation of superspace field theories utiyama theorems and conservation laws related to local gauge and general invariance of the theory are proved

consistent superspace integration the standard berezin method for integration over odd variables is combined in a new way with de witt s contour method for integration over even grassmann variables to give a new method of superspace integration it is shown that this integral unlike the standard superspace integral is invariant under coordinate transformations in superspace the relation between the new method and the standard method is discussed

numerical study of time splitting spectral discretizations of nonlinear schrodinger equations in the semiclassical regimes in this paper we study the performance of time splitting spectral approximations for general nonlinear schrodinger equations nls in the semiclassical regimes where the planck constant varepsilon is small the time splitting spectral approximation under study is explicit unconditionally stable and conserves the position density in l 1 moreover it is time transverse invariant and time reversible when the corresponding nls is extensive numerical tests are presented for weak strong focusing defocusing nonlinearities for the gross pitaevskii equation and for current relaxed quantum hydrodynamics the tests are geared towards the understanding of admissible meshing strategies for obtaining correct physical observables in the semiclassical regimes furthermore comparisons between the solutions of the nls and its hydrodynamic semiclassical limit are presented

long time dynamics of von karman semi flows with non linear boundary interior damping abstract we study von karman evolution equations with non linear dissipation and with partially clamped and partially free boundary conditions two distinctive mechanisms of dissipation are considered i internal dissipation generated by non linear operator and ii boundary dissipation generated by shear forces friction acting on a free part of the boundary the main emphasis is given to the effects of boundary dissipation under suitable hypotheses we prove existence of a compact global attractor and finiteness of its fractal dimension we also show that any solution is stabilized to an equilibrium and estimate the rate of the convergence which in turn depends on the behaviour at the origin of the functions describing the dissipation

order estimates in time of splitting methods for the nonlinear schrodinger equation in this paper we consider the nonlinear schrodinger equation u t i delta u f u 0 in two dimensions we show by an operator theoretic proof that the well known lie and strang formulae which are splitting methods are approximations of the exact solution of order 1 and 2 in time

global attractors for von karman evolutions with a nonlinear boundary dissipation abstract dynamic von karman equations with a nonlinear boundary dissipation are considered questions related to long time behaviour existence and structure of global attractors are studied it is shown that a nonlinear boundary dissipation with a large damping parameter leads to an existence of global compact attractor for all weak finite energy solutions this result has been known in the case of full interior dissipation but it is new in the case when the boundary damping is the main dissipative mechanism in the system in addition we prove that fractal dimension of the attractor is finite the proofs depend critically on the infinite speed of propagation associated with the von karman model considered

numerical methods for computing ground states and dynamics of nonlinear relativistic hartree equation for boson stars efficient and accurate numerical methods are presented for computing ground states and dynamics of the three dimensional 3d nonlinear relativistic hartree equation both without and with an external potential this equation was derived recently for describing the mean field dynamics of boson stars in its numerics due to the appearance of pseudodifferential operator which is defined in phase space via symbol spectral method is more suitable for the discretization in space than other numerical methods such as finite difference method etc for computing ground states a backward euler sine pseudospectral besp method is proposed based on a gradient flow with discrete normalization and respectively for computing dynamics a time splitting sine pseudospectral tssp method is presented based on a splitting technique to decouple the nonlinearity both besp and tssp are efficient in computation via discrete sine transform and are of spectral accuracy in spatial discretization tssp is of second order accuracy in temporal discretization and conserves the normalization in discretized level in addition when the external potential and initial data for dynamics are spherically symmetric the original 3d problem collapses to a quasi 1d problem for which both besp and tssp methods are extended successfully with a proper change of variables finally extensive numerical results are reported to demonstrate the spectral accuracy of the methods and to show very interesting and complicated phenomena in the mean field dynamics of boson stars

an attractor for a nonlinear dissipative wave equation of kirchhoff type abstract in this paper we prove the existence and some absorbing properties of an attractor in a local sense for the initial boundary value problem of a quasilinear wave equation of kirchhoff type with a standard dissipation u t

a projection gradient method for computing ground state of spin 2 bose einstein condensates in this paper a projection gradient method is presented for computing ground state of spin 2 bose einstein condensates bec we first propose the general projection gradient method for solving energy functional minimization problem under multiple constraints in which the energy functional takes real functions as independent variables we next extend the method to solve a similar problem where the energy functional now takes complex functions as independent variables we finally employ the method into finding the ground state of spin 2 bec the key of our method is by constructing continuous gradient flows cgfs the ground state of spin 2 bec can be computed as the steady state solution of such cgfs we discretized the cgfs by a conservative finite difference method along with a proper way to deal with the nonlinear terms we show that the numerical discretization is normalization and magnetization conservative and energy diminishing numerical results of the ground state and their energy of spin 2 bec are reported to demonstrate the effectiveness of the numerical method

sharp threshold of blow up and scattering for the fractional hartree equation abstract we consider the fractional hartree equation in the l 2 supercritical case and find a sharp threshold of the scattering versus blow up dichotomy for radial data if m u 0 s s c s c e u 0 m q s s c s c e q and m u 0 s s c s c u 0 h s 2 m q s s c s c q h s 2 then the solution u t is globally well posed and scatters if m u 0 s s c s c e u 0 m q s s c s c e q and m u 0 s s c s c u 0 h s 2 m q s s c s c q h s 2 the solution u t blows up in finite time this condition is sharp in the sense that the solitary wave solution e i t q x is global but not scattering which satisfies the equality in the above conditions here q is the ground state solution for the fractional hartree equation

integration on supermanifolds and a generalized cartan calculus a suggestion by berezin for a method of integration on supermanifolds is given a precise differential geometric meaning by assuming that a supermanifold is the total space of a fibre bundle with connection the relevant objects for integration are identified as suitable horizontal vertical projections of hyperforms the latter are generalizations of differential forms having both covariant and contravariant indices the exterior calculus of these projected hyperforms is developed analogously to the cartan calculus by introducing appropriate derivations and determining their commutators respectively anticommutators

computing the ground state solution of bose einstein condensates by a normalized gradient flow in this paper we present a continuous normalized gradient flow cngf and prove its energy diminishing property which provides a mathematical justification of the imaginary time method used in the physics literature to compute the ground state solution of bose einstein condensates bec we also investigate the energy diminishing property for the discretization of the cngf two numerical methods are proposed for such discretizations one is the backward euler centered finite difference befd method the other is an explicit time splitting sine spectral tssp method energy diminishing for befd and tssp for the linear case and monotonicity for befd for both linear and nonlinear cases are proven comparison between the two methods and existing methods e g crank nicolson finite difference cnfd or forward euler finite difference fefd shows that befd and tssp are much better in terms of preserving the energy diminishing property of the cngf numerical results in one two and three dimensions with magnetic trap confinement potential as well as a potential of a stirrer corresponding to a far blue detuned gaussian laser beam are reported to demonstrate the effectiveness of befd and tssp methods furthermore we observe that the cngf and its befd discretization can also be applied directly to compute the first excited state solution in bec when the initial data is chosen as an odd function

exponential attractors for weakly damped wave equation with sub quintic nonlinearity abstract in this paper we study the existence of exponential attractors for the weakly damped wave equation u t t u t u f u g in a bounded smooth domain of r 3 based on the recent extension of the strichartz estimate for the case of a bounded domain we establish an exponential attractor in a natural energy space for the case of sub quintic nonlinearity the results obtained essentially improve and complete some previous works

on splitting methods for schrodinger poisson and cubic nonlinear schrodinger equations we give an error analysis of strang type splitting integrators for nonlinear schrodinger equations for schrodinger poisson equations with an h 4 regular solution a first order error bound in the h 1 norm is shown and used to derive a second order error bound in the l 2 norm for the cubic schrodinger equation with an h 4 regular solution first order convergence in the h 2 norm is used to obtain second order convergence in the l 2 norm basic tools in the error analysis are lie commutator bounds for estimating the local error and h m conditional stability for error propagation where m 1 for the schrodinger poisson system and m 2 for the cubic schrodinger equation

on time splitting spectral approximations for the schrodinger equation in the semiclassical regime in this paper we study time splitting spectral approximations for the linear schrodinger equation in the semiclassical regime where the planck constant e is small in this regime the equation propagates oscillations with a wavelength of o e and finite difference approximations require the spatial mesh size h o e and the time step k o e in order to obtain physically correct observables much sharper mesh size constraints are necessary for a uniform l2 approximation of the wave function the spectral time splitting approximation under study will be proved to be unconditionally stable time reversible and gauge invariant it conserves the position density and gives uniform l2 approximation of the wave function for k o e and h o e extensive numerical examples in both one and two space dimensions and analytical considerations based on the wigner transform even show that weaker constraints e g k independent of e and h o e are admissible for obtaining correct observables finally we address the application to nonlinear schrodinger equations and conduct some numerical experiments to predict the corresponding admissible meshing strategies

two level discretization techniques for ground state computations of bose einstein condensates this work presents a new methodology for computing ground states of bose einstein condensates based on finite element discretizations on two different scales of numerical resolution in a preprocessing step a low dimensional coarse generalized finite element space is constructed it is based on a local orthogonal decomposition of the solution space and exhibits high approximation properties the nonlinear eigenvalue problem that characterizes the ground state is solved by some suitable iterative solver exclusively in this low dimensional space without significant loss of accuracy when compared with the solution of the full fine scale problem the preprocessing step is independent of the types and numbers of bosons a postprocessing step further improves the accuracy of the method we present rigorous a priori error estimates that predict convergence rates h 3 for the ground state eigenfunction and h 4 for the corresponding eigenvalue without pre asymptotic effects h being the coarse scale discretization parameter numerical experiments indicate that these high rates may still be pessimistic

on the existence of global integral forms on supermanifolds it is shown that the berezin approach to integration on supermanifolds can be applied to cases where the supermanifold is a twisted extension of a real manifold this is done by showing that supermanifolds admit a subatlas of coordinate charts with transition functions of a quite restricted kind

an analysis of schwarz waveform relaxation domain decomposition methods for the imaginary time linear schrodinger and gross pitaevskii equations the aim of this paper is to derive and numerically validate some asymptotic estimates of the convergence rate of classical and quasi optimal schwarz waveform relaxation swr domain decomposition methods applied to the computation of the stationary states of the one dimensional linear and nonlinear schrodinger equations with a space dependent potential although swr methods are currently used for efficiently solving high dimensional partial differential equations their convergence analysis and most particularly obtaining expressions of their convergence rate remains largely open even in one dimension except in simple cases in this work we tackle this problem for linear and nonlinear one dimensional schrodinger equations by developing techniques which can be extended to higher dimensional problems and other types of pdes the approach combines the method developed in gander and halpern siam j num anal 45 2 666 697 2007 for the linear advection reaction diffusion equation and the theory of inhomogeneous pseudodifferential operators in conjunction with the associated symbolic asymptotic expansions for computing the stationary states we consider the imaginary time formulation of the schrodinger equation based on the normalized gradient flow ngf method and use a semi implicit euler scheme for the discretization some numerical results in the one dimensional case illustrate the analysis for both the linear schrodinger and gross pitaevskii equations

efficient numerical methods for computing ground states and dynamics of dipolar bose einstein condensates new efficient and accurate numerical methods are proposed to compute ground states and dynamics of dipolar bose einstein condensates becs described by a three dimensional 3d gross pitaevskii equation gpe with a dipolar interaction potential due to the high singularity in the dipolar interaction potential it brings significant difficulties in mathematical analysis and numerical simulations of dipolar becs in this paper by decoupling the two body dipolar interaction potential into short range or local and long range interactions or repulsive and attractive interactions the gpe for dipolar becs is reformulated as a gross pitaevskii poisson type system based on this new mathematical formulation we prove rigorously existence and uniqueness as well as nonexistence of the ground states and discuss the existence of global weak solution and finite time blow up of the dynamics in different parameter regimes of dipolar becs in addition a backward euler sine pseudospectral method is presented for computing the ground states and a time splitting sine pseudospectral method is proposed for computing the dynamics of dipolar becs due to the adoption of new mathematical formulation our new numerical methods avoid evaluating integrals with high singularity and thus they are more efficient and accurate than those numerical methods currently used in the literatures for solving the problem extensive numerical examples in 3d are reported to demonstrate the efficiency and accuracy of our new numerical methods for computing the ground states and dynamics of dipolar becs

short note a short note on simplified pseudospectral methods for computing ground state and dynamics of spherically symmetric schrodinger poisson slater system in a recent paper we proposed and compared various approaches to compute the ground state and dynamics of the schrodinger poisson slater sps system for general external potential and initial condition concluding that the methods based on sine pseudospectral discretization in space are the best candidates this note is concerned with the case that the external potential and initial condition are spherically symmetric for the sps system with spherical symmetry via applying a proper change of variables into the reduced quasi 1d model we simplify the methods proposed for the general 3d case such that both the memory and computational load are significantly reduced

efficient numerical methods for computing ground states of spin 1 bose einstein condensates based on their characterizations in this paper we propose efficient numerical methods for computing ground states of spin 1 bose einstein condensates becs with without the ioffe pritchard magnetic field b x when b x 0 a numerical method is introduced to compute the ground states and it is also applied to study properties of ground states numerical results suggest that the densities of m f 1 components in ground states are identical for any nonzero b x in particular if b x b 0 is a constant the ground states satisfy the single mode approximation when b x 0 efficient and simpler numerical methods are presented to solve the ground states of spin 1 becs based on their ferromagnetic antiferromagnetic characterizations numerical simulations show that our methods are more efficient than those in the literature in addition some conjectures are made from our numerical observations

improved numerical approach for the time independent gross pitaevskii nonlinear schrodinger equation in the present work we improve a numerical method developed to solve the gross pitaevkii nonlinear schrodinger equation a particular scaling is used in the equation which permits us to evaluate the wave function normalization after the numerical solution we have a two point boundary value problem where the second point is taken at infinity the differential equation is solved using the shooting method and runge kutta integration method requiring that the asymptotic constants for the function and its derivative be equal for large distances in order to obtain fast convergence the secant method is used s1063 651x 99 04608 5 pacs number s 02 60 lj 71 10 2w 03 75 fi 11 10 lm precise and fast numerical solutions to nonlinear equa tions have become considerably important in computational physics so the numerical procedures are relevant also to be described when treating such problems considering the fact that the computation is time consuming in the present work we pay attention especially to this problem proposing an alternative approach to a method recently described in ref 1 which was used to solve the gross pitaevskii 2 nonlin ear schrodinger equation nlse for trapped neutral atoms with positive two body scattering lengths in ref 3 the nlse treated in ref 1 was extended to a time dependent one for both positive and negative two body scattering lengths where the crank nicolson algorithm appropriate for time evolution was considered this approach however has the disadvantage that it can only reach stable solutions in case one needs to add other nonlinear terms of higher order in the original equation 4 it is not feasible to reach another region of stable solutions if in between there is an unstable region this implies that it should be appropriate to combine a static method such as the one used in ref 1 with the method used in ref 3 when we are interested in obtain all the stable and unstable solutions and also the cor responding time evolution so in this perspective any im provement of the method considered in ref 1 would be relevant in the following we briefly describe the physics related to the nlse considered in 1 and the numerical procedure used to solve it then we present an alternative approach which can considerably reduce the time needed to search for the solutions and the normalizations the nonlinear schrodinger equation which describes the condensed wave function in the mean field approximation can be written as 1

existence of a global attractor for semilinear dissipative wave equations on rn abstract we consider the semilinear hyperbolic problem utt ut x u f u x x r n t 0 with the initial conditions u x 0 u0 x and ut x 0 u1 x in the case where n 3 and x 1 g x lies in ln 2 r n the energy space x 0 d 1 2 r n l2g r n is introduced to overcome the difficulties related with the noncompactness of operators which arise in unbounded domains we derive various estimates to show local existence of solutions and existence of a global attractor in x 0 the compactness of the embedding d 1 2 r n l2g r n is widely applied

a generalized laguerre hermite pseudospectral method for computing symmetric and central vortex states in bose einstein condensates a generalized laguerre hermite pseudospectral method is proposed for computing symmetric and central vortex states in bose einstein condensates becs in three dimensions with cylindrical symmetry the new method is based on the properly scaled generalized laguerre hermite functions and a normalized gradient flow it enjoys three important advantages i it reduces a three dimensional 3d problem with cylindrical symmetry into an effective two dimensional 2d problem ii it solves the problem in the whole space instead of in a truncated artificial computational domain and iii it is spectrally accurate extensive numerical results for computing symmetric and central vortex states in becs are presented for one dimensional 1d bec 2d bec with radial symmetry and 3d bec with cylindrical symmetry

split step alternating direction implicit difference scheme for the fractional schrodinger equation in two dimensions in this paper we propose a conservative and effective difference scheme for solving the two dimensional nonlinear space fractional schrodinger equation with the riesz fractional derivative the scheme is constructed by combining the split step method for handling the nonlinearity with the alternating direction implicit adi method for resolving the multi dimensions difficulty the riesz space fractional derivative is approximated by the second order accurate fractional centered difference based on matrix analysis we show that in the discrete sense the scheme conserves the mass and energy for linear problems and conserves the mass for nonlinear problems the unconditional convergence is proved rigorously in the linear case numerical tests are performed to support our theoretical results and show the efficiency of the proposed scheme

existence and nonexistence of global solutions for a nonlinear hyperbolic system with damping abstract this paper concerns with the cauchy problem for the damped nonlinear hyperbolic system u t t u u t v p t x r 1 r n v t t v v t u q t x r 1 r n u 0 x u 0 x u t 0 x u 1 x x r n v 0 x v 0 x v t 0 x v 1 x x r n where p q 1 and satisfy p q 1 we shall prove that if max 1 p p q 1 1 q p q 1 n 2 for n 1 or 3 then the solution with small initial data exists globally in time and satisfies the decay estimates u t c 1 t and v t c 1 t for some positive constants and while if max 1 p p q 1 1 q p q 1 n 2 for n 1 then every solution with initial data having positive average value does not exist globally

berezin integration on general fermionic supermanifolds recent results on the global structure of supermanifolds are used to define a notion of berezin integration on any purely fermionic rogers supermanifold this leads to an integration theory on a large class of supermanifolds having both bosonic and fermionic coordinates the existence of global functions and forms on such supermanifolds is discussed as is some elementary cohomology of supermanifolds

a finite element method with mesh adaptivity for computing vortex states in fast rotating bose einstein condensates numerical computations of stationary states of fast rotating bose einstein condensates require high spatial resolution due to the presence of a large number of quantized vortices in this paper we propose a low order finite element method with mesh adaptivity by metric control as an alternative approach to the commonly used high order finite difference or spectral approximation methods the mesh adaptivity is used with two different numerical algorithms to compute stationary vortex states an imaginary time propagation method and a sobolev gradient descent method we first address the basic issue of the choice of the variable used to compute new metrics for the mesh adaptivity and show that refinement using simultaneously the real and imaginary part of the solution is successful mesh refinement using only the modulus of the solution as adaptivity variable fails for complicated test cases then we suggest an optimized algorithm for adapting the mesh during the evolution of the solution towards the equilibrium state considerable computational time saving is obtained compared to uniform mesh computations the new method is applied to compute difficult cases relevant for physical experiments large nonlinear interaction constant and high rotation rates

computing the ground state and dynamics of the nonlinear schrodinger equation with nonlocal interactions via the nonuniform fft we present efficient and accurate numerical methods for computing the ground state and dynamics of the nonlinear schrodinger equation nlse with nonlocal interactions based on a fast and accurate evaluation of the long range interactions via the nonuniform fast fourier transform nufft we begin with a review of the fast and accurate nufft based method in 29 for nonlocal interactions where the singularity of the fourier symbol of the interaction kernel at the origin can be canceled by switching to spherical or polar coordinates we then extend the method to compute other nonlocal interactions whose fourier symbols have stronger singularity at the origin that cannot be canceled by the coordinate transform many of these interactions do not decay at infinity in the physical space which adds another layer of complexity since it is more difficult to impose the correct artificial boundary conditions for the truncated bounded computational domain the performance of our method against other existing methods is illustrated numerically with particular attention on the effect of the size of the computational domain in the physical space finally to study the ground state and dynamics of the nlse we propose efficient and accurate numerical methods by combining the nufft method for potential evaluation with the normalized gradient flow using backward euler fourier pseudospectral discretization and time splitting fourier pseudospectral method respectively extensive numerical comparisons are carried out between these methods and other existing methods for computing the ground state and dynamics of the nlse with various nonlocal interactions numerical results show that our scheme performs much better than those existing methods in terms of both accuracy and efficiency

on the blow up solutions for the nonlinear fractional schrodinger equation abstract this paper is dedicated to the blow up solutions for the nonlinear fractional schrodinger equation arising from pseudorelativistic boson stars first we compute the best constant of a gg n inequality by the profile decomposition theory and variational arguments then we find the sharp threshold mass of the existence of finite time blow up solutions finally we study the dynamical properties of finite time blow up solutions around the sharp threshold mass by giving a refined compactness lemma

efficient and spectrally accurate numerical methods for computing ground and first excited states in bose einstein condensates in this paper we present two efficient and spectrally accurate numerical methods for computing the ground and first excited states in bose einstein condensates becs we begin with a review on the gradient flow with discrete normalization gfdn for computing stationary states of a nonconvex minimization problem and show how to choose initial data effectively for the gfdn for discretizing the gradient flow we use sine pseudospectral method for spatial derivatives and either backward euler scheme besp or backward forward euler schemes for linear nonlinear terms bfsp for temporal derivatives both besp and bfsp are spectral order accurate for computing the ground and first excited states in bec of course they have their own advantages i for linear case besp is energy diminishing for any time step size where bfsp is energy diminishing under a constraint on the time step size ii at every time step the linear system in bfsp can be solved directly via fast sine transform fst and thus it is extremely efficient and in besp it needs to be solved iteratively via fst by introducing a stabilization term and thus it could be efficient too comparisons between besp and bfsp as well as other existing numerical methods are reported in terms of accuracy and total computational time our numerical results show that both besp and bfsp are much more accurate and efficient than those existing numerical methods in the literature finally our new numerical methods are applied to compute the ground and first excited states in bec in one dimension 1d 2d and 3d with a combined harmonic and optical lattice potential for demonstrating their efficiency and high resolution

global and blow up solutions for a system of nonlinear hyperbolic equations with dissipative terms abstract in this paper we study a nonlinear hyperbolic system with strong damping firstly by use of the successive approximation method and a series of classical estimates we prove the local existence and uniqueness of weak solution secondly via some inequalities applying the potential method and the concave method we discuss the asymptotic and blow up behavior of weak solution with different conditions

invariant manifolds for flows in banach spaces on presente une theorie des varietes invariantes lisses basee sur la methode classique de lyapunov penon pour des semi flots continus dans des espaces de banach