one dimensional transport equations with discontinuous coefficients we consider one dimensional linear transport equations with bounded but possibly discontinuous coefficient a the cauchy problem is studied from two different points of view in the first case we assume that a is piecewise continuous we give an existence result and a precise description of the solutions on the lines of discontinuity in the second case we assume that a satisfies a one sided lipschitz condition we give existence uniqueness and general stability results for backward lipschitz solutions and forward measure solutions by using a duality method we prove that the flux associated to these measure solutions is a product by some canonical representative of a key words linear transport equations discontinuous coefficients weak stability duality product of a measure by a discontinuous function nonnegative solutions 1991 mathematics subject classification primary 35f10 35b35 34a12 to appear in nonlinear analysis tma departement de mathematiques et applications umr cnrs 8553 ecole normale superieure et cnrs 45 rue d ulm 75230 paris cedex 05 france francois bouchut ens fr mathematiques applications et physique mathematique d orleans umr cnrs 6628 universite d orleans 45067 orleans cedex 2 france james math cnrs fr

the gap lemma and geometric criteria for instability of viscous shock profiles an obstacle in the use of evans function theory for stability analysis of traveling waves occurs when the spectrum of the linearized operator about the wave accumulates at the imaginary axis since the evans function has in general been constructed only away from the essential spectrum a notable case in which this difficulty occurs is in the stability analysis of viscous shock profiles here we prove a general theorem the gap lemma concerning the analytic continuation of the evans function associated with the point spectrum of a traveling wave into the essential spectrum of the wave this allows geometric stability theory to be applied in many cases where it could not be applied previously we demonstrate the power of this method by analyzing the stability of certain undercompressive viscous shock waves a necessary geometric condition for stability is determined in terms of the sign of a certain melnikov integral of the associated viscous profile this sign can easily be evaluated numerically we also compute it analytically for solutions of several important classes of systems in particular we show for a wide class of systems that homoclinic solitary waves are linearly unstable confirming these as the first known examples of unstable viscous shock waves we also show that strong heteroclinic undercompressive waves are sometimes unstable similar stability conditions are also derived for lax and overcompressive shocks and for nn

asymptotic behavior near planar transition fronts for the cahn hilliard equation abstract we consider the asymptotic behavior of perturbations of planar wave solutions arising in the cahn hilliard equation in space dimensions d 2 such equations are well known to arise in the study of spinodal decomposition a phenomenon in which rapid cooling of a homogeneously mixed binary alloy causes separation to occur resolving the mixture into regions in which one component or the other is dominant with these regions separated by steep transition layers a critical feature of the cahn hilliard equation in one space dimension is that the linear operator that arises upon linearization of the equation about a standing wave solution has essential spectrum extending onto the imaginary axis a feature that is known to complicate the step from spectral to nonlinear stability the analysis of planar waves in multiple space dimensions is further complicated by the fact that the leading eigenvalue for this linearized operator leading in the case of stability moves into the negative real half plane with cubic scaling 3 where r d 1 denotes a fourier variable associated with spatial components transverse to the planar wave under the assumption of spectral stability described in terms of an appropriate evans function we develop detailed asymptotics for perturbations from planar wave solutions establishing asymptotic stability for initial perturbations decaying at an appropriate algebraic rate in an l 1 norm of the transverse variables

einstein metrics on s 2 bundles new einstein metrics are constructed on the associated bbb r bbb p 2 s 2 and bbb r 2 bundles of principal circle bundles with base a product of kahler einstein manifolds with positive first chern class and with euler class a rational linear combination of the first chern classes these einstein metrics represent different generalizations of the well known einstein metrics found by berard bergery d page c pope n koiso and y sakane corresponding new einstein weyl structures are also constructed

multidimensional viscous shocks ii the small viscosity limit in this paper we prove the existence of curved multidimensional viscous shocks and also justify the small viscosity limit starting with a curved multidimensional inviscid shock solution to a system of hyperbolic conservation laws we show that the shock can be obtained as a small viscosity limit of solutions to an associated parabolic problem viscous shocks the two main hypotheses are a natural evans function assumption on the viscous profile together with a restriction on how much the shock can deviate from flatness the main tools are a conjugation lemma that removes xn dependence from the linearization of the parabolic problem about the viscous profile new degenerate kreiss type symmetrizers used to prove an l2 estimate for the linearized problem and a finite regularity calculus of semiclassical and mixed type classical semiclassical pseudodifferential operators 2003 wiley periodicals inc

the classification of 2 2 systems of non strictly hyperbolic conservation laws with application to oil recovery on considere un systeme 2 2 a une variable d espace u f u x 0 0 ou u u x t r 2 et f r 2 r 2 on montre que pour des equations hyperboliques des points ombilicaux sont typiquement isoles et que en un point ombilical isole df est typiquement diagonalisable on donne une classification des equations hyperboliques a points ombilicaux isoles

ion acoustic shock in a collisional plasma abstract the paper is concerned with the propagation of ion acoustic shock waves in a collision dominated plasma whose equations of motion are described by the one dimensional isothermal navier stokes poisson system for ions with the electron density determined by the boltzmann relation the main results include three parts a we establish the existence and uniqueness of a small amplitude smooth traveling wave by solving a 3 d ode in terms of the center manifold theorem b we study the shock structure in a specific asymptotic regime where the viscosity coefficient and the shock strength are proportional to e and the debye length is proportional to e 1 2 with two parameters e and and show that in the limit e 0 shock profiles obtained in a can be approximated by the profiles of kdv burgers uniformly for 0 0 with some 0 0 the proof is based on the suitable construction of the kdv burgers shock profiles together with the delicate analysis of a linearized variable coefficient system in exponentially weighted sobolev spaces involving parameters e and c we also prove the large time asymptotic stability of traveling waves under suitably small smooth zero mass perturbations note that the ions temperature is allowed to be zero in parts a and b but necessarily required to be strictly positive in the proof of part c

asymptotic behavior near transition fronts for equations of generalized cahn hilliard form we consider the asymptotic behavior of perturbations of standing wave solutions arising in evolutionary pde of generalized cahn hilliard form in one space dimension such equations are well known to arise in the study of spinodal decomposition a phenomenon in which the rapid cooling of a homogeneously mixed binary alloy causes separation to occur resolving the mixture into its two components with their concentrations separated by sharp transition layers motivated by work of bricmont kupiainen and taskinen 5 we regard the study of standing waves as an interesting step toward understanding the dynamics of these transitions a critical feature of the cahn hilliard equation is that the linear operator that arises upon linearization of the equation about a standing wave solution has essential spectrum extending onto the imaginary axis a feature that is known to complicate the step from spectral to nonlinear stability under the assumption of spectral stability described in terms of an appropriate evans function we develop detailed asymptotics for perturbations from standing wave solutions establishing phase asymptotic orbital stability for initial perturbations decaying with appropriate algebraic rate

nonlinear stability of an undercompressive shock for complex burgers equation though there is strong numerical evidence for the stability of undercompressive shocks their stability has not been verified analytically in particular the energy methods used to analyze stability of standard shocks do not apply here we present the first proof of stability for a particular undercompresive shock the real burgers shock considered as a solution of the complex burgers equation our analysis is by direct calculation of the green s function for the linearized equations combined with pointwise estimates of nonlinear effects a benefit of this method is to obtain fairly detailed information about the solution includingl1 behavior and rates of decay in different regions of space

homogeneous contact manifolds and resolutions of calabi yau cones in the present work we provide a constructive method to describe contact structures on compact homogeneous contact manifolds the main feature of our approach is to describe the cartan ehresmann connection gauge field for principal u 1 bundles over complex flag manifolds by using elements of representation theory of simple lie algebras this description allows us to compute explicitly the expression of the contact form for any boothby wang fibration over complex flag manifolds boothby and wang in ann math 68 721 734 1958 as well as their underlying sasaki structures by following conlon and hein duke math j 162 2855 2902 2013 van coevering math ann 2009 https doi org 10 1007 s00208 009 0446 1 and goto j math soc jpn 64 1005 1052 2012 as an application of our results we use the cartan remmert reduction grauert in math ann 146 331 368 1962 and the calabi ansatz technique calabi in ann sci ecole norm sup 4 12 269 294 1979 to provide many explicit examples of crepant resolutions of calabi yau cones with certain homogeneous sasaki einstein manifolds realized as links of isolated singularities these concrete examples illustrate the existence part of the conjecture introduced in martelli and sparks phys rev d 79 6 065009 2009

hamiltonian 2 forms in kahler geometry ii global classification we present a classification of compact kahler manifolds admitting a hamiltonian 2 form which were classified locally in part i of this work this involves two components of independent interest the first is the notion of a rigid hamiltonian torus action this natural condition for torus actions on a kahler manifold was introduced locally in part i but such actions turn out to be remarkably well behaved globally leading to a fairly explicit classification up to a blow up compact kahler manifolds with a rigid hamiltonian torus action are bundles of toric kahler manifolds the second idea is a special case of toric geometry which we call orthotoric we prove that orthotoric kahler manifolds are diffeomorphic to complex projective space but we extend our analysis to orthotoric orbifolds where the geometry is much richer we thus obtain new examples of kahler einstein 4 orbifolds combining these two themes we prove that compact kahler manifolds with hamiltonian 2 forms are covered by blow downs of projective bundles over kahler products and we describe explicitly how the kahler metrics with a hamiltonian 2 form are parameterized we explain how this provides a context for constructing new examples of extremal kahler metrics in particular a subclass of such metrics which we call weakly bochner flat we also provide a self contained treatment of the theory of compact toric kahler manifolds since we need it and find the existing literature incomplete

a momentum construction for circle invariant kahler metrics examples of kahler metrics of constant scalar curvature are relatively scarce over the past two decades several workers in geometry and physics have used symmetry reduction to construct complete kahler metrics of constant scalar curvature by ode methods one fruitful idea the calabi ansatz is to begin with an hermitian line bundle p l h m g m over a kahler manifold and to search for kahler forms p m dd c f t in some disk subbundle where t is the logarithm of the norm function and f is a function of one variable our main technical result theorem a is the calculation of the scalar curvature for an arbitrary kahler metric g arising from the calabi ansatz this suggests geometric hypotheses which we call a constancy to impose upon the base metric g m and hermitian structure h in order that the scalar curvature of g be specified by solving an ode we show that constancy is necessary and sufficient for the calabi ansatz to work in the following sense under the assumption of constancy the disk bundle admits a one parameter family of complete kahler metrics of constant scalar curvature that restrict to g m on the zero section theorems b and d an analogous result holds for the punctured disk bundle theorem c a simple criterion determines when such a metric is einstein conversely in the absence of constancy the calabi ansatz yields at most one metric of constant scalar curvature in either the disk bundle or the punctured disk bundle theorem e many of the metrics constructed here seem to be new including a complete negative einstein kahler metric on the disk subbundle of a stable vector bundle over a riemann surface of genus at least two and a complete scalar flat kahler metric on c 2

dynamical stability of phase transitions in the p system with viscosity capillarity we show for the p system with viscosity capillarity that all heteroclinic phase transitions lying sufficiently close to the maxwell line are linearly dynamically stable under localized perturbations independent of the form of the flux on the other hand we show that nearby homoclinic phase transitions involving two or more changes in phase are exponentially unstable these results together imply that stationary and slowly moving phase transitions are dynamically stable if and only if they are monotone in analogy with results obtained by carr gurtin and slemrod in the variational setting arch rational mech anal 86 1984 pp 317 351 this research is significant as the first proof of stability of a strong viscous shock wave for systems our analysis uses the technical framework developed in r gardner and k zumbrun comm pure appl math 51 1998 pp 797 855 k zumbrun and p howard indiana univ math j 47 1998 pp 741 871 together with a nonstandard energy estimate

on the dynamic exponent of the two dimensional ising model the authors use the method of damage spreading to measure the time taken for the equilibrium damage the magnetization to disappear in finite systems 10 or 2 27

stability of travelling waves for non convex scalar viscous conservation laws travelling waves of a viscous conservation law so called viscous profiles are shown to be stable in polynomialy weighted l spaces there is no assumption of convexity on the nonlinear term and thus earlier results of il in and ole nik sattinger and kawashima and matsumura are generalized the method uses the semigroup of the linearized equation with solutions of the full problem expressed by the variation of constants formula estimates are derived for the semigroup through a new technique for estimating the resolvent 1993 john wiley sons inc

on nonlinear stability of general undercompressive viscous shock waves we study the nonlinear stability of general undercompressive viscous shock waves previously the authors showed stability in a special case when the shock phase shift can be determined a priori from the total mass of the perturbation using new pointwise methods by examining time invariants associated with the linearized equations we can now overcome a new difficulty in the general case namely nonlinear movement of the shock we introduce a coordinate transformation suitable to treat this new aspect and demonstrate our method by analyzing a model system of generic type we obtain sharp pointwise bounds andlp behavior of the solution for allp 1 p

center manifolds for a class of degenerate evolution equations and existence of small amplitude kinetic shocks abstract we construct center manifolds for a class of degenerate evolution equations including the steady boltzmann equation and related kinetic models establishing in the process existence and behavior of small amplitude kinetic shock and boundary layers notably for boltzmann s equation we show that elements of the center manifold decay in velocity at near maxwellian rate in accord with the formal chapman enskog picture of near equilibrium flow as evolution along the manifold of maxwellian states or grad moment approximation via hermite polynomials in velocity our analysis is from a classical dynamical systems point of view with a number of interesting modifications to accommodate ill posedness of the underlying evolution equation

on the stability of traveling waves in weighted l spaces abstract i present new techniques and results concerning the stability of travelling waves to semilinear parabolic equations i am concerned with those problems in which the essential spectrum of the linearization about the wave contains 0 in these circumstances the stability results of d henry in lecture notes in mathematics vol 840 springer verlag new york 1981 and p bates and c jones dynamics rep 2 1989 1 38 are not applicable as the spectral information does not yield an exponential time decay estimate on the semigroup i show that under certain relatively easily computable conditions the wave is stable in polynomially weighted l spaces furthermore the rate at which the perturbation decays to the wave as t depends on the growth rate of the polynomial near infinity the method of proof uses the semigroup of the linearized equation with solutions of the full problem expressed by the variation of constants formula estimates for the semigroup are derived through a new technique for estimating the resolvent

critical dynamics of the potts model in several dimensionalities abstract we study critical dynamics of general q state potts models on d dimensional hypercubic lattices the master equation is formulated according to a theory recently presented by the author a simple bond moving technique followed by decimation is used to obtain the dynamical exponents although this approximation yields poor results for the static exponents the dynamic behaviour is closer to monte carlo simulations we compare our results with those obtained with a different formulation of dynamics a final discussion is included

counter examples to the generalized positive action conjecture we give examples of complete locally asymptotically flat riemannian 4 manifolds with zero scalar curvature and negative mass the generalized positive action conjecture of hawking and pope 5 is therefore false

calabi yau metrics on canonical bundles of complex flag manifolds abstract in the present paper we provide a description of complete calabi yau metrics on the canonical bundle of generalized complex flag manifolds by means of lie theory we give an explicit description of complete ricci flat kahler metrics obtained through the calabi ansatz technique we use this approach to provide several explicit examples of noncompact complete calabi yau manifolds these examples include canonical bundles of non toric flag manifolds e g grassmann manifolds and full flag manifolds

ale scalar flat kahler metrics on non compact weighted projective spaces we construct new explicit toric scalar flat kahler ale metrics with orbifold singularities which we use to obtain smooth extremal kahler metrics on appropriate resolutions of orbifolds

conditional stability of unstable viscous shock waves in compressible gas dynamics and mhd extending our previous work in the strictly parabolic case we show that a linearly unstable lax type viscous shock solution of a general quasilinear hyperbolic parabolic system of conservation laws possesses a translation invariant center stable manifold within which it is nonlinearly orbitally stable with respect to small l 1 h 3 perturbations converging time asymptotically to a translate of the unperturbed wave that is for a shock with p unstable eigenvalues we establish conditional stability on a codimension p manifold of initial data with sharp rates of decay in all l p for p 0 we recover the result of unconditional stability obtained by mascia and zumbrun the main new difficulty in the hyperbolic parabolic case is to construct an invariant manifold in the absence of parabolic smoothing

domain growth kinetics and symmetry breaking in a thermally quenched system the authors simulate thermal quenches on a two dimensional hard core model with symmetry breaking phase transition by dividing large configurations into many small sub squares lack of self averaging is avoided and statistics are improved dynamic scaling is observed during more than two decades

pointwise green s function bounds for multidimensional scalar viscous shock fronts we determine detailed pointwise bounds for the green s function of the linearized operator about a multidimensional scalar viscous shock front these extend the pointwise semigroup methods introduced by howard and zumbrun in the one dimensional case to multidimensions sharpening lp estimates obtained by goodman and miller using a weighted norm approach moreover our results apply to shocks of arbitrary strength as previous results did not as described in a companion paper the bounds we obtain are sufficient to give a straightforward treatment of the nonlinear lp asymptotic behavior of the front under small perturbation the analysis of the multidimensional case involves several new features not found in the one dimensional case concerned with the geometry of propagating signals

existence and stability of multidimensional shock fronts in the vanishing viscosity limit in this paper we present a new approach to the study of linear and nonlinear stability of inviscid multidimensional shock waves under small viscosity perturbation yielding optimal estimates and eventually an extension to the viscous case of the celebrated theorem of majda on existence and stability of multidimensional shock waves more precisely given a curved lax shock solution u0 to a hyperbolic system of conservation laws we construct nearby viscous shock solutions ue to a parabolic viscous perturbation of the hyperbolic system which converge to u0 as viscosity e 0 and satisfy an appropriate conormal version of majda s stability estimate

dynamics of the formation of two dimensional ordered structures the growth of ordered domains in lattice gas models which occurs after the system is quenched from infinite temperature to a state below the critical temperaturet c is studied by monte carlo simulation for a square lattice with repulsion between nearest and next nearest neighbors which in equilibrium exhibits fourfold degenerate 2 1 superstructures the time dependent energy e t domain size l t and structure functions q t are obtained both for glauber dynamics no conservation law and the case with conserved density kawasaki dynamics at late times the energy excess and halfwidth of the structure factor decrease proportional tot x whilel t t x where the exponent x 1 2 for glauber dynamics and x 1 3 for kawasaki dynamics in addition the structure factor satisfies a scaling laws k t t 2xs ktx the smaller exponent for the conserved density case is traced back to the excess density contained in the walls between ordered domains which must be redistributed during growth quenches tot t c t tc where we estimate dynamic critical exponents andt 0 are also considered in the latter case the system becomes frozen in a glasslike domain pattern far from equilibrium when using kawasaki dynamics the generalization of our results to other lattices and structures also is briefly discussed

stability of post fertilization traveling waves article history this paper studies the stability of a family of traveling wave solutions to the system proposed by lane et al d c lane j d murray v s manoranjan analysis of wave phenomena in a morphogenetic mechanochemical model and an application to post fertilization waves on eggs ima j math appl med biol 4 4 1987 309 331 to model a pair of mechanochemical phenomena known as post fertilization waves on eggs the waves consist of an elastic deformation pulse on the egg s surface and a free calcium concentration front the family is indexed by a coupling parameter measuring contraction stress effects on the calcium concentration this work establishes the spectral linear and nonlinear orbital stability of these post fertilization waves for small values of the coupling parameter the usual methods for the spectral and evolution equations cannot be applied because of the presence of mixed partial derivatives in the elastic equation nonetheless exponential decay of the directly constructed semigroup on the complement of the zero eigenspace is established we show that small perturbations of the waves yield solutions to the nonlinear equations decaying exponentially to a phase modulated traveling

locus of the tricritical transition in a two dimensional q state potts model a simple two dimensional potts model with vacancies is introduced the phase transition between paramagnetic and ferromagnetic is located by simple symmetry arguments it is shown that the transition has a continuous and a first order segment separated by a tricritical transition the locus of the tricritical point is determined by a transformation of the partition sum into that of an o n model at the tricritical point and at some of the critical points this potts model is soluble

pointwise green s function estimates toward stability for multidimensional fourth order viscous shock fronts abstract for the case of multidimensional viscous conservation laws with fourth order smoothing only we develop detailed pointwise estimates on the green s function for the linear fourth order convection equation that arises upon linearization of the conservation law about a viscous planar wave solution as in previous analyses in the case of second order smoothing our estimates are sufficient to establish that spectral stability implies nonlinear stability though the full development of this result will be considered in a companion paper

central kahler metrics with non constant central curvature the central curvature of a riemannian metric is the determinant of its ricci endomorphism while the scalar curvature is its trace a kahler metric is called central if the gradient of its central curvature is a holornorphic vector field such metrics may be viewed as analogs of the extremal kahler metrics defined by calabi in this work central metrics of non constant central curvature are constructed on various ruled surfaces most notably the first hirzebruch surface this is achieved via the momentum construction of hwang and singer a variant of an ansatz employed by calabi 1979 and by koiso and sakane 1986 non existence real analyticity and positivity properties of central metrics arising in this ansatz are also established

linear stability of shock profiles for systems of conservation laws with semi linear relaxation the evans function theory which has recently been applied to the study of linear stability of viscous shock profiles is developed below for semi linear relaxation we study the linear stability of shock profiles in the lax undercompressive and overcompressive cases the results we obtain are similar to those found for viscous approximations by gardner and zumbrun commun pure appl math 51 7 1998 797

stability of periodic solutions of conservation laws with viscosity analysis of the evans function nonclassical conservation laws with viscosity arising in multiphase fluid and solid mechanics exhibit a rich variety of traveling wave phenomena including homoclinic pulse type and periodic solutions along with the standard heteroclinic shock or front type solutions here we investigate stability of periodic traveling waves within the abstract evans function framework established by r a gardner our main result is to derive a useful stability index analogous to that developed by gardner and zumbrun in the traveling front or pulse context giving necessary conditions for stability with respect to initial perturbations that are periodic on the same period t as the traveling wave moreover we show that the periodic stability index has an interpretation analogous to that of the traveling front or pulse index in terms of well posedness of an associated riemann problem for an inviscid medium now to be interpreted as allowing a wider class of measure valued solutions or alternatively in terms of existence and nonsingularity of a local mass map from perturbation mass to potential time asymptotic t periodic states a closely related calculation yields also a complementary long wave stability criterion necessary for stability with respect to periodic perturbations of arbitrarily large period nt n we augment these analytical results with numerical investigations analogous to those carried out by brin in the traveling front or pulse case approximating the spectrum of the linearized operator about the wave

the cauchy problem for 1 d linear nonconservative hyperbolic systems with possibly expansive discontinuity of the coefficient a viscous approach abstract in this paper we consider linear nonconservative cauchy systems with discontinuous coefficients across the noncharacteristic hypersurface x 0 for the sake of simplicity we restrict ourselves to piecewise constant hyperbolic operators of the form t a x x with a x a 1 x 0 a 1 x 0 where a m n r under assumptions incorporating a sharp spectral stability assumption we prove that a unique solution is successfully singled out by a vanishing viscosity approach due to our framework which includes systems with expansive discontinuities of the coefficient the selected small viscosity solution satisfies an unusual hyperbolic problem which is well posed even though it does not satisfy in general a uniform lopatinski condition in addition based on a detailed analysis of our stability assumption explicit examples of 2 2 systems checking our assumptions are given our result is new and contains both a stability result and a description of the boundary layers forming at any order two kinds of boundary layers form each polarized on specific linear subspaces in direct sum

dynamical properties of the two and three dimensional ising models by damage spreading the method of damage spreading is used to measure the dynamic exponent z for both the two and three dimensional ising models with heat bath dynamics the authors also measure a new exponent alpha describing the rate at which a damage cloud grows they report values of z 2 24 or 0 04 and alpha 0 77 or 0 11 in two dimensions and z 2 202 or 0 03 and alpha 0 92 or 0 10 in three dimensions

spectral stability of small shock waves this paper is the first in a series of articles that study the eigenvalue problem for small amplitude shock waves in systems of conservation laws with viscosity or relaxation the papers show in various contexts that in the zero amplitude limit appropriately scaled versions of the associated evans bundles converge to suspensions of evans bundles for fixed shock waves in related reduced systems with lower dimensional state space in this article the new geometric framework establishing this connection is introduced in the simplest context that of shock waves associated with a simple genuinely nonlinear mode in systems with identity viscosity in one space dimension

ordinary differential equations transport theory and sobolev spaces we obtain some new existence uniqueness and stability results for ordinary differential equations with coefficients in sobolev spaces these results are deduced from corresponding results on linear transport equations which are analyzed by the method of renormalized solutions

hamiltonian constructions of kahler einstein metrics and kahler metrics of constant scalar curvature assuming the existence of a real torus acting through holomorphic isometries on a kahler manifold we construct an ansatz for kahler einstein metrics and an ansatz for kahler metrics with constant scalar curvature using this hamiltonian approach we solve the differential equations in special cases and find in particular a family of constant scalar curvature kahler metrics describing a non linear superposition of the bergman metric the calabi metric and a higher dimensional generalization of the lebrun kahler metric the superposition contains kahler einstein metrics and all the geometries are complete on the open disk bundle of some line bundle over the complex projective spacepn we also build such kahler geometries on kahler quotients of higher cohomogeneity

stability of travelling wave solutions of diffusive predator prey systems the stability of travelling wave solutions of singularly perturbed diffusive predator prey systems is proved by showing that the linearized operator about such a solution has no unstable spectrum and that the translation eigenvalue at a 0 is simple the proof illustrates the application of some recently developed geometric and topological methods for counting eigenvalues

stability of bright solitary wave solutions to perturbed nonlinear schrodinger equations abstract the propagation of pulses in ideal nonlinear optical fibers without loss is governed by the nonlinear schrodinger equation nls when considering realistic fibers one must examine perturbed nls equations with the particular perturbation depending on the physical situation that is being modeled a common example is the complex ginzburg landau equation cgl which is a dissipative perturbation it is known that some of the stable bright solitons of the nls survive a dissipative perturbation such as the cgl given that a wave persists it is then important to determine its stability with respect to the perturbed nls a major difficulty in analyzing the stability of solitary waves upon adding dissipative terms is that eigenvalues may bifurcate out of the essential spectrum since the essential spectrum of the nls is located on the imaginary axis such eigenvalues may lead to an unstable wave in fact eigenvalues can pop out of the essential spectrum even if the unperturbed problem has no eigenvalue embedded in the essential spectrum here we present a technique which can be used to track these bifurcating eigenvalues as a consequence we are able to locate the spectrum for bright solitary wave solutions to various perturbed nonlinear schrodinger equations and determine precise conditions on parameters for which the waves are stable in addition we show that a particular perturbation the parametrically forced nls equation supports stable multi bump solitary waves the technique for tracking eigenvalues which bifurcate from the essential spectrum is very general and should therefore be applicable to a larger class of problems than those presented here

stability of noncharacteristic boundary layers in the standing shock limit we investigate one and multi dimensional stability of noncharacteristic boundary layers in the limit approaching a standing planar shock wave u x1 x 1 0 obtaining necessary conditions of i weak stability of the limiting shock ii weak stability of the constant layer u u lim z z and iii nonnegativity of a modified lopatinski determinant similar to that of the inviscid shock case for lax 1 shocks we obtain equally simple sufficient conditions for p shocks p 1 the situation appears to be more complicated using these results we determine the stability of certain gas dynamical boundary layers generalizing earlier work of serre zumbrun and costanzino humphreys nguyen zumbrun

a lax shock profile satisfying a sufficient condition of spectral instability abstract the aim of this paper is to construct a spectrally unstable lax shock profile of the semi linear approximation of a 2 2 system of conservation laws and to show numerically the instability thanks to an appropriate scheme

on the stability of periodic travelling waves with large spatial period abstract in many circumstances a pulse to a partial differential equation pde on the real line is accompanied by periodic wave trains that have arbitrarily large period it is then interesting to investigate the pde stability of the periodic wave trains given that the pulse is stable using the evans function gardner has demonstrated that every isolated eigenvalue of the linearization about the pulse generates a small circle of eigenvalues for the linearization about the periodic waves in this article the precise location of these circles is determined it is demonstrated that the stability properties of the periodic waves depend on certain decay and oscillation properties of the tails of the pulse as a consequence periodic waves with long wavelength typically destabilize at homoclinic bifurcation points at which multi hump pulses are created that is in contrast to the situation for the underlying pulses whose stability properties are not affected by these bifurcations the proof uses lyapunov schmidt reduction and relies on the existence of exponential dichotomies the approach is also applicable to periodic waves with large spatial period of elliptic problems on r n or on unbounded cylinders r with bounded

stability of isentropic navier stokes shocks in the high mach number limit by a combination of asymptotic ode estimates and numerical evans function calculations we establish stability of viscous shock solutions of the isentropic compressible navier stokes equations with law pressure i in the limit as mach number m goes to infinity for any 1 proved analytically and ii for m 2 500 gamma in 1 2 5 or m 13 000 gamma in 2 5 3 demonstrated numerically this builds on and completes earlier studies by matsumura nishihara and barker humpherys rudd zumbrun establishing stability for low and intermediate mach numbers respectively indicating unconditional stability independent of shock amplitude of viscous shock waves for law gas dynamics in the range gamma in 1 3 other values may be treated similarly but have not been checked numerically the main idea is to establish convergence of the evans function in the high mach number limit to that of a pressureless or infinitely compressible gas with additional upstream boundary condition determined by a boundary layer analysis recall that low mach number behavior is formally incompressible

einstein metrics spinning top motions and monopoles on considere la construction de solutions pour l equation d einstein autoduale on applique la construction de lebrun a la sphere de berger la 3 sphere de metrique 1 2 2 2 i 3 3 2 ou 1 2 3 est une base de 1 formes invariante a gauche satisfaisant d i j k e ijk j k et i 3 est une constante

stability of the travelling wave solution of the fitzhugh nagumo system on demontre la stabilite relative au systeme complet d equations aux derivees partielles de ces ondes de propagation

pointwise convergence to shock waves for viscous conservation laws we are interested in the pointwise behavior of the perturbations of shock waves for viscous conservation laws it is shown that besides a translation of the shock waves and of linear and nonlinear diffusion waves of heat and burgers equations a perturbation also gives rise to algebraically decaying terms which measure the coupling of waves of different characteristic families our technique is a combination of time asymptotic expansion construction of approximate green functions and analysis of nonlinear wave interactions the pointwise estimates yield optimal lp convergence of the perturbation to the shock and diffusion waves 1 p the new approach of obtaining pointwise estimates based on the green functions for the linearized system and the analysis of nonlinear wave interactions is also useful for studying the stability of waves of distinct types and nonclassical shocks these are being explored elsewhere 1997 john wiley sons inc

stable viscosity matrices for systems of conservation laws abstract a natural class of appropriate viscosity matrices for strictly hyperbolic systems of conservation laws in one space dimension u 1 f u x 0 u r m is studied these matrices are admissible in the sense that small amplitude shock wave solutions of the hyperbolic system are shown to be limits of smooth traveling wave solutions of the parabolic system u t f u x v du x x as ifv 0 if d is in this class the class is determined by a linearized stability requirement the cauchy problem for the equation u 1 f u 0 u x vdu xx should be well posed in l 2 uniformly in v as v 0 previous examples of inadmissible viscosity matrices are accounted for through violation of the stability criterion

efficient computation of analytic bases in evans function analysis of large systems in evans function computations of the spectra of asymptotically constant coefficient linearized operators of large systems a problem that becomes important is the efficient computation of global analytically varying bases for invariant subspaces of the limiting coefficient matrices in the case that the invariant subspace is spectrally separated from its complementary invariant subspace we propose an efficient numerical implementation of a standard projection based algorithm of kato for which the key step is the solution of an associated sylvester problem this may be recognized as the analytic cousin of a c k algorithm developed by dieci and collaborators based on orthogonal projection rather than eigenprojection as in our case for a one dimensional subspace it reduces essentially to an algorithm of bridges derks and gottwald based on path finding and continuation methods

evans functions for periodic waves on infinite cylindrical domains abstract using galerkin approximations an evans function for spatially periodic waves on infinite cylindrical domains is constructed it is also shown that the evans function can be used to define a parity index for periodic waves that detects whether the wave admits an odd number of real unstable eigenvalues this parity index depends only on local information for the existence problem of the wave in particular it uses information about the linear dispersion relation near zero and the orientability of the unstable and stable manifolds along the nonlinear wave the results are applied to small amplitude wave trains for a scalar equation on an infinite strip

stability of viscous detonations for majda s model abstract using analytical and numerical evans function techniques we examine the spectral stability of strong detonation wave solutions of a version of majda s scalar model for a reacting gas mixture with an arrhenius type ignition function we introduce an energy estimate to limit possible unstable eigenvalues to a compact region in the unstable complex half plane and we use a numerical approximation of the evans function to search for possible unstable eigenvalues in this region our results show for the parameter values tested that these waves are spectrally stable combining these numerical results with the pointwise green function analysis of lyng raoofi texier zumbrun g lyng m raoofi b texier k zumbrun pointwise green function bounds and stability of combustion waves j differential equations 233 2 2007 654 698 we conclude that these waves are nonlinearly stable this represents the first demonstration of nonlinear stability for detonation wave solutions of a majda type model without a smallness assumption notably our results indicate that for this simplified scalar model there does not occur either in a normal parameter range or in the limit of high activation energy hopf bifurcation to galloping or pulsating solutions as is observed in the full reactive navier stokes equations this answers in the negative for this model a question posed by majda as to whether such scalar detonation models capture this aspect of detonation behavior

bifurcation of nonclassical viscous shock profiles from the constant state we determine the bifurcation from the constant solution of nonclassical transitional and overcompressive viscous shock profiles in regions of strict hyperbolicity whereas classical shock waves in systems of conservation laws involve a single characteristic field nonclassical waves involve two fields in an essential way this feature is reflected in the viscous profile differential equation which undergoes codimension three bifurcation of the kind studied by dumortier et al as opposed to the codimension one bifurcation occurring in the classical case we carry out a complete bifurcation analysis for systems of two quadratic conservation laws with constant strictly parabolic viscosity matrices by reducing to a canonical form introduced by fiddelaers we show that all such systems except possibly those on a codimension one variety in parameter space give rise to nonclassical shock waves and we classify the number and types of their bifurcation points one consequence of our analysis is that weak transitional waves arise in pairs with profiles forming a 2 cycle configuration previously shown to lead to nonuniqueness of riemann solutions and to nontrivial asymptotic dynamics of the conservation laws another consequence is that appearance of weak nonclassical waves is necessarily associated with change of stability in constant solutions of the parabolic system of conservation laws rather than with change of type in the associated hyperbolic system