the units of a ring spectrum and a logarithmic cohomology operation recall that if r is a commutative ring then the set r r of invertible elements of r is naturally an abelian group under multiplication this construction is a functor from commutative rings to abelian groups in general there is no obvious relation between the additive group of a ring r and the multiplicative group of units r however under certain circumstances one can define a homomorphism from a subgroup of r to a suitable completion of r e g the natural logarithm q 0 r or the p adic logarithm 1 pzp zp the logarithmic cohomology operation is a homotopy theoretic analogue of the above where r is a commutative s algebra and completion is bousfield localization with respect to a morava k theory the purpose of this paper is to give a formula for the logarithmic operation in certain contexts in terms of power operations before giving our results we briefly explain some of the concepts involved

diffractive geometric optics for bloch wave packets we study for times of order 1 varepsilon solutions of wave equations which are fancyscript o varepsilon 2 modulations of an varepsilon periodic wave equation the solutions are of slowly varying amplitude type built on bloch plane waves with wavelength of order varepsilon we construct accurate approximate solutions of the three scale wkb type the leading profile is both transported at the group velocity and dispersed by a schrodinger equation given by the quadratic approximation of the bloch dispersion relation at the plane wave a ray average hypothesis of the small divisor type guarantees stability we introduce techniques related to those developed in nonlinear geometric optics which lead to new results even on time scales t fancyscript o 1 a pair of asymptotic solutions yield accurate approximate solutions of oscillatory initial value problems the leading term yields h1 asymptotics when the envelopes are only h1

high frequency nonlinear optics from the nonlinear schrodinger approximation to ultrashort pulses equations after a brief introduction and physical motivation we show how the nonlinear schrodinger nls equation can be derived from a general class of nonlinear hyperbolic systems its purpose is to describe the behaviour of high frequency oscillating wave packets over a large time scale that requires us to take into account diffractive effects we then show that the nls approximation fails for short pulses and propose some alternative models including a modified schrodinger equation with improved frequency dispersion it turns out that these models have better properties and are quite accurate for short pulses for ultrashort pulses however they must also be abandoned for more complex approaches we give the main steps for such an analysis and explain one striking fact about ultrashort pulses their dynamics in dispersive media is linear

asymptotic of the solutions of hyperbolic equations with a skew symmetric perturbation using methods introduced by s schochet inj differential equations114 1994 476 512 we compute the first term of an asymptotic expansion of the solutions of hyperbolic equations perturbated by a skew symmetric linear operator that result is first applied to two systems describing the motion of geophysic fluids the rotating euler equations and the primitive system of the quasigeostrophic equations finally in the last part we study the slightly compressible euler equations by application of that same result

dispersive nonlinear geometric optics we construct infinitely accurate approximate solutions to systems of hyperbolic partial differential equations which model short wavelength dispersive nonlinear phenomena the principal themes are the following 1 the natural framework for the study of dispersion is wavelength e solutions of systems of partial differential operators in e the natural e characteristic equation and e eikonal equations are not homogeneous this corresponds exactly to the fact that the speeds of propagation which are called group velocities depend on the length of the wave number 2 the basic dynamic equations are expressed in terms of the operator e t as a result growth or decay tends to occur at the catastrophic rate ect e the analysis is limited to conservative or nearly conservative models 3 if a phase x e satisfies the natural e eikonal equation the natural harmonic phases n x e generally do not one needs to impose a coherence hypothesis for the harmonics 4 in typical examples the set of harmonics

a numerical model for the raman amplification for laser plasma interaction in this paper we continue the study of the raman amplification initiated in m colin t colin on a quasi linear zakharov system describing laser plasma interactions differential integral equations 17 3 4 2004 297 330 we use a dispersive quasi linear system the quasi linear part is not hyperbolic and this difficulty is overcome using the dispersion we give an asymptotic result on a reduced system we then introduce a simple robust and efficient numerical scheme on the whole system that takes into account the non hyperbolicity of the quasi linear part as well as the nonlinear saturation of the raman growth the scheme is validated thanks to the asymptotic result finally we present 1 d and 2 d simulations

the malliavin calculus a functional analytic approach in his groundbreaking articles s 91 malliavin introduced a technique for obtaining elliptic regularity results using function space calculus in his formulation the function space calculus on which his theory rests is the martingale calculus coming from the ornstein uhlenbeck process on wiener space cf 121 for a detailed discussion of malliavin s theory from this point of view although this formulation has a great deal to recommend it and is particularly pleasing to afficionados of the so called brownian sheet most analysts including those who are reasonably facile with the machinery of probability theory are unlikely to find the technical difficulties inherent in this approach outweighed by the eventual rewards shigekawa 111 discovered that there is an alternative formulation of malliavin s theory in shigekawa s formulation no mention need be made of the ornstein uhlenbeck process instead what he relies on is a sobolev type extension of the frechet dervative with wiener measure playing the role played by lebesgue measure in the finite dimensional context a third formulation of malliavin s theory was recently provided by bismut l bismut s idea is to exploit the quasi invariance of wiener measure as it is manifested in a beautiful relation due to haussmann 4 the present article has two aims in the first place section 1 2 and 3 are devoted to yet another formulation of malliavin s calculus although the end results are precisely the same as those obtained in 121 no reference is made to the ornstein uhlenbeck process on which the original theory depended instead the machinery is entirely that of elementary functional analysis and is devoid of any sophisticated martingale calculus it has been my intent that these three sections should be readily accessible to any analyst who has had some acquintance with wiener measure

justification of the nls approximation for a quasilinear water wave model abstract we show how for a quasilinear water wave model the nls approximation can be justified the model presents several new difficulties due to the quadratic terms which have to be eliminated by a normal form transformation due to the quasilinearity of the problem there is some loss of regularity associated with the normal form transformation and there is a nontrivial resonance present in the problem the loss of regularity is dealt with by using a cauchy kowalevskaya like method to treat the initial value problem and the nontrivial resonance is dealt with via a rescaling argument

hyperbolic boundary problems with large oscillatory coefficients multiple amplification abstract we study weakly stable hyperbolic boundary problems with highly oscillatory coefficients that are large o 1 compared to the small wavelength of oscillations such problems arise for example in the study of classical questions concerning the stability of mach stems and compressible vortex sheets for such applications one seeks to prove energy estimates that are in an appropriate sense uniform with respect to the small wavelength but the large oscillatory coefficients are a formidable obstacle to obtaining such estimates in this paper we analyze a simplified form of the linearized problems that are relevant to the above stability questions and obtain results that are both positive and negative on the one hand we identify favorable structural conditions under which it is possible to prove uniform estimates and then do so by a new approach we also construct examples showing that large oscillatory coefficients can give rise to an instantaneous multiple amplification of the amplitude of solutions relative to data for example oscillatory boundary data of a given amplitude o 1 can immediately give rise to a solution of amplitude o 1 k where k 2 1 we use the examples of multiple amplification to confirm the optimality of our uniform estimates when the favorable structural conditions hold when those conditions do not hold we explain how multiple amplification of infinite order may rule out useful estimates

existence of long time solutions and validity of the nonlinear schrodinger approximation for a quasilinear dispersive equation abstract we consider a nonlinear dispersive equation with a quasilinear quadratic term we establish two results first we show that solutions to this equation with initial data of order o e in sobolev norms exist for a time span of order o e 2 for sufficiently small e secondly we derive the nonlinear schrodinger nls equation as a formal approximation equation describing slow spatial and temporal modulations of the envelope of an underlying carrier wave and justify this approximation with the help of error estimates in sobolev norms between exact solutions of the quasilinear equation and the formal approximation obtained via the nls equation the proofs of both results rely on estimates of appropriate energies whose constructions are inspired by the method of normal form transforms to justify the nls approximation we have to overcome additional difficulties caused by the occurrence of resonances we expect that the method developed in the present paper will also allow to prove the validity of the nls approximation for a larger class of quasilinear dispersive systems with resonances

singular pseudodifferential operators symmetrizers and oscillatory multidimensional shocks abstract we introduce a calculus of singular pseudodifferential operators spos depending on wavelength e and use them to solve three different types of singular quasilinear hyperbolic systems such systems arise in nonlinear geometric optics and also for example in the study of incompressible limits and of nonlinear wave equations with small nonlinear terms or small data the spos act in both slow and fast variables and are singular not only because their symbols have finite regularity and depend on 1 e but also because their derivatives fail to decay in the usual way in the dual variables there is a necessarily crude calculus with large parameter e g residual operators are just bounded on l2 but the calculus admits the proof of garding inequalities and enables us to symmetrize and sometimes even diagonalize the singular systems being considered by microlocalizing simultaneously in both slow and fast variables the paper culminates in a proof of the existence of oscillatory multidimensional shocks on a fixed time interval independent of the wavelength e as e 0 the use of spos allows us to eliminate the small divisor assumptions made in earlier work and also to construct more general oscillatory solutions in which elliptic boundary layers are present on one or both sides of the shock

wavepacket preservation under nonlinear evolution we study nonlinear systems of hyperbolic pde s in mathbb r d the hyperbolicity is understood in a wider sense namely multiple roots of the characteristic equation are allowed and dispersive equations are permitted they describe wave propagation in dispersive nonlinear media such as for example electromagnetic waves in nonlinear photonic crystals the initial data is assumed to be a finite sum of wavepackets referred to as a multi wavepacket the wavepackets and the medium nonlinearity are characterized by two principal small parameters and varrho where i frac 1 beta is a factor describing spatial extension of involved wavepackets ii frac 1 varrho is a factor describing the relative magnitude of the linear part of the evolution equation compared to its nonlinearity a key element in our approach is a proper definition of a wavepacket remarkably the introduced definition has a flexibility sufficient for a wavepacket to preserve its defining properties under a general nonlinear evolution for long times in particular the corresponding wave vectors and the band numbers of involved wavepackets are conserved quantities we also prove that the evolution of a multi wavepacket is described with high accuracy by a properly constructed system of envelope equations with a universal nonlinearity the universal nonlinearity is obtained by a time averaging applied to the original nonlinearity in simpler cases the averaged system turns into a system of nonlinear schrodinger equations

higher order resonances and instability of high frequency wkb solutions abstract this paper focuses on the destabilizing role of resonances in high frequency wkb solutions specifically we study higher order resonances associated with higher order harmonics generated by nonlinearities we give examples of systems and solutions for which such resonances generate instantaneous instabilities even though the equations linearized around the leading wkb terms are initially stable meaning in particular that the key destabilizing terms are not present in the data

on the boyd kadomstev system for a three wave coupling problem and its asymptotic limit we consider the boyd kadomstev system which is in particular a model for the brillouin backscattering in laser plasma interaction it couples the propagation of two laser beams the incoming and the backscattered waves with an ion acoustic wave which propagates at a much slower speed the ratio varepsilon between the plasma sound velocity and the group velocity of light is small with typical value of order 10 3 in this paper we make a rigorous analysis of the behavior of solutions as varepsilon to 0 this problem can be cast in the general framework of fast singular limits for hyperbolic systems the main new point which is addressed in our analysis is that the singular relaxation term present in the equation is a nonlinear first order system

semiclassical and spectral analysis of oceanic waves in this work we prove that the shallow water flow subject to strong wind forcing and linearized around an adequate stationary profile develops for large times closed trajectories due to the propagation of rossby waves while poincare waves are shown to disperse the methods used in this paper involve semi classical analysis and dynamical systems for the study of rossby waves while some refined spectral analysis is required for the study of poincare waves due to the large time scale involved which is of diffractive type

the mach stem equation and amplification in strongly nonlinear geometric optics we study highly oscillating solutions to a class of weakly well posed hyperbolic initial boundary value problems weak well posedness is associated with an amplification phenomenon of oscillating waves on the boundary in the previous works cgw14 cw14 we have rigorously justified a weakly nonlinear regime for semilinear problems in that case the forcing term on the boundary has amplitude o e 2 and oscillates at a frequency o 1 e the corresponding exact solution which has been shown to exist on a time interval that is independent of e 0 1 has amplitude o e in this paper we deal with the exact same scaling namely o e 2 forcing term on the boundary and o e solution for quasilinear problems in analogy with cgm03 this corresponds to a strongly nonlinear regime and our main result proves solvability for the corresponding wkb cascade of equations which yields existence of approximate solutions on a time interval that is independent of e 0 1 existence of exact solutions close to approximate ones is a stability issue which as shown in cgm03 highly depends on the hyperbolic system and on the boundary conditions we do not address that question here this work encompasses previous formal expansions in the case of weakly stable shock waves mr83 and two dimensional compressible vortex sheets am87 in particular we prove well posedness for the leading amplitude equation the mach stem equation of mr83 and generalize its derivation to a large class of hyperbolic boundary value problems and to periodic forcing terms the latter case is solved under a crucial nonresonant assumption and a small divisor condition

modeling the coastal ocean over a time period of several weeks from a scale analysis of hydrodynamic phenomena having a significant action on the drift of an object in coastal ocean waters we deduce equations modeling the associated hydrodynamic fields over a time period of several weeks these models are essentially non linear hyperbolic systems of pde involving a small parameter then from the models we extract a simplified and nevertheless typical one for which we prove that its classical solution exists on a time interval which is independent of the small parameter we then show that the solution weak converges as the small parameter goes to zero and we characterize the equation satisfied by the weak limit

highly oscillatory waves in quasilinear hyperbolic parabolic coupled equations in this paper we study the cauchy problem for a two speed quasi linear hyperbolic parabolic coupled system in several space variables with highly oscillatory initial data and small viscosity by means of nonlinear geometric optics we derive the asymptotic expansions of oscillatory waves and deduce that the leading oscillation profiles satisfy quasilinear hyperbolic parabolic coupled equations with integral terms from which we obtain that the oscillations of the solutions to the hyperbolic parabolic equations are propagated along the characteristics of the hyperbolic operators and partial profiles of oscillations are dissipated by the parabolic effect of the system furthermore by using the energy method in weighted spaces we rigorously justify the asymptotic expansion and obtain the existence of the highly oscillatory solutions in a time interval independent of the wavelength finally we use this general result to study the behavior of oscillatory waves in the one dimensional compressible viscous f

buildings elliptic curves and the k 2 local sphere we investigate a dense subgroup of the second morava stabilizer group given by a certain group of quasi isogenies of a supersingular elliptic curve in characteristic p the group acts on the bruhat tits building for gl2 through its action on the adic tate module this action has finite stabilizers giving a small resolution for the homotopy fixed point spectrum eh 2 hgal by spectra of topological modular forms here e2 is a version of morava e theory and gal gal

supercritical geometric optics for nonlinear schrodinger equations we consider the small time semi classical limit for nonlinear schrodinger equations with defocusing smooth nonlinearity for a super cubic nonlinearity the limiting system is not directly hyperbolic due to the presence of vacuum to overcome this issue we introduce new unknown functions which are defined nonlinearly in terms of the wave function itself this approach provides a local version of the modulated energy functional introduced by y brenier the system we obtain is hyperbolic symmetric and the justification of wkb analysis follows

global solutions for the gravity water waves equation in dimension 3 we show existence of global solutions for the gravity water waves equation in dimension 3 in the case of small data the proof combines energy estimates which yield control of l 2 related norms with dispersive estimates which give decay in l 8 to obtain these dispersive estimates we use an analysis in fourier space the study of space and time resonances is then the crucial point

a deterministic model for the propagation of turbulent oscillations abstract the deterministic point of view on turbulent fluid motion is to consider the cauchy problem for equations of navier stokes type associated with large reynolds numbers and with singular initial data although the corresponding mathematical study has a lot progressed it remains limited by fundamental difficulties related to the presence of instabilities precisely the purpose of this article is to show on a realistic two dimensional model that up to some extent such instabilities can be managed this is achieved in the framework of a supercritical nonlinear geometric optics the aim is to provide a theory allowing to take into account the interaction of a large amplitude monophase oscillating wave with waves oscillating at smaller frequencies in the other direction the effect is that very complicated phenomena can occur in the inertial range including for instance the production of new scales

the validity of nonlinear geometric optics for weak solutions of conservation laws the method of weakly nonlinear geometric optics is one of the main formal perturbation techniques used in analyzing nonlinear wave motion for hyperbolic systems the tacit assumption in using such perturbation methods is that the corresponding solutions of the hyperbolic system remain smooth since shock waves typically form in such solutions these assumptions are rarely satisfied in practice nevertheless in a variety of applied contexts these methods give qualitatively reliable answers for discontinuous weak solutions here we give a rigorous proof for the validity of nonlinear geometric optics for general weak solutions of systems of hyperbolic conservation laws in a single space variable the methods of proof do not mimic the formal construction of weakly nonlinear asymptotics but instead rely on structural symmetries of the approximating equations stability estimates for intermediate asymptotic times and the rapid decay in variation of weak solutions for large asymptotic times

two asymptotic problems for a singular nonlinear schrodinger system in this paper we continue our investigation of the relation between various systems that can be derived from the klein gordon zakharov system in the high frequency and subsonic limits in this paper we start from the singular nonlinear schr odinger system which was derived in a previous work and derive the classical nonlinear schr odinger system in two different limit cases these two nonlinear schr odinger systems have different coefficients in the nonlinearity

weakly nonlinear high frequency waves abstract in this paper we derive a method for finding small amplitude high frequency solutions to hyperbolic systems of quasilinear partial differential equations our solution is the sum of two parts i a superposition of small amplitude high frequency waves ii a slowly varying mean solution each high frequency wave displays nonlinear distortion of the wave profile and shocks may form shock conditions are derived for conservative systems different high frequency waves do not interact provided the frequencies and wave numbers of two waves are not linearly related to those of a third the mean solution is found by solving a linear partial differential equation this method generalizes whitham s nonlinearization technique 9 for single waves to problems where many waves are present we obtain these results by generalizing a scheme first proposed by choquet bruhat 1 which employs the method of multiple scales author

hyperbolic conservation laws with stiff relaxation terms and entropy we study the limiting behavior of systems of hyperbolic conservation laws with stiff relaxation terms reduced systems inviscid and viscous local conservation laws and weakly nonlinear limits are derived through asymptotic expansions an entropy condition is introduced for n n systems that ensures the hyperbolicity of the reduced inviscid system the resulting characteristic speeds are shown to be interlaced with those of the original system moreover the first correction to the reduced system is shown to be dissipative a partial converse is proved for 2 2 systems this structure is then applied to study the convergence to the reduced dynamics for the 2 2 case 1994 john wiley sons inc

simultaneous kummer congruences and mathbb e _ infty e orientations of ko and tmf building on results of ando hopkins and rezk we show the existence of uncountably many mathbb e infty string orientations of real k theory ko and of topological modular forms tmf generalizing the hat a resp the witten genus furthermore the obstruction to lifting an mathbb e infty string orientations from ko to tmf is identified with a classical iwasawa theoretic condition the common key to all these results is a precise understanding of the classical kummer congruences imposed for all primes simultaneously this result is of independent arithmetic interest

realizability of algebraic galois extensions by strictly commutative ring spectra we discuss some of the basic ideas of galois theory for commutative s algebras originally formulated by john rognes we restrict our attention to the case of finite galois groups and to global galois extensions we describe parts of the general framework developed by rognes central roles are played by the notion of strong duality and a trace mapping constructed by greenlees and may in the context of generalized tate cohomology we give some examples where algebraic data on coefficient rings ensures strong topological consequences we consider the issue of passage from algebraic galois extensions to topological ones by applying obstruction theories of robinson and goerss hopkins to produce topological models for algebraic galois extensions and the necessary morphisms of commutative s algebras examples such as the complex k theory spectrum as a ko algebra indicate that more exotic phenomena occur in the topological setting we show how in certain cases topological abelian galois extensions are classified by the same harrison groups as algebraic ones and this leads to computable harrison groups for such spectra we end by proving an analogue of hilbert s theorem 90 for the units associated with a galois extension

finiteness results for modular curves of genus at least 2 a curve x over inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i is modular if it is dominated by x 1 n for some n if in addition the image of its jacobian in j 1 n is contained in the new subvariety of j 1 n then x is called a new modular curve we prove that for each g 2 the set of new modular curves over inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 02i of genus g is finite and computable for the computability result we prove an algorithmic version of the de franchis severi theorem similar finiteness results are proved for new modular curves of bounded gonality for new modular curves whose jacobian is a quotient of j 0 n new with n divisible by a prescribed prime and for modular curves new or not with levels in a restricted set we study new modular hyperelliptic curves in detail in particular we find all new modular curves of genus 2 explicitly and construct what might be the complete list of all new modular hyperelliptic curves of all genera finally we prove that for each field k of characteristic zero and g 2 the set of genus g curves over k dominated by a fermat curve is finite and computable

justification of multidimensional single phase semilinear geometric optics for semilinear strictly hyperbolic systems lu f x u we construct and justify high frequency nonlinear asymptotic expansions of the form u e x e j u j x x e lu e f x u e 0 the study of the principal term of such expansions is called nonlinear geometric optics in the applied literature we show 1 formal expansions with periodic profiles u j can be computed to all orders ii the equations for the profiles from i are solvable and iii there are solutions of the exact equations which have the formal series as high frequency asymptotic expansion

a cameron martin type quasi invariance theorem for brownian motion on a compact riemannian manifold abstract let m be a compact manifold without boundary o be a fixed base point in m g be a riemannian metric on m and be a g compatible covariant derivative on tm the tangent space to m assume the torsion t of satisfies the skew symmetry condition g t x y y 0 for all vector fields x and y on m for example take to be the levi civita covariant derivative on m g also let denote the wiener measure on w0 m c 0 1 m 0 o and let h s denote stochastic parallel translation relative to along the path w0 m up to time s 0 1 given a c1 function h 0 1 t0m it is shown that the differential equation t h t h with initial condition 0 id w m w m has a solution r maps w m w m the measurable maps from w m to w m this function is a flow on w m i e for all t r t t furthermore t has the quasi invariance property the law t of t with respect to the wiener measure is equivalent to for all t r this result is used to prove an integration by parts formula for the h derivative hf defined by h f d dt 0 f t where f is a c2 cylinder function on w m

long wave approximations for water waves in this paper we obtain new nonlinear systems describing the interaction of long water waves in both two and three dimensions these systems are symmetric and conservative rigorous convergence results are provided showing that solutions of the complete free surface euler equations tend to associated solutions of these systems as the amplitude becomes small and the wavelength large using this result as a tool a rigorous justification of all the two dimensional approximate systems recently put forward and analysed by bona chen and saut is obtained in the two dimensional context our methods also allows a significant improvement of the convergence estimate obtained by schneider and wayne in their justification of the decoupled korteweg de vries approximation of the two dimensional euler equations it also follows from our theory that coupled models provide a better description than the decoupled ones over short time scales results are obtained both on an unbounded domain for solutions that evanesce at infinity as well as for solutions that are spatially periodic

topological modular forms with level structure the cohomology theory known as mathrm tmf for topological modular forms is a universal object mapping out to elliptic cohomology theories and its coefficient ring is closely connected to the classical ring of modular forms we extend this to a functorial family of objects corresponding to elliptic curves with level structure and modular forms on them along the way we produce a natural way to restrict to the cusps providing multiplicative maps from mathrm tmf with level structure to forms of k theory in particular this allows us to construct a connective spectrum mathrm tmf 0 3 consistent with properties suggested by mahowald and rezk this is accomplished using the machinery of logarithmic structures we construct a presheaf of locally even periodic elliptic cohomology theories equipped with highly structured multiplication on the log etale site of the moduli of elliptic curves evaluating this presheaf on modular curves produces mathrm tmf with level structure

justification and failure of the nonlinear schrodinger equation in case of non trivial quadratic resonances abstract the nonlinear schrodinger nls equation can be derived as an amplitude equation describing slow modulations in time and space of an underlying spatially and temporarily oscillating wave packet the purpose of this paper is to prove estimates between the formal approximation obtained via the nls equation and true solutions of the original system in case of non trivial quadratic resonances it turns out that the approximation property app holds if the approximation is stable in the system for the three wave interaction twi associated to the resonance we construct a counterexample showing that the nls equation can fail to approximate the original system if instability occurs for the approximation in the twi system in the unstable case we give some arguments why the validity of the app can be expected for spatially localized solutions and why it cannot be expected for non localized solutions although we restrict ourselves to a nonlinear wave equation as original system we believe that the results hold in more general situations too

the cauchy problem for semilinear hyperbolic systems with discontinuous data on demontre l existence a t court des solutions pour le probleme de cauchy strictement hyperbolique lu t u l 1 n a l t x xt u f t x u t x u t 0 g ou g est lisse par morceaux avec des sauts seulement sur une hypersurface t r n

interaction of nonlinear progressing waves for semilinear wave equations on etudie l interaction transversale de 3 ondes planes progressives pour une equation d onde semi lineaire a 3 dimensions

global wellposedness of the 3 d full water wave problem we consider the problem of global in time existence and uniqueness of solutions of the 3 d infinite depth full water wave problem in the setting that the interface tends to the horizontal plane the velocity and acceleration on the interface tend to zero at spatial infinity we show that the nature of the nonlinearity of the water wave equation is essentially of cubic and higher orders for any initial interface that is sufficiently small in its steepness and velocity we show that there exists a unique smooth solution of the full water wave problem for all time and the solution decays at the rate 1 t

bilinear dispersive estimates via space time resonances dimensions two and three consider a bilinear interaction between two linear dispersive waves with a generic resonant structure roughly speaking space and time resonant sets intersect transversally we derive an asymptotic equivalent of the solution for data in the schwartz class and bilinear dispersive estimates for data in weighted lebesgue spaces an application to water waves with infinite depth gravity and surface tension is also presented

propagation of oscillations in real vanishing viscosity limit in this paper we study viscous perturbations of quasilinear hyperbolic systems in several space dimensions the equations involve a singular parameter which goes to zero they arise in realistic models of compressible flow the large scale motions in the atmosphere 12 they come also from theoretical considerations in non linear geometric optics 4 5 we prove that solutions u exist on a domain of space time independent on open image in new window

k 1 local topological modular forms we construct the witten orientation of the topological modular forms spectrum tmf in the k 1 local setting by attaching e cells to the bordism theory mo we also identify the ko homology of tmf with the ring of divided congruences

almost global wellposedness of the 2 d full water wave problem we consider the problem of global in time existence and uniqueness of solutions of the 2 d infinite depth full water wave equation it is known that this equation has a solution for a time period 0 t e for initial data of the form e where t depends only on in this paper we show that for such data there exists a unique solution for a time period 0 e t e this is achieved by better understandings of the nature of the nonlinearity of the full water wave equation

long wave short wave resonance for nonlinear geometric optics the aim of this paper is to study oscillatory solutions of nonlinear hyperbolic systems in the framework developped during the last decade by joly metivier and rauch here we mainly focus on rectification effects that is the interaction of oscillations with a mean field created by the nonlinearity a real interaction can occur only under some geometric conditions described in jmr1 and l1 that are generally not satisfied by the physical models except in the 1 d case we introduce here a new type of ansatz that allows us to obtain rectification effects under weaker asumptions we obtain a new class of profile equations and construct solutions for a subclass finally the stability of the asymptotic expansion is proved in the context of maxwell bloch type systems

large time existence for 3d water waves and asymptotics we rigorously justify in 3d the main asymptotic models used in coastal oceanography including shallow water equations boussinesq systems kadomtsev petviashvili kp approximation green naghdi equations serre approximation and full dispersion model we first introduce a variable nondimensionalized version of the water waves equations which vary from shallow to deep water and which involves four dimensionless parameters using a nonlocal energy adapted to the equations we can prove a well posedness theorem uniformly with respect to all the parameters its validity ranges therefore from shallow to deep water from small to large surface and bottom variations and from fully to weakly transverse waves

a time splitting spectral scheme for the maxwell dirac system we present a time splitting spectral scheme for the maxwell dirac system and similar time splitting methods for the corresponding asymptotic problems in the semi classical and the non relativistic regimes the scheme for the maxwell dirac system conserves the lorentz gauge condition is unconditionally stable and highly efficient as our numerical examples show in particular we focus in our examples on the creation of positronic modes in the semi classical regime and on the electron positron interaction in the non relativistic regime furthermore in the non relativistic regime our numerical method exhibits uniform convergence in the small parameter d which is the ratio of the characteristic speed and the speed of light

counter examples to concentration cancellation we study the existence and the asymptotic behavior of large amplitude high frequency oscillating waves subjected to the two dimensional burger equation this program is achieved by developing tools related to supercritical wkb analysis by selecting solutions which are divergence free we show that incompressible or compressible two dimensional euler equations are not locally closed for the weak l2 topology

resonant one dimensional nonlinear geometric optics abstract in this paper we study the existence and resonant interaction of oscillatory wave trains in one space dimension giving a rigorous proof of the validity of the corresponding expansions of weakly nonlinear optics we consider both semilinear and quasilinear systems the latter before shock formation some important features of the study are the following 1 we prove the existence of families of exact solutions which have asymptotic expansions governed by weakly nonlinear optics equations with variable coefficients nonconstant background fields and nonlinear phases are permitted our weak transversality hypotheses allow us to justify expansions where even a formal theory did not exist before 2 we make a detailed study of resonances the geometry associated with such resonances is related to the theory of planar webs 3 we study the smoothness of the profiles their regularity is ruled by a sum law analogous to that describing the propagation of singularities in one dimension 4 the expansions are justified up to the breakdown of the profiles which coincides with a suitably defined breakdown for exact solutions

derivation of the planetary geostrophic equations in this paper we justify mathematically the derivation of the planetary geostrophic equations pge from the hydrostatic boussinesq equations with coriolis force usually named the primitive equations pe the planetary geostrophic equations which are a classical model of thermohaline circulation are obtained from the primitive equations as the froude number fr the rossby number varepsilon and the burger number bu go to 0 these numbers are supposed to satisfy fr o varepsilon 1 2 and bu o varepsilon which is relevant to the thermohaline planetary dynamics the analysis performed here does not follow the same lines as previous asymptotic studies on rotating fluids it involves a singular operator which is not skew symmetric and prevents classical energy estimates to handle such operator requires to put the primitive equations under normal form together with an appropriate use of the viscous terms

energy convergence for singular limits of zakharov type systems we prove existence and uniqueness of solutions to the klein gordon zakharov system in the energy space h1 l2 on some time interval which is uniform with respect to two large parameters c and these two parameters correspond to the plasma frequency and the sound speed in the simultaneous high frequency and subsonic limit we recover the nonlinear schrodinger system at the limit we are also able to say more when we take the limits separately

derivation of the zakharov equations this article continues the study initiated in 27 7 of the validity of the zakharov model which describes langmuir turbulence we give an existence theorem for a class of singular quasilinear equations this theorem is valid for prepared initial data we apply this result to the euler maxwell equations which describes laser plasma interactions to obtain in a high frequency limit an asymptotic estimate that describes solutions of the euler maxwell equations in terms of wkb approximate solutions the leading terms of which are solutions of the zakharov equations due to the transparency properties of the euler maxwell equations evidenced in 27 this study is carried out in a supercritical highly nonlinear regime in such a regime resonances between plasma waves electromagnetric waves and acoustic waves could create instabilities in small time the key of this work is the control of these resonances the proof involves the techniques of geometric optics of joly metivier and rauch 12 13 recent results by lannes on norms of pseudodifferential operators 14 and a semiclassical paradifferential calculus

orbifold genera product formulas and power operations we generalize the definition of orbifold elliptic genus and introduce orbifold genera of chromatic level h using h tuples rather than pairs of commuting elements we show that our genera are in fact orbifold invariants and we prove integrality results for them if the genus arises from an h8 map into the morava lubin tate theory eh then we give a formula expressing the orbifold genus of the symmetric powers of a stably almost complex manifold m in terms of the genus of m itself our formula is the p typical analogue of the dijkgraaf moore verlinde verlinde formula for the orbifold elliptic genus r dijkgraaf et al elliptic genera of symmetric products and second quantized strings comm math phys 185 1 1997 197 209 it depends only on h and not on the genus

the long wave limit for the water wave problem i the case of zero surface tension the korteweg de vries equation boussinesq equation and many other equations can be formally derived as approximate equations for the two dimensional water wave problem in the limit of long waves here we consider the classical problem concerning the validity of these equations for the water wave problem in an infinitely long canal without surface tension we prove that the solutions of the water wave problem in the long wave limit split up into two wave packets one moving to the right and one to the left where each of these wave packets evolves independently as a solution of a korteweg de vries equation our result allows us to describe the nonlinear interaction of solitary waves 2000 john wiley sons inc

the sigma orientation is an h map in an earlier paper the authors constructed a natural map called the sigma orientation from the thom spectrum mu 6 to any elliptic spectrum mu 6 is an h ring spectrum and in this paper we show that if e c t is the elliptic spectrum associated to the universal deformation of a supersingular elliptic curve over a perfect field of characteristic p 0 then the sigma orientation is a map of h ring spectra

power operations in elliptic cohomology and representations of loop groups the first part describes power operations in elliptic cohomology in terms of isogenies of the underlying elliptic curve the second part discusses a relationship between equivariant elliptic cohomology and representations of loop groups the third part investigates the representation theoretic considerations which give rise to the power operations discussed in the first part

multifrequency nls scaling for a model equation of gravity capillary waves this paper is the first in a series papers devoted to the study of the rigorous derivation of the nonlinear schrodinger nls equation as well as other related systems starting from a model coming from the gravity capillary water wave system in the long wave limit our main goal is to understand resonances and their effects on having the nonlinear schrodinger approximation or modification of it or having other models to describe the limit equation in this first paper our goal is not to derive nls but to allow the presence of an arbitrary sequence of frequencies around which we have a modulation and prove local existence on a uniform time this yields a new class of large data for which we have a large time of existence 2012 wiley periodicals inc

justification of the 2d nls equation for a fourth order nonlinear wave equation quadratic resonances do not matter much in case of analytic initial conditions abstract the 2d nonlinear schrodinger nls equation can be derived by multiple scaling analysis in order to describe slow temporal and spatial modulations of an oscillating wave packet in general dispersive wave systems it is the purpose of this paper to establish estimates for the difference between the so obtained approximation and true solutions of the original system in case that the original system possesses nontrivial quadratic resonances we explain for a fourth order nonlinear wave equation that quadratic resonances do not matter much in case of analytic initial conditions for the nls equation in the sense that independently of whether the resonance is stable or unstable an approximation property can be established on the natural time scale of the nls equation if the set of resonant wave numbers is bounded away from the integer multiples of the basic wave number of the underlying wave packet finally we explain that in case of an approximation by a so called four wave interaction fwi system the same approach works and an even stronger result can be established

renormalization group method applied to the primitive equations abstract in this article we study the limit as the rossby number e goes to zero of the primitive equations of the atmosphere and the ocean from the mathematical viewpoint we study the averaging of a penalization problem displaying oscillations generated by an antisymmetric operator and by the presence of two time scales