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

unifying derived deformation theories abstract we develop a framework for derived deformation theory valid in all characteristics this gives a model category reconciling local and global approaches to derived moduli theory in characteristic 0 we use this to show that the homotopy categories of dglas and shlas l algebras considered by kontsevich hinich and manetti are equivalent and are compatible with the derived stacks of toen vezzosi and lurie another application is that the cohomology groups associated to any classical deformation problem in any characteristic admit the same operations as andre quillen cohomology

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

numerical simulation of singular solutions to the two dimensional cubic schrodinger equation numerical simulations of the cubic nonlinear schrodinger equation are presented in dimension n 2 the emphasis is on a detailed mechanism of blow up the numerical results indicate that the blow up is not restricted to the case where the problem is posed in the entire r2 space but also occurs with periodic boundary conditions the structure of the solution as the singular time t is approached has been investigated using examples with and without radial symmetry and or periodicity in most cases we observe that the amplitude develops something like the peaked self similar profile predicted by zakharov and synakh 38 which has a t t 2 3 maximum amplitude and a t t 2 3 width

exponential time differencing crank nicolson method with a quartic spline approximation for nonlinear schrodinger equations abstract this paper studies a central difference and quartic spline approximation based exponential time differencing crank nicolson etd cn method for solving systems of one dimensional nonlinear schrodinger equations and two dimensional nonlinear schrodinger equations a local extrapolation is employed to achieve a fourth order accuracy in time the numerical method is proven to be highly efficient and stable for long range soliton computations numerical examples associated with dirichlet neumann and periodic boundary conditions are provided to illustrate the accuracy efficiency and stability of the method proposed

nonlinear schrodinger equations and sharp interpolation estimates a sharp sufficient condition for global existence is obtained for the nonlinear schrodinger equation begin array 20 c nls 2i phi t delta phi left phi right 2 sigma phi 0 x in mathbb r n t in mathbb r end array in the case 2 n this condition is in terms of an exact stationary solution nonlinear ground state of nls it is derived by solving a variational problem to obtain the best constant for classical interpolation estimates of nirenberg and gagliardo

a rescaling algorithm for the numerical calculation of blowing up solutions on presente un algorithme a invariance d echelle pour le calcul numerique des solutions a singularites explosives des equations d evolution non lineaires

computer simulation of wave collapses in the nonlinear schro dinger equation abstract the self tuning numerical scheme for simulation of collapse phenomena in the framework of the nonlinear schrodinger equation is developed the scheme allows to reach extremely high field amplitudes 2 0 2 10 15 without lack of validity numerical results are compared with analytical predictions

numerical studies on the split step finite difference method for nonlinear schrodinger equations in this study we use the split step finite difference method to solve various nonlinear schrodinger equations including coupled ones we study the numerical accuracy of the method detailed numerical results including higher dimensions show that the method provides accurate and stable solutions for nonlinear schrodinger equations

tracing complex singularities with spectral methods abstract a numerical method for investigating the possibility of blow up after a finite time is introduced for a large class of nonlinear evolution problems with initial data analytic in the space variable s the solutions have for any t 0 complex space singularities at the edge of an analyticity strip of width t loss of regularity corresponds to the vanishing of t numerical integration by high resolution spectral methods reveals the large wavenumber behavior of the fourier transform of the solutions from which t is readily obtained its time evolution can be traced down to about one mesh length by extrapolation of t such numerical experiments provide evidence suggesting finite time blow up or all time regularity the method is tested on the inviscid and viscous burgers equations and is applied to the one dimensional nonlinear schrodinger equation with quartic potential and to the two dimensional incompressible euler equation all with periodic boundary conditions in the latter case evidence is found suggesting that existing all time regularity results can be substantially sharpened

higher kac moody algebras and moduli spaces of g bundles abstract we provide a generalization to the higher dimensional case of the construction of the current algebra g z its kac moody extension g and of the classical results relating them to the theory of g bundles over a curve for a reductive algebraic group g with lie algebra g we define a dg lie algebra g n of n dimensional currents in g for any symmetric g invariant polynomial p on g of degree n 1 we get a higher kac moody algebra g n p as a central extension of g n by the base field k further for a smooth projective variety x of dimension n 2 we show that g n acts infinitesimally on the derived moduli space r bun g rig x x of g bundles over x trivialized at the neighborhood of a point x x finally for a representation g g l r we construct an associated determinantal line bundle on r bun g rig x x and prove that the action of g n extends to an action of g n p on such bundle for p the n 1 th chern character of

solitary waves of nonlinear dispersive evolution equations with critical power nonlinearities we study the nonlinear schrodinger nls generalized korteweg de vries gkdv and generalized benjamin ono equations gbo with critical power nonlinearities the ground state solitary wave is conditionally stable modulo the group of translations in space and phase and dilations this is a natural extension of the result for subcritical nonlinearities where solitary waves are stable modulo translations alone to a case where solutions may blow up we also present results on global existence for gbo and the structure of singularities for certain solutions of gbo which are analogues of previous results by the author for nls and gkdv

algebras and modules in monoidal model categories in recent years the theory of structured ring spectra formerly known as a infty and e infty ring spectra has been simplified by the discovery of categories of spectra with strictly associative and commutative smash products now a ring spectrum can simply be defined as a monoid with respect to the smash product in one of these new categories of spectra in this paper we provide a general method for constructing model category structures for categories of ring algebra and module spectra this provides the necessary input for obtaining model categories of symmetric ring spectra functors with smash product gamma rings and diagram ring spectra algebraic examples to which our methods apply include the stable module category over the group algebra of a finite group and unbounded chain complexes over a differential graded algebra 1991 mathematics subject classification primary 55u35 secondary 18d10

focusing and multi focusing solutions of the nonlinear schro dinger equation abstract a method of dynamic rescaling of variables is used to investigate numerically the nature of the focusing singularities of the cubic and quintic schrodinger equations in two and three dimensions and describe their universal properties the same method is applied to simulate the multi focusing phenomena produced by simple models of saturating nonlinearities

overcomplete basis for one dimensional hamiltonians it is demonstrated for one dimensional hamiltonians and under a special hypothesis on the ground state wave function that it is possible to build an analytical overcomplete basis that generalizes the coherent state basis of the harmonic oscillator without any reference to group theory the semi classical consequences of this formalism are developed and the usual ideas of normal and antinormal expansions for operators are extended

on the stability of collapse in the critical case collapsing solutions in the cauchy problem of the nonlinear schrodinger equation i t 2 p 0 x rd are considered in the so called critical case pd 4 where d is the spatial dimension a stability theorem for radial collapse is presented which proves that the formation of the singularity remains close to the self similar collapsing solution with a spatial profile given by the ground state solitary wave provided the energy h 0 an analogous result is given for the scalar zakharov equations and a generalized korteweg de vries equation tu p 1 up xu 2 xu 0

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

blowup in diffusion equations a survey this paper deals with quasilinear reaction diffusion equations for which a solution local in time exists if the solution ceases to exist for some finite time we say that it blows up in contrast to linear equations blowup can occur even if the data are smooth and well defined for all times depending on the equation either the solution or some of its derivatives become singular we shall concentrate on those cases where the solution becomes unbounded in finite time this can occur in quasilinear equations if the heat source is strong enough there exist many theoretical studies on the question on the occurrence of blowup in this paper we shall recount some of the most interesting criteria and most important methods for analyzing blowup the asymptotic behavior of solutions near their singularities is only completely understood in the special case where the source is a power a better knowledge would be useful also for their numerical treatment thus not surprisingly the numerical analysis of this type of problems is still at a rather early stage the goal of this paper is to collect some of the known results and algorithms and to direct the attention to some open problems

on fully discrete galerkin methods of second order temporal accuracy for the nonlinear schrodinger equation we approximate the solutions of an initial and boundary value problem for nonlinear schrodinger equations with emphasis on the cubic nonlinearity by two fully discrete finite element schemes based on the standard galerkin method in space and two implicit crank nicolson type second order accurate temporal discretizations for both schemes we study the existence and uniqueness of their solutions and provel 2 error bounds of optimal order of accuracy for one of the schemes we also analyze one step of newton s method for solving the nonlinear systems that arise at every time step we then implement this scheme using an iterative modification of newton s method that at each time stept n requires solving a number of sparse complex linear systems with a matrix that does not change withn the effect of this inner iteration is studied theoretically and numerically

an efficient time splitting compact finite difference method for gross pitaevskii equation abstract we propose an efficient time splitting compact finite difference method for gross pitaevskii equation gpe in our method we solve the gpe in time with time splitting technique and in space by the compact finite difference method to find the numerical solution of the resulting discretized system in one dimension 1d two dimensions 2d and three dimensions 3d we apply the fast discrete sine transform in 1d 2d and 3d respectively and get an efficient solver for the discretized system in 1d 2d and 3d respectively our numerical algorithm at every time step does not need linear algebraic equations solver whose computation cost will be much higher when the spatial dimension increases the method also has the merit that it is unconditionally stable and conservative moreover the method can achieve spectral like accuracy in space when high order compact finite difference method is applied extensive numerical tests for the gpe in 1d 2d and 3d are presented to demonstrate the power and accuracy of the proposed numerical method

from classical to quantum mechanics how to translate physical ideas into mathematical language following previous works by e prugovecki physica a 91a 202 1978 and stochastic quantum mechanics and quantum space time reidel dordrecht 1986 on common features of classical and quantum mechanics we develop a unified mathematical framework for classical and quantum mechanics based on l2 spaces over classical phase space in order to investigate to what extent quantum mechanics can be obtained as a simple modification of classical mechanics on both logical and analytical levels to obtain this unified framework we split quantum theory in two parts i general quantum axiomatics a system is described by a state in a hilbert space observables are self adjoints operators and so on and ii quantum mechanics proper that specifies the hilbert space as l2 rn the heisenberg rule pi qj i ij with p i the free hamiltonian h 2 2m and so on we show that general quantum axiomatics up to a supplementary axiom of classicity can be used as a nonstandard mathematical ground to formula

a physical origin for singular support conditions in geometric langlands theory we explain how the nilpotent singular support condition introduced into the geometric langlands conjecture by arinkin and gaitsgory arises naturally from the point of view of 4 dimensional mathcal n 4 supersymmetric gauge theory we define what it means in topological quantum field theory to restrict a category of boundary conditions to the full subcategory of objects compatible with a fixed choice of vacuum both in functorial field theory and in the language of factorization algebras for b twisted mathcal n 4 gauge theory with gauge group g the moduli space of vacua is equivalent to mathfrak h w and the nilpotent singular support condition arises by restricting to the vacuum 0 in mathfrak h w we then investigate the categories obtained by restricting to points in larger strata and conjecture that these categories are equivalent to the geometric langlands categories with gauge symmetry broken to a levi subgroup and furthermore that by assembling such for the groups mathrm gl n with n ge 1 one finds a hidden factorization algebra structure for the geometric langlands theory

the numerical study of blowup with application to a nonlinear schrodinger equation abstract we discuss the use of numerical methods in the study of the solutions of evolution problems which exhibit finite time unbounded growth we first examine a naive approach in which the growth rate of the numerical solution is accepted as an approximation of the true growth rate as we shall demonstrate for a radial nonlinear schrodinger equation this approach is inadequate since different discretizations exhibit different growth rates the spurious behaviour of discretizations in the neighbourhood of the singularity is discussed a reliable procedure for the estimation of the blowup parameters is considered which eliminates the discrepancies between different numerical methods

koszul duality for lie algebroids abstract this paper studies the role of dg lie algebroids in derived deformation theory more precisely we provide an equivalence between the homotopy theories of formal moduli problems and dg lie algebroids over a commutative dg algebra of characteristic zero at the level of linear objects we show that the category of representations of a dg lie algebroid is an extension of the category of quasi coherent sheaves on the corresponding formal moduli problem we describe this extension geometrically in terms of pro coherent sheaves

numerical approximation of blow up of radially symmetric solutions of the nonlinear schrodinger equation we consider the initial value problem for the radially symmetric nonlinear schrodin ger equation with cubic nonlinearity nls in d 2 and 3 space dimensions to approximate smooth solutions of this problem we construct and analyze a numerical method based on a standard galerkin finite element spatial discretization with piecewise linear continuous functions and on an implicit crank nicolson type time stepping procedure we then equip this scheme with an adaptive spatial and temporal mesh refinement mechanism that enables the numerical technique to approximate well singular solutions of the nls equation that blow up at the origin as the temporal variable t tends from below to a finite value t star for the blow up of the amplitude of the solution we recover numerically the well known rate t star t 1 2 for d 3 for d 2 our numerical evidence supports the validity of the log log law ln ln frac 1 t star t t star t 1 2 for t extremely close to t star the scheme also approximates well the details of the blow up of the phase of the solution at the origin as t to t star

supersymmetry factorisation of the schrodinger equation and a hamiltonian hierarchy the author presents a systematic procedure for constructing a hierarchy of non relativistic hamiltonians with the property that the adjacent members of the hierarchy are supersymmetric partners i e they share the same eigenvalue spectrum except for the missing ground state and the eigenvectors are simply related