a boundary value problem for the minimum time function a natural boundary value problem for the dynamic programming partial differential equation associated with the minimum time problem is proposed the minimum time function is shown to be the unique viscosity solution of this boundary value problem

to the theory of viscosity solutions for uniformly elliptic isaacs equations we show how a theorem about the solvability in c 1 1 c 1 1 of special isaacs equations can be used to obtain existence and uniqueness of viscosity solutions of general uniformly nondegenerate isaacs equations we apply it also to establish the c 1 c 1 regularity of viscosity solutions and show that finite difference approximations have an algebraic rate of convergence the main coefficients of the isaacs equations are supposed to be in c c with slightly less than 1 2

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

the equivalence problem of multitape finite automata abstract using a result of b h neumann we extend eilenberg s equality theorem to a general result which implies that the multiplicity equivalence problem of two nondeterministic multitape finite automata is decidable as a corollary we solve a long standing open problem in automata theory namely the equivalence problem for multitape deterministic finite automata the main theorem states that there is a finite test set for the multiplicity equivalence of finite automata over conservative monoids embeddable in a fully ordered group

a spectral galerkin continuation method using chebyshev polynomials for the numerical solutions of the gross pitaevskii equation we study an efficient spectral galerkin continuation method sgcm and two grid centered difference approximations for the numerical solutions of the gross pitaevskii equation gpe where the second kind chebyshev polynomials are used as the basis functions for the trial function space some basic formulae for the sgcm are derived so that the eigenvalues of the associated linear eigenvalue problems can be easily computed the sgcm is implemented to investigate the ground and first excited state solutions of the gpe both the parabolic and quadruple well trapping potentials are considered we also study bose einstein condensates bec in optical lattices where the periodic potential described by the sine or cosine functions is imposed on the gpe of particular interest here is the investigation of symmetry breaking solutions sample numerical results are reported

on dol power series abstract we study d0l power series we show how elementary morphisms introduced by ehrenfeucht and rozenberg can be used in connection with power series characterize the sequences of rational numbers and integers which can appear as coefficients in d0l power series and establish various decidability results

symplectic integrator for nonlinear high order schrodinger equation with a trapped term in the manuscript we discuss the symplectic integrator for the numerical solution of a kind of high order schrodinger equation with trapped terms the hamiltonian formulism is discovered for it we first discretize the hamiltonian system in space to reduce it to a finite dimensional one then the symplectic midpoint scheme is applied to the temporal discretization the symplectic scheme we devise is of second order accuracy in time and 2lth order accuracy in space it is proved that it preserves the charge of the original equation veraciously the energy is not preserved explicitly for the exception of the linear case however after some computation the energy transit formula in the temporal direction is obtained in the numerical part we compare our scheme with some existing schemes including the leap frog scheme and the energy preserving scheme from the numerical evidence we find that our numerical schemes are efficient and available the numerical results are consistent with the theoretical analysis

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

harnack inequality for quasilinear elliptic equations on riemannian manifolds abstract we study viscosity solutions to degenerate and singular elliptic equations div f u u u h of p laplacian type on riemannian manifolds where an even function f c 1 r c 2 0 is supposed to be strictly convex on 0 under the assumption that either f c 2 r or its convex conjugate f c 2 r with some structural condition we establish a locally uniform abp type estimate and the krylov safonov type harnack inequality on riemannian manifolds with the use of an intrinsic geometric quantity to the operator here the c 2 regularities of f and f account for degenerate and singular operators respectively

regularity and large time behaviour of solutions of a conservation law without convexity a l aide de la methode des caracteristiques generalisees on etudie la regularite et le comportement a grand des solutions faibles admissibles d une seule loi de conservation a 1 variable d espace avec un point d inflexion

aleksandrov bakelman pucci maximum principles for a class of uniformly elliptic and parabolic integro pde abstract we prove generalized aleksandrov bakelman pucci maximum principles for elliptic and parabolic integro pdes of hamilton jacobi bellman isaacs types whose pde parts are either uniformly elliptic or uniformly parabolic the proofs of these results are based on the classical aleksandrov bakelman pucci maximum principles for the elliptic and parabolic pdes and an iteration procedure using solutions of pucci extremal equations we also provide proofs of nonlocal versions of the classical aleksandrov bakelman pucci maximum principles for elliptic and parabolic integro pdes

geometric properties of solutions of hamilton jacobi equations abstract the method of moving parallel planes previously used for elliptic and parabolic pde is adapted to study solutions of the cauchy problem for hamilton jacobi equations this is possible in the framework of the theory of viscosity solutions using the comparison theorem for such solutions as a kind of maximum principle one of the main results states that if the initial data are nonnegative and compact supported the hamiltonian radial and the level sets expanding then the level sets become asymptotically spherical as t the convergence taking place in the lipschitz norm

on the differentiability of the solution to the hamilton jacobi equation with critical fractional diffusion abstract we prove that the hamilton jacobi equation for an arbitrary hamiltonian h locally lipschitz but not necessarily convex and fractional diffusion of order one critical has classical c 1 solutions the proof is achieved using a new holder estimate for solutions of advection diffusion equations of order one with bounded vector fields that are not necessarily divergence free

a comparison among various notions of viscosity solution for hamilton jacobi equations on networks abstract three definitions of viscosity solutions for hamilton jacobi equations on networks recently appeared in literature achdou et al 2013 1 imbert et al 2013 4 schieborn and camilli 2013 6 being motivated by various applications they appear to be considerably different the aim of this note is to establish their equivalence

finite difference solutions of a non linear schrodinger equation abstract this paper describes a finite difference method to approximate a schrodinger equation with a power non linearity a special case of this equation is currently used to model the propagation of a laser beam in a plasma the main feature of the method we present is that it satisfies a discrete analogue of some important conservation laws of the equations we present numerical results which show in particular the propagation and the formation of solitions in the one dimensional case

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

a space time finite element method for the nonlinear schro dinger equation the discontinuous galerkin method the convergence of the discontinuous galerkin method for the nonlinear cubic schrodinger equation is analyzed in this paper we show the existence of the resulting approximations and prove optimal order error estimates in l l 2 these estimates are valid under weak restrictions on the space time mesh

symplectic partitioned runge kutta methods with the phase lag property abstract in this work specially tuned symplectic partitioned runge kutta sprk methods with minimum phase lag and phase fitted have been considered the general framework for constructing sprk methods with minimum phase lag is given

a stochastic control approach to the pricing of options we construct a stochastic control model of a portfolio in which the investor may invest in stock call options on the stock or risk free bonds he may also borrow funds we assume that the investor is attempting to maximize expected utility of total wealth at expiration of the options where the utility function is given as wealth to the power 0 1 this includes both the risk averse and risk neutral cases we allow the borrowing interest rate r to be distinct from the lending interest rate r we make two fundamental assumptions i the investor cannot achieve infinite expected utility of wealth in finite time stable markets and ii the market will set the option price to minimize the investor s maximal expected utility of wealth i e options are priced using a minimax strategy the market is stable only when r r we prove in the case r r and 1 that the minimax european call option price ft s is given as the solution of the black scholes equation ft i i 2s2fss i s fs i f 0 where i r when r i 2 1 i r when i 2 1 r and i i 2 1 if r i 2 1 r the parameter is the stock return drift and i is the stock return diffusion if r r then the call option price must by stable markets always be the black scholes price with rate r in the risk neutral 1 case we must have r r for a stable market and the option price given by the black scholes equation but with i explicit solutions are presented and the optimal portfolios and returns are also derived comparable results are obtained for american call options which can be exercised at any time up to and including the expiration date in this case the option pricing function is shown to be the solution of an obstacle problem

analytical and numerical aspects of certain nonlinear evolution equations ii numerical nonlinear schrodinger equation abstract various numerical methods are employed in order to approximate the nonlinear schrodinger equation namely i the classical explicit method ii hopscotch method iii implicit explicit method iv crank nicolson implicit scheme v the ablowitz ladik scheme vi the split step fourier method f tappert and vii pseudospectral fourier method fornberg and whitham comparisons between the ablowitz ladik scheme which was developed using notions of the inverse scattering transform and the other utilized schemes are obtained

fourier methods for the perturbed harmonic oscillator in linear and nonlinear schrodinger equations we consider the numerical integration of the gross pitaevskii equation with a potential trap given by a time dependent harmonic potential or a small perturbation thereof splitting methods are frequently used with fourier techniques since the system can be split into the kinetic and remaining part and each part can be solved efficiently using fast fourier transforms splitting the system into the quantum harmonic oscillator problem and the remaining part allows us to get higher accuracies in many cases but it requires us to change between hermite basis functions and the coordinate space and this is not efficient for time dependent frequencies or strong nonlinearities we show how to build methods that combine the advantages of using fourier methods while solving the time dependent harmonic oscillator exactly or with a high accuracy by using a magnus integrator and an appropriate decomposition

pseudo spectral solution of nonlinear schro dinger equations abstract we compare four pseudo spectral split step methods for solving a class of nonlinear schrodinger nls equations the importance of observing the l 2 invariance of the continuous problem is demonstrated through numerical experiments the best performance is obtained by transforming the given equation to an nls equation where two of the coefficients satisfy a simple algebraic relationship the problem can be solved efficiently in terms of the new variables and the l 2 norm of the computed solution is time invariant

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 two step high order compact scheme for the laplacian operator and its implementation in an explicit method for integrating the nonlinear schrodinger equation we describe and test an easy to implement two step high order compact 2shoc scheme for the laplacian operator and its implementation into an explicit finite difference scheme for simulating the nonlinear schrodinger equation nlse our method relies on a compact double differencing which is shown to be computationally equivalent to standard fourth order non compact schemes through numerical simulations of the nlse using fourth order runge kutta we confirm that our scheme shows the desired fourth order accuracy a computation and storage requirement comparison is made between the 2shoc scheme and the non compact equivalent scheme for both the laplacian operator alone as well as when implemented in the nlse simulations stability bounds are also shown in order to get maximum efficiency out of the method we conclude that the modest increase in storage and computation of the 2shoc schemes is well worth the advantages of having the schemes compact and their ease of implementation makes their use very useful for practical implementations

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

regularity of solutions to a class of variable exponent fully nonlinear elliptic equations abstract in the present paper we propose the investigation of variable exponent degenerate singular elliptic equations in non divergence form this current endeavor parallels the by now well established theory of functionals satisfying nonstandard growth condition which in particular encompasses problems ruled by the p x laplacian operator under rather general conditions we prove viscosity solutions to variable exponent fully nonlinear elliptic equations are locally of class c 1 for a universal constant 0 1 a key feature of our estimates is that they do not depend on the modulus of continuity of exponent coefficients and thus may be employed to investigate a variety of problems whose ellipticity degenerates and or blows up in a discontinuous fashion

nonlinear h control and estimation of optimal h gain abstract the structure of nonlinear h controller and the estimation of optimal h gain are investigated in this paper the essential problem boils down to the existence of semipositive solution to the hamilton jacobi isaacs inequality further the solvability of this first order fully nonlinear differential inequality is discussed we look for one kind of special semipositive radial solution to the hamilton jacobi isaacs inequality an explicit estimation of optimal h gain and explicit formulas of semipositive radial solutions to the hamilton jacobi isaacs inequality are obtained the results are quite simple and intuitive they even shed a new insight on the linear h theory

practical symplectic partitioned runge kutta and runge kutta nystrom methods we present new symmetric fourth and sixth order symplectic partitioned runge kutta and runge kutta nystrom methods we studied compositions using several extra stages optimising the efficiency an effective error ef is defined and an extensive search is carried out using the extra parameters the new methods have smaller values of ef than other methods found in the literature when applied to several examples they perform up to two orders of magnitude better than previously known method which is in very good agreement with the values of ef

fully nonlinear neumann type boundary conditions for first order hamilton jacobi equations we consider problems of the following type h x u u 0 in where is a smooth bounded open set in r n u is a scalar function from into r u denotes its frechet differential at x and h is a given continuous function on xr r n this equation is nothing but a general form of the classical first order hamilton jacobi equation in order to investigate existence and uniqueness questions the equation has to be complemented with a boundary condition on or on a part of we consider here for reasons described below a fully nonlinear boundary condition of the form f x u u 0 on that we describe as being of the neumann type

on the aleksandrov bakel man pucci estimate for some elliptic and parabolic nonlinear operators in this work we prove the aleksandrov bakel man pucci estimate for possibly degenerate nonlinear elliptic and parabolic equations of the form rm div left f left nabla u x right right f left x right quad rm in omega subset mathbb r n and u t x t rm div left f left nabla u x t right right f left x t right quad rm in q subset mathbb r n 1 for f a fancyscript c 1 monotone field under some suitable conditions examples of applications such as the p laplacian and the mean curvature flow are considered as well as extensions of the general results to equations that are not in divergence form such as the m curvature flow

optimizing schrodinger functionals using sobolev gradients applications to quantum mechanics and nonlinear optics in this paper we study the application of the sobolev gradients technique to the problem of minimizing several schrodinger functionals related to timely and difficult nonlinear problems in quantum mechanics and nonlinear optics we show that these gradients act as preconditioners over traditional choices of descent directions in minimization methods and show a computationally inexpensive way to obtain them using a discrete fourier basis and a fast fourier transform we show that the sobolev preconditioning provides a great convergence improvement over traditional techniques for finding solutions with minimal energy as well as stationary states and suggest a generalization of the method using arbitrary linear operators

fourier splitting methods for the dynamics of rotating bose einstein condensates abstract we present a new method to propagate rotating bose einstein condensates subject to explicitly time dependent trapping potentials using algebraic techniques we combine magnus expansions and splitting methods to yield any order methods for the multivariate and nonautonomous quadratic part of the hamiltonian that can be computed using only fourier transforms at the cost of solving a small system of polynomial equations the resulting scheme solves the challenging component of the nonlinear hamiltonian and can be combined with optimized splitting methods to yield efficient algorithms for rotating bose einstein condensates the method is particularly efficient for potentials that can be regarded as perturbed rotating and trapped condensates e g for small nonlinearities since it retains the near integrable structure of the problem for large nonlinearities the method remains highly efficient if higher order p 2 is sought furthermore we show how it can be adapted to the presence of dissipation terms numerical examples illustrate the performance of the scheme

risk sensitive control with hara utility a control methodology based on the hyperbolic absolute risk averse hara utility function is presented as an alternative to the exponential of an integral approach to finding robust controllers the work is inspired by the intuition that hara controllers while being robust may give better performance than exponential controllers in normal situations the hara problem is shown to be equivalent to a certain differential game and the asymptotic properties of the hara problem and this differential game are studied as an example a linear quadratic hara problem is studied where the problem of finding a robust hara controller is proved to be equivalent to solving a standard linear quadratic problem for a system with a higher noise intensity this reveals an interesting relationship between robustness and uncertainty

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

high order compact splitting multisymplectic method for the coupled nonlinear schrodinger equations in this paper we develop a new kind of multisymplectic integrator for the coupled nonlinear schrodinger cnls equations the cnls equations are cast into multisymplectic formulation then it is split into a linear multisymplectic formulation and a nonlinear hamiltonian system the space of the linear subproblem is approximated by a high order compact hoc method which is new in multisymplectic context the nonlinear subproblem is integrated exactly for splitting and approximation we utilize an hoc sms integrator its stability and conservation laws are investigated in theory numerical results are presented to demonstrate the accuracy conservation laws and to simulate various solitons as well for the hoc sms integrator they are consistent with our theoretical analysis

the maximum principle for viscosity solutions of fully nonlinear second order partial differential equations we prove that viscosity solutions in w1 of the second order fully nonlinear equation f d2u du u 0 are unique when i f is degenerate elliptic and decreasing in u or ii f is uniformly elliptic and nonincreasing in u we do not assume that f is convex the method of proof involves constructing nonlinear approximation operators which map viscosity subsolutions and supersolutions onto viscosity subsolutions and supersolutions respectively this method is completely different from that used in lions 8 9 for second order problems with f convex in d 2 u and from that used by crandall lions 3 and crandall evans lions 2 for fully nonlinear first order problems

symplectic structure preserving integrators for the two dimensional gross pitaevskii equation for bec a b s t r a c t symplectic integrators have been developed for solving the two dimensional gross pitaevskii equation the equation is transformed into a hamiltonian form with symplectic structure then symplectic integrators including the midpoint rule and a splitting symplectic scheme are developed for treating this equation it is shown that the proposed codes fulfill the discrete charge conservation law furthermore the global error of the numerical solution is theoretically estimated the theoretical analysis is supported by some numerical simulations

a regularity result for viscosity solutions of hamilton jacobi equations in one space dimension on demontre que dans le cas particulier d un probleme stationnaire a une dimension sous des hypotheses tres generales les solutions viscosite continues de lipschitz ont des derivees a droite et a gauche en chaque point

numerical methods for the simulation of trapped nonlinear schrodinger systems we propose analyze and compare the efficiency and accuracy of different numerical schemes for the solution of the nonlinear schrodinger equation with a trapping potential in particular we study schemes of finite difference pseudospectral and spectral types for the space discretization together with explicit symplectic multistep split step and standard variable step integrators to solve the time evolution all of these schemes are evaluated comparatively and some recommendations based on their accuracy and computational efficiency are made

a numerical study of adaptive space and time discretisations for gross pitaevskii equations as a basic principle benefits of adaptive discretisations are an improved balance between required accuracy and efficiency as well as an enhancement of the reliability of numerical computations in this work the capacity of locally adaptive space and time discretisations for the numerical solution of low dimensional nonlinear schrodinger equations is investigated the considered model equation is related to the time dependent gross pitaevskii equation arising in the description of bose einstein condensates in dilute gases the performance of the fourier pseudo spectral method constrained to uniform meshes versus the locally adaptive finite element method and of higher order exponential operator splitting methods with variable time stepsizes is studied numerical experiments confirm that a local time stepsize control based on a posteriori local error estimators or embedded splitting pairs respectively is effective in different situations with an enhancement either in efficiency or reliability as expected adaptive time splitting schemes combined with fast fourier transform techniques are favourable regarding accuracy and efficiency when applied to gross pitaevskii equations with a defocusing nonlinearity and a mildly varying regular solution however the numerical solution of nonlinear schrodinger equations in the semi classical regime becomes a demanding task due to the highly oscillatory and nonlinear nature of the problem the spatial mesh size and the time increments need to be of the size of the decisive parameter 0 e 1 especially when it is desired to capture correctly the quantitative behaviour of the wave function itself the required high resolution in space constricts the feasibility of numerical computations for both the fourier pseudo spectral and the finite element method nevertheless for smaller parameter values locally adaptive time discretisations facilitate to determine the time stepsizes sufficiently small in order that the numerical approximation captures correctly the behaviour of the analytical solution further illustrations for gross pitaevskii equations with a focusing nonlinearity or a sharp gaussian as initial condition respectively complement the numerical study

a nonconforming morley finite element method for the fully nonlinear monge ampere equation in this paper we study finite element approximations of the viscosity solution of the fully nonlinear monge ampere equation det d 2 u f 0 using the well known nonconforming morley element our approach is based on the vanishing moment method which was recently proposed as a constructive way to approximate fully nonlinear second order equations by the author and feng j sci comput 38 1 74 98 2009 the vanishing moment method approximates the monge ampere equation by the fourth order quasilinear equation epsilon delta 2u epsilon rm det d 2u epsilon f with appropriate boundary conditions we develop a finite element scheme using the n dimensional morley element introduced in wang and xu numer math 103 155 169 2006 to approximate the regularized fourth order problem in two and three dimensions and then derive optimal order error estimates

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

on lindenmayerian algebraic sequences abstract we define and study lindenmayerian algebraic sequences these sequences are a generalization of algebraic sequences k regular sequences and automatic sequences

stochastic target games and dynamic programming via regularized viscosity solutions we study a class of stochastic target games where one player tries to find a strategy such that the state process almost surely reaches a given target no matter which action is chosen by the opponent our main result is a geometric dynamic programming principle which allows us to characterize the value function as the viscosity solution of a non linear partial differential equation because abstract measurable selection arguments cannot be used in this context the main obstacle is the construction of measurable almost optimal strategies we propose a novel approach where smooth supersolutions are used to define almost optimal strategies of markovian type similarly as in verification arguments for classical solutions of hamilton jacobi bellman equations the smooth supersolutions are constructed by an extension of krylov s method of shaken coefficients we apply our results to a problem of option pricing under model uncertainty with different interest rates for borrowing and lending

a time splitting spectral method for coupled gross pitaevskii equations with applications to rotating bose einstein condensates we propose a time splitting spectral method for the coupled gross pitaevskii equations which describe the dynamics of rotating two component bose einstein condensates at a very low temperature the new numerical method is explicit unconditionally stable time reversible time transverse invariant and of spectral accuracy in space and second order accuracy in time moreover it conserves the position densities in the discretized level numerical applications on studying the generation of topological modes and the vortex lattice dynamics for the rotating two component bose einstein condensates are presented in detail

a time splitting pseudospectral method for the solution of the gross pitaevskii equations using spherical harmonics with generalised laguerre basis functions we present a method for numerically solving a gross pitaevskii system of equations with a harmonic and a toroidal external potential that governs the dynamics of one and two component bose einstein condensates the method we develop maintains spectral accuracy by employing fourier or spherical harmonics in the angular coordinates combined with generalised laguerre basis functions in the radial direction using an error analysis we show that the method presented leads to more accurate results than one based on a sine transform in the radial direction when combined with a time splitting method for integrating the equations forward in time in contrast to a number of previous studies no assumptions of radial or cylindrical symmetry is assumed allowing the method to be applied to 2d and 3d time dependent simulations this is accomplished by developing an efficient algorithm that accurately performs the generalised laguerre transforms of rotating bose einstein condensates for different orders of the laguerre polynomials using this spatial discretisation together with a second order strang time splitting method we illustrate the scheme on a number of 2d and 3d computations of the ground state of a non rotating and rotating condensate comparisons between previously derived theoretical results for these ground state solutions and our numerical computations show excellent agreement for these benchmark problems the method is further applied to simulate a number of time dependent problems including the kelvin helmholtz instability in a two component rotating condensate and the motion of quantised vortices in a 3d condensate

an investigation into the effect of product approximation in the numerical solution of the cubic nonlinear schro dinger equation abstract we study the effect of product approximation on the galerkin solutions of the one dimensional cubic schrodinger equation a crank nicolson scheme is used to discretize in time the paper describes two numerical experiments in the first we examine the approximation obtained by the standard galerkin method and discuss the possibility of enforcing discrete analogs of the conservation laws satisfied by the exact solution in the second experiment numerical results obtained by the product approximation version of the galerkin method are compared and the effectiveness of the method for different combinations of test and trial functions is also investigated

symbolic calculation in development of algorithms split step methods for the gross pitaevskii equation we employ symbolic calculation to perform a systematic study of the accuracy of split step fourier transform methods for the time dependent gross pitaevskii equation gpe provided the most recent approximation for the wave function is always used in the nonlinear atom atom interaction energy every split step algorithm we have tried has the same order time stepping error for the nonlinear gpe and for the linear schroedinger equation

high order split step exponential methods for solving coupled nonlinear schrodinger equations numerical integration schemes for coupled time dependent nonlinear schrodinger equations are examined using exponential splitting step methods exponentiation of the nonlinear potential term is reduced to the exponential of a kinetic energy term which can be calculated by fast fourier transforms high order iteration schemes involving a minimum number of product operators are shown to yield highly accurate amplitude and phase these new splitting methods are shown to be highly efficient both with respect to accuracy and integration time

recognizable formal power series on trees trees are a very basic object in computer science they intervene in nearly a ly domain and they are studied for their own or used to represent conveniently a given situation there are at least three directions where investigations on trees themselves are motivated and this for different reasons first the notion of tree is the basis of algebraic semantics nivat 19 rosen 2 2 etc in this context the study of special languages of trees i e forests their classification and their behaviour under various types of transformations are of great importance arnold fl dauchet 8 j l ilin 1161 mongy 18 by essence work in this area is an extension of the algebraic theory of languages trees and ianguages are in fact directly related via the derivation trees of an algebraic grammar thatcher 25 a second topic heavily related to trees concerns dzra structures trees mainly binary trees and its variants constitute one of the most widely known data structures see e g hlnuth 15 the analysis of the worst cale expected or average running time t ehzvgrr nfi certain algorithms requires sometimes long and delicate computations flajolet lo kemp 14 flajolet and steyaert 12 finally trees occupy a distinguished place in the enumeration of graphs and maps both because of the simplicity of their structure and for the relationship between their el lcodings and aigebraic languages the nature of the enumerating series and especially the question whether they are algebraic or not is one of the central problems in this domain cori 7 chottin 4 we propose here a theory of formal power series on trees and present some of their basic properties together with various examples of applications which as we hope will show the interest of its development within the framework we just sketched a formal power series on trees is a function which gssociates a number to esch tree thus we could also have called them tree functions ii in analogy with the term word function used by several authors paz and salomaa zo cobham 5 as an equivalent denomination for formal power series on words the main goal of a formal power series is to count or to represent the result of some computation on

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

generalized one sided estimates for solutions of hamilton jacobi equations and applications on etudie les proprietes locales des solutions de l equation de hamilton jacobi u t h t x u 0 et plus generalement de l equation u t h t x u u 0 ou t o t et x r n

on lindenmayerian algebraic power series abstract we define lindenmayerian algebraic power series and study their properties these series generalize both algebraic series and etol languages

some properties of viscosity solutions of hamilton jacobi equations abstract recently m g crandall and p l lions introduced the notion of viscosity solutions of scalar nonlinear first order partial differential equations viscosity solutions need not be differentiable anywhere and thus are not sensitive to the classical problem of the crossing of characteristics the value of this concept is established by the fact that very general existence uniqueness and continuous dependence results hold for viscosity solutions of many problems arising in fields of application the notion of a viscosity solution admits several equivalent formulations here we look more closely at two of these equivalent criteria and exhibit their virtues by both proving several new facts and reproving various known results in a simpler manner moreover by forsaking technical generality we hereby provide a more congenial introduction to this subject than the original paper author

finite element approximation of a nonlinear heat conduction problem in anisotropic media abstract this paper is a survey of results which we have obtained in solving a stationary nonlinear heat conduction problem by the finite element method in particular we present uniqueness theorems for the classical and weak solutions a comparison theorem existence theorems for the weak and finite element solutions approximation of a curved boundary and numerical integration a discrete maximum principle convergence without any regularity assumptions a priori error estimates nonlinear boundary conditions and numerical algorithms

automata and languages generalized to ω continuous semirings abstract we generalize the following two language and automata theoretic results to continuous semirings i the family of languages accepted by finite automata is the smallest class containing all finite languages and closed under union product and star kleene s theorem ii the family of languages accepted by pushdown automata is the family ofcontext free languages