the renyi power means of matrices abstract in this article we introduce a new family of power means for m positive definite matrices called renyi power means these means are defined in terms of the matrix function q z a b b 1 2 z a z b 1 2 z z related to the renyi divergence we study several properties of these means and some related norm inequalities

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

attractors and error estimates for discretizations of incompressible navier stokes equations by embedding sets of nodal values to function spaces variational arguments are applied to the finite difference approximations to the two dimensional incompressible navier stokes equations the error estimates for trajectories and the it time free error estimates with tolerance for attractors are proved the upper semicontinuity of the attractors with respect to the finite difference discretizations are also proved arguments of applying the techniques to the finite element approximations are also given

on the toda and kac van moerbeke systems given a solution of the toda lattice we explicitly construct a solution of the kac van moerbeke system related to each other by a miura type transformation as an illustration of our method we derive the n soliton solutions of the kac van moerbeke lattice

riemannian geometry and matrix geometric means the geometric mean of two positive definite matrices has been defined in several ways and studied by several authors including pusz and woronowicz and ando the characterizations by these authors do not readily extend to three matrices and it has been a long standing problem to define a natural geometric mean of three positive definite matrices in some recent papers new understanding of the geometric mean of two positive definite matrices has been achieved by identifying the geometric mean of a and b as the midpoint of the geodesic with respect to a natural riemannian metric joining a and b this suggests some natural definitions for a geometric mean of three positive definite matrices we explain the necessary geometric background and explore the properties of some of these candidates

on hamiltonian perturbations of hyperbolic systems of conservation laws ii universality of critical behaviour hamiltonian perturbations of the simplest hyperbolic equation u t a u u x 0 are studied we argue that the behaviour of solutions to the perturbed equation near the point of gradient catastrophe of the unperturbed one should be essentially independent on the choice of generic perturbation neither on the choice of generic solution moreover this behaviour is described by a special solution to an integrable fourth order ode

why nonconservative schemes converge to wrong solutions error analysis this paper attempts to give a qualitative and quailtitative description of the numerical error introduced by using finite difference schemes in nonconservative form for scalar conservation laws we show that these schemes converge strongly in li norm to the solution of an inhomogeneous conservation law containing a borel measure source term moreover we analyze the properties of this borel measure and derive a sharp estimate for the l1 error between the limit function given by the scheme and the correct solution in general the measure source term is of the order of the entropy dissipation measure associated with the scheme in certain cases the error can be small for short times which makes it difficult to detect numerically but generically such an error will grow in time and this would lead to a large error for large time calculations finally we show that a local correction of any high order accurate scheme in nonconservative form is sufficient to ensure its convergence to the correct solution

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

dissipative behavior of some fully non linear kdv type equations the kdv equation can be considered as a special case of the general equation ut cf u x g uxx x d 0 0 where f is non linear and g is linear namely f u d u 2 2 and g v d v as the parameter tends to 0 the dispersive behavior of the kdv equation has been throughly investigated see e g p g drazin solitons london math soc lect note ser 85 cambridge university press cambridge 1983 p d lax c d levermore the small dispersion limit of the korteweg de vries equation iii commun pure appl math 36 1983 809 829 g b whitham linear and nonlinear waves wiley interscience new york 1974 and the references therein we show through numerical evidence that a completely different dissipative behavior occurs when g is non linear namely when g is an even concave function such as g v dj vj or g v d v 2 in particular our numerical results hint that as 0 the solutions strongly converge to the unique entropy solution of the formal limit equation in total contrast with the solutions of the kdv equation 2000 elsevier science b v all rights reserved

an effective matrix geometric mean satisfying the ando li mathias properties we propose a new matrix geometric mean satisfying the ten properties given by ando li and mathias linear alg appl 2004 this mean is the limit of a sequence which converges superlinearly with convergence of order 3 whereas the mean introduced by ando li and mathias is the limit of a sequence having order of convergence 1 this makes this new mean very easily computable we provide a geometric interpretation and a generalization which includes as special cases our mean and the ando li mathias mean

lexicographic improvement of the target values in convex goal programming the aim of this paper is to carry out an exhaustive post optimization analysis in a convex goal programming problem so as to study the possible existence of satisfying solutions for different levels of the target values to this end an interactive algorithm is proposed which allows us to improve the values of the objective functions after obtaining a satisfying solution if such a solution exists in such a way that a pareto optimal solution is finally reached through a successive actualization of such target values this way the target values are lexicographically improved according to the priority order previously given by the decision maker in an attempt to harmonize the concepts of satisfying and efficient solutions which have traditionally been in conflict

log majorization related to renyi divergences abstract for z 0 with 1 motivated by comparison between different kinds of renyi divergences in quantum information we consider log majorization between the matrix functions p a b b 1 2 b 1 2 a b 1 2 b 1 2 q z a b b 1 2 z a z b 1 2 z z of two positive semi definite matrices a b we precisely determine the parameters z for which p a b log q z a b and q z a b log p a b hold respectively

new classes of toda soliton solutions we provide a detailed investigation of limits of n soliton solutions of the toda lattice as n tends to infinity our principal results yield new classes of toda solutions including in particular new kinds of soliton like i e reflectionless solutions as a byproduct we solve an inverse spectral problem for one dimensional jacobi operators and explicitly construct tri diagonal matrices that yield a purely absolutely continuous spectrum in 1 1 and give rise to an eigenvalue spectrum that includes any prescribed countable and bounded subset of mr bs 1 1

binary and ternary oscillations in a cubic numerical scheme abstract we study a central difference semi discretization of the cubic scalar conservation law both spatial period 2 binary and period 3 ternary oscillations are stationary solutions of this scheme and we find where each type is linearly stable we observe numerically the formation of ternary oscillations to the left of riemann shock initial data with ur 0 while binary oscillations form to the right of riemann rarefaction data having ul 0 we derive modulation equations for both wave patterns using them to show that binary oscillations generated by the scheme are numerical artifacts while computing an explicit solution for the ternary modulation equations for positive initial data the ternary modulation equations remain hyperbolic while the solutions enter an elliptic region for data of both signs conditions under which solutions of the ternary modulation equations can be inverted to yield period 3 oscillations are also discussed

retrieval and use of the balance set in multiobjective global optimization abstract it is shown on examples how to compute the balance set and the balance number in vector optimization problems vops of different nature new developments are presented concerning possible interrelation between the balance set and the balance number a new notion of the projection of the balance set onto the parameter space new approaches for solving vops with unbounded objective functions and some approximation techniques in determining the balance set

experiments with classification based scalarizing functions in interactive multiobjective optimization in multiobjective optimization methods the multiple conflicting objectives are typically converted into a single objective optimization problem with the help of scalarizing functions and such functions may be constructed in many ways we compare both theoretically and numerically the performance of three classification based scalarizing functions and pay attention to how well they obey the classification information in particular we devote special interest to the differences the scalarizing functions have in the computational cost of guaranteeing pareto optimality it turns out that scalarizing functions with or without so called augmentation terms have significant differences in this respect we also collect a set of mostly nonlinear benchmark test problems that we use in the numerical comparisons

existence and modulation of traveling waves in particles chains we consider an infinite particle chain whose dynamics are governed by the following system of differential equations where qn t is the displacement of the nth particle at time t along the chain axis and denotes differentiation with respect to time we assume that all particles have unit mass and that the interaction potential v between adjacent particles is a convex c function for this system we prove the existence of c time periodic traveling wave solutions of the form qn t q wt kn where q is a periodic function q z q z 1 the period is normalized to equal 1 and k are respectively the frequency and the wave number is the mean particle spacing and can be chosen to be an arbitrary parameter we present two proofs one based on a variational principle and the other on topological methods in particular degree theory for small amplitude waves based on perturbation techniques we describe the form of the traveling waves and we derive the weakly nonlinear dispersion relation for the fully nonlinear case when the amplitude of the waves is high we use numerical methods to compute the traveling wave solution and the non linear dispersion relation we finally apply whitham s method of averaged lagrangian to derive the modulation equations for the wave parameters k and 1999 john wiley sons inc

the toda rarefaction problem in the toda shock problem see 7 1 11 s and also 131 one considers a driving particle moving with a fixed velocity 2a and impinging on a one dimensional semi infinite lattice of particles initially equally spaced and at rest and interacting with exponential forces in this paper we consider the related toda rarefaction problem in which the driving particle now moves away from the lattice at fixed speed in analogy with a piston being withdrawn as it were from a container filled with gas we make use of the riemann hilbert factorization formulation of the related inverse scattering problem in the case where the speed 21al of the driving particle is sufficiently large la1 l we show that the particle escapes from the lattice which then executes a free motion of the type studied for example in 5 in other words in analogy with a piston being withdrawn too rapidly from a container filled with gas cavitation develops 0 1996 john wiley sons inc

on some inequalities for the matrix power and karcher means abstract we establish a useful inequality for matrix power means of positive definite matrices it eventually provides not only a new characterization of the karcher mean cartan centroid riemannian geometric mean but also a simple proof of the convergence of power means to the karcher mean some new matrix mean inequalities on the karcher and power means are presented

high order schemes entropy inequalities and nonclassical shocks we are concerned with the approximation of undercompressive regularization sensitive nonclassical solutions of hyperbolic systems of conservation laws by high order accurate conservative and semidiscrete finite difference schemes nonclassical shock waves can be generated by diffusive and dispersive terms kept in balance particular attention is given here to a class of systems of conservation laws including the scalar equations and the system of nonlinear elasticity and to linear diffusion and dispersion in either the conservative or the entropy variables first we investigate the existence and the properties of entropy conservative schemes a notion due to tadmor math comp 49 1987 pp 91 103 in particular we exhibit a new five point scheme which is third order accurate at least second we study a class of entropy stable and high order accurate schemes satisfying a single cell entropy inequality they are built from any high order entropy conservative scheme by adding to it a mesh independent numerical viscosity which preserves the order of accuracy of the base scheme these schemes can only converge to solutions of the system of conservation laws satisfying the entropy inequality these entropy stable schemes exhibit mild oscillations near shocks and interestingly may converge to classical or nonclassical entropy solutions depending on the sign of their dispersion coefficient then based on a third order entropy conservative scheme we propose a general scheme for the numerical computation of nonclassical shocks importantly our scheme satisfies a cell entropy inequality following hayes and lefloch siam j numer anal 35 1998 pp 2169 2194 we determine numerically the kinetic function which uniquely characterizes the dynamics of nonclassical shocks for each regularization of the conservation laws our results compare favorably with previous analytical and numerical results finally we prove that there exists no fully discrete and entropy conservative scheme and we investigate the entropy stability of a class of fully discrete lax wendroff type schemes

the stochastic burgers equation we study burgers equation perturbed by a white noise in space and time we prove the existence of solutions by showing that the cole hopf transformation is meaningful also in the stochastic case the problem is thus reduced to the anaylsis of a linear equation with multiplicative half white noise an explicit solution of the latter is constructed through a generalized feynman kac formula typical properties of the trajectories are then discussed a technical result concerning the regularizing effect of the convolution with the heat kernel is proved for stochastic integrals

shock waves in the dissipative toda lattice we consider the propagation of a shock wave sw in the damped toda lattice the sw is a moving boundary between two semi infinite lattice domains with different densities a steadily moving sw may exist if the damping in the lattice is represented by an inner friction which is a discrete analogue of the second viscosity in hydrodynamics the problem can be considered analytically in the continuum approximation and the analysis produces an explicit relation between the sw s velocity and the densities of the two phases numerical simulations of the lattice equations of motion demonstrate that a stable sw establishes if the initial velocity is directed towards the less dense phase in the opposite case the wave gradually spreads out the numerically found equilibrium velocity of the sw turns out to be in very good agreement with the analytical formula even in a strongly discrete case if the initial velocity is essentially different from the one determined by the densities but has the correct sign the velocity does not alter significantly but instead the sw adjusts itself to the given velocity by sending another sw in the opposite direction

the zero dispersion limit of the korteweg de vries equation with periodic initial data on etudie le probleme aux valeurs initiales pour l equation de korteweg de vries u t 6uu x e 2 u xxx 0 dans la limite e 0 on calcul la limite faible de la solution pour des donnees initiales periodiques quand e 0

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

two results concerning asymptotic behavior of solutions of the burgers equation with force we consider the burgers equation with an external force for the case of the force periodic in space and time we prove the existence of a solution periodic in space and time which is the limit of a wide class of solutions ast if the force is the product of a periodic function ofx and white noise in time we prove the existence of an invariant distribution concentrated on the space of space periodic functions which is the limit of a wide class of distributions ast

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

on the exponential metric increasing property a short and simple proof is given for the inequality that shows that positive definite matrices constitute a riemannian manifold of negative curvature the idea of the proof leads to generalisations to non riemannian metrics and to connections with some well known inequalities of mathematical physics

matrix power means and the karcher mean we define a new family of matrix means pt a t 1 1 where and a vary over all positive probability vectors in rn and n tuples of positive definite matrices resp each of these means except t 0 arises as a unique positive definite solution of a non linear matrix equation satisfies all desirable properties of power means of positive real numbers and interpolates between the weighted harmonic and arithmetic means the main result is that the karcher mean coincides with the limit of power means as t 0 this provides not only a sequence of matrix means converging to the karcher mean but also a simple proof of the monotonicity of the karcher mean conjectured by bhatia and holbrook and other new properties which have recently been established by lawson and lim and also bhatia and karandikar using probabilistic methods on the metric structure of positive definite matrices equipped with the trace metric

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

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

on a conjecture of bhatia lim and yamazaki abstract in this short note we prove a conjecture due to bhatia lim and yamazaki on the matrix power means as a consequence we obtain a related inequality for the matrix heron mean and its non kubo ando extension conjectured by bhatia et al

dispersive regularizations and numerical discretizations for the inviscid burgers equation we study centred second order in time and space discretizations of the inviscid burgers equation although this equation in its continuum formulation supports non smooth shock wave solutions the discrete equation generically supports smooth solitary wave solutions using backward error analysis we derive the modified equation associated with the numerical scheme we identify three different equations the korteweg de vries kdv equation the camassa holm ch equation and the b 0 member of the b family solutions of the first two equations are solitary waves and do not converge to the shock solutions of the burgers equation the third equation however supports solutions which strongly approximate weak solutions of the burgers equation we corroborate our analytical results with numerical simulations

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

wavelet methods to eliminate resonances in the galerkin truncated burgers and euler equations it is well known that solutions to the fourier galerkin truncation of the inviscid burgers equation and other hyperbolic conservation laws do not converge to the physically relevant entropy solution after the formation of the first shock this loss of convergence was recently studied in detail ray et al phys rev e 84 016301 2011 and traced back to the appearance of a spatially localized resonance phenomenon perturbing the solution in this work we propose a way to remove this resonance by filtering a wavelet representation of the galerkin truncated equations a method previously developed with a complex valued wavelet frame is applied and expanded to embrace the use of real valued orthogonal wavelet bases which we show to yield satisfactory results only under the condition of adding a safety zone in wavelet space we also apply the complex valued wavelet based method to the two dimensional euler equation problem showing that it is able to filter the resonances in this case as well

a comparison of two reference point methods in multiple objective mathematical programming abstract when making decisions with multiple criteria a decision maker often thinks in terms of an aspiration point or levels of achievement for the criteria in multiple objective mathematical programming solution methods based on aspiration points can generate nondominated solutions using a variety of scalarizing functions these reference point solution methods commonly use a scalarizing function that reaches down from the ideal solution in a direction specified by the aspiration point conversely a similar scalarizing function can push out from the nadir point toward a specified aspiration point these scalarizing functions are similar in structure but diametrically opposed in their reference points in this paper we examine how these approaches behave from a technical point of view and conduct an experiment to understand better the human behaviour of users of these approaches insights we gain on the evenness of dispersion are relevant when attempting to construct a representation of the nondominated set further the technical characteristics of the two formulations solutions combined with behavioural tendencies allow us to comment on the implications for their use in interactive multiple objective methods

operator splitting methods for generalized korteweg de vries equations we apply the method of operator splitting on the generalized korteweg de vries kdv equation ut f u x uxxx 0 by solving the nonlinear conservation law ut f u x 0 and the linear dispersive equation ut uxxx 0 sequentially we prove that if the approximation obtained by operator splitting converges then the limit function is a weak solution of the generalized kdv equation convergence properties are analyzed numerically by studying the effect of combining different numerical methods for each of the simplified problems

dispersive regularization of the whitham equation for the toda lattice in this paper the averaged whitham equations for the slow modulations of multi phase wavetrains for the toda lattice are studied our main result is showing how the solution with step initial conditions may be regularized by choosing an appropriate riemann surface on which the whitham equation is defined

b spline quasi interpolation based numerical methods for some sobolev type equations in this article we intend to use quadratic and cubic b spline quasi interpolants to develop higher order numerical methods for some sobolev type equations in one space dimension our aim is also to compare the performance of the proposed methods in terms of the accuracy and the rate of convergence we also discuss another approach to the cubic b spline quasi interpolation based method where we achieve fourth order of accuracy in space we theoretically establish the order of accuracy for the three proposed methods and also establish the l 2 stability in the linear case using von neumann analysis as a particular case of the sobolev type equations we take the equal width and the benjamin bona mahony burgers equations and perform several numerical experiments to support our theoretical results

long time behavior of the continuum limit of the toda lattice and the generation of infinitely many gaps from c initial data we analyze a continuum limit of the finite non periodic toda lattice through an associated constrained maximization problem over spectral density functions the maximization problem was derived by deift and mclaughlin using the lax levermore approach initially developed for the zero dispersion limit of the kdv equation it encodes the evolution of the continuum limit for all times including evolution through shocks the formation of gaps in the support of the maximizer is indicative of oscillations in the toda lattice and the lack of strong convergence of the continuum limit for large times the maximizer tends to have zero gaps which is the continuum analogue of the sorting property of the finite lattice using methods from logarithmic potential theory we show that this behavior depends crucially on the initial data we exhibit initial data for which the zero gap ansatz holds uniformly in the spatial parameter at large times and other initial data for which this uniformity fails at all times we then construct an example of c smooth initial data generating at a later time infinitely many gaps in the support of the maximizer while for even larger times all gaps have closed

on a nonlinear hyperbolic variational equation ii the zero viscosity and dispersion limits we consider viscosity and dispersion regularizations of the nonlinear hyperbolic partial differential equation u t uux x 1 2u x 2 with the simplest initial data such that u x blows up in finite time we prove that the zero viscosity limit selects a unique global weak solution of the partial differential equation without viscosity we also present numerical experiments which indicate that the zero dispersion limit selects a different global weak solution of the same initial value problem

resonance phenomenon for the galerkin truncated burgers and euler equations it is shown that the solutions of inviscid hydrodynamical equations with suppression of all spatial fourier modes having wave numbers in excess of a threshold k mathrm g exhibit unexpected features the study is carried out for both the one dimensional burgers equation and the two dimensional incompressible euler equation for large k mathrm g and smooth initial conditions the first symptom of truncation a localized short wavelength oscillation which we call a tyger is caused by a resonant interaction between fluid particle motion and truncation waves generated by small scale features shocks layers with strong vorticity gradients etc these tygers appear when complex space singularities come within one galerkin wavelength ensuremath lambda mathrm g 2 ensuremath pi k mathrm g from the real domain and typically arise far away from preexisting small scale structures at locations whose velocities match that of such structures tygers are weak and strongly localized at first in the burgers case at the time of appearance of the first shock their amplitudes and widths are proportional to k mathrm g ensuremath 2 3 and k mathrm g ensuremath 1 3 respectively but grow and eventually invade the whole flow they are thus the first manifestations of the thermalization predicted by t d lee q j appl math 10 69 1952 the sudden dissipative anomaly the presence of a finite dissipation in the limit of vanishing viscosity after a finite time t ensuremath star which is well known for the burgers equation and sometimes conjectured for the three dimensional euler equation has as counterpart in the truncated case the ability of tygers to store a finite amount of energy in the limit k mathrm g ensuremath rightarrow ensuremath infty this leads to reynolds stresses acting on scales larger than the galerkin wavelength and thus prevents the flow from converging to the inviscid limit solution there are indications that it may eventually be possible to purge the tygers and thereby to recover the correct inviscid limit behavior

an interactive fuzzy satisficing method for multiobjective nonconvex programming problems through floating point genetic algorithms in this paper we focus on multiobjective nonconvex nonlinear programming problems and present an interactive fuzzy satisficing method through floating point genetic algorithms after determining the fuzzy goals of the decision maker if the decision maker specifies the reference membership values the corresponding pareto optimal solution can be obtained by solving the augmented minimax problems for which the floating point genetic algorithm called genocop iii is applicable in order to overcome the drawbacks of genocop iii we propose the revised genocop iii by introducing a method for generating an initial feasible point and a bisection method for generating a new feasible point efficiently then an interactive fuzzy satisficing method for deriving a satisficing solution for the decision maker efficiently from a pareto optimal solution set is presented together with an illustrative numerical example

an interactive weighted tchebycheff procedure for multiple objective programming the procedure samples the efficient set by computing the nondominated criterion vector that is closest to an ideal criterion vector according to a randomly weighted tchebycheff metric using filtering techniques maximally dispersed representatives of smaller and smaller subsets of the set of nondominated criterion vectors are presented at each iteration the procedure has the advantage that it can converge to non extreme final solutions especially suitable for multiple objective linear programming the procedure is also applicable to integer and nonlinear multiple objective programs

binary modulated oscillations in a semi discrete version of burgers equation abstract in this paper we study both analytically and numerically the binary modulated oscillations which arise in an integrable semi discrete approximation to inviscid burgers explicit solutions of the binary modulation equations derived in goodman and lax 1988 are found for several choices of initial data observations based on numerical experiments are presented as to which initial data lead to solutions containing purely binary oscillations comparisons are made to the integrable theory of the kac van moerbeke and toda lattices non integrable schemes for which the binary oscillations also remain hyperbolic are presented and explicit solutions of their binary modulation equations are found

advection of passive and reactive tracers in multi dimensional burgers velocity field abstract we find exact solutions to a number of equations related to the passive tracer transport problem in a multi dimensional burgers velocity field convenient approximations are also proposed physical characteristics and other features of solutions of the concentration field equation are discussed in addition an exactly solvable model of a nonlinear reaction diffusion equation is studied for continuity equations involving generalized velocity fields the concept of a supersingular distribution is found useful

kinetic decomposition for singularly perturbed higher order partial differential equations abstract in this paper we consider singularly perturbed higher order partial differential equations we establish the condition under which the approximate solutions converge in a strong topology to the entropy solution of a scalar conservation laws using methodology developed in hwang and tzavaras comm partial differential equations 27 2002 1229 first we obtain the approximate transport equation for the given dispersive equations then using the averaging lemma we obtain the convergence

long time behaviour of an integrable heisenberg spin chain we consider an integrable modification of the standard heisenberg spin chain which is known to be a discretization of the continuous heisenberg magnet model and indeed equivalent to a discretization of the nonlinear schrodinger equation in particular we investigate the long time behaviour of the system we are mostly interested in the non soliton part thus completing the discussion in papanicolaou 1987 of the solitonic solutions our method makes use of the related inverse scattering problem and its formulation as a riemann hilbert factorization problem we show that the problem can be reduced to the analogous one for the toda lattice and provide explicit asymptotic formulae

kinetic decomposition for kinetic models of bgk type abstract in this paper we consider kinetic models of bgk type which describe the scalar conservation law at the microscopic scale we use new technique developed in comm partial differential equations 27 2002 1229 in order to get the convergence first we obtain the approximate transport equation for the given kinetic models of bgk type then using the averaging lemma we obtain the convergence this paper shows how to relate the given kinetic model with the averaging lemma to get the convergence

long time asymptotics of the nonlinear schrodinger equation shock problem the long time asymptotics of two colliding plane waves governed by the focusing nonlinear schrodinger equation are analyzed via the inverse scattering method we find three asymptotic regions in space time a region with the original wave modified by a phase perturbation a residual region with a one phase wave and an intermediate transition region with a modulated two phase wave the leading order terms for the three regions are computed with error estimates using the steepest descent method for riemann hilbert problems the nondecaying initial data requires a new adaptation of this method a new breaking mechanism involving a complex conjugate pair of branch points emerging from the real axis is observed between the residual and transition regions also the effect of the collision is felt in the plane wave state well beyond the shock front at large times 2007 wiley periodicals inc

kinetic formulation for systems of two conservation laws and elastodynamics for scalar conservation laws the kinetic formulation makes it possible to generate all the entropies from a simple kernel we show how this concept replaces and simplifies greatly the concept of young measures avoiding the difficulties encountered when working in lp the general construction of the two kinetic functions that generate the entropies of 2 2 strictly hyperbolic systems is also developed here we show that it amounts to building a universal entropy i e one that can be truncated by a kinetic value along riemann invariants for elastodynamics this construction can be completed and specialized using the additional galilean invariance this allows a full characterization of convex entropies it yields a kinetic formulation consisting of two semi kinetic equations which as usual are equivalent to the infinite family of all the entropy inequalities

dispersive shock waves and modulation theory there is growing physical and mathematical interest in the hydrodynamics of dissipationless dispersive media since g b whitham s seminal publication fifty years ago that ushered in the mathematical study of dispersive hydrodynamics there has been a significant body of work in this area however there has been no comprehensive survey of the field of dispersive hydrodynamics utilizing whitham s averaging theory as the primary mathematical tool we review the rich mathematical developments over the past fifty years with an emphasis on physical applications the fundamental large scale coherent excitation in dispersive hydrodynamic systems is an expanding oscillatory dispersive shock wave or dsw both the macroscopic and microscopic properties of dsws are analyzed in detail within the context of the universal integrable and foundational models for uni directional korteweg de vries equation and bi directional nonlinear schrodinger equation dispersive hydrodynamics a dsw fitting procedure that does not rely upon integrable structure yet reveals important macroscopic dsw properties is described dsw theory is then applied to a number of physical applications superfluids nonlinear optics geophysics and fluid dynamics finally we survey some of the more recent developments including non classical dsws dsw interactions dsws in perturbed and inhomogeneous environments and two dimensional oblique dsws

the zero dispersion limit a deterministic analogue of turbulence weak limits of oscillatory sequences of dispersive compressible flows show a remarkable resemblance to the ensemble averages of classical turbulence theory how much relevance such ideas have for incompressible flows is not clear

diffusion dispersion limits for multidimensional scalar conservation laws with source terms abstract in this paper we consider conservation laws with diffusion and dispersion terms we study the convergence for approximation applied to conservation laws with source terms the proof is based on the hwang and tzavaras s new approach seok hwang athanasios e tzavaras kinetic decomposition of approximate solutions to conservation laws application to relaxation and diffusion dispersion approximations comm partial differential equations 27 5 6 2002 1229 1254 and the kinetic formulation developed by lions perthame and tadmor p l lions b perthame e tadmor a kinetic formulation of multidimensional scalar conservation laws and related equations j amer math soc 7 1 1994 169 191

analysis and computation of a discrete kdv burgers type equation with fast dispersion and slow diffusion the long time behavior of the dynamics of an example of a fast slow system of ordinary differential equations is examined the system is derived from a spatial discretization of a korteweg de vries burgers type equation with fast dispersion and slow diffusion the discretization is based on a model developed by goodman and lax that is composed of a fast system drifted by a slow forcing term a difficulty to invoke available multiscale methods arises since the underlying system does not possess a natural split to fast and slow state variables our approach depicts the limit behavior as a young measure with values being invariant measures of the fast contribution to the flow the slow contribution to the dynamics causes these invariant measures to drift we keep track of this drift via slowly evolving observables averaging equations for the latter lead to computation of characteristic features of the motion and the location the invariant measures such computations are presented in the paper

toda soliton limits on general backgrounds abstract starting from an arbitrary background solution of the toda lattice we study limits of n soliton solutions on this given background as n tends to infinity this yields a new class of solutions of the toda lattice simultaneously we solve an inverse spectral problem for one dimensional jacobi operators we explicitly construct jacobi operators whose spectrum contains a given countable bounded set of eigenvalues and whose absolutely continuous spectrum coincides with that of a given background operator

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

non classical shocks and kinetic relations scalar conservation laws this paper analyzes the non classical shock waves which arise as limits of certain diffusive dispersive approximations to hyperbolic conservation laws such shocks occur for non convex fluxes and connect regions of different convexity they have negative entropy dissipation for a single convex entropy function but not all convex entropies and do not obey the classical oleinik entropy criterion we derive necessary conditions for the existence of non classical shock waves and construct them as limits of traveling wave solutions for several diffusive dispersive approximations

a note on the numerical integration of the kdv equation via isospectral deformations the main purpose of this paper is to test the performance of lie group integrators as applied to a semi discrete version of the kdv equation

the whitham equations for optical communications mathematical theory of nrz we present a model of optical communication system for high bit rate data transmission in the nonreturn to zero nrz format over transoceanic distance the system operates in a small group velocity dispersion regime and the model equation is given by the well known whitham equations describing the slow modulation of multiphase wavetrains of the defocusing nonlinear schrodinger nls equation the model equation is of hyperbolic type and nrz pulse with certain initial phase modulation develops a shock we then show how one can obtain a global solution by choosing an appropriate riemann surface on which the whitham equation is defined we also discuss the effect of third order dispersion by using an integrable hierarchy of the nls equation and we give a condition to avoid a shock formation

the korteweg de vries equation with small dispersion higher order lax levermore theory we utilize the inverse scattering transformation to study the rapid oscillations which arise in the solution of the initial value problem begin gathered u t 6u u x in 2 u xxx 0 hfill u left x 0 right v left x right hfill end gathered when 0 we refine the lax levermore theory which gives the weak limit of the solution as 0 by introducing a quantum condition this allows us to calculate the waveform of the local oscillations up to phase shifts when is small

commutation methods for jacobi operators abstract we offer two methods of inserting eigenvalues into spectral gaps of a given background jacobi operator the single commutation method which introduces eigenvalues into the lowest spectral gap of a given semi bounded background jacobi operator and the double commutation method which inserts eigenvalues into arbitrary spectral gaps moreover we prove unitary equivalence of the commuted operators restricted to the orthogonal complement of the eigenspace corresponding to the newly inserted eigenvalues with the original background operator in addition we compute the matrix valued weyl m functions of the commuted operator in terms of the background weyl m functions finally we show how to iterate the above methods and give explicit formulas for various quantities such as eigenfunctions and spectra of the iterated operators in terms of the corresponding background quantities and scattering matrix concrete applications include an explicit realization of the isospectral torus for algebro geometric finite gap jacobi operators and the n soliton solutions of the toda and kac van moerbeke lattice equations with respect to arbitrary background solutions

asymptotic stability of singularly perturbed differential equations abstract asymptotic stability is examined for singularly perturbed ordinary differential equations that may not possess a natural split into fast and slow motions rather the right hand side of the equation is comprised of a singularly perturbed component and a regular one the limit dynamics consists then of young measures with values being invariant measures of the fast contribution drifted by the slow one relations between the asymptotic stability of the perturbed system and the limit dynamics are examined and a lyapunov functions criterion based on averaging is established