describing 1 d intensity transitions with gaussian derivatives at the resolutions matching the transition widths a full description of image edges requires a complete characterization of their local intensity transitions the spatial structure of those transitions and a description of adjacent image regions the authors propose as a step toward this end a 1 d algorithm for describing local intensity transitions by their gaussian derivatives at a resolution where the support of the gaussian smoothing matches their widths blur the algorithm estimates the transition width from the second derivative of 1 d gaussian response zero crossing slope and leads to a characterization of the transition with its first three derivatives at the resolution matching the width the authors describe how this algorithm can be applied to images and give an example

almost sure well posedness of the cubic nonlinear schrodinger equation below l 2 mathbb t we consider the cauchy problem for the one dimensional periodic cubic non linear schrodinger equation nls with initial data below l 2 in particular we exhibit nonlinear smoothing when the initial data are randomized then we prove local well posedness of nls almost surely for the initial data in the support of the canonical gaussian measures on h s t for each s 1 and global well posedness for each s 1 12

a generalized depth estimation algorithm with a single image a depth estimation algorithm proposed by a p pentland 1987 is generalized in the proposed algorithm the raw image data in the vicinity of the edge is used to estimate the depth from defocus since no differentiation operation on the image data is required before the optimization process the method is less sensitive to the noise disturbance of measurements furthermore the edge orientation that was critical in pentland s approach will not be required in the case this algorithm is then applied to synthetic images containing various amounts of noise to test its performance experimental results indicate that the depth estimation errors are kept within 5 of true values on the average when it is applied to real images

well posedness for the schrodinger korteweg de vries system we study well posedness of the cauchy problem associated to the schrodinger korteweg de vries system we obtain local well posedness for weak initial data where the best result obtained is for data in the sobolev space l 2 r x h 3 4 this result implies in particular the global well posedness in the energy space h 1 r x h 1 r both results considerably improve the previous ones by bekiranov ogawa and ponce 1997 guo and miao 1999 and tsutsumi 1993

global well posedness and scattering for the defocusing l 2 critical nonlinear schrodinger equation when d 2 in this article we prove that the defocusing cubic nonlinear schrodinger initial value problem is globally well posed and scattering for u0 l2 r2 the proof uses the bilinear estimates of planchon and vega and a frequency localized interaction morawetz estimate similar to the high frequency estimate of colliander keel staffilani takaoka and tao and especially the low frequency estimate of dodson

the asymptotic optimal frequency domain filter for edge detection in an earlier paper by shanmugam dickey and green an edge detection filter was derived which maximized the energy within a specified interval about an edge feature the initial expression of this filter involved a prolate spheroidal wave function however a careful analysis of the application of an asymptotic approximation to this function uncovered a major dimensional error the corrected derivation of the asymptotic optimal filter forms the subject of this paper to verify the results the filter found is compared to a similar filter developed independently by marr and hildreth

the holder continuity of the solution map to the b family equation in weak topology we prove that the solution map of the b family equation is holder continuous as a map from a bounded set of h s mathbb r s frac 3 2 with h r mathbb r 0 le r 0 moreover we show that the obtained exponent of the holder continuity is optimal when s 1 r s

multilinear estimates for periodic kdv equations and applications we prove an endpoint multilinear estimate for the xs b spaces associated to the periodic airy equation as a consequence we obtain sharp local well posedness results for periodic generalized kdv equations as well as some global well posedness results below the energy norm

the initial value problem for a novikov system it is shown that the initial value problem for an integrable novikov system is well posed in sobolev spaces hs s 3 2 in the sense of hadamard furthermore it is proved that the dependence on initial data is sharp i e the data to solution map is continuous but not uniformly continuous also peakon traveling wave solutions are used to prove that the solution map is not uniformly continuous in hs for s 3 2

non uniform dependence on initial data for the camassa holm equation in besov spaces abstract in the paper we consider the initial value problem to the camassa holm equation in the real line case based on the local well posedness result and the lifespan we proved that the data to solution map of this problem is not uniformly continuous in nonhomogeneous besov spaces in the sense of hadamard this result improves considerably the previous work given by himonas kenig 23

a strong type 2 2 estimate for a maximal operator associated to the schrodinger equation soit t f x sup t 0 t t f x ou t t f 1 exp it 2 f 1 4 on montre que etant donne un intervalle fini i i t f 2 dx c i r f x 2 dx

a priori bounds for kdv equation below h 34 abstract we consider the korteweg de vries equation kdv on the real line and prove that the smooth solutions satisfy a priori local in time h s bound in terms of the h s size of the initial data for s 4 5

discrete conservation laws and the convergence of long time simulations of the mkdv equation pseudospectral collocation methods and finite difference methods have been used for approximating an important family of soliton like solutions of the mkdv equation these solutions present a structural instability which make difficult to approximate their evolution in long time intervals with enough accuracy the standard numerical methods do not guarantee the convergence to the proper solution of the initial value problem and often fail by approaching solutions associated to different initial conditions in this frame the numerical schemes that preserve the discrete invariants related to some conservation laws of this equation produce better results than the methods which only take care of a high consistency order pseudospectral spatial discretization appear as the most robust of the numerical methods but finite difference schemes are useful in order to analyze the rule played by the conservation of the invariants in the convergence

global well posedness of the derivative nonlinear schrodinger equation with periodic boundary condition in h12 abstract we establish the global well posedness of the derivative nonlinear schrodinger equation with periodic boundary condition in the sobolev space h 1 2 provided that the mass of initial data is less than 4 this result matches the one by miao wu and xu and its recent mass threshold improvement by guo and wu in the non periodic setting below h 1 2 we show that the uniform continuity of the solution map on bounded subsets of h s does not hold for any gauge equivalent equation

self similar planar curves related to modified korteweg de vries equation we exhibit a time reversible geometric flow of planar curves which can develop singularities in finite time within the uniform topology the example is based on the construction of selfsimilar solutions of modified korteweg de vries equation of a given small mean

stability of periodic traveling waves for complex modified korteweg de vries equation abstract we study the existence and stability of periodic traveling wave solutions for complex modified korteweg de vries equation we also discuss the problem of uniform continuity of the data solution mapping

on the cauchy problem of fourth order nonlinear schrodinger equations abstract in this paper we study the cauchy problem of the fourth order schrodinger equation i t u a u b 2 u u p for dimension 4 where p is an integer greater than 1 with initial data in besov spaces we prove that for any 4 p 2 1 4 n p 4 n q the cauchy problem of this equation is locally well posed in b 2 q s p r n and b 2 q s r n where s p n 2 4 p 1 and s s p and for any 1 q almost global well posedness holds in these spaces if initial data are small we also prove that if a 0 then global well posedness holds in these spaces for small initial data

non uniqueness for the fokas olver rosenau qiao equation abstract for the fokas olver rosenau qiao equation forq on both the line and the circle it is proved that there exist initial data in the sobolev spaces h s with s 3 2 which have nonunique solutions this phenomenon is demonstrated by constructing a 2 peakon solution with an asymmetric peakon antipeakon initial profile the peakon and antipeakon eventually collide and the solution at the collision time coincides with a second solitary peakon solution using the time reversibility of the equation this scenario proves nonuniqueness

autonomous ovsyannikov theorem and applications to nonlocal evolution equations and systems abstract this work presents an ovsyannikov type theorem for an autonomous abstract cauchy problem in a scale of decreasing banach spaces which in addition to existence and uniqueness of solution provides an estimate about the analytic lifespan of the solution then using this theorem it studies the cauchy problem for camassa holm type equations and systems with initial data in spaces of analytic functions on both the circle and the line which is the main goal of this paper finally it studies the continuity of the data to solution map in spaces of analytic functions

global attractor for weakly damped nonlinear schrodinger equations in l2 r abstract we prove that the weakly damped nonlinear schrodinger flow in l 2 r provides a dynamical system which possesses a global attractor the proof relies on the continuity of the schrodinger flow for the weak topology in l 2 r

a priori bounds and weak solutions for the nonlinear schrodinger equation in sobolev spaces of negative order solutions to the cauchy problem for the one dimensional cubic nonlinear schrodinger equation on the real line are studied in sobolev spaces hs for s negative but close to 0 for smooth solutions there is an a priori upper bound for the hs norm of the solution in terms of the hs norm of the datum for arbitrarily large data for sufficiently short time weak solutions are constructed for arbitrary initial data in hs

an improved local well posedness result for the one dimensional zakharov system abstract the 1d cauchy problem for the zakharov system is shown to be locally well posed for low regularity schrodinger data u 0 h k p and wave data n 0 n 1 h l p h l 1 p under certain assumptions on the parameters k l and 1 p 2 where u 0 h k p k u 0 l p generalizing the results for p 2 by ginibre tsutsumi and velo especially we are able to improve the results from the scaling point of view and also allow suitable k 0 l 1 2 i e data u 0 l 2 and n 0 n 1 h 1 2 h 3 2 which was excluded in the case p 2

non uniform dependence on initial data of solutions to the euler equations of hydrodynamics we show that continuous dependence on initial data of solutions to the euler equations of incompressible hydrodynamics is optimal more precisely we prove that the data to solution map is not uniformly continuous in sobolev h s topology for any s in mathbb r if the domain is the flat torus mathbb t n mathbb r n 2 pi mathbb z n and for any s 0 if the domain is the whole space mathbb r n

on the cauchy problem for a coupled system of third order nonlinear schrodinger equations abstract we investigate some well posedness issues for the initial value problem ivp associated with the system 2 i t u q x 2 u i x 3 u f 1 u w 2 i t w q x 2 w i x 3 w f 2 u w where f 1 and f 2 are polynomials of degree 3 involving u w and their derivatives this system describes the dynamics of two nonlinear short optical pulse envelopes u x t and w x t in fibers porsezian et al 1994 1 and hasegawa kodama 1987 2 we prove sharp local well posedness result for the ivp with data in sobolev spaces h s r h s r s 1 4 and global well posedness result with data in sobolev spaces h s r h s r s 3 5

on the location error of curved edges in low pass filtered 2 d and 3 d images the authors study the location error of curved edges in two and three dimensional images after analog and digital low pass filtering the zero crossing of a second derivative filter is a well known edge localization criterion the second derivative in gradient direction sdgd produces a predictable bias in edge location towards the centers of curvature while the linear laplace filter produces a shift in the opposite direction their sum called plus plus laplace sdgd leads to an edge detector that finds curved edges one order more accurately than its constituents this argument holds irrespective of the dimension the influence of commonly used low pass filters such as the psf originating from diffraction limited optics using incoherent light 2 d the gaussian filter with variable cutoff point d d and the isotropic uniform filter d d is studied

on the maximal bochner riesz conjecture in the plane for p 2 we give a new estimate on the maximal bochner riesz operator in the plane for p 2 as a corollary we obtain an almost everywhere convergence result for certain bochner riesz means this work was inspired by discussions with michael christ and chris sogge

on the classification of image regions by colour texture and shape abstract a unified approach to how colour texture and shape can be encoded in a single spatio chromatic feature space is developed in particular it is shown how such features can be extracted using multi scaled filtering and correlation methods which capture the variations of colour over space in ways which encode important image features not extracted by techniques which separate colour texture and shape into separate channels finally it is shown how such spatio chromatic features can be used in difficult classification problems

low regularity solutions for a 2d quadratic nonlinear schrodinger equation we establish that the initial value problem for the quadratic nonlinear schrodinger equation iu t u u 2 where u r 2 x r c is locally well posed in h s r 2 when s 1 the critical exponent for this problem is s c 1 and previous work by colliander delort kenig and staffilani 2001 established local well posedness for s 3 4

a higher dispersion kdv equation on the line abstract the cauchy problem for a korteweg devries equation with dispersion of order m 2 j 1 where j is a positive integer kdvm is studied with data in sobolev and analytic spaces first optimal bilinear estimates in bourgain spaces are proved and using them well posedness in sobolev spaces h s s j 1 4 is established then well posedness in analytic gevrey spaces g s 0 is proved by using an analytic version of the bilinear estimates this implies that the uniform radius of analyticity persist for some time for the later times a lower bound for the radius of spacial analyticity is derived which is given by t c t with 4 3 e for any e 0 when j 1 and 1 when j 2 finally it is shown that the regularity of the solution in the time variable is gevrey of order m and this is optimal

optimal edge detector design i parameter selection and noise effects this paper comments on the optimality of the laplacian of a gaussian edge detection filter which localizes edges through zero crossings in the filtered image the arguments of both marr and hildreth and dickey and shanmugam are reviewed to establish that the filter is optimal in the sense of maximizing output image energy near edge features this filter s principal advantage over other edge detectors is that its response is user adjustable through selection of a single parameter the gaussian standard deviation however no clear method for the selection of this parameter has been provided the problem is addressed here by applying the filter to two ideal periodic edge models blurred by a gaussian distribution point spread function the observed response to the edge spacing and blur standard deviation is then translated into a filter parameter design procedure the problems of optimum filter performance in the presence of additive gaussian noise are then addressed the problem of selecting the sampled filter s coefficient word size is dealt with in a companion paper

on the stability of self similar solutions of 1d cubic schrodinger equations in this paper we will study the stability properties of self similar solutions of 1 d cubic nls equations with time dependent coefficients of the form begin aligned displaystyle iu t u xx frac u 2 left u 2 frac a t right 0 quad a in mathbb r end aligned 0 1 the study of the stability of these self similar solutions is related through the hasimoto transformation to the stability of some singular vortex dynamics in the setting of the localized induction equation lie an equation modeling the self induced motion of vortex filaments in ideal fluids and superfluids we follow the approach used by banica and vega that is based on the so called pseudo conformal transformation which reduces the problem to the construction of modified wave operators for solutions of the equation begin aligned iv t v xx frac v 2t v 2 a 0 end aligned as a by product of our results we prove that eq 0 1 is well posed in appropriate function spaces when the initial datum is given by u 0 x z 0 mathrm p v frac 1 x for some values of z 0 in mathbb c setminus 0 and a is adequately chosen this is in deep contrast with the case when the initial datum is the dirac delta distribution

sharp well posedness and ill posedness results for a quadratic non linear schrodinger equation abstract we establish that the quadratic non linear schrodinger equation iu t u xx u 2 where u r r c is locally well posed in h s r when s 1 and ill posed when s 1 previous work in c kenig g ponce l vega quadratic forms for the 1 d semilinear schrodinger equation trans amer math soc 346 1996 3323 3353 had established local well posedness for s 3 4 the local well posedness is achieved by an iteration using a modification of the standard x s b spaces the ill posedness uses an abstract and general argument relying on the high to low frequency cascade present in the non linearity and a computation of the first non linear iterate

nonuniform continuity of the solution map to the two component camassa holm system abstract we prove that the solution map of the two component camassa holm system is not uniformly continuous as a map from a bounded subset of the sobolev space h s t h r t to c 0 1 h s t h r t when s 1 and r 0 we also demonstrate the nonuniform continuous property in the continuous function space c 1 t c 1 t

on the well posedness of higher order viscous burgers equations abstract we consider higher order viscous burgers equations with generalized nonlinearity and study the associated initial value problems for given data in the l 2 based sobolev spaces we introduce appropriate time weighted spaces to derive multilinear estimates and use them in the contraction mapping principle argument to prove local well posedness for data with sobolev regularity below l 2 we also prove ill posedness for this type of models and show that the local well posedness results are sharp in some particular cases viz when the orders of dissipation p and nonlinearity k 1 satisfy a relation p 2 k 1

color edge extraction using orthogonal polynomials based zero crossings scheme abstract a new color edge extraction scheme based on identifying the zero crossings in the second directional derivatives in terms of orthogonal polynomials has been proposed in this paper the proposed scheme supports a complete set of finite difference operators which are easily configured from orthogonal polynomials a color image region in presence of gaussian noise is represented as a linear combination of these operators towards color edges and noise statistical hypothesis testing procedures are employed to separate out the responses towards noise from those towards color edges the usage of the proposed framework is highlighted in devising a color edge detector based on identifying the zero crossings in the second directional derivatives of the color image region the proposed color edge extraction scheme is also compared with two existing color edge extraction schemes

on the variational structure of breather solutions i sine gordon equation abstract in this paper we describe stability properties of the sine gordon breather solution these properties are first described by suitable variational elliptic equations which also implies that the stability problem reduces in some sense to i the study of the spectrum of explicit linear systems and ii the understanding of how bad directions if any can be controlled using low regularity conservation laws then we discuss spectral properties of a fourth order linear matrix system using numerical methods we confirm that all spectral assumptions leading to the h 2 h 1 stability of sg breathers are numerically satisfied even in the ultra relativistic singular regime

non uniform dependence on initial data for the modified μ camassa holm equation considered herein is the initial value problem for the modified camassa holm equation it is shown that the solution map of this problem is not uniformly continuous in sobolev spaces hs s for s 5 2 on the circle based on the well posedness result and the lifespan for this problem the method of approximate solutions is utilized

multilinear weighted convolution of l 2 functions and applications to nonlinear dispersive equations the x s b spaces as used by beals bourgain kenig ponce vega klainerman machedon and others are fundamental tools to study the low regularity behavior of nonlinear dispersive equations it is of particular interest to obtain bilinear or multilinear estimates involving these spaces by plancherel s theorem and duality these estimates reduce to estimating a weighted convolution integral in terms of the l 2 norms of the component functions in this paper we systematically study weighted convolution estimates on l 2 as a consequence we obtain sharp bilinear estimates for the kdv wave and schrodinger x s b spaces

the generalized localization for multiple fourier integrals in this paper we investigate almost everywhere convergence properties of the bochner riesz means of n fold fourier integrals under summation over domains bounded by the level surfaces of the elliptic polynomials it is proved that if the order of the bochner riesz means s n 1 1 p 1 2 then the bochner riesz means of a function f lp rn 1 p 2 converge to zero almost everywhere on rn supp f

ill posedness of the navier stokes equations in a critical space in 3d abstract we prove that the cauchy problem for the three dimensional navier stokes equations is ill posed in b 1 in the sense that a norm inflation happens in finite time more precisely we show that initial data in the schwartz class s that are arbitrarily small in b 1 can produce solutions arbitrarily large in b 1 after an arbitrarily short time such a result implies that the solution map itself is discontinuous in b 1 at the origin

holder continuity on μ b equation abstract this paper is concerned with the holder continuous on initial data for the b equation on the circle it is shown that the data to solution for the b equation is holder continuous from bounded set of sobolev spaces with exponent s 3 2 measured in a weaker sobolev norm with index r s in periodic case

on the continuity of the solution operator to the wave map system we investigate the continuity properties of the solution operator to the wave map system from n to a general nonflat target of arbitrary dimension and we prove by an explicit class of counterexamples that this map is not uniformly continuous in the critical norms on any neighborhood of 0 2003 wiley periodicals inc

a remark on admissible triples for the generalized kdv equation abstract in this paper we give a condition of type l x p l t q under which the solution u t of the generalized kdv equation satisfies for any 1 1 for any admissible triples p 1 q 1 1 the following inequality d x 1 u t l x p 1 l t q 1 c d x u 0 l 2 we also present a global well posedness result in some spaces of admissible triples

a bilinear estimate with applications to the kdv equation u x 0 u0 x where u0 h r our principal aim here is to lower the best index s for which one has local well posedness in h r i e existence uniqueness persistence and continuous dependence on the data for a finite time interval whose size depends on u0 hs equation in 1 1 was derived by korteweg and de vries 21 as a model for long wave propagating in a channel a large amount of work has been devoted to the existence problem for the ivp 1 1 for instance see 9 10 the inverse scattering method applies to this problem and under appropriate decay assumptions on the data several existence results have been established see 5 6 14 28 33 another approach inherited from hyperbolic problems relies on the energy estimates and in particular shows that 1 1 is locally well posed in h r for s 3 2 see 2 3 12 29 30 31 using these results and conservation laws global in time well posedness in h r s 2 was established see 3 12 30 also global in time weak solutions in the energy space h r were constructed in 34 in 13 and 22 a local smoothing effect for solutions of 1 1 was discovered this combined with the conservation laws was used in 13 and 22 to construct global in time weak solutions with data in h r and even in l r in 16 we introduced oscillatory integral techniques to establish local well posedness of 1 1 in h r s 3 4 and hence global in time well posedness in h r s 1 in 16 we showed how to obtain the above mentioned result by picard iteration in an appropriate function space in 4 j bourgain introduced new function spaces adapted to the linear operator t 3 x for which there are good bilinear estimates for the nonlinear term x u 2 using these spaces bourgain was able to establish local well posedness of 1 1 in h r l r and hence by a conservation

well posedness and ill posedness of kdv equation with higher dispersion abstract first the cauchy problem for kdv equation with 2 n 1 order dispersion is studied and the local well posedness result for the initial data in sobolev spaces h s r with s n 1 4 is established via the fourier restriction norm method second we prove that the kdv equation with 2 n 1 order dispersion is ill posed for the initial data in h s r with s n 1 4 n 2 n n if the flow map is c 2 differentiable at zero form h s r to c 0 t h s r finally we obtain the sharp regularity requirement for the kdv equation with 2 n 1 order dispersion s n 1 4

on the ill posedness of the gardner equation abstract we present ill posedness results for the initial value problem ivp for the gardner equation we measure the regularity of the cauchy problem in the classical sobolev spaces h s and show the critical sobolev index under which the local well posedness of the problem is not present in the sense that the dependence of solutions upon initial data fails to be continuous

detection localization and estimation of edges a method to detect locate and estimate edges in a one dimensional signal is presented it is inherently more accurate than all previous schemes as it explicitly models and corrects interaction between nearby edges the method is iterative with initial estimation of edges provided by the zero crossings of the signal convolved with laplacian of gaussian log filter the necessary computations necessitate knowledge of this convolved output only in a neighborhood around each zero crossing and in most cases could be performed locally by independent parallel processors results on one dimensional slices extracted from real images and on images which have been proposed independently in the row and column directions are shown an analysis of the method is provided including issues of complexity and convergence and directions of future research are outlined

local well posedness for the modified kdv equation in almost critical widehat h r_s spaces we study the cauchy problem for the modified kdv equation u t u xxx u 3 x 0 u 0 u 0 for data u 0 in the space h r s defined by the norm u 0 h r s s u 0 lr local well posedness of this problem is established in the parameter range 2 r 1 s 1 2 1 2r so the case s r 0 1 which is critical in view of scaling considerations is almost reached to show this result we use an appropriate variant of the fourier restriction norm method as well as bi and trilinear estimates for solutions of the airy equation

nonlinear smoothing for dispersive pde a unified approach abstract in several cases of nonlinear dispersive pdes the difference between the nonlinear and linear evolutions with the same initial data i e the integral term in duhamel s formula exhibits improved regularity this property is usually called nonlinear smoothing we employ the method of infinite iterations of normal form reductions to obtain a very general theorem yielding existence of nonlinear smoothing for dispersive pdes contingent only on establishing two particular bounds we then apply this theorem to show that the nonlinear smoothing property holds depending on the regularity of the initial data for five classical dispersive equations in r the cubic nonlinear schrodinger the korteweg de vries the modified korteweg de vries and the derivative schrodinger equations in r 2 the modified zakharov kuznetsov equation for the aforementioned one dimensional equations this unifying methodology matches or improves the existing nonlinear smoothing results while enlarging the ranges of sobolev regularities where the property holds

bilinear eigenfunction estimates and the nonlinear schrodinger equation on surfaces we study the cubic non linear schrodinger equation nls on compact surfaces on the sphere mathbb s 2 and more generally on zoll surfaces we prove that for s 1 4 nls is uniformly well posed in hs which is sharp on the sphere the main ingredient in our proof is a sharp bilinear estimate for laplace spectral projectors on compact surfaces

the principle of general localization on unit sphere abstract in this paper we study the general localization principle for fourier laplace series on unit sphere s n r n 1 weak type 1 1 property of maximal functions is used to establish the estimates of the maximal operators of riesz means at critical index n 1 2 the properties jacobi polynomials are used in estimating the maximal operators of spectral expansions in l 2 s n for extending positive results on critical line n 1 1 p 1 2 1 p 2 we apply interpolation theorem for the family of the linear operators of weak types the generalized localization principle is established by the analysis of spectral expansions in l 2 we have proved the sufficient conditions for the almost everywhere convergence of fourier laplace series by riesz means on the critical line

the cauchy problem for a higher order shallow water type equation on the circle abstract in this paper we investigate the cauchy problem for a higher order shallow water type equation u t u t x x x 2 j 1 u x 2 j 3 u 3 u u x 2 u x u x x u u x x x 0 where x t r 2 and j n firstly we prove that the cauchy problem for the shallow water type equation is locally well posed in h s t with s j 2 2 for arbitrary initial data by using the i method we prove that the cauchy problem for the shallow water type equation is globally well posed in h s t with 2 j 1 j 2 2 j 1 s 1 our results improve the results of a a himonas g misiolek himonas and misiolek 1998 11 2000 12

gevrey regularity of the periodic gkdv equation abstract using the multilinear estimates which were derived for proving well posedness of the generalized korteweg de vries gkdv equation it is shown that if the initial data belongs to gevrey space g 1 in the space variable then the solution to the corresponding cauchy problem for gkdv belongs also to g in the space variable moreover the solution is not necessarily g in the time variable however it belongs to g 3 near 0 when 1 these are analytic regularity results for gkdv

ill posedness for a system of quadratic nonlinear schrodinger equations in two dimensions abstract this paper concerns the ill posedness issue for a system of quadratic nonlinear schrodinger equations in two dimensions from previous studies for the large time behavior of the solution to the system one may expect that the critical regularity to show the well posedness depends on the dispersion coefficients to prove the ill posedness we show the failure of the uniform continuity of the data solution map for the mass resonance case and show the norm inflation for other cases

local and global well posedness for the 2d zakharov kuznetsov burgers equation in low regularity sobolev space abstract in the present paper we consider the cauchy problem of the 2d zakharov kuznetsov burgers zkb equation which has the dissipative term x 2 u this is known that the 2d zakharov kuznetsov equation is well posed in h s r 2 for s 1 2 and the 2d nonlinear parabolic equation with quadratic derivative nonlinearity is well posed in h s r 2 for s 0 by using the fourier restriction norm with dissipative effect we prove the well posedness for zkb equation in h s r 2 for s 1 2

nonlinear stability of gardner breathers abstract we show that breather solutions of the gardner equation a natural generalization of the kdv and mkdv equations are h 2 r stable through a variational approach we characterize gardner breathers as minimizers of a new lyapunov functional and we study the associated spectral problem through i the analysis of the spectrum of explicit linear systems spectral stability and i i controlling degenerated directions by using low regularity conservation laws

on optimal exponential decay properties of solutions to the korteweg de vries equation abstract we consider the cauchy problem associated to the korteweg de vries equation kdv and study the preservation of exponential decay of order 3 2 on the right of the x axis as time evolves more precisely for a solution of the equation which decays at t 0 as e a 0 x 3 2 for x 0 we find an optimal function a t with a 0 a 0 such that the solution decays as e a t x 3 2 for x t 0

analysis of multichannel narrow band filters for image texture segmentation a model for texture analysis and segmentation using multiple oriented channel filters is analyzed in the general framework several different arguments are applied leading to the conclusion that the two dimensional gabor filters possess strong optimality properties for this task properties of the multiple channel segmentation approach are analyzed in particular perturbations of textures from an ideal model are found to have important effects on the segmentation that can usually be ameliorated by simply preceding the segmentation process by a logarithmic operation and using a low pass postfilter prior to making region assignments the difficult problems of space variant textures and multiple component textures are also considered local spatial frequency estimation approaches are suggested that use the responses as constraints in estimating the locally emergent texture frequencies complex texture aggregates containing multiple shared frequency components can be analyzed if the textures are distinct and few in number

the cauchy problem of a periodic higher order kdv equation in analytic gevrey spaces abstract this paper studies the periodic cauchy problem for a kdv equation whose dispersion is of order m 2 j 1 where j is a positive integer kdvm using bourgain gevrey type analytic spaces and appropriate bilinear estimates it is shown that local in time well posedness holds when the initial data belong to an analytic gevrey spaces of order this implies that in the space variable the regularity of the solution remains the same with that of the initial data it also implies that the size of the uniform radius of analyticity is preserved moreover the solution is not necessarily g in time however it belongs to g m r near zero for every x on the circle

the gardner equation and the l 2 stability of the n soliton solution of the korteweg de vries equation multi soliton solutions of the korteweg de vries equation kdv are shown to be globally l2 stable and asymptotically stable in the sense of martel and merle the proof is surprisingly simple and combines the gardner transform which links the gardner and kdv equations together with the martel merle tsai and martel merle recent results on stability and asymptotic stability in the energy space applied this time to the gardner equation as a by product the results of maddocks sachs and merle vega are improved in several directions

non uniform dependence and well posedness for the hyperelastic rod equation abstract it is shown that the solution map for the hyperelastic rod equation is not uniformly continuous on bounded sets of sobolev spaces with exponent greater than 3 2 in the periodic case and greater than 1 in the non periodic case the proof is based on the method of approximate solutions and well posedness estimates for the solution and its lifespan

asymptotics frequency modulation and low regularity ill posedness for canonical defocusing equations in a recent paper kenig ponce and vega study the low regularity behavior of the focusing nonlinear schrodinger nls focusing modified korteweg de vries mkdv and complex korteweg de vries kdv equations using soliton and breather solutions they demonstrate the lack of local well posedness for these equations below their respective endpoint regularities in this paper we study the defocusing analogues of these equations namely defocusing nls defocusing mkdv and real kdv all in one spatial dimension for which suitable soliton and breather solutions are unavailable we construct for each of these equations classes of modified scattering solutions which exist globally in time and are asymptotic to solutions of the corresponding linear equations up to explicit phase shifts these solutions are used to demonstrate lack of local well posedness in certain sobolev spaces in the sense that the dependence of solutions upon initial data fails to be uniformly continuous in particular we show that the mkdv flow is not uniformly continuous in the l 2 topology despite the existence of global weak solutions at this regularity finally we investigate the kdv equation at the endpoint regularity h and construct solutions for both the real and complex kdv equations the construction provides a nontrivial time interval t t and a locally lipschitz continuous map taking the initial data in h to a distributional solution u c 0 t t h which is uniquely defined for all smooth data the proof uses a generalized miura transform to transfer the existing endpoint regularity theory for mkdv to kdv

on the localization performance measure and optimal edge detection the localization performance measure of edge detectors is addressed a one dimensional formulation of the problem is considered a linear space invariant filter is used for the detection the locations of local maxima in the thresholded output of the filter are declared to be the edges the limitations of conventional performance measures are shown and a localization performance measure for edge detection is suggested this performance measure is based on the theory of zero crossings of stochastic processes it is shown that the derivative of a gaussian is the optimal edge detector for the measure

solvability of the cauchy problem of nonlinear beam equations in besov spaces abstract in this paper we study solvability of the cauchy problem of the nonlinear beam equation t 2 u 2 u u p with initial data in besov spaces we prove that for any 1 q the cauchy problem of this equation is locally well posed in the besov spaces b 2 q s p r n and b 2 q s r n where s p n 2 4 p 1 and s s p and globally well posed in these spaces if initial data are small moreover we obtain scattering results in b 2 q s p r n

global well posedness of the one dimensional cubic nonlinear schrodinger equation in almost critical spaces abstract in this paper we first introduce a new function space m h p whose norm is given by the l p sum of modulated h norms of a given function in particular when 1 2 we show that the space m h p agrees with the modulation space m 2 p r on the real line and the fourier lebesgue space f l p t on the circle we use this equivalence of the norms and the galilean symmetry to adapt the conserved quantities constructed by killip visan zhang to the modulation space and fourier lebesgue space setting by applying the scaling symmetry we then prove global well posedness of the one dimensional cubic nonlinear schrodinger equation nls in almost critical spaces more precisely we show that the cubic nls on r is globally well posed in m 2 p r for any p while the renormalized cubic nls on t is globally well posed in f l p t for any p in appendix we also establish analogous global in time bounds for the modified kdv equation mkdv in the modulation spaces on the real line and in the fourier lebesgue spaces on the circle an additional key ingredient of the proof in this case is a galilean transform which converts the mkdv to the mkdv nls equation

global wellposedness of kdv in h 1 t r by the inverse method we show that the korteweg de vries equation kdv tv x t x3v x t 6v x t xv x t x t t r is globally in time wellposed in the sobolev space of distributions h t r for any 1

on the well posedness of the schrodinger korteweg de vries system abstract we prove that the cauchy problem for the schrodinger korteweg de vries system is locally well posed for the initial data belonging to the sobolev spaces l 2 r h 3 4 r and h s r h 3 4 r s 1 16 for the resonant case the new ingredient is that we use the f s type space introduced by the first author in guo 2009 10 to deal with the kdv part of the system and the coupling terms in order to overcome the difficulty caused by the lack of scaling invariance we prove uniform estimates for the multiplier this result improves the previous one by corcho and linares 2007 6

the influence of edge direction on the estimation of edge contrast and orientation abstract this paper concerns the influence of edge direction on the estimation of edge contrast and orientation we present an analytical study of general first order detectors we show that the gradient estimated using radial filters is not affected by edge orientation for non radial filters the gradient may be affected by edge orientation for instance we find that the estimated edge orientation using a non radial filter may be biased even if the image is noise free however there are non radial filters for which gradient is unaffected by edge orientation as in the case of radial filters the properties of these functions are given in this paper the results are illustrated by studying the detectors of canny deriche and shen castan we take into account discretization effects such as subpixel and tesselation of the image plane the quantitative results provide insights into the estimation of gradient by these commonly used detectors taken together the theoretical and experimental results give a clear indication of the effect of the rotation invariance property of an edge detector on its response thus providing a more precise meaning for this property in edge detection

quadratic forms for the 1 d semilinear schrodinger equation this paper is concerned with 1 d quadratic semilinear schr6dinger equations we study local well posedness in classical sobolev space hs of the associated initial value problem and periodic boundary value problem our main interest is to obtain the lowest value of s which guarantees the desired local well posedness result we prove that at least for the quadratic cases these values are negative and depend on the structure of the nonlinearity considered

well posedness and weak rotation limit for the ostrovsky equation we consider the cauchy problem of the ostrovsky equation we first prove the time local well posedness in the anisotropic sobolev space hs a with s a 2 3 4 and 0 a 1 by the fourier restriction norm method this result include the time local well posedness in hs with s 3 4 for both positive and negative dissipation namely for both 0 and 0 we next consider the weak rotation limit we prove that the solution of the ostrovsky equation converges to the solution of the kdv equation when the rotation parameter goes to 0 and the initial data of the kdv equation is in l2 to show this result we prove a bilinear estimate which is uniform with respect to

the cauchy problem for higher order modified camassa holm equations on the circle abstract in this paper we investigate the cauchy problem for the shallow water type equation u t x 2 n 1 u 1 2 x u 2 x 1 x 2 1 u 2 1 2 u x 2 0 with low regularity data in the periodic settings firstly we proved that the bilinear estimate related to the nonlinear term of the equation in space w s defined in page 5 is invalid with s n 2 1 then the locally well posed of the cauchy problem for the periodic shallow water type equation is obtained in h s t with s n 3 2 n 2 for arbitrary initial data thus our result improves the result of himonas and misiolek commun partial differ equ 23 1998 123 139 where they have proved that the problem is locally well posed for small initial data in h s t with s n 2 1 n n with the aid of the standard fourier restriction norm method