finite volume modeling of poroelastic fluid wave propagation with mapped grids in this work we develop a high resolution mapped grid finite volume method code to model wave propagation in two dimensions in systems of multiple orthotropic poroelastic media and or fluids with curved interfaces between different media we use a unified formulation to simplify modeling of the various interface conditions open pores imperfect hydraulic contact or sealed pores that may exist between such media our numerical code is based on the clawpack framework but in order to obtain correct results at a material interface we use a modified transverse riemann solution scheme and at such interfaces are forced to drop the second order correction term typical of high resolution finite volume methods we verify our code against analytical solutions for reflection and transmission of waves at a material interface and for scattering of an acoustic wave train around an isotropic poroelastic cylinder for reflection and transmission at a flat interface we achieve second order convergence in the 1 no

analysis of the linearization around a critical point of an infinite dimensional hamiltonian system partial differential equations that conserve energy can often be written as infinite dimensional hamiltonian systems of the following general form du dt je u where j x x is a symplectic matrix i e jj 1 and e x r is a c 2 functional defined on some hilbert space x

dispersive and dissipative behaviour of high order discontinuous galerkin finite element methods the dispersive and dissipative properties of hp version discontinuous galerkin finite element approximation are studied in three different limits for the small wave number limit hk 0 we show the discontinuous galerkin gives a higher order of accuracy than the standard galerkin procedure thereby confirming the conjectures of hu and atkins j comput phys 182 2 2002 516 if the mesh is fixed and the order p is increased it is shown that the dissipation and dispersion errors decay at a super exponential rate when the order p is much larger than hk finally if the order is chosen so that 2p 1 hk for some fixed constant 1 then it is shown that an exponential rate of decay is obtained

graphical krein signature theory and evans krein functions two concepts evidently very different in nature have proved to be useful in analytical and numerical studies of spectral stability in nonlinear wave theory i the krein signature of an eigenvalue a quantity usually defined in terms of the relative orientation of certain subspaces that is capable of detecting the structural instability of imaginary eigenvalues and hence their potential for moving into the right half plane leading to dynamical instability under perturbation of the system and ii the evans function an analytic function detecting the location of eigenvalues one might expect these two concepts to be related but unfortunately examples demonstrate that there is no way in general to deduce the krein signature of an eigenvalue from the evans function for example by studying derivatives of the latter the purpose of this paper is to recall and popularize a simple graphical interpretation of the krein signature well known in the spectral theory of polynomial operator pencils once esta

tensored polynomial models the theme of the present paper is the study of two different versions of tensor products of functional models and present some applications to various problems related to system theory among those are stability of higher order systems tangent spaces spaces of intertwining maps invariant factors of tensor products of linear transformations and the solvability of sylvester equations

nodal high order methods on unstructured grids we present a convergent high order accurate scheme for the solution of linear conservation laws in geometrically complex domains as our main example we include a detailed development and analysis of a scheme for the time domain solution of maxwell s equations in a three dimensional domain the fully unstructured spatial discretization is made possible by the use of a high order nodal basis employing multivariate lagrange polynomials defined on the triangles and tetrahedra while the equations themselves are satisfied in a discontinuous galerkin form with the boundary conditions being enforced weakly through a penalty term accuracy stability and convergence of the semidiscrete approximation to maxwell s equations is established rigorously and bounds on the growth of the global divergence error are provided concerns related to efficient implementations are discussed in detail this sets the stage for the presentation of examples verifying the theoretical results and illustrating the versatility flexibility and robustness when solving two and three dimensional benchmark problems in computational electromagnetics pure scattering as well as penetration is discussed and high parallel performance of the scheme is demonstrated

stability theory for high order equations a liapunov type stability theory for high order systems of differential equations is developed this is done by reduction to the classical case using the theory of polynomial models

dynamics of optical pulses in randomly birefringent fibers abstract a nonlinear optical fiber with random birefringence is modeled by a system of nonlinear schrodinger nls equations with stochastic coupling an average variational principle is used to obtain the approximate evolution of the parameters describing pulses with nls soliton like shapes then in the diffusion limit the stochastic problem is reduced to a partial differential equation of the poincare sphere solutions to this equation give probability densities for the polarization state of a propagating pulse

factoring and decomposing a class of linear functional systems abstract within a constructive homological algebra approach we study the factorization and decomposition problems for a class of linear functional determined over determined under determined systems using the concept of ore algebras of functional operators e g ordinary partial differential operators shift operators time delay operators we first concentrate on the computation of morphisms from a finitely presented left module m over an ore algebra to another one m where m resp m is a module intrinsically associated with the linear functional system ry 0 resp r z 0 these morphisms define applications sending solutions of the system r z 0 to solutions of r y 0 we explicitly characterize the kernel image cokernel and coimage of a general morphism we then show that the existence of a non injective endomorphism of the module m is equivalent to the existence of a non trivial factorization r r 2 r 1 of the system matrix r the corresponding system can then be integrated in cascade under certain conditions we also show that the system ry 0 is equivalent to a system r z 0 where r is a block triangular matrix of the same size as r we show that the existence of idempotents of the endomorphism ring of the module m allows us to reduce the integration of the system ry 0 to the integration of two independent systems r 1 y 1 0 and r 2 y 2 0 furthermore we prove that under certain conditions idempotents provide decompositions of the system ry 0 i e they allow us to compute an equivalent system r z 0 where r is a block diagonal matrix of the same size as r applications of these results in mathematical physics and control theory are given finally the different algorithms of the paper are implemented in the maple package m orphisms based on the library oremodules

lie algebra representations nilpotent matrices and the c numerical range abstract in this paper we survey some applications of the representation theory of lie algebras to linear algebra this includes the derivation of the jordan form for tensor products of invertible matrices the study of normal form problems for nilpotent matrices as well as the derivation of explicit formulas for the c numerical radius of certain nilpotent block shift matrices arising in quantum mechanics and nuclear magnetic resonance nmr spectroscopy in this latter case a conjecture concerning an explicit formula for the c numerical radius is stated we show the existence of unitary transformations that realize the prospective maxima our approach depends on the clebsch gordan decomposition for unitary representations of su 2 c

structural stability of non ground state traveling waves of coupled nonlinear schrodinger equations abstract we consider the linear stability and structural stability of non ground state traveling waves of a pair of coupled nonlinear schrodinger equations cnls which describe the evolution of co propagating polarized pulses in the presence of birefringence viewing the cnls equations as a hamiltonian perturbation of the manakov equations we find parameter regimes in which there are two stable families of traveling waves the usual variational methods for stability analysis of ground states do not apply instead we employ a liapunov schmidt type reduction to detect eigenvalues bifurcating from the imaginary axis we also demonstrate the instability of a family of vector solitons

stability of multi parameter solitons asymptotic approach a general asymptotic approach to the stability problem of multi parameter solitons in hamiltonian systems has been developed the presented approach gives an analytical criterion for oscillatory instability and also predicts novel stationary instability of solitons higher order approximations allow one to calculate corresponding eigenvalues with an arbitrary accuracy it is also shown that asymptotic study of the soliton stability reduces to the calculation of a certain sequence of the determinants where the famous determinant of the matrix consisting from the derivatives of the system invariants is just the first in the series

on quadratic differential forms this paper develops a theory around the notion of quadratic differential forms in the context of linear differential systems in many applications we need to not only understand the behavior of the system variables but also the behavior of certain functionals of these variables the obvious cases where such functionals are important are in lyapunov theory and in lq and h infty optimal control with some exceptions these theories have almost invariably concentrated on first order models and state representations in this paper we develop a theory for linear time invariant differential systems and quadratic functionals we argue that in the context of systems described by one variable polynomial matrices the appropriate tool to express quadratic functionals of the system variables are two variable polynomial matrices the main achievement of this paper is a description of the interaction of one and two variable polynomial matrices for the analysis of functionals and for the application of higher order lyapunov functionals

an analysis of the discontinuous galerkin method for a scalar hyperbolic equation we prove lp stability and error estimates for the discontinuous galerkin method when applied to a scalar linear hyperbolic equation on a convex polygonal plane domain using finite element analysis techniques we obtain l2 estimates that are valid on an arbitrary locally regular triangulation of the domain and for an arbitrary degree of polynomials lp estimates for p 2 are restricted to either a uniform or piecewise uniform triangulation and to polynomials of not higher than first degree the latter estimates are proved by combining finite difference and finite element analysis techniques

low storage implicit explicit runge kutta schemes for the simulation of stiff high dimensional ode systems implicit explicit imex runge kutta rk schemes are effective for time marching ode systems with both stiff and nonstiff terms on the rhs such schemes implement an often a stable or better implicit rk scheme for the stiff part of the ode which is often linear and simultaneously a more convenient explicit rk scheme for the nonstiff part of the ode which is often nonlinear low storage rk schemes are especially effective for time marching high dimensional ode discretizations of pde systems on modern cache based computational hardware in which memory management is often the most significant computational bottleneck in this paper we develop and characterize eight new low storage implicit explicit rk schemes which have higher accuracy and better stability properties than the only low storage implicit explicit rk scheme available previously the venerable second order crank nicolson runge kutta wray cn rkw3 algorithm that has dominated the dns les literature for the last 25 years while requiring similar storage two three or four registers of length n and comparable floating point operations per timestep

on the elementary divisors of the sylvester and lyapunov maps the paper addresses the problem of computing the elementary divisors of the tensor product of linear transformations using the analysis of the tensor products of polynomial models as developed in fuhrmann and helmke 5 we use this to study the elementary divisors of the lyapunov and complementary lyapunov maps

on bezoutians van der monde matrices and the lienard chipart stability criterion we show how the bezoutian can be reduced by congruence to diagonal form with the congruence given by a van der monde matrix this result is applied to obtain new proofs of some classical stability criteria and in particular of the lienardchipart theorem

stability theory of solitary waves in the presence of symmetry ii abstract consider an abstract hamiltonian system which is invariant under a group of operators we continue to study the effect of the group invariance on the stability of solitary waves applications are given to bound states and traveling wave solutions of nonlinear wave equations

on tangential matrix interpolation abstract the paper presents an algebraic approach using polynomial and rational models over an arbitrary field to tangential interpolation problems both by polynomial as well as rational matrix functions appropriate extensions of scalar problems associated with the names of lagrange first order hermite high order and newton recursive are derived the relation of interpolation problems to the matrix chinese remainder theorem are clarified some two sided interpolation problems are discussed using the theory of tensored functional models

a study of behaviors abstract the paper presents the results of the study of behavior theory as developed by j c willems from the point of view of polynomial and rational models considering behaviors in the discrete time case to be generalizations of rational models a natural focal point becomes the concept of a behavior homomorphism we give a characterization of behavior homomorphisms and analyze their invertibility properties in terms of embeddings in unimodular polynomial matrices these results which are of intrinsic interest are then applied to the uniform derivation of a large number of results for equivalence in different classes of behavior representations to a certain extent these are generalizations of the strict system equivalence concept for the class of polynomial matrix description of systems in the style of rosenbrock a study of behavioral controllability is undertaken and gives some new insights into connections with geometric control theory

a discontinuous galerkin method for poroelastic wave propagation the two dimensional case abstract in this paper we consider a high order discontinuous galerkin dg method for modelling wave propagation in coupled poroelastic elastic media the upwind numerical flux is derived as an exact solution for the riemann problem including the poroelastic elastic interface attenuation mechanisms in both biot s low and high frequency regimes are considered the current implementation supports non uniform basis orders which can be used to control the numerical accuracy element by element in the numerical examples we study the convergence properties of the proposed dg scheme and provide experiments where the numerical accuracy of the scheme under consideration is compared to analytic and other numerical solutions

the krein matrix general theory and concrete applications in atomic bose einstein condensates when finding the nonzero eigenvalues for hamiltonian eigenvalue problems it is especially important to locate not only the unstable eigenvalues i e those with positive real part but also those which are purely imaginary but have negative krein signature these latter eigenvalues have the property that they can become unstable upon collision with other purely imaginary eigenvalues i e they are a necessary building block in the mechanism leading to the so called hamiltonian hopf bifurcation in this paper we review a general theory for constructing a meromorphic matrix valued function the so called krein matrix which has the property of not only locating the unstable eigenvalues but also those with negative krein signature these eigenvalues are realized as zeros of the determinant the resulting finite dimensional problem obtained by setting the determinant of the krein matrix to zero presents a valuable simplification in this paper the usefulness of the technique is illustrated through prototypi

a normal form for nonlinear resonance of embedded solitons a normal form for nonlinear resonance of embedded solitons is derived for a coupled two wave system that generalizes the second harmonic generating model this wave system is non hamiltonian in general an embedded soliton is a localized mode of the nonlinear system that coexists with the linear wave spectrum it occurs as a result of a codimension one bifurcation of non local wave solutions nonlinearity couples the embedded soliton and the linear wave spectrum and induces a one sided radiation driven decay of embedded solitons the normal form shows that the embedded soliton is semi stable i e it survives under perturbations of one sign but is destroyed by perturbations of the opposite sign when a perturbed embedded soliton sheds continuous wave radiation the radiation amplitude is generally not minimal even if the wave system is hamiltonian the results of the analytical theory are confirmed by numerical computations

a functional approach to the stein equation abstract this paper applies the theory of tensor products of functional models to the study of the stein equation we analyse the connections between the sylvester and stein equations as well as those between the anderson jury bezoutian associated with the polynomial sylvester equation and the t bezoutian associated with the homogeneous polynomial stein equation the functional setting allows us to extend the theory of finite section toeplitz matrices and their inversion

generalized bezoutian and matrix equations abstract a natural generalization of the classical bezout matrix of two polynomials is introduced for a family of several matrix polynomials the main aim of the paper is to show that this generalized bezoutian serves as an adequate connecting link between the class of equations in matrix polynomials m y z l r and the class of linear matrix equations ax xb c each equation in one of these classes is coupled with a certain equation in the other class so that for each couple the generalized bezoutian corresponding to a solution y z of the equation in matrix polynomials is a solution of the matrix equation and conversely any solution x of the matrix equation is a generalized bezoutian corresponding to a certain solution of the equation in matrix polynomials in particular either both equations are solvable or both have no solutions explicit formulas connecting the solutions of the two equations are given also various representation formulas for the generalized bezoutian are derived and its relation to the resultant matrix and the greatest common divisor of several matrix polynomials is discussed

discontinuous galerkin methods for wave propagation in poroelastic media we have developed a new numerical method to solve the heterogeneous poroelastic wave equations in bounded three dimensional domains this method is a discontinuous galerkin method that achieves arbitrary high order accuracy on unstructured tetrahedral meshes for the low frequency range and the inviscid case by using biot s equations and darcy s dynamic laws we have built a scheme that can successfully model wave propagation in fluid saturated porous media when anisotropy of the pore structure is allowed zero inflow fluxes are used as absorbing boundary conditions a continuous arbitrary high order derivatives time integration is used for the high frequency inviscid case whereas a space time discontinuous scheme is applied for the low frequency case we conducted a numerical convergence test of the proposed methods we used a series of examples to quantify the quality of our numerical results comparing them to analytic solutions as well as numerical solutions obtained by other methodologies in partic

lyapunov stability of ground states of nonlinear dispersive evolution equations on presente une nouvelle demonstration de la stabilite orbitale des solitons d etat fondamental de l equation de schrodinger non lineaire pour une grande classe de non linearites on demontre la stabilite de l onde solitaire pour l equation de korteweg de vries generalisee

tangent spaces of rational matrix functions we consider the task of determining tangent spaces for classes of rational matrix valued functions our analysis is based on methods from control theory and in particular the theory of polynomial models explicit descriptions of tangent spaces of rational transfer functions stable rational transfer functions rational inner functions and symmetric rational transfer functions are obtained moreover a new proof of delchamps s decomposition formula for the tangent bundle of rational transfer functions is given a riemannian metric as well as a symplectic structure is defined 0 1998 elsevier science inc

duality in polynomial models with some applications to geometric control theory duality is studied in the context of polynomial models for linear systems the output injection group the dual of the feedback group is studied and a polynomial characterization of c a invariant subspaces as well as of the maximal reachability subspaces contained in ker c is given

generalized polynomial bases and the bezoutian abstract a foundation polynomial is used to induce polynomial bases for fn 1 x the vector space of polynomials of degree less than n over an arbitrary field f the associated bases are then used to block diagonalize the bezout matrix of two polynomials under congruence

controllability observability and the solution of ax xb c a closed form finite series representation of the unique solution x of the matrix equation ax xb c is developed using this representation the image kernel and rank of x are related to the controllable and unobservable subspaces of the a c and c b pairs respectively bounds on the rank of x are obtainedin terms of the dimensions of these subspaces in the case that c has unitary rank an exact calculation of rank x is made the generic rank of x with a b fixed and c generic is evaluated

a space time discontinuous galerkin trefftz method for time dependent maxwell s equations we consider the discretization of electromagnetic wave propagation problems by a discontinuous galerkin method based on trefftz polynomials this method fits into an abstract framework for space time discontinuous galerkin methods for which we can prove consistency stability and energy dissipation without the need to completely specify the approximation spaces in detail any method of such a general form results in an implicit time stepping scheme with some basic stability properties for the local approximation on every space time element we then consider trefftz polynomials i e the subspace of polynomials that satisfy maxwell s equations exactly on the respective element we present an explicit construction of a basis for the local trefftz spaces in two and three dimensions and summarize some of their basic properties using local properties of the trefftz polynomials we can establish the well posedness of the resulting discontinuous galerkin trefftz method consistency stability and energy dis

linearized instability for nonlinear schrodinger and klein gordon equations in this paper i am introducing a new technique for proving instability of bound states for hamiltonian systems there are already two disparate types of instability results in the literature the approach developed by strauss shatah 20 gave an instability criterion coming from the variational structure of the problem on the other hand jones approach 11 produced a complementary criterion related to the difference between the number of negative eigenvalues of two selfadjoint operators using quite different techniques it turns out that with the methods developed in this paper these two criteria can be derived within a single framework that also leads to a generalization of the previous results finally in order to demonstrate how this method works i apply the instability criterion in some specific examples