the stabilized self starter a stability problem is eliminated from the self starter which is a parabolic equation technique for generating initial conditions the stability problem is associated with evanescent modes that correspond to nearly vertical propagation it causes the solution to break down at certain frequencies for problems involving relatively low attenuation the error is restricted to relatively short ranges unless the medium is strongly range dependent near the source the stability problem is eliminated by placing the pole of the smoothing operator in the self starter away from the eigenvalues of the depth operator

the l 2 index theorem for homogeneous spaces of lie groups the l2 index theorem for covering spaces of atiyah 3 and singer 20 asserts that given a discrete group f acting smoothly and freely on a manifold m with compact quotient m f m and an elliptic differential operator d on m which is f invariant and thus descends to an elliptic differential operator d on the compact manifold m then the p index indrd dimrkerd dimrkerd of d coincides with the ordinary index ind d dim ker d dim ker d of d here ker d is the space of l2 solutions of du 0 and dim r denotes the dimension function corresponding to the trace on the commutant of f acting on the hilbert space of l2 sections over m naturally associated to p the importance of this theorem lies in the fact that indd 0 implies the existence of nontrivial l2 solutions for the equation du 0 and as such it was used in a crucial way by atiyah and schmid 5 to construct explicit realizations of the discrete series representations for semisimple lie groups indeed if g is a lie group which possesses a discrete torsion free cocompact subgroup f and h is a compact subgroup of g then the l2 index theorem applied to the covering space m g h of m f g h combined with the index formula of atiyah singer 6 yields existence results for l2 solutions of g invariant elliptic equations on the homogeneous space g h it is relevant to note that with d denoting this time a g invariant elliptic differential operator on g h the ratio between the f index of d and the covolume of f gives a number independent of f this real number indg d can be in fact intrinsically defined as the difference of the two formal degrees corresponding to the representations of g on ker d and ker d respectively and hence makes sense for any unimodular lie group g even when it has no discrete cocompact subgroups which outside the semisimple case is the generic

exponential times in the one dimensional gross pitaevskii equation with multiple well potential we consider the gross pitaevskii equation in 1 space dimension with a n well trapping potential we prove in the semiclassical limit that the finite dimensional eigenspace associated to the lowest n eigenvalues of the linear operator is slightly deformed by the nonlinear term into an almost invariant manifold mathcal m precisely one has that solutions starting on mathcal m or close to it will remain close to mathcal m for times exponentially long with the inverse of the size of the nonlinearity as heuristically expected the effective equation on mathcal m is a perturbation of a discrete nonlinear schrodinger equation we deduce that when the size of the nonlinearity is large enough then tunneling among the wells essentially disappears that is for almost all solutions starting close to mathcal m their restriction to each of the wells has norm approximatively constant over the considered time scale in the particular case of a double well potential we give a more precise result showing persistence or destruction of the beating motions over exponentially long times the proof is based on canonical perturbation theory surprisingly enough due to the gauge invariance of the system no non resonance condition is required

bifurcation and stability for nonlinear schrodinger equations with double well potential in the semiclassical limit we consider the stationary solutions for a class of schrodinger equations with a symmetric double well potential and a nonlinear perturbation here in the semiclassical limit we prove that the reduction to a finite mode approximation give the stationary solutions up to an exponentially small term and that symmetry breaking bifurcation occurs at a given value for the strength of the nonlinear term the kind of bifurcation picture only depends on the nonlinearity power we then discuss the stability instability properties of each branch of the stationary solutions finally we consider an explicit one dimensional toy model where the double well potential is given by means of a couple of attractive dirac s delta pointwise interactions

radiowave propagation analysis with a bidirectional 3 d vector parabolic equation method a 3 d bidirectional solution to the parabolic approximation of the wave equation is investigated by using a vector field representation the backward propagating wave is integrated to the classical parabolic equation approach which represents the forward propagating wave propagation over flat terrain in the presence of knife edges is considered as well as over irregular terrain consisting of hills modeled by the succession of knife edges at each knife edge appropriate boundary conditions are enforced and the wave is partly reflected in the backward direction the wave is marched in both directions by using the split step algorithm different tests are conducted in order to analyze and validate the results obtained by the proposed algorithm comparisons with results from both 2 d parabolic equation based algorithm and 3 d finite difference time domain based algorithm are presented in this paper

a split step pade solution for the parabolic equation method a split step pade solution is derived for the parabolic equation pe method higher order pade approximations are used to reduce both numerical errors and asymptotic errors e g phase errors due to wide angle propagation this approach is approximately two orders of magnitude faster than solutions based on pade approximations that account for asymptotic errors but not numerical errors in contrast to the split step fourier solution which achieves similar efficiency for some problems the split step pade solution is valid for problems involving very wide propagation angles large depth variations in the properties of the waveguide and elastic ocean bottoms the split step pade solution is practical for global scale problems

nonlinear double well schrodinger equations in the semiclassical limit we consider time dependent schrodinger equations with a double well potential and an external nonlinear both local and non local perturbation in the semiclassical limit the finite dimensional eigenspace associated to the lowest eigenvalues of the linear operator is almost invariant for times of the order of the beating period and the dominant term of the wavefunction is given by means of the solutions of a finite dimensional dynamical system in the case of local nonlinear perturbation we assume the spatial dimension d 1 or d 2

rational square root approximations for parabolic equation algorithms in this article stable pade approximations to the function 1 z are derived by choosing a branch cut in the negative half plane the pade coefficients are complex and may be derived analytically to arbitrary order from the corresponding real coefficients associated with the principal branch defined by z 1 i z 0 i z denotes the imaginary part of z the characteristics of the corresponding square root approximation are illustrated for various segments of the complex plane in particular for waveguide problems it is shown that an increasingly accurate representation may be obtained of both the evanescent part of the mode spectrum for the acoustic case and the complex mode spectrum for the elastic case an elastic parabolic equation algorithm is used to illustrate the application of the new pade approximations to a realistic ocean environment including elasticity in the ocean bottom

a nonlinear schrodinger equation with two symmetric point interactions in one dimension we consider a time dependent one dimensional nonlinear schrodinger equation with a symmetric double well potential represented by two dirac s among our results we give an explicit formula for the integral kernel of the unitary semigroup associated with the linear part of the hamiltonian then we establish the corresponding strichartz type estimate and we prove local existence and uniqueness of the solution to the original nonlinear problem

destruction of the beating effect for a non linear schrodinger equation we consider a non linear perturbation of a symmetric double well potential as a model for molecular localization in the semiclassical limit we prove the existence of a critical value of the perturbation parameter giving the destruction of the beating effect this value is twice the one corresponding to the first bifurcation of the fundamental state here we make use of a particular projection operator introduced by g nenciu in order to extend to an infinite dimensional space some known results for a two level system

a two way parabolic equation that accounts for multiple scattering a two way parabolic equation that accounts for multiple scattering is derived and tested a range dependent medium is divided into a sequence of range independent regions the field is decomposed into outgoing and incoming fields in each region the conditions between vertical interfaces are implemented using rational approximations for the square root of an operator rational approximations are also used to relate fields between neighboring interfaces an iteration scheme is used to solve for the outgoing and incoming fields at the vertical interfaces the approach is useful for solving problems involving scattering from waveguide features and compact objects

normal form for the symmetry breaking bifurcation in the nonlinear schrodinger equation abstract we derive and justify a normal form reduction of the nonlinear schrodinger equation for a general pitchfork bifurcation of the symmetric bound state that occurs in a double well symmetric potential we prove persistence of normal form dynamics for both supercritical and subcritical pitchfork bifurcations in the time dependent solutions of the nonlinear schrodinger equation over long but finite time intervals

a two way parabolic equation for acoustic backscattering in the ocean the parabolic equation pe method is generalized to handle backscattered acoustic energy in the ocean the two way pe is based on the single scattering approximation and the approach of two way coupled modes in which range dependent environments are approximated by a sequence of range independent regions at the vertical boundaries between regions the solution of the two way pe is required to satisfy two continuity conditions the range derivative in one of the conditions is replaced by a higher order pe depth operator the reflected and transmitted fields that satisfy these conditions are computed with an efficient iteration scheme the outgoing and incoming fields are propagated by two way range marching the two way pe which is presently implemented for two dimensional problems is a practical method for solving large scale reverberation problems the accuracy of the two way pe is demonstrated by comparisons with reference solutions the two way pe is applied to simulate the localization of a source of backscattering using the method of back propagation

propagation modeling over irregular terrain by the improved two way parabolic equation method the two way split step parabolic equation 2w sspe method is improved to be able to model both forward and backward waves propagation over irregular perfectly electrical conducting pec surfaces and irregular lossy surfaces with impedance boundary conditions ibc the discrete mixed fourier transform dmft is applied in both forward and backward direction to deal with ibc and the boundary shift bs method is used to dispose of the irregular terrain effects numerical results obtained by the improved 2w sspe algorithm for several typical scenarios with different ground types are compared with those got by the previous 2w sspe method and the one way parabolic equation method

the discrete spectrum of the spinless one dimensional salpeter hamiltonian perturbed by δ interactions we rigorously define the self adjoint one dimensional salpeter hamiltonian perturbed by an attractive of strength centred at the origin by explicitly providing its resolvent our approach is based on a coupling constant renormalization a technique used first heuristically in quantum field theory and implemented in the rigorous mathematical construction of the self adjoint operator representing the negative laplacian perturbed by the in two and three dimensions we show that the spectrum of the self adjoint operator consists of the absolutely continuous spectrum of the free salpeter hamiltonian and an eigenvalue given by a smooth function of the parameter the method is extended to the model with two twin attractive deltas symmetrically situated with respect to the origin in order to show that the discrete spectrum of the related self adjoint hamiltonian consists of two eigenvalues namely the ground state energy and that of the excited antisymmetric state we investigate in detail the dependence of these two eigenvalues on the two parameters of the model that is to say both the aforementioned strength and the separation distance with regard to the latter a remarkable phenomenon is observed differently from the well behaved schrodinger case the 1d salpeter hamiltonian with two identical dirac distributions symmetrically situated with respect to the origin does not converge as the separation distance shrinks to zero to the one with a single centred at the origin having twice the strength however the expected behaviour in the limit in the norm resolvent sense can be achieved by making the coupling of the twin deltas suitably dependent on the separation distance itself

on the variation in the cohomology of the symplectic form of the reduced phase space is called the momentum mapping of the hamiltonian t action given 1 1 the condition 1 2 just means that t acts along the fibers of j for the basic definitions and properties of non commutative hamiltonian group actions see am the results of this paper can easily be extended to hamiltonian actions of arbitrary compact connected lie groups by applying our results to the action of its maximal toms and using the equivariance of the momentum mapping for some more details see the remarks at the end of sect 2 we will assume throughout this paper that the momentum map is proper that is j i u is compact for each compact subset u of t now let r be a regular value of j that is t j tmm t is surjective for all meye j 1 4 then ye is a smooth submanifold of m compact because

supersymmetry and morse theory it is shown that the morse inequalities can be obtained by consideration of a certain supersymmetric quantum mechanics hamiltonian some of the implications of modern ideas in mathematics for supersymmetric theories are discussed

a two way parabolic equation for elastic media the parabolic equation pe method is extended to handle wave propagation and scattering in range dependent elastic waveguides which are approximated by a sequence of range independent regions the two way elastic pe involves an efficient pe based scattering method for computing reflected and transmitted fields at the vertical interfaces separating range independent regions the one way elastic pe is used to propagate the outgoing and incoming fields through the range independent regions with two way range marching in addition to providing the backscattered field the two way pe provides a correction to the outgoing field the self starter is generalized to handle a source in a solid layer