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

high order exponential operator splitting methods for time dependent schrodinger equations in this paper we deduce high order error bounds for exponential operator splitting methods the employed techniques are specific to linear differential equations of the form u t a u t b u t t geq 0 involving an unbounded operator a in particular evolutionary schrodinger equations with sufficiently regular initial values are included in the analysis

positive entire solutions of semilinear elliptic equations on considere l equation elliptique semi lineaire u m 2 u f x u 0 x r n n 2 ou m 0 est une constante et f x u est une fonction localement continue de holder dans r n 0 on etudie l existence et le comportement asymptomatique des solutions entieres positives

invariant measures for cherry flows we investigate the invariant probability measures for cherry flows i e flows on the two torus which have a saddle a source and no other fixed points closed orbits or homoclinic orbits in the case when the saddle is dissipative or conservative we show that the only invariant probability measures are the dirac measures at the two fixed points and the dirac measure at the saddle is the physical measure in the other case we prove that there exists also an invariant probability measure supported on the quasi minimal set we discuss some situations when this other invariant measure is the physical measure and conjecture that this is always the case the main techniques used are the study of the integrability of the return time with respect to the invariant measure of the return map to a closed transversal to the flow and the study of the close returns near the saddle

positive solutions of a class of second order semilinear elliptic equations in the plane on considere l equation elliptique a 2 dimensions u f x u u x r 2 ou x x 1 x 2 et f x u p est une fonction continue non negative sur r 2 r xr 2 on considere l existence de solutions positives definies dans tout r 2

synchronization properties of random piecewise isometries we study the synchronization properties of the random double rotations on tori we give a criterion that show when synchronization is present in the case of random double rotations on the circle and prove that it is always absent in dimensions two and higher

pathological foliations and removable zero exponents the ergodic theory of uniformly hyperbolic or axiom a diffeomorphisms has been studied extensively beginning with the pioneering work of anosov sinai ruelle and bowen an si ru bo while uniformly hyperbolic systems enjoy strong mixing properties they are not dense among c diffeomorphisms sm as using the concept of lyapunov exponents pesin introduced a weaker form of hyperbolicity which he termed nonuniform hyperbolicity nonuniformly hyperbolic diffeomorphisms share several mixing properties with uniformly hyperbolic ones our construction of the diffeomorphisms in this paper was motivated by the question of whether nonuniform hyperbolicity is dense among a large class of diffeomorphisms as a curious by product of our construction we prove that a pathological feature of central foliations the complete failure of absolute continuity can exist in a c open set of volume preserving diffeomorphisms we recall the definition of a nonuniformly hyperbolic diffeomorphism a real number is a lyapunov exponent of the diffeomorphism g m m if there exists a nonzero vector v tm such that

absolutely singular dynamical foliations let a 3 be the product of the automorphism of t 2 and of the identity on t 1 a small perturbation g of a 3 among volume preserving diffeomorphisms will have an invariant family of smooth circles forming a continuous foliation of t 3 corresponding to the vector bundle tangent to the circles there is a central lyapunov exponent of g volume which is nonzero for an open set of ergodic g s this surprising result of shub and wilkinson is complemented here by showing that the volume on t 3 has atomic conditional measures on the s there is a finite k such that almost every carries k atoms of mass 1 k

convergence analysis of high order time splitting pseudospectral methods for nonlinear schrodinger equations in this work the issue of favorable numerical methods for the space and time discretization of low dimensional nonlinear schrodinger equations is addressed the objective is to provide a stability and error analysis of high accuracy discretizations that rely on spectral and splitting methods as a model problem the time dependent gross pitaevskii equation arising in the description of bose einstein condensates is considered for the space discretization pseudospectral methods collocated at the associated quadrature nodes are analyzed for the time integration high order exponential operator splitting methods are studied where the decomposition of the function defining the partial differential equation is chosen in accordance with the underlying spectral method the convergence analysis relies on a general framework of abstract nonlinear evolution equations and fractional power spaces defined by the principal linear part essential tools in the derivation of a temporal global error estimate are furthe

unbounded positive entire solutions of semilinear elliptic equations on etudie l existence et le comportement asymptotique des solutions positives non bornees d equations elliptiques semilineaires du type u f x u dans r n on etablit des resultats globaux

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

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

theoretical study and numerical simulation of pattern formation in the deterministic and stochastic gray scott equations abstract mathematical models based on systems of reaction diffusion equations provide fundamental tools for the description and investigation of various processes in biology biochemistry and chemistry in specific situations an appealing characteristic of the arising nonlinear partial differential equations is the formation of patterns reminiscent of those found in nature the deterministic gray scott equations constitute an elementary two component system that describes autocatalytic reaction processes depending on the choice of the specific parameters complex patterns of spirals waves stripes or spots appear in the derivation of a macroscopic model such as the deterministic gray scott equations from basic physical principles certain aspects of microscopic dynamics e g fluctuations of molecules are disregarded an expedient mathematical approach that accounts for significant microscopic effects relies on the incorporation of stochastic processes and the consideration of stochastic partial differential equations the present work is concerned with a theoretical and numerical study of the stochastic gray scott equations driven by independent spatially time homogeneous wiener processes under suitable regularity assumptions on the prescribed initial states existence as well as the uniqueness of the solution processes is proven numerical simulations based on the application of a time adaptive first order operator splitting method and the fast fourier transform illustrate the formation of patterns in the deterministic case and their variation under the influence of stochastic noise

foliations the ergodic theorem and brownian motion regarding the leaves of a compact foliated riemannian manifold as the orbits of a dynamical system we will prove a birkhoff ergodic theorem asserting that the spatial average of a function equals its time average along random paths in the leaves there are several unexpected developments in the foundations first there always are measures on any compact foliated riemannian manifold m f with respect to which an adequate brownian motion along the leaves ergodic theorem holds second relative to these measures which are called harmonic there is a liouville type theorem for any bounded bore1 function on the manifold which is harmonic on each leaf namely for almost all leaves relative to any one of these harmonic measures this leaf harmonic function is constant on each leaf third the harmonic measures have a very reasonable local characterization in any foliation coordinate system they are transversal sums of h l dx l where h l is a positive harmonic function on the leaf l and dx l is the riemannian measure on l these statements comprise theorem 1 which is stated more precisely later in the introduction the first fact is surprising because up to now the only measures attached to foliations were the holonomy invariant transverse measures of plante 141 and ruelle sullivan 151 and some foliations do not admit these the second fact is unexpected because there are foliations where all the leaves are hyperbolic planes and each hyperbolic plane being conformal to the disk admits 285 0022 1236183 3 00

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

lyapunov exponents and relative entropy for a stochastic flow of diffeomorphisms the lyapunov exponents 1 2 d for a stochastic flow of diffeomorphisms of a d dimensional manifold m with a strongly recurrent one point motion describe the almost sure limiting exponential growth rates of tangent vectors under the flow this paper shows how the lyapunov exponents are related to measure preserving properties of the stochastic flow on m and of the induced stochastic flow on the projective bundle pm relative entropy is used to quantify the extent to which a measure fails to be invariant under the flow the results include the following if m is compact and if the one point motion on m is a non degenerate diffusion with stationary probability measure then 1 d 0 with equality if and only if the flow preserves almost surely if in addition the induced one point motion on pm satisfies a weak non degeneracy condition then 1 d if and only if there is a smooth riemannian structure on m with respect to which the flow is conformal almost surely

a wigner measure analysis of the dufort frankel scheme for the schrodinger equation we apply wigner transform techniques to the analysis of the dufort frankel difference scheme for the schrodinger equation and to the continuous analogue of the scheme in the case of a small scaled planck constant semiclassical regime in this way we are able to obtain sharp conditions on the spatial temporal grid which guarantee convergence for average values of observables as the planck constant tends to zero the theory developed in this paper is not based on local and global error estimates and does not depend on whether or not caustics develop numerical test examples are presented to help interpret the theory and to compare the dufort frankel scheme to other difference schemes for the schr odinger equation

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

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

defect based local error estimators for splitting methods with application to schrodinger equations part ii higher order methods for linear problems in this work defect based local error estimators for higher order exponential operator splitting methods are constructed and analyzed in the context of time dependent linear schrodinger equations the technically involved procedure is carried out in detail for a general three stage third order splitting method and then extended to the higher order case asymptotical correctness of the a posteriori local error estimator is proven under natural commutator bounds for the involved operators and along the way the known non stiff order conditions and a priori convergence bounds are recovered the theoretical error estimates for higher order splitting methods are confirmed by numerical examples for a test problem of schrodinger type further numerical experiments for a test problem of parabolic type complement the investigations

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

positive solutions of semilinear elliptic problems in unbounded domains di a ax i l n each a e c yq b e cg q h x 3 b 0 0 ci 1 and f x u satisfies assumptions f fs below in particular it is assumed that f x 0 0 for all x e 52 implying that the boundary value problem 1 1 always has the trivial solution our main purpose is to establish the existence of a positive solution of 1 1 throughout 1 2 in cases for which the nonlinearity in 1 1 is unbounded above i e f x r t co as t co locally uniformly in q we treat the case of bounded nonlinearities elsewhere 20 the main theorem 4 5

hyperbolicity sinks and measure in one dimensional dynamics letf be ac2 map of the circle or the interval and let f denote the complement of the basins of attraction of the attracting periodic orbits we prove that f is a hyperbolic expanding set if and obviously only if every periodic point is hyperbolic and f doesn t contain the critical point this is the real one dimensional version of fatou s hyperbolicity criteria for holomorphic endomorphisms of the riemann sphere we also explore other applications of the techniques used for the result above proving for instance that for everyc2 immersionf of the circle i e a map of the circle onto itself without critical points either its julia set has measure zero or it is the whole circle and thenf is ergodic i e positively invariant borel sets have zero or full measure

discrete transparent boundary conditions for wide angle parabolic equations in underwater acoustics this paper is concerned with transparent boundary conditions tbcs for wide angle parabolic equations wapes in the application to underwater acoustics assuming cylindrical symmetry existing discretizations of these tbcs introduce slight numerical reflections at this artificial boundary and also render the overall crank nicolson finite difference method only conditionally stable here a novel discrete tbc is derived from the fully discretized whole space problem that is reflection free and yields an unconditionally stable scheme while we shall assume a uniform discretization in range the interior depth discretization i e in the water column may be nonuniform and we shall discuss strategies for the best exterior discretization i e in the sea bottom the superiority of the new discrete tbc over existing discretizations is illustrated on several benchmark problems in the literature different wapes or wape and the standard parabolic equation have been coupled in the water and the sea bottom we analyze under which conditions this yields a hybrid model that is conservative for the acoustic field

defect based local error estimators for splitting methods with application to schrodinger equations part i the linear case we introduce a defect correction principle for exponential operator splitting methods applied to time dependent linear schrodinger equations and construct a posteriori local error estimators for the lie trotter and strang splitting methods under natural commutator bounds on the involved operators we prove asymptotical correctness of the local error estimators and along the way recover the known a priori convergence bounds numerical examples illustrate the theoretical local and global error estimates

entropie geometrique des feuilletages on definit et on etudie l entropie geometrique d un feuilletage generalisant directement l entropie topologique d un flot

the harmonic measures of lucy garnett abstract this paper presents an improved approach to the theory of harmonic measures for foliated spaces introduced by garnett this approach is based on a method for solving elliptic equations on foliated spaces and on the hille yosida theory the diffusion semigroup of a general laplacian and its infinitesimal generator are made explicit applications of the path space to the dynamical study of a foliated space are described in particular the final section studies cocycles on foliated spaces a formula for their asymptotic limit and some analytic and geometric consequences

numerical approximation of quadratic observables of schrodinger type equations in the semi classical limit we apply wigner transform techniques to the analysis of difference methods for schrodinger type equations in the case of a small planck constant in this way we are able to obtain sharp conditions on the spatial temporal grid which guarantee convergence for average values of observables as the planck constant tends to zero the theory developed in this paper is not based on local and global error estimates and does not depend on whether caustics develop or not numerical test examples are presented to help interpret the theory

a numerical scheme for the simulation of blow up in the nonlinear schrodinger equation we present a self adaptive multigrid version of a conservative finite difference scheme useful for the study of collapse processes in nonlinear schrodinger equations nlses as an example we study the character of the focusing singularity of the two dimensional critical nlse

a convergent adaptive algorithm for poisson s equation we construct a converging adaptive algorithm for linear elements applied to poisson s equation in two space dimensions starting from a macro triangulation we describe how to construct an initial triangulation from a priori information then we use a posteriors error estimators to get a sequence of refined triangulation and approximate solutions it is proved that the error measured in the energy norm decreases at a constant rate in each step until a prescribed error bound is reached extension to higher order elements in two space dimension and numerical results are included

random walks on homeo s 1 in this paper we study random walks g n f n 1 ldots f 0 on the group homeo s 1 of the homeomorphisms of the circle where the homeomorphisms f k are chosen randomly independently with respect to a same probability measure nu we prove that under the only condition that there is no probability measure invariant by nu almost every homeomorphism the random walk almost surely contracts small intervals it generalizes what has been known on this subject until now since various conditions on nu were imposed in order to get the phenomenon of contractions moreover we obtain the surprising fact that the rate of contraction is exponential even in the lack of assumptions of smoothness on the f k s we deduce various dynamical consequences on the random walk g n finiteness of ergodic stationary measures distribution of the trajectories asymptotic law of the evaluations etc the proof of the main result is based on a modification of the avila viana s invariance principle working for continuous cocycles on a space fibred in circles

sur la dynamique unidimensionnelle en regularite intermediaire using probabilistic methods we prove new rigidity results for groups and pseudo groups of diffeomorphisms of one dimensional manifolds with intermediate regularity class i e between c 1 and c 2 in particular we show some generalizations of denjoy theorem and the classical kopell lemma for abelian groups these techniques are also applied to the study of codimension 1 foliations for instance we obtain several generalized versions of sacksteder theorem in class c 1 we conclude with some remarks about the stationary measure

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

the poisson formula for groups with hyperbolic properties the poisson boundary of a group g with a probability measure is the space of ergodic components of the time shift in the path space of the associated random walk via a generalization of the classical poisson formula it gives an integral representation of bounded harmonic functions on g in this paper we develop a new method of identifying the poisson boundary based on entropy estimates for conditional random walks it leads to simple purely geometric criteria of boundary maximality which bear hyperbolic nature and allow us to identify the poisson boundary with natural topological boundaries for several classes of groups word hyperbolic groups and discontinuous groups of isometries of gromov hyperbolic spaces groups with inflnitely many ends cocompact lattices in cartan hadamard manifolds discrete subgroups of semisimple lie groups