forced lattice vibrations part ii this is the first part of a two part series on forced lattice vibrations in which a semi infinite lattice of one dimensional particles xn n 1 is driven from one end by a particle x0 this particle undergoes a given periodically perturbed uniform motion x0 t at h yt where a and are constants and h has period 2 for a wide variety of restoring forces f i e f 0 numerical calculations indicate the existence of a sequence of thresholds 1 1 a h f 2 2 a h f k k a h f k 0 as k if k k 1 a k phase wave that is well described by the wave form emerges and travels through the lattice the goal of this series is to describe the emergence and calculate some properties of these wave forms in part i the authors first consider the case where f x ex i e toda forces but h is arbitrary and show how to compute a basic diagnostic see j formula 1 26 for the system in terms of the solution of an associated scalar riemann hilbert problem once a certain finite set of numbers is known in another direction the authors consider the case where f x is restoring but arbitrary and h is small here the authors prove a general result asserting that if there exists a sufficiently ample family of traveling wave solutions of the doubly infinite lattice then it is possible to construct time periodic k phase wave solutions with asymptotics in n of type iii for the driven system i in part ii the authors prove that sufficiently ample families of traveling wave solutions of the system iv exist in the cases 1 and 1 2 for general restoring forces f in the case with toda forces f x ex the authors prove that sufficiently ample families of traveling wave solutions

wkb analysis for nonlinear schrodinger equations with potential we justify the wkb analysis for the semiclassical nonlinear schrodinger equation with a subquadratic potential this concerns subcritical critical and supercritical cases as far as the geometrical optics method is concerned in the supercritical case this extends a previous result by e grenier we also have to restrict to nonlinearities which are defocusing and cubic at the origin but besides subquadratic potentials we consider initial phases which may be unbounded for this we construct solutions for some compressible euler equations with unbounded source term and unbounded initial velocity

a weighted occupation time for a class of measured valued branching processes a weighted occupation time is defined for measure valued processes and a representation for it is obtained for a class of measure valued branching random motions on rd considered as a process in its own right the first and second order asymptotics are found as time t specifically the finiteness of the total weighted occupation time is determined as a function of the dimension d and when infinite a central limit type renormalization is considered yielding gaussian or asymmetric stable generalized random fields in the limit in one gaussian case the results are contrasted in high versus low dimensions

geometry of the modulational instability iii homoclinic orbits for the periodic sine gordon equation in this paper the homoclinic geometric structure of the integrable sine gordon equation under periodic boundary conditions is developed specifically focus is given to orbits homoclinic to n tori simple examples of such homoclinic orbits are constructed and a physical interpretation of these states is given a labeling is provided which identifies and catalogues all such orbits these orbits are related in a one to one manner to linearized instabilities explicit formulas for all homoclinic orbits are given in terms of backlund transformations

geometric optics and instability for semi classical schrodinger equations we prove some instability phenomena for semi classical linear or nonlinear schrodinger equations for some perturbations of the data we show that for very small times we can neglect the laplacian and the mechanism is the same as for the corresponding ordinary differential equation our approach allows smaller perturbations of the data where the instability occurs for times such that the problem cannot be reduced to the study of an ordinary differential equation

the semiclassical limit of the defocusing nls hierarchy we establish the semiclassical limit of the one dimensional defocusing cubic nonlinear schrodinger nls equation complete integrability is exploited to obtain a global characterization of the weak limits of the entire nls hierarchy of conserved densities as the field evolves from reflectionless initial data under all the associated commuting flows consequently this also establishes the zero dispersion limit of the modified korteweg de vries equation that resides in that hierarchy we have adapted and clarified the strategy introduced by lax and levermore to study the zero dispersion limit of the korteweg de vries equation expanding it to treat entire integrable hierarchies and strengthening the limits obtained a crucial role is played by the convexity of the underlying log determinant with respect to the times associated with the commuting flows c 1999 john wiley sons inc

loss of regularity for supercritical nonlinear schrodinger equations we consider the nonlinear schrodinger equation with defocusing smooth nonlinearity below the critical sobolev regularity it is known that the cauchy problem is ill posed we show that this is even worse namely that there is a loss of regularity in the spirit of the result due to g lebeau in the case of the wave equation as a consequence the cauchy problem for energy supercritical equations is not well posed in the sense of hadamard we reduce the problem to a supercritical wkb analysis for super cubic smooth nonlinearity this analysis is new and relies on the introduction of a modulated energy functional a la brenier

one dimensional stochastic partial differential equations and the branching measure diffusion an existence theorem in the spirit of keisler ke is proved for the simple one dimensional diffusion equation driven by white noise modulated by a non linear function of the solution this is used to obtain a density and a stochastic partial differential equation in one dimension for the critical branching diffusion studied by dawson et al

multiplicative and transitive symplectic groups classification des sous groupes transitifs et multiplicatifs de sp v decomposition de sp v quand i ou r est stable

supercritical geometric optics for nonlinear schrodinger equations we consider the small time semi classical limit for nonlinear schrodinger equations with defocusing smooth nonlinearity for a super cubic nonlinearity the limiting system is not directly hyperbolic due to the presence of vacuum to overcome this issue we introduce new unknown functions which are defined nonlinearly in terms of the wave function itself this approach provides a local version of the modulated energy functional introduced by y brenier the system we obtain is hyperbolic symmetric and the justification of wkb analysis follows

semi classical limit of schrodinger poisson equations in space dimension n 3 abstract we prove the existence of solutions to the schrodinger poisson system on a time interval independent of the planck constant when the doping profile does not necessarily decrease at infinity in the presence of a subquadratic external potential the lack of integrability of the doping profile is resolved by working in zhidkov spaces in space dimension at least three we infer that the main quadratic quantities position density and modified momentum density converge strongly as the planck constant goes to zero when the doping profile is integrable we prove pointwise convergence

the small dispersion limit of the korteweg de vries equation ii in part i the scattering transform method is used to study the weak limit of solutions to the initial value problem for the korteweg devries kdv equation as the dispersion tends to zero in that limit the associated schrodinger operator becomes semiclassical so the exact scattering data is replaced by its corresponding wkb expressions only nonpositive initial data are considered in that case the limiting reflection coefficient vanishes the explicit solution of kay and moses for the reflectionless inverse transform is then analyzed and the weak limit valid for all time is characterized by a quadratic minimum problem with constraints this minimum problem is reduced to a riemann hilbert problem in function theory in parts ii and iii we use function theoretical methods to solve the riemann hilbert problem in terms of solutions to an auxiliary initial value problem the weak limit satisfies the kdv equation with the dispersive term dropped until its derivatives become infinite up to that time the weak limit is a strong l2 limit at later times the weak limit is locally described by whitham s averaged equations or more generally by the equations found by flaschka et al using multiphase averaging for large times behavior of the weak limit is studied directly from the minimum problem

optimal local holder index for density states of superprocesses with 1 β branching mechanism for 0 0 if d 1 but locally unbounded otherwise moreover in the case of continuity we determine the optimal local holder index

smoothing property for schrodinger equations with potential superquadratic at infinity we prove a smoothing property for one dimensional time dependent schrodinger equations with potentials which satisfy at infinity k 2 as an application we show that the initial value problem for certain nonlinear schrodinger equations with such potentials is l2 well posed we also prove a sharp asymptotic estimate of the lp norm of the normalized eigenfunctions of h v for large energy

a local time correspondence for stochastic partial differential equations it is frequently the case that a white noise driven parabolic and or hyperbolic stochastic partial differential equation spde can have random field solutions only in spatial dimension one here we show that in many cases where the spatial operator is the l generator of a levy process x a linear spde has a random field solution if and only if the symmetrization of x possesses local times this result gives a probabilistic reason for the lack of existence of random field solutions in dimensions strictly bigger than one in addition we prove that the solution to the spde is holder continuous in its spatial variable if and only if the said local time is holder continuous in its spatial variable we also produce examples where the random field solution exists but is almost surely unbounded in every open subset of space time our results are based on first establishing a quasi isometry between the linear l space of the weak solutions of a family of linear spdes on one hand and the dirichlet space generated by the symmetrization of x on the other hand we study mainly linear equations in order to present the local time correspondence at a modest technical level however some of our work has consequences for nonlinear spdes as well we demonstrate this assertion by studying a family of parabolic spdes that have additive nonlinearities for those equations we prove that if the linearized problem has a random field solution then so does the nonlinear spde moreover the solution to the linearized equation is holder continuous if and only if the solution to the nonlinear equation is and the solutions are bounded and unbounded together as well finally we prove that in the cases that the solutions are unbounded they almost surely blow up at exactly the same points date october 7 2007 2000 mathematics subject classification primary 60h15 60j55 secondary 35r60 35d05

large oscillations arising in a dispersive numerical scheme abstract we study the oscillatory behavior that arises in solutions of a dispersive numerical scheme for the hopf equation whenever the classical solution of the equation develops a singularity modulation equations are derived that describe period two oscillations so long as the solution of those equations takes values for which the equations are hyperbolic however those equations have an elliptic region that may be entered by its solutions in a finite time after which the corresponding period two oscillations are seen to break down this kind of phenomenon has not been observed for integrable schemes the generation and propagation of period two oscillations are asymptotically analyzed and a matching formula is found for the transition between oscillatory and nonoscillatory regions modulation equations are also presented for period three oscillations numerical experiments are carried out that illustrate our analysis

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

macroscopic lattice dynamics abstract fully nonlinear modulation equations are presented that govern the slow evolution of single phase harmonic wavetrains under a large family of spatially discrete nonintegrable flows among which is the discrete nonlinear schrodinger equation dnls these modulation equations are a pair of partial differential equations in conservation form that are hyperbolic for some data and elliptic for other data in some cases these equations are capable of dynamically changing type from hyperbolic to elliptic a phenomenon that has been associated with the modulational instability of the underlying wavetrain by putting the modulation equations in riemann invariant form one can select initial data that avoid this dynamic change of type numerical experiments demonstrating the theoretical results are presented

on semiclassical zero dispersion limit solutions of the focusing nonlinear schrodinger equation we calculate the leading order term of the solution of the focusing nonlinear cubic schrodinger equation nls in the semiclassical limit for a certain one parameter family of initial conditions this family contains both solitons and pure radiation in the pure radiation case our result is valid for all times t 0 we utilize the riemann hilbert problem formulation of the inverse scattering problem to obtain the leading order term of the solution error estimates are provided 2004 wiley periodicals inc

the behavior of the weyl function in the zero dispersion kdv limit the moment formulas that globally characterize the zero dispersion limit of the korteweg devries kdv equation are known to be expressed in terms of the solution of a maximization problem here we establish a direct relation between this maximizer and the zero dispersion limit of the logarithm of the jost functions associated with the inverse spectral transform all the kdv conserved densities are encoded in the spatial derivative of these functions known as weyl functions we show the weyl functions are densities of measures that converge in the weak sense to a limiting measure this limiting measure encodes all of the weak limits of the kdv conserved densities moreover we establish the weak limit of spectral measures associated with the dirichlet problem

wkb analysis of bohmian dynamics we consider a semiclassically scaled schrodinger equation with wkb initial data we prove that in the classical limit the corresponding bohmian trajectories converge locally in measure to the classical trajectories before the appearance of the first caustic in a second step we show that after caustic onset this convergence in general no longer holds in addition we provide numerical simulations of the bohmian trajectories in the semiclassical regime that illustrate the above results 2014 wiley periodicals inc

stochastic partial differential equation driven by stable noise we construct a weak solution to the stochastic partial differential equation driven by a one sided stable noise without negative jumps we prove the weak existence of the solution for parameters the facts that for is not a lipschitz function and is not a gaussian noise require the development of new methods which we believe are of independent interest we also show that when the above equation gives an alternative description of super brownian motion with stable branching mechanism

dispersive approximations in fluid dynamics we have compared the results of a von neumann richtmyer calculation with the weak limit of calculations performed by von neumann s original method although there are discernible differences they are surprisingly small we analyze von newmann s algorithm for solving the initial value problem for the lagrangian equations of compressible flow

stochastic integral representation and regularity of the density for the exit measure of super brownian motion this paper studies the regularity properties of the density of the exit measure for super brownian motion with 1 stable branching mechanism it establishes the continuity of the density in dimension d 2 and the unboundedness of the density in all other dimensions where the density exists an alternative description of the exit measure and its density is also given via a stochastic integral representation results are applied to the probabilistic representation of nonnegative solutions of the partial differential equation u u 1

local smoothing property and strichartz inequality for schrodinger equations with potentials superquadratic at infinity abstract we study smoothing properties for time dependent schrodinger equations i u t 1 2 u v x u u 0 l 2 r n with potentials which satisfy v x o x m at infinity m 2 we show that the solution u t x is 1 m times differentiable with respect to x at almost all t r and explain that this is the result of the fact that the sojourn time of classical particles with energy in arbitrary compact set is less than ct 1 m during 0 t when is very large we also show strichartz s inequality with derivative loss for such potentials and give its application to nonlinear schrodinger equations

on the stochastic heat equation with spatially colored random forcing we consider the stochastic heat equation of the following form t ut x lut x b ut x ut x t x for t 0 x r where l is the generator of a levy process and is a spatially colored temporally white gaussian noise we will be concerned mainly with the long term behavior of the mild solution to this stochastic pde for the most part we work under the assumptions that the initial data u0 is a bounded and measurable function and is nonconstant and lipschitz continuous in this case we find conditions under which the preceding stochastic pde admits a unique solution which is also weakly intermittent in addition we study the same equation in the case that lu is replaced by its massive dispersive analogue lu u where r and we describe accurately the effect of the parameter on the intermittence of the solution in the case that u is proportional to u the parabolic anderson model we also look at the linearized version of our stochastic pde that is the case when is identically equal to one any other constant works also in this case we study not only the existence and uniqueness of a solution but also the regularity of the solution when it exists and is unique

bohmian measures and their classical limit abstract we consider a class of phase space measures which naturally arise in the bohmian interpretation of quantum mechanics we study the classical limit of these so called bohmian measures in dependence on the scale of oscillations and concentrations of the sequence of wave functions under consideration the obtained results are consequently compared to those derived via semi classical wigner measures to this end we shall also give a connection to the theory of young measures and prove several new results on wigner measures themselves our analysis gives new insight on oscillation and concentration effects in the semi classical regime

equations d ondes quasilineaires et estimations de strichartz dans cet article notre but est de resoudre des equations d ondes quasilineaires pour des donnees initiales moins regulieres que ce qu imposent les methodes d energie ceci impose de demontrer des estimees de type strichartz pour des operateurs d ondes a coefficients seulement lipschitziens

morse and melnikov functions for nls pde s the theory of the focusing nls equation under periodic boundary conditions together with the floquet spectral theory of its associated zakharov shabat linear operator hat l is developed in sufficient detail for later use in studies of perturbations of the nls equation counting lemmas for the non selfadjoint operator hat l are established which control its spectrum and show that all of its eccentricities are finite in number and must reside within a finite discd in the complex eigenvalue plane the radius of the discd is controlled by theh 1 norm of the potential overrightarrow q for this integrable nls hamiltonian system unstable tori are identified and backlund transformations are then used to construct global representations of their stable and unstable manifolds whiskered tori for the nls pde the floquet discriminant delta lambda overrightarrow q is used to introduce a natural sequence of nls constants of motion f j overrightarrow q equiv delta lambda lambda j c overrightarrow q overrightarrow q where denotes thej th critical point of the floquet discriminant a taylor series expansion of the constants f j overrightarrow q with explicit representations of the first and second variations is then used to study neighborhoods of the whiskered tori in particular critical tori with hyperbolic structure are identified through the first and second variations of f j overrightarrow q which themselves are expressed in terms of quadratic products of eigenfunctions of hat l the second variation permits identification within the disc d of important bifurcations in the spectral configurations of the operator hat l the constant f j overrightarrow q as the height of the floquet discriminant over the critical point admits a natural interpretation as a morse function for nls isospectral level sets this morse interpretation is studied in some detail it is valid globally for the infinite tail f j overrightarrow q j n which is associated with critical points outside the discd within this disc the interpretation is only valid locally with the same obstruction to its global validity as to a global ordering of the spectrum nevertheless this local morse theory together with the backlund representations of the whiskered tori produces extremely clear pictures of the stratification of nls invariant sets near these whiskered tori pictures which are useful in the study of perturbations of nls finally a natural connection is noted between the constants f j overrightarrow q of the integrable theory and melnikov functions for the theory of perturbations of the nls equation this connection generates a simple but general representations of the melnikov functions

regularity and irregularity of bolds 1 beta stable super brownian motion this paper establishes the continuity of the density of 1 stable super brownian motion 0 1 for fixed times in d 1 and local unboundedness of the density in all higher dimensions where it exists we also prove local unboundedness of the density in time for a fixed spatial parameter in any dimension where the density exists and local unboundedness of the occupation density the local time in the spatial parameter for dimensions d 2 where the local time exists

on dispersive difference schemes i etude du comportement oscillatoire et de la convergence faible des solutions des approximations aux differences dispersives de u t 1 2 u 2 x 0

unitary groups with excellent s and entire e u l abstract unitary groups of hermitian forms with a hyperbolic rank at least one over local rings have been studied by d g james j algebra 52 1978 354 363 using his methods we extend some of his results i e generators congruence subgroups and the classification of subgroups normalized by the eichler subgroups to more general rings which contain semilocal rings