variational properties and orbital stability of standing waves for nls equation on a star graph we study standing waves for a model of nonlinear schr odinger equation on a graph the graph is obtained joining n halflines at a vertex i e it is a star graph at the vertex an interaction occurs due to a boundary condition of delta type with strength alpha leqslant 0 which includes the free or kirchhoff case alpha 0 there imposed the nonlinearity is of focusing power type the dynamics is given by an equation of the form i frac d dt psi t h psi t psi t 2 mu psi t where h is the s a operator which generates the linear schr odinger dynamics on the graph in the hilbert space l 2 gg we show the existence of several families of standing waves solutions of the form phi t e i omega t psi omega where psi omega is a vector amplitude parametrized by a frequency omega they exist for every sign of the coupling at the vertex attractive alpha 0 and for every omega frac alpha 2 n 2 the number of the families depends on the number n of the edges of the graph we moreover determine the ground state which is the standing wave the amplitude of which minimizes the action on the natural nehari constraint and order the various families of standing waves according to their increasing action thus determining a nonlinear spectrum of the problem finally we show that the ground state is orbitally stable for every allowed omega if the nonlinearity is subcritical or critical for supercritical nonlinearities there exists an omega frac alpha 2 n 2 such that the ground state is orbitally stable for frac alpha 2 n 2 omega omega action minimization requires technical subtleties in particular an adaptation of decreasing rearrangements on a star graph is developed with a generalization of the p olya szeg h o inequality

instability of standing waves for klein gordon zakharov equations with different propagation speeds in three space dimensions abstract this paper is concerned with the standing wave for klein gordon zakharov equations with different propagation speeds in three space dimensions the existence of standing wave with the ground state is established by applying an intricate variational argument and the instability of the standing wave is shown by applying pagne and sattinger s potential well argument and levine s concavity method

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

stability and instability of solitary waves of korteweg de vries type considered herein are the stability and instability properties of solitary wave solutions of a general class of equations that arise as mathematical models for the unidirectional propagation of weakly nonlinear dispersive long waves special cases for which our analysis is decisive include equations of the korteweg de vries and benjamin ono type necessary and sufficient conditions are formulated in terms of the linearized dispersion relation and the nonlinearity for the solitary waves to be stable

error driven dynamical hp meshes with the discontinuous galerkin method for three dimensional wave propagation problems an hp adaptive discontinuous galerkin method for electromagnetic wave propagation phenomena in the time domain is proposed the method is highly efficient and allows for the first time the adaptive full wave simulation of large time dependent problems in three dimensional space refinement is performed anisotropically in the approximation order p and the mesh step size h regardless of the resulting level of hanging nodes for guiding the adaptation process a variant of the concept of reference solutions with largely reduced computational costs is proposed the computational mesh is adapted such that a given error tolerance is respected throughout the entire time domain simulation

the hp local discontinuous galerkin method for low frequency time harmonic maxwell equations the local discontinuous galerkin method for the numerical approximation of the time harmonic maxwell equations in a low frequency regime is introduced and analyzed topologically nontrivial domains and heterogeneous media are considered containing both conducting and insulating materials the presented method involves discontinuous galerkin discretizations of the curl curl and grad div operators derived by introducing suitable auxiliary variables and so called numerical fluxes an hp analysis is carried out and error estimates that are optimal in the meshsize h and slightly suboptimal in the approximation degree p are obtained

periodic solutions of the intermediate long wave equation a nonlinear superposition principle periodic stationary wave solutions of the intermediate long wave ilw equation are derived using the bilinear transformation method and a new expression for the dispersion relation is obtained the class of physically important real valued solutions is identified these solutions may be represented as an infinite superposition of solitary wave profiles a property shared by the related korteweg de vries kdv and benjamin ono bo equation this nonlinear superposition principle which has been the subject of various interpretations in the literature is discussed the ilw periodic solution approximates to a sinusoidal wave and a solitary wave in the limits of small and large amplitudes respectively for intermediate amplitudes the solution can be well approximated by either a sine wave or solitary wave in the shallow water kdv limit the ilw periodic solution leads to the familiar cnoidal wave whereas the deep water bo limit yields benjamin s periodic wave a previously unknown expression for the cnoidal wave dispersion relation in terms of theta functions is obtained

on the solitary wave dynamics under slowly varying medium for nonlinear schrodinger equations we consider the problem of a solitary wave propagation in a slowly varying medium for a variable coefficients nonlinear schrodinger equation we prove global existence and uniqueness of solitary wave solutions for a large class of slowly varying media moreover we describe for all time the behavior of these solutions which include refracted and reflected solitary waves depending on the initial energy

intuitive formulation of discontinuous galerkin surface integral equations for electromagnetic scattering problems an intuitive formulation of discontinuous galerkin surface integral equation dg sie method is proposed in this paper which is established by performing singularity extraction and singling out the infinitely large term in the entries of the impedance matrices associated with the sies which are popular in solving electromagnetic scattering problems nonoverlapping trial and test functions defined on triangles are employed without requirements of the surface current normal continuity across their internal boundary contours the proposed sie dg formulation is very flexible and is validated by examples of perfect electrically conducting targets in both conformal and nonconformal meshes

finite element study of the lorentz model in metamaterials metamaterials are materials with electromagnetic properties that originate from artificial sub wavelength structures although the lorentz metamaterial model has been used by physicists and electrical engineers its mathematical properties have not been investigated yet also more robust and efficient algorithms for time domain simulations are needed in this paper we initiate our work in this respect by developing some finite element schemes for solving the lorentz model stability and error analysis are proved for these schemes and 2 d numerical results justifying our theoretical results are presented

linear stability of perturbed hamiltonian systems theory and a case example in this paper we present a general theory based on a lyapunov schmidt reduction for the linearized stability of a perturbed hamiltonian system with a number of symmetries the reduction leads to an eigenvalue problem in a small subspace of the original system whose eigenvalues and eigenvectors can be used to derive the ones of the perturbed original system the krein signature of the eigenvalues can also be obtained in this framework the method is applied to the case example of the coupled nonlinear schr dinger equations giving excellent agreement with the full numerical linear stability results

numerical analysis of a pml model for time dependent maxwell s equations in this paper we study a perfectly matched layer model for the three dimensional time dependent maxwell s equations we develop both semi and fully discrete finite element methods for solving the truncated pml problem by nedelec edge elements optimal convergence rates are proved for both semi and fully discrete schemes to our knowledge this is the first error analysis obtained for time domain finite element method for pml models

on asymptotic stability of stationary solutions to nonlinear wave and klein gordon equations we consider nonlinear wave and klein gordon equations with general nonlinear terms localized in space conditions are found which provide asymptotic stability of stationary solutions in local energy norms these conditions are formulated in terms of spectral properties of the schrodinger operator corresponding to the linearized problem they are natural extensions to partial differential equations of the known lyapunov condition for the nonlinear wave equation in three dimensional space we find asymptotic expansions as t of the solutions which are close enough to a stationary asymptotically stable solution

a priori and posteriori error analysis for time dependent maxwell s equations abstract we consider time dependent maxwell s equations discretized by variable time steps in time domain and edge elements in spatial domain first the stability and optimal a priori error estimate are proved for both semi and fully discrete schemes then a posteriori error analysis is carried out for both schemes

an instability mechanism for radially symmetric standing waves of a nonlinear schrodinger equation abstract a condition is proved for the spectrum of nonlinear schrodinger equations linearised at a standing wave to have a positive eigenvalue the standing waves considered are radially symmetric ones in higher space dimensions the instability result is applied to show that if there are multiple asymptotically positive nondegenerate waves with a fixed number of zeroes then there is an unstable one the techniques used are dynamical systems arguments and involve a shooting argument in the space of lagrangian planes

lattice soliton solutions to the kadomtsev petviashvili equation with positive dispersion lattice soliton solutions that have doubly periodic array of the localized structures in the plane are presented to the kadomtsev petviashvili equation with the positive dispersion by using the superposition of the rational functions each structure is similar to the rational soliton the existence condition and some properties are also presented

the special modular transformation for polycnoidal waves of the korteweg de vries equation the modular transformation of the riemann theta function is used to show that the implicit dispersion relation for the n polycnoidal waves of the korteweg de vries equation has a countable infinity of branches for n 2 although the transformation also implies that each branch or mode can be written in a countable infinity of ways it is also shown that there is a unique physical representation for each mode such that the parameters of the theta function can be interpreted as wavenumbers and amplitudes in the limit of either very small or very large amplitude unfortunately the small amplitude physical representation is different by a modular transformation from the large amplitude physical representation for a given mode but this difference explains an apparent paradox as described in the text the general modular transformation expresses the theta function in terms of complex wavenumbers phase speeds and coordinates that have no physical relevance to the korteweg de vries equation but it is shown that for n 2 there is a subgroup here dubbed the special modular transformation which gives a real result this subgroup is explicitly constructed for general n and presented as a table for n 2

stable standing waves of nonlinear klein gordon equations in this paper we give sufficient conditions for the stability of the standing waves of least energy for nonlinear klein gordon equations

on the dynamics of solitons in the nonlinear schrodinger equation we study the behavior of the soliton solutions of the equation i frac partial psi partial t frac 1 2m delta psi frac 1 2 w varepsilon prime psi v x psi where w varepsilon prime is a suitable nonlinear term which is singular for varepsilon 0 we use the strong nonlinearity to obtain results on existence shape stability and dynamics of the soliton the main result of this paper theorem 1 shows that for varepsilon rightarrow 0 the orbit of our soliton approaches the orbit of a classical particle in a potential v x

superconvergence analysis for maxwell s equations in dispersive media in this paper we consider the time dependent maxwell s equations in dispersive media on a bounded three dimensional domain global superconvergence is obtained for semi discrete mixed finite element methods for three most popular dispersive media models the isotropic cold plasma the one pole debye medium and the two pole lorentz medium global superconvergence for a standard finite element method is also presented to our best knowledge this is the first superconvergence analysis obtained for maxwell s equations when dispersive media are involved

stability and folds it is known that when one branch of a simple fold in a bifurcation diagram represents linearly stable solutions the other branch represents unstable solutions the theory developed here can predict instability of some branches close to folds without knowledge of stability of the adjacent branch provided that the underlying problem has a variational structure first one particular bifurcation diagram is identified as playing a special role the relevant diagram being specified by the choice of functional plotted as ordinate the results are then stated in terms of the shape of the solution branch in this distinguished bifurcation diagram in many problems arising in elasticity the preferred bifurcation diagram is the loaddisplacement graph the theory is particularly useful in applications where a solution branch has a succession of folds

heavy traffic limits for queues with many exponential servers two different kinds of heavy traffic limit theorems have been proved for s server queues the first kind involves a sequence of queueing systems having a fixed number of servers with an associated sequence of traffic intensities that converges to the critical value of one from below the second kind which is often not thought of as heavy traffic involves a sequence of queueing systems in which the associated sequences of arrival rates and numbers of servers go to infinity while the service time distributions and the traffic intensities remain fixed with the traffic intensities being less than the critical value of one in each case the sequence of random variables depicting the steady state number of customers waiting or being served diverges to infinity but converges to a nondegenerate limit after appropriate normalization however in an important respect neither procedure adequately represents a typical queueing system in practice because in the heavy traffic limit an arriving customer is either a

a priori and computable a posteriori error estimates for an hdg method for the coercive maxwell equations abstract in this paper we present and analyze a hybridizable discontinuous galerkin hdg method for a mixed curl curl formulation of the steady state coercive maxwell equations with a discrete sobolev embedding type estimates for the discontinuous polynomials we provide a priori error estimates for the electric field and the lagrange multiplier in the energy norm with the smooth or minimal regularity assumption on the exact solution we have optimal convergence rate for the electric field and the lagrange multiplier in the energy norm the a priori error estimate for the electric field in the l 2 norm is also obtained by the duality argument and the approximation is also optimal for the electric field in the l 2 norm moreover by employing suitable helmholtz decompositions of the error together with the upper bound estimate for the lagrange multiplier we provide a computable residual based a posteriori error estimator which is derived based on the error measured in terms of a mesh dependent energy norm the efficiency of the a posteriori error estimator is also established three dimensional numerical results testing the performance of the a priori and a posteriori error estimates for the maxwell equations are given

theta functions gaussian series and spatially periodic solutions of the korteweg de vries equation it has been shown by novikov funct anal appl 8 236 1974 dubrovin et al russian math surveys 31 59 1976 lax commun pure appl math 28 141 1975 mckean and van moerbeke inv math 30 217 1975 and others that the nonlinear evolution equations which admit solitary waves also have spatially periodic exact solutions polycnoidal waves which can be expressed in terms of multidimensional riemann theta functions here it is shown that via poisson summation the fourier series that define the theta functions can be transformed into an infinite series of gaussian functions because the lowest terms of the gaussian series generate the usual solitary waves it is possible to intimately explore the relationship between solitary waves and these spatially periodic polycnoidal waves also by using the gaussian series one can perturbatively calculate phase velocities and wave structure for the polycnoidal wave even in the strongly nonlinear regime for which the soliton or mult

on exact solutions of the regularized long wave equation a direct approach to partially integrable equations ii periodic solutions in this second of two articles designated i and ii the bilinear transformation method is used to obtain stationary periodic solutions of the partially integrable regularized long wave rlw equation these solutions are expressed in terms of riemann theta functions and this approach leads to a new and compact expression for the important dispersion relation the periodic solution or cnoidal wave can be represented as an infinite sum of sech2 solitary waves this remarkable property may be interpreted in the context of a nonlinear superposition principle the rlw cnoidal wave approximates to a sinusoidal wave and a solitary wave in the limits of small and large amplitudes respectively analytic approximations and error estimates are given which shed light on the character of the cnoidal wave in the different parameter regimes similar results are presented in brief for the related rlw boussinesq rlwb equation

nonlinear stability of periodic traveling wave solutions to the schrodinger and the modified korteweg de vries equations abstract this work is concerned with stability properties of periodic traveling waves solutions of the focusing schrodinger equation i u t u x x u 2 u 0 posed in r and the modified korteweg de vries equation u t 3 u 2 u x u x x x 0 posed in r our principal goal in this paper is the study of positive periodic wave solutions of the equation 3 0 called dnoidal waves a proof of the existence of a smooth curve of solutions with a fixed fundamental period l 2 2 l 2 h per 0 l is given it is also shown that these solutions are nonlinearly stable in the energy space h per 1 0 l and unstable by perturbations with period 2l in the case of the schrodinger equation

a multiscale discontinuous galerkin method with the computational structure of a continuous galerkin method proliferation of degrees of freedom has plagued discontinuous galerkin methodology from its inception over 30 years ago this paper develops a new computational formulation that combines the advantages of discontinuous galerkin methods with the data structure of their continuous galerkin counterparts the new method uses local element wise problems to project a continuous finite element space into a given discontinuous space and then applies a discontinuous galerkin formulation the projection leads to parameterization of the discontinuous degrees of freedom by their continuous counterparts and has a variational multiscale interpretation this significantly reduces the computational burden and at the same time little or no degradation of the solution occurs in fact the new method produces improved solutions compared with the traditional discontinuous galerkin method in some situations

local well posedness and wave breaking results for periodic solutions of a shallow water equation for waves of moderate amplitude abstract we study the local well posedness of a periodic nonlinear equation for surface waves of moderate amplitude in shallow water we use an approach proposed by kato which is based on semigroup theory for quasi linear equations we also show that singularities for the model equation can occur only in the form of wave breaking in particular surging breakers

mixed discontinuous galerkin approximation of the maxwell operator we introduce and analyze a discontinuous galerkin discretization of the maxwell operator in mixed form here all the unknowns of the underlying system of partial differential equations are approximated by discontinuous finite element spaces of the same order for piecewise constant coefficients the method is shown to be stable and optimally convergent with respect to the mesh size numerical experiments highlighting the performance of the proposed method for problems with both smooth and singular analytical solutions are presented

relaxation of solitons in nonlinear schrodinger equations with potential abstract in this paper we study dynamics of solitons in the generalized nonlinear schrodinger equation nls with an external potential in all dimensions except for 2 for a certain class of nonlinearities such an equation has solutions which are periodic in time and exponentially decaying in space centered near different critical points of the potential we call those solutions which are centered near the minima of the potential and which minimize energy restricted to l 2 unit sphere trapped solitons or just solitons in this paper we prove under certain conditions on the potentials and initial conditions that trapped solitons are asymptotically stable moreover if an initial condition is close to a trapped soliton then the solution looks like a moving soliton relaxing to its equilibrium position the dynamical law of motion of the soliton i e effective equations of motion for the soliton s center and momentum is close to newton s equation but with a dissipative term due to radiation of the energy to infinity

control of trunk line systems in heavy traffic the paper deals with the heavy traffic modelling and optimal control of trunk line networks the controls concern either acceptance or rerouting both discounted and ergodic cost functions are used it is shown that the sequence of optimal value functions of the physical process converges to the optimal value function for the limit model also good controls for the limit system are good for the physical system single trunk systems with both priority and other inputs as well as networks are addressed for the network case the limit model is nonstandard since the control occurs only on the boundary and is a control over the direction of reflection

on the periodic solution of the burgers equation a unified approach for the most part the various known expressions for the periodic solution to the burgers equation have been derived by a mixture of ad hoc argument and approximation methods these include the well known solutions due to r d fay and d f parker remarkably these solutions were obtained without recourse to the celebrated hopf cole transformation which linearizes the burgers equation to the heat conduction equation we present here an alternative approach to the periodic problem which uses the fact that the riemann theta functions solve the burgers equation via the hopf cole transformation and thereby provide a unifying framework in which all previous representations of the periodic solution find their natural place yet another expression for the periodic solution in the form of an amplitude modulated sine wave is given and would appear to be new the limiting forms of the periodic solution as t rightarrow 0 embryonic sawtooth profile and t rightarrow infty old age sinusoidal profile are examined in detail the representation due to d f parker which expresses the periodic wave as a superposition of spreading taylor shock profiles is discussed within the context of a nonlinear superposition principle the composite asymptotic expansion due to m b lesser and d g crighton is examined in the light of our results the curious behaviour of this expansion in the embryonic region t ll 1 whereby a first order approximation yields an exact periodic solution the fay solution is explained

dynamic scheduling of a multiclass queue in the halfin whitt heavy traffic regime we consider a markovian model of a multiclass queueing system in which a single large pool of servers attends to the various customer classes customers waiting to be served may abandon the queue and there is a cost penalty associated with such abandonments service rates abandonment rates and abandonment penalties are generally different for the different classes the problem studied is that of dynamically scheduling the various classes we consider the halfin whitt heavy traffic regime where the total arrival rate and the number of servers both become large in such a way that the system s traffic intensity parameter approaches one an approximating diffusion control problem is described and justified as a purely formal that is nonrigorous heavy traffic limit the hamilton jacobi bellman equation associated with the limiting diffusion control problem is shown to have a smooth classical solution and optimal controls are shown to have an extremal or bang bang character several useful qualitative insights are derived from the mathematical analysis including a square root rule for sizing large systems and a sharp contrast between system behavior in the halfin whitt regime versus that observed in the conventional heavy traffic regime the latter phenomenon is illustrated by means of a numerical example having two customer classes

instability of nonlinear bound states we establish a sharp instability theorem for the bound states of lowest energy of the nonlinear klein gordon equation utt u f u 0 and the nonlinear schrodinger equation iut u f u 0

orbital stability of double solitons for the benjamin ono equation in this paper we prove the orbital stability of double solitons for the benjamin ono equation in the case of the kdv equation this stability has been proved in 17 parts of the proof given there rely on the fact that the euler lagrange equations for the conserved quantities of the kdv equation are ordinary differential equations since this is not the case for the benjamin ono equation new methods are required our approach consists in using a new invariant for multi solitons and certain new identities motivated by the sylvester law of inertia

unstable ground state of nonlinear klein gordon equations on demontre l instabilite de l etat fondamental des equations de klein gordon non lineaires a n dimensions d espace n 3

a centre stable manifold for the focussing cubic nls in mathbb r 1 3 consider the focussing cubic nonlinear schrodinger equation in mathbb r 3 i psi t delta psi psi 2 psi quad 0 1 it admits special solutions of the form eit where phi in mathcal s mathbb r 3 is a positive 0 solution of delta phi alpha phi phi 3 quad 0 2 the space of all such solutions together with those obtained from them by rescaling and applying phase and galilean coordinate changes called standing waves is the 8 dimensional manifold that consists of functions of the form e i v cdot gamma phi cdot y alpha we prove that any solution starting sufficiently close to a standing wave in the sigma w 1 2 mathbb r 3 cap x 1 l 2 mathbb r 3 norm and situated on a certain codimension one local lipschitz manifold exists globally in time and converges to a point on the manifold of standing waves furthermore we show that mathcal n is invariant under the hamiltonian flow locally in time and is a centre stable manifold in the sense of bates jones batjon the proof is based on the modulation method introduced by soffer and weinstein for the l2 subcritical case and adapted by schlag to the l2 supercritical case an important part of the proof is the keel tao endpoint strichartz estimate in mathbb r 3 for the nonselfadjoint schrodinger operator obtained by linearizing 0 1 around a standing wave solution all results in this paper depend on the standard spectral assumption that the hamiltonian mathcal h left begin array cc delta 2 phi cdot alpha 2 alpha quad phi cdot alpha 2 phi cdot alpha 2 quad delta 2 phi cdot alpha 2 alpha end array right quad 0 3 has no embedded eigenvalues in the interior of its essential spectrum infty alpha cup alpha infty

a finite volume method for the approximation of maxwell s equations in two space dimensions on arbitrary meshes a new finite volume method is presented for discretizing the two dimensional maxwell equations this method may be seen as an extension of the covolume type methods to arbitrary possibly non conforming or even non convex n sided polygonal meshes thanks to an appropriate choice of degrees of freedom an equivalent formulation of the scheme is given in terms of discrete differential operators obeying discrete duality principles the main properties of the scheme are its energy conservation its stability under a cfl like condition and the fact that it preserves gauss law and divergence free magnetic fields second order convergence is demonstrated numerically on non conforming and distorted meshes

lattice solitons directly by the bilinear method recently a doubly periodic solution of the kadomtsev petviashvili equation was deduced by summing over component solitons the same solution was derived directly by the hirota bilinear method this alternate route enables one to obtain a new solution such a mechanism can also be applied to wavepacket dynamics e g the davey stewartson equation

stability and instability of the kdv solitary wave under the kp i flow we consider the kp i and gkp i equations in mathbb r times mathbb r 2 pi mathbb z we prove that the kdv soliton with subcritical speed 0 c in the spirit of the work by duyckaerts and merle gafa 18 1787 1840 2009 we sharpen our previous instability result and construct a global solution which is different from the solitary wave and its translates and which converges to the solitary wave as time goes to infinity this last result also holds for the gkp i equation

efficient large scale electromagnetic simulations using dynamically adapted meshes with the discontinuous galerkin method a framework for performing dynamic mesh adaptation with the discontinuous galerkin method dgm is presented adaptations include modifications of the local mesh step size h adaptation and the local degree of the approximating polynomials p adaptation as well as their combination the computation of the approximation within locally adapted elements is based on projections between finite element spaces fes which are shown to preserve an upper limit of the electromagnetic energy the formulation supports high level hanging nodes and applies precomputation of surface integrals for increasing computational efficiency error and smoothness estimates based on interface jumps are presented and applied to the fully hp adaptive simulation of two examples in one dimensional space a full wave simulation of electromagnetic scattering from a radar reflector demonstrates the applicability to large scale problems in three dimensional space

the double cnoidal wave of the korteweg de vries equation an overview earlier work of the author on the spatially periodic solutions of the korteweg de vries equation is here extended via an in depth treatment of a special case the double cnoidal wave is the simplest generalization of the ordinary cnoidal wave discovered by korteweg and de vries in 1895 in the limit of small amplitude the double cnoidal wave is the sum of two noninteracting linear sine waves in the oppositie limit of large amplitude it is the sum of solitary waves of two different heights repeated periodically over all space although special the double cnoidal wave is important because it is but the particular case n 2 of a broad family of solutions known variously as n polycnoidal waves finite gap finite zone solutions waves on a circle or n phase wave trains it has been shown by others that the set of n polycnoidal waves gives the general initial value solution to the korteweg de vries equation this present work is the core of a three part treatment of the double cnoidal wave this part the overview presents graphic examples in all the important parameter regimes explains how collision phase shifts alter the average speed of the two wave phases from the free velocities of the two solitary waves describes the different branches or modes of the double cnoidal wave it is possible to have many solitary waves on each spatial period provided they are of only two distinct sizes and contrasts the results of this work with the very limited numerical calculations of previous authors the second part describes how the problem of numerically calculating the double cnoidal wave can be reduced down to solving four algebraic equations by perturbation theory the third part explains how the so called modular transformation of the riemann theta functions is important in interpreting n polycnoidal waves

orbital stability of one parameter periodic traveling waves for dispersive equations and applications abstract this paper establishes sufficient conditions for the orbital stability of one parameter spatially periodic traveling wave solutions for one dimensional dispersive equations our method of proof combines known techniques with some new ideas as a consequence of our result we give several applications for well known dispersive equations extension of the theory to regularized equations is also established

the korteweg de vries equation and its symmetry preserving discretization the korteweg de vries equation is one of the most important nonlinear evolution equations in the mathematical sciences in this article invariant discretization schemes are constructed for this equation both in the lagrangian and in the eulerian form we also propose invariant schemes that preserve the momentum numerical tests are carried out for all invariant discretization schemes and related to standard numerical schemes we find that the invariant discretization schemes give generally the same level of accuracy as the standard schemes with the added benefit of preserving galilean transformations which is demonstrated numerically as well

diffusion approximations for a single node accessed by congestion controlled sources we consider simple models of congestion control in high speed networks and develop diffusion approximations which could be useful for resource allocation we first show that if the sources are on off type with exponential on and off times then under a certain scaling the steady state distribution of the number of active sources can be described by a combination of two appropriately truncated and renormalized normal distributions for the case where the source arrival process is poisson and the service times are exponential the steady state distribution consists of appropriately normalized and truncated gaussian and exponential distributions we then consider the case where the arrival process is a general renewal process with finite coefficient of variation and service time distributions that are phase type and show the impact of these distributions on the steady state distribution of the number of sources in the system we also establish an insensitivity to service time distribution when the arrival process is poisson we use these results to relate the capacity of a bottleneck node to performance measures of interest for best effort traffic

isoinertial family of operators and convergence of kdv cnoidal waves to solitons abstract in this paper we show the convergence of korteweg de vries cnoidal waves to the limit soliton it is proved that the convergence is uniform and in h 2 norm as the period of the solutions tends to infinity families of hill operators are also studied we obtain a condition under which families of operators are isoinertial this condition is satisfied for classes of hill operators that are obtained by linearization our application is to the family of linearized operators at the kdv cnoidal waves it is proved that this family is isoinertial and also the value of the inertial index is calculated

double cnoidal waves of the korteweg de vries equation a boundary value approach double cnoidal waves of the korteweg de vries equation are studied by direct solution of the nonlinear boundary value problem these double cnoidal waves which are the spatially periodic generalization of the well known double soliton are exact solutions with two independent phase speeds the equation is written in terms of two phase variables and expanded in two dimensional fourier series the small amplitude solution is obtained via the stokes perturbation expansion this solution is numerically extended to larger amplitude by employing a newton kantorovich continuation in amplitude galerkin algorithm the crests of the finite amplitude solution closely match the sech 2 solitary wave form and the three cases of solitary wave interaction described by lax are identified for the double cnoidal waves this simple approach reproduces specific features such as phase shift upon collision distinction between instantaneous and average phase speeds and a paradox of wavenumbers

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

orbital stability of periodic traveling wave solutions for the log kdv equation abstract in this paper we establish the orbital stability of periodic waves related to the logarithmic korteweg de vries equation our motivation is inspired in the recent work 3 in which the authors established the well posedness and the linear stability of gaussian solitary waves by using the approach put forward recently in 20 to construct a smooth branch of periodic waves as well as to get the spectral properties of the associated linearized operator we apply the abstract theories in 13 and 25 to deduce the orbital stability of the periodic traveling waves in the energy space

convergence analysis of an adaptive interior penalty discontinuous galerkin method we study the convergence of an adaptive interior penalty discontinuous galerkin ipdg method for a two dimensional model second order elliptic boundary value problem based on a residual type a posteriori error estimator we prove that after each refinement step of the adaptive scheme we achieve a guaranteed reduction of the global discretization error in the mesh dependent energy norm associated with the ipdg method in contrast to recent work on adaptive ipdg methods o karakashian and f pascal convergence of adaptive discontinuous galerkin approximations of second order elliptic problems preprint university of tennessee knoxville tn 2007 the convergence analysis does not require multiple interior nodes for refined elements of the triangulation in fact it will be shown that bisection of the elements is sufficient the main ingredients of the proof of the error reduction property are the reliability and a perturbed discrete local efficiency of the estimator a bulk criterion that takes care of a proper selection of edges and elements for refinement and a perturbed galerkin orthogonality property with respect to the energy inner product the results of numerical experiments are given to illustrate the performance of the adaptive method

construction of lighthill s unitary functions the imbricate series of unity the poisson sum theorem implies that every spatially periodic function f x can be written as a so called imbricate series that is as the superposition of an infinite number of evenly space copies of a pattern function a x the pattern is the fourier transform of a function a n that specifies the usual fourier coefficients because there are many a n which coincide with a given set of fourier coefficients aj when n is an integer the pattern function a x is not unique in this work we show how to construct pattern functions of the form a x f x w x where f x is the periodic function itself the imbrication of a x and where w x is what lighthill called a unitary function we derive an explicit formula to construct all possible unitary functions and tabulate eight examples we also prove that all unitary functions are pattern functions of unity that is for the imbricate series that sums to f x 1 x

on the asymptotic stability of solitary waves in the discrete schrodinger equation coupled to a nonlinear oscillator abstract the long time asymptotics is analyzed for finite energy solutions of the 1d discrete schrodinger equation coupled to a nonlinear oscillator the coupled system is invariant with respect to the phase rotation group u 1 for initial states close to a solitary wave the solution converges to a sum of another solitary wave and a dispersive wave which is a solution to the free schrodinger equation the proofs use the strategy of buslaev perelman v s buslaev g s perelman on the stability of solitary waves for nonlinear schrodinger equations amer math soc trans 2 164 1995 75 98 the linearization of the dynamics on the solitary manifold the symplectic orthogonal projection method of majorants etc

a class of exact periodic solutions of nonlinear envelope equations a class of periodic solutions of nonlinear envelope equations e g the nonlinear schrodinger equation nls is expressed in terms of rational functions of elliptic functions the hirota bilinear transformation and theta functions are used to extend and generalize this class of solutions first reported for nls earlier in the literature in particular a higher order nls and the davey stewartson ds equations are treated doubly periodic standing waves solutions are obtained for both the dsi and dsii equations a symbolic manipulation software is used to confirm the validity of the solutions independently

a posteriori error estimator based on gradient recovery by averaging for discontinuous galerkin methods we consider some anisotropic and piecewise constant diffusion problems in domains of r 2 approximated by a discontinuous galerkin method with polynomials of any fixed degree we propose an a posteriori error estimator based on gradient recovery by averaging it is shown that this estimator gives rise to an upper bound where the constant is one up to some additional terms that guarantee reliability the lower bound is also established moreover these additional terms are negligible when the recovered gradient is superconvergent the reliability and efficiency of the proposed estimator is confirmed by some numerical tests

cnoidal waves as exact sums of repeated solitary waves new series for elliptic functions via poisson summation of the corresponding fourier series it is shown that the cnoidal wave of the korteweg de vries equation can be written as an infinite sum of periodically repeated solitary waves a result first proved by toda through a different method similar series of hyperbolic secant or tangent functions are also derived for the elliptic dn sn and cn functions which make it possible to express the cnoidal wave of almost any evolution equation in terms of a series of the corresponding solitary waves as a by product new series are also obtained for the complete elliptic integral k and modulus k

the regularized benjamin ono and bbm equations well posedness and nonlinear stability abstract nonlinear stability of nonlinear periodic solutions of the regularized benjamin ono equation and the benjamin bona mahony equation with respect to perturbations of the same wavelength is analytically studied these perturbations are shown to be stable we also develop a global well posedness theory for the regularized benjamin ono equation in the periodic and in the line setting in particular we show that the cauchy problem in both periodic and nonperiodic case cannot be solved by an iteration scheme based on the duhamel formula for negative sobolev indices

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

energy norm a posteriori error estimation for discontinuous galerkin methods in this paper we present a residual based a posteriori error estimate of a natural mesh dependent energy norm of the error in a family of discontinuous galerkin approximations of elliptic problems the theory is developed for an elliptic model problem in two and three spatial dimensions and general nonconvex polygonal domains are allowed we also present some illustrating numerical examples

orbital stability of periodic standing waves for the logarithmic klein gordon equation abstract the main goal of this paper is to present orbital stability results of periodic standing waves for the one dimensional logarithmic klein gordon equation to do so we first use compactness arguments and a non standard analysis to obtain the existence and uniqueness of weak solutions for the associated cauchy problem in the energy space second we prove the orbital stability of standing waves using a stablity analysis of conservative systems

the wkb approximation to the g m m queue g m m queues with a large number of servers m gg 1 and an interarrival time density that is sharply peaked near the origin i e typical interarrival times are short of the order o 1 m are considered using the wkb method asymptotic approximations to the stationary queue length distribution for m gg 1 are constructed some numerical comparisons illustrate the quality of the approximations

dispersive analysis of charge transfer models we prove dispersive estimates for the time dependent schrodinger equation with a charge transfer hamiltonian as a by product we also obtain another proof of asymptotic completeness of the wave operators for a charge transfer model established earlier by k yajima and j m graf we also consider a more general matrix non self adjoint charge transfer problem this model appears naturally in the study of nonlinear multisoliton systems and is specifically motivated by the problem of asymptotic stability of multisoliton states of a nonlinear schrodinger equation 2004 wiley periodicals inc

semiclassical quantization of the periodic toda chain gutzwiller s semiclassical quantization scheme for the trace of green s function is applied to the periodic toda chain we obtain a set of algebraic equations that determine the energy levels arising from a special periodic orbit namely the single cnoidal wave solution our formulae show a simple dependence on the number of particles n in the chain n merely occurs as a parameter we perform the soliton limit of our equations and get a semiclassical correction to first order in h to the dispersion relation e e p of a soliton on the infinite chain which is in remarkable agreement with the bethe ansatz result the classical data which enter into the semiclassical quantization formula are of interest in their own right we give a complete treatment of the linear stability analysis of a single cnoidal wave and also some new expressions for its dispersion relation which expresses the frequency v as a function of the wavenumber k

sharp threshold of global existence and instability of standing wave for a davey stewartson system this paper concerns the sharp threshold of blowup and global existence of the solution as well as the strong instability of standing wave phi t x e i omega t u x for the system i phi t delta phi a phi p 1 phi b e 1 phi 2 phi 0 quad t geq 0 quad x in bf r n quad quad quad rm ds where a 0 b 0 1 0 by combining the former results

a discontinuous galerkin surface integral equation method for electromagnetic wave scattering from nonpenetrable targets we present a discontinuous galerkin surface integral equation method herein referred to as iedg for time harmonic electromagnetic wave scattering from nonpenetrable targets the proposed iedg algorithm allows the implementation of the combined field integral equation cfie using square integrable trial and test functions without any considerations of continuity requirements across element boundaries due to the local characteristics of basis functions it is possible to employ nonconformal surface discretizations of the targets furthermore it enables the possibility to mix different types of elements and employ different order of basis functions within the same discretization therefore the proposed iedg method is highly flexible to apply adaptation techniques numerical results are included to validate the accuracy and demonstrate the versatility of the proposed iedg method in addition a complex large scale simulation is conducted to illustrate the potential benefits offered by the proposed method for modeling multiscale electrically large targets

solitons from sine waves analytical and numerical methods for non integrable solitary and cnoidal waves abstract the fkdv equation u t uu x u xxxxx 0 is used as a testbed for a variety of analytical and numerical methods that can be applied to solitary waves and cnoidal waves of non integrable differential equations that is to say to equations which cannot be solved by the inverse scattering transform the basic tools are i pade approximants formed from power series in the amplitude ii a newton kantorovich pseudospectral fourier continuation numerical method iii singular perturbation theory for two interacting solitons of almost identical phase speed iv bifurcation and branch switching methods v the imbricate soliton series a number of new results for the fkdv equation are obtained including extensive numerical calculations of the spatially periodic solutions with one peak cnoidal wave and two peaks bicnoidal wave per period an analytical expression for the double peaked soliton bion calculation of both the limit and bifurcation points for the bicnoidal wave and finally the computation of accurate analytical approximations to the cnoidal wave for all amplitudes more important all of these analytical and numerical tools are highly effective for this equation in spite of the fact that it cannot be solved by the inverse scattering transform work now in progress will apply these methods to non integrable equations in two space dimensions

stability properties of periodic waves for the klein gordon equation with quintic nonlinearity in this paper we prove results of existence and orbital instability of periodic waves for the klein gordon equation with quintic nonlinearity the main tool used here is a recent computational approach in order to decide about the nonpositive spectrum of the linearized operator combined with the grillakis shatah and strauss stability theory

long time oscillation of solutions of nonlinear schrodinger equations near minimal mass ground state abstract in this paper we consider the long time dynamics of radially symmetric solutions of nonlinear schrodinger equations nls having a minimal mass ground state in particular we show that there exist solutions with initial data near the minimal mass ground state that oscillate for long time more precisely we introduce a coordinate defined near the minimal mass ground state which consists of finite and infinite dimensional part associated to the discrete and continuous part of the linearized operator then we show that the finite dimensional 2 d part approximately obeys newton s equation of motion for a particle in an anharmonic potential well showing that the infinite dimensional part is well separated from the finite dimensional part we will have long time oscillation

perturbation series for the double cnoidal wave of the korteweg de vries equation by means of the theorems proved earlier by the author the problem of the double cnoidal wave of the korteweg de vries equation is reduced to four algebraic equations in four unknowns two of the unknowns are the nonlinear phase speeds c1 and c2 another is a physically irrelevant integration constant the fourth unknown is the off diagonal element of the symmetric 2 2 theta matrix which in turn gives the explicit coefficients of the riemann theta function the double cnoidal wave u x t is then obtained by taking the second x derivative of the logarithm of the theta function two separate forms of these four nonlinear residual equations are given one is obtained from the fourier series of the theta function and is useful for small wave amplitude the other is based on the gaussian series of the theta function and is highly efficient in the large amplitude regime where the double cnoidal wave is the sum of two solitary waves both sets of residual equations can be solved via perturbation theory and

variance reduction in simulations of loss models we propose a new estimator of steady state blocking probabilities for simulations of stochastic loss models that can be much more efficient than the natural estimator ratio of losses to arrivals the proposed estimator is a convex combination of the natural estimator and an indirect estimator based on the average number of customers in service obtained from little s law l w it exploits the known offered load product of the arrival rate and the mean service time the variance reduction is dramatic when the blocking probability is high and the service times are highly variable the advantage of the combination estimator in this regime is partly due to the indirect estimator which itself is much more efficient than the natural estimator in this regime and partly due to strong correlation most often negative between the natural and indirect estimators in general when the variances of two component estimators are very different the variance reduction from the optimal convex combination is abo

locking free adaptive discontinuous galerkin fem for linear elasticity problems an adaptive discontinuous galerkin finite element method for linear elasticity problems is presented we develop an a posteriori error estimate and prove its robustness with respect to nearly incompressible materials absence of volume locking furthermore we present some numerical experiments which illustrate the performance of the scheme on adaptively refined meshes

energy norm a posteriori error estimation for mixed discontinuous galerkin approximations of the maxwell operator in this paper we develop the a posteriori error estimation of mixed discontinuous galerkin finite element approximations of the maxwell operator in particular by employing suitable helmholtz decompositions of the error together with the conservation properties of the underlying method computable upper bounds on the error measured in terms of a natural mesh dependent energy norm are derived numerical experiments testing the performance of our a posteriori error bounds for problems with both smooth and singular analytical solutions are presented

numerical convergence and physical fidelity analysis for maxwell s equations in metamaterials abstract in this paper we develop a leap frog mixed finite element method for solving maxwell s equations resulting from metamaterials our scheme is similar to the popular yee s fdtd scheme used in electrical engineering community and is preferable for three dimensional large scale modeling since no storage of the large coefficient matrix is needed our scheme is proved to obey the gauss s law automatically if the initial fields satisfy that furthermore the conditional stability and optimal error estimate for the proposed scheme are proved to our best knowledge we are unaware of any other publications devoted to the convergence analysis of this leap frog explicit scheme for maxwell s equations even in a simple medium while our results for metamaterials automatically reduce to the standard maxwell s equations in vacuum by dropping some terms resulting from the constitutive equations numerical results confirming our analysis are presented

numerical inverse spectral transform for the periodic sine gordon equation theta function solutions and their linearized stability abstract the inverse spectral transform for integrable nonlinear ordinary and partial differential equations such as the toda lattice korteweg de vries sine gordon and nonlinear schrodinger equations provides explicit algorithms to generate exact solutions under periodic or quasiperiodic boundary conditions these oscillatory wavetrains may be prescribed a priori to consist of a nonlinear superposition of n phases j x t j x j t 0 j j 1 n where the wave is 2 periodic independently in each phase this paper exhibits the numerical implementation of the inverse spectral solution of the sine gordon equation the general construction is outlined and then implemented for n 1 2 and 3 we compute 1 the exact theta function solutions 2 the floquet spectrum of x periodic solutions 3 the labelling of linearized instabilities of n phase solutions in terms of spectral data and 4 the linearized growth rate in each unstable mode the associated surfaces q n x t are displayed to illustrate a variety of spatial and dynamical phenomena in the oscillatory solution space of this integrable system

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

new series for the cosine lemniscate function and the polynomialization of the lemniscate integral we discuss the numerical computation of the cosine lemniscate function and its inverse the lemniscate integral these were previously studied by bernoulli euler gauss abel jacobi and ramanujan we review general elliptic formulas for this special case and provide some novelties we show that a fourier series by ramanujan converges twice as fast as the standard elliptic cosine fourier series specialized to this case the so called imbricate series however converges geometrically fast over the entire complex plane we derive two new expansions the rational plus fourier series converges much faster than ramanujan s for real z each term is asymptotically 12 400 times smaller than its immediate predecessor coslem z 4b q 1 q cos bz 1 q 2 4qcos 2 bz n 1 q n 1 2 1 1 q 2 n 1 1 cos 2n 1 bz where q exp p is the elliptic nome k 1 85 is the complete elliptic integral of the first kind for a modulus m 1 2 and b p k2 the rational imbricate series is uniformly valid over the complex plane but converges twice as fast as the sech imbricate series coslem z 4bq 1 q j q 1 q cos bz 1 q 2 4qcos 2 b z jpi where p 4 2 k is the period in both the real and imaginary directions we devise a new approximation for the lemniscate integral for real argument as the arccosine of a chebyshev series and show that 17 terms yield about 15 digits of accuracy for complex argument we show that the lemniscate integral can be found to near machine precision assumed as sixteen decimal digits by computing the roots of a polynomial of degree thirteen alternatively newton s iteration converges in three iterations with an initialization accurate to four decimal places that is the chosen root of a cubic equation