justification of the nonlinear schrodinger approximation for a quasilinear klein gordon equation we consider a nonlinear klein gordon equation with a quasilinear quadratic term the nonlinear schrodinger nls equation can be derived as a formal approximation equation describing the evolution of the envelopes of slowly modulated spatially and temporarily oscillating wave packet like solutions to the quasilinear klein gordon equation it is the purpose of this paper to present a method that allows one to prove error estimates in sobolev norms between exact solutions of the quasilinear klein gordon equation and the formal approximation obtained via the nls equation the paper contains the first validity proof of the nls approximation of a nonlinear hyperbolic equation with a quasilinear quadratic term by error estimates in sobolev spaces we expect that the method developed in the present paper will allow an answer to the relevant question of the validity of the nls approximation for other quasilinear hyperbolic systems

reflection about k jets and holomorphic extension of cr mappings abstract let f m m 1 be a cr mapping between real analytic generic submanifolds m m 1 of c n and c n 1 respectively according to webster s theory proc amer math soc 86 1982 236 240 and its further developments f has holomorphic extension to a full neighborhood of m in c n when the following requirements are fulfilled f extends to a wedge w continuous up to m f is of class c k f t c m t c m 1 where t c denotes the complex tangent bundle m 1 is k nondegenerate we deal here with the case where f t c m is strictly smaller than t c m 1 but is still real analytic in suitable sense we show that a suitably refined condition of k nondegeneracy still entails holomorphic extension of f

normal forms and quadratic nonlinear klein gordon equations on construit une transformation qui change l equation a non linearite cubique que l on sait resoudre a 3 1 dimensions on demontre que la solution du probleme original a le meme comportement asymptotique que l equation de klein gordon non lineaire

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

approximation of the korteweg de vries equation by the nonlinear schrodinger equation abstract in 1968 zakharov derived the nonlinear schrodinger equation as an approximation equation for the water wave problem it is still an open problem if there are really solutions of the water wave problem which can be approximated by the solutions of the nonlinear schrodinger equation in answering this question we made some progress for a model problem the approximation of the korteweg de vries equation by the nonlinear schrodinger equation

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

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

the validity of the kdv approximation in case of resonances arising from periodic media abstract it is the purpose of this short note to discuss some aspects of the validity question concerning the korteweg de vries kdv approximation for periodic media for a homogeneous model possessing the same resonance structure as it arises in periodic media we prove the validity of the kdv approximation with the help of energy estimates

on the estimation of the structural index from low pass filtered magnetic data abstractthe estimation of the structural index and of the depth to the source is the main task of many popular methods used to analyze potential field data such as euler deconvolution however these estimates are unstable even in the presence of a weak amount of noise and euler deconvolution of noisy data leads to an underestimation of structural index and depth we have studied how the structural index and depth estimates are affected by applying low pass filtering to the data physically based low pass filters such as upward continuation and integration have been shown to be the best choice over a range of altitudes upward continuation or orders integration filters mainly because their outputs have a well defined physical meaning in contrast mathematical low pass filters require that the filter parameters be tuned carefully by means of several trial tests to produce optimally smoothed fields the c norm criterion is a reliable strategy to produce a stabilized vertical derivative and we disc

interaction of modulated gravity water waves of finite depth abstract we consider the capillary gravity water wave problem of finite depth with a flat bottom of one or two horizontal dimensions we derive the modulation equations of leading and next to leading order in the hyperbolic scaling for three weakly amplitude modulated plane wave solutions of the linearized problem in the absence of quadratic and cubic resonances we justify the derived system of macroscopic equations in the case of gravity waves using the stability of the finite depth water wave problem on the time scale o 1

depth estimation from downward continuation an entropy based approach to normalized full gradient the normalized full gradient is based on the noticeable stability of the modulus of the analytic signal of downward potential fields we propose a new approach to the study of the normalized full gradient based on assessing the entropy of the normalized modulus of the analytic signal at each level of continuation the increased disorder of progressively downward continued fields implies an increase of the computed entropy however a local decrease of the entropy is expected at the source level where the field gets singular as entropy decreases when the information is concentrated thus our method is based on a simple search for a minimum of the computed entropy vs depth curve and the estimated depth will be that at which the minimum is attained the method is sensitive to interference and other types of noise and specific strategies to deal with these limitations are defined and tested on synthetic data the depth estimate is obtained without the assumption of a specific source shape the depth co

global well posedness of 2d non focusing schrodinger equations via rigorous modulation approximation abstract we consider the long time well posedness of the cauchy problem with large sobolev data for a class of nonlinear schrodinger equations nls on r 2 with power nonlinearities of arbitrary odd degree specifically the method in this paper applies to those nls equations having either elliptic signature with a defocusing nonlinearity or else having an indefinite signature by rigorously justifying that these equations govern the modulation of wave packet like solutions to an artificially constructed equation with an advantageous structure we show that a priori every subcritical inhomogeneous sobolev norm of the solution increases at most polynomially in time global well posedness follows by a standard application of the subcritical local theory

a rigorous justification of the matthews cox approximation for the nikolaevskiy equation abstract the nikolaevskiy equation is an example of a pattern forming system with marginally stable long modes it has the unusual property that the typical ginzburg landau scaling ansatz for the description of propagating patterns does not yield asymptotically consistent amplitude equations instead another scaling proposed by matthews and cox can be used to formally derive a consistent system of modulation equations we give a rigorous proof that this system makes correct predictions about the dynamics of the nikolaevskiy equation

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

maps from the two ball to the three ball in 4 webster proved the following theorem let b z e c 7 2 3 is a proper holomorphic map which is c a up to the boundary of b then f is linear fractional the purpose of this paper is to extend this result to the case n 2 however the theorem is false when n 2 f need not be linear fractional so instead of trying to prove this false theorem we shall instead try to make a list of maps f bz b a which are c 3 up to the boundary of course given such an f we can get another such by composing f with an automorphism of b 2 and an automorphism of b a so we make the

mappings of real algebraic hypersurfaces a holomorphic function h z defined in a neighborhood of a point z0 e cn is algebraic if there are polynomials qj z j 0 k not all identically zero such that qk z h z k qo z 0 a holomorphic mapping is algebraic if all its components are algebraic a smooth real hypersurface m in cn is algebraic if it is contained in the zero set of a nontrivial real valued polynomial in z and z we assume throughout this paper that n 1 in wi webster proved the following celebrated result if m and m are algebraic hypersurfaces in cn with nondegenerate levi forms and if h is a biholomorphism defined in an open neighborhood of m and mapping m into m then h is algebraic previously it had long been known that if m and m are open subsets of spheres then the components of h are in fact rational functions poincare p tanaka t in this paper we go a step further and give a complete characterization of algebraic hypersurfaces in cn such that any holomorphic mapping with nonvanishing jacobian determinant between two such hypersurfaces must be algebraic our main result is stated in theorem 1 below n by a germ at p0 of a holomorphic vector field in c we shall mean a

lie group structures on automorphism groups of real analytic cr manifolds given any real analytic cr manifold m we provide general conditions on m guaranteeing that the group of all its global real analytic cr automorphisms aut cr m is a lie group in an appropriate topology in particular we obtain a lie group structure for aut cr m when m is an arbitrary compact real analytic hypersurface embedded in some stein manifold

real hypersurfaces orders of contact and applications a fundamental problem in the modem theory of several complex variables concerns the boundary behavior of the cauchy riemann equations suppose that d is an open domain in cn and that its boundary m is a smooth real submanifold of cn how does the geometry of m influence the function theory on d one approach to this question is the a neumann problem 6 there one studies the inhomogeneous cauchy riemann equations au a where a is a 0 q form with distribution coefficients on the closed domain d u m satisfying the necessary compatibility condition a a 0 the a neumann problem constructs a particular solution u which is orthogonal to the null space of the complex laplacian kohn 10a has solved this problem and has shown that good local regularity properties for u in terms of a follow from so called subelliptic estimates it is natural to seek geometric conditions on m that are necessary and sufficient for these estimates in this paper we will be concerned with a certain geometric condition on m that we feel is the right one for subellipticity although the motivation for the questions discussed here comes from partial differential equations the techniques come from algebraic geometry to see why we state theorems of kohn greiner and catlin concerning subellipticity kohn 1oa has discovered a sufficient condition for a subelliptic estimate near a fixed point p in m in case d is pseudoconvex and is contained in c2 this condition is equivalent to the finiteness of order of tangency of every complex analytic one dimensional manifold with m at p greiner 13 established the necessity of this condition in

on the third gap for proper holomorphic maps between balls let f be a proper rational map from the complex ball mathbb b n into mathbb b n with n 7 and 3n 1 le n le 4n 7 then f is equivalent to a map g 0 dots 0 where g is a proper holomorphic map from mathbb b n into mathbb b 3n

the validity of phase diffusion equations and of cahn hilliard equations for the modulation of pattern in reaction diffusion systems abstract so called phase diffusion equations and cahn hilliard equations can be derived via multiple scaling analysis in order to describe slow modulations in time and space of stable and slightly unstable spatially periodic pattern it is the purpose of this paper to explain the extent to which these formal approximations are valid by proving estimates between the formal approximations and true solutions of the original system the proof is given for an abstract reaction diffusion system as original system

justification of the nonlinear schrodinger equation for the evolution of gravity driven 2d surface water waves in a canal of finite depth in 1968 v e zakharov derived the nonlinear schrodinger equation for the two dimensional water wave problem in the absence of surface tension that is for the evolution of gravity driven surface water waves in order to describe slow temporal and spatial modulations of a spatially and temporarily oscillating wave packet in this paper we give a rigorous proof that the wave packets in the two dimensional water wave problem in a canal of finite depth can be approximated over a physically relevant timespan by solutions of the nonlinear schrodinger equation

multiridge analysis of potential fields geometric method and reduced euler deconvolution a new method based on 3d multiridge analysis of potential fields assumes a 3d subset in the harmonic region and studies the behavior of potential field ridges which are built by joining extreme points of the analyzed field computed at different altitudes three types of ridges are formed by searching for the zeros of the first horizontal and first vertical derivatives of the potential field types i and ii respectively and the zeros of the potential field itself type iii this method uses a redundant set of ridges called a multiridge set to determine source type and location for homogeneous potential fields generated by simple sources all of the ridges are straight lines converging to the source position this method analyzes the multiridges by using a geometric criterion to find the source position at the intersection of the multiridge set and by solving the three reduced euler equations associated with ridge types i ii and iii the reduced euler type i and ii equations are used to obtain the structural index and the vertical and horizontal source positions equation type iii estimates the horizontal and vertical source positions tests on synthetic as well as the bishop model field yield good results even with noise corrupted data results obtained using magnetic data collected over the wreck of a military ship in the tyrrhenian sea successfully determine its vertical and horizontal positions and the structural index

extending proper holomorphic mappings of positive codimension in this paper we obtain results on holomorphic continuation of proper holomorphic mappings between pseudoconvex domains with real analytic boundaries in complex spaces of different dimensions equivalently we obtain results concerning the analyticity of cauchy riemann mappings between real analytic pseudoconvex hypersurfaces in complex spaces of different dimensions to begin with we recall the corresponding results for mappings of equidimensional domains let d and d be bounded pseudoconvex domains with smooth boundaries in c if the boundaries of d and d are strictly pseudoconvex or more generally of finite type in the sense of d angelo 15 then every proper holomorphic map of d onto d extends smoothly t o according to the results of bell and catlin 7 8 and diederich and forn ess 19 for mappings between strictly pseudoconvex domains this was proved by fefferman 26 and nirenberg webster and yang 37 if the boundaries of the pseudoconvex domains d d c c are real analytic then every proper holomorphic mapping of d onto d extends holomorphically to a neighborhood of 3 according to baouendi and rothschild 2 and diederich and forn ess 21 this reflection principle was first discovered by lewy 34 and pin6uk 38 for mappings between strictly pseudoconvex domains in the case of biholomorphic mappings between weakly pseudoconvex domains with realanalytic boundaries the result follows from the work of baouendi jacobowitz and treves 5 results in this direction were obtained in recent years by several authors see the papers 4 6 18 22 33 47 49 in this paper we are treating the case when the domains d and d have different dimensions to be specific we assume that d e c and d c c n are bounded pseudoconvex domains with real analytic boundaries and n n l in this situation a proper holomorphic map f d d need not be regular at the boundary for instance there exist proper holomorphic maps of balls of different dimensions

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

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

the validity of modulation equations for extended systems with cubic nonlinearities modulation equations play an essential role in the understanding of complicated systems near the threshold of instability here we show that the modulation equation dominates the dynamics of the full problem locally at least over a long time scale for systems with no quadratic interaction term we develop a method which is much simpler than previous ones it involves a careful bookkeeping of errors and an estimate of gronwall type as an example for the dissipative case we find that the ginzburg landau equation is the modulation equation for the swift hohenberg problem moreover the method also enables us to handle hyperbolic problems the nonlinear schrodinger equation is shown to describe the modulation of wave packets in the sine gordon equation

determination of formal cr mappings by a finite jet abstract we prove that if m is a connected real analytic holomorphically nondegenerate hypersurface in c n 1 then for any point p m there exists an integer k such that any two germs at p of local biholomorphic mappings h 1 h 2 c n 1 p c n 1 p that send m into itself and whose k jets agree at p are identical the above is a special case of a more general theorem stated for formal hypersurfaces that gives a finite jet determination result for the class of formal mappings whose jacobian determinant does not vanish identically