a non local elliptic hyperbolic system related to the short pulse equation abstract in this paper we prove the well posedness of a non local elliptic hyperbolic system related to the short pulse equation it is a model which describes the evolution of the electrical field of linearly polarized continuum spectrum pulses in optical waveguides including fused silica telecommunication type or photonic crystal fibers as well as hollow capillaries filled with transparent gases

fourier transform inequalities with measure weights abstract fourier transform norm inequalities f q p are proved for measure weights on moment subspaces of lp r n density theorems are established to extend the inequalities to all of lp r n in both cases the conditions for validity are computable for n 2 and are radial and the results are applied to prove spherical restriction theorems which include power weights t t n p 1 p n p 1

a nonlinear parallel algorithm with application to the stefan problem the algebraic problem b gamma e a beta e eta which may evolve from elliptic or parabolic partial differential equations is studied conditions are given so that a parallel algorithm of a nonlinear gauss seidel type converges to the unique solution an application is given to the solidification of a channel in which the continuum problem is discretized by the finite element method these computations show a significant decrease in computing times as compared to serial algorithms

new exact solutions for the kdv equation with higher order nonlinearity by using the variational method the korteweg de vries kdv equation with higher order nonlinearity models the wave propagation in one dimensional nonlinear lattice a higher order extension of the familiar kdv equation is produced for internal solitary waves in a density and current stratified shear flow with a free surface the variational approximation method is applied to obtain the solutions for the well known kdv equation explicit solutions are presented and compared with the exact solutions very good agreement is achieved demonstrating the high efficiency of variational approximation method the existence of a lagrangian and the invariant variational principle for the higher order kdv equation are discussed the simplest version of the variational approximation based on trial functions with two free parameters is demonstrated the jost functions by quadratic cubic and fourth order polynomials are approximated also we choose the trial jost functions in the form of exponential and sinh solutions all solutions are exact and stable and have applications in physics

isometries fock spaces and spectral analysis of schrodinger operators on trees abstract we construct conjugate operators for the real part of a completely non unitary isometry and we give applications to the spectral and scattering theory of a class of operators on complete fock spaces natural generalizations of the schrodinger operators on trees we consider c algebras generated by such hamiltonians with certain types of anisotropy at infinity we compute their quotient with respect to the ideal of compact operators and give formulas for the essential spectrum of these hamiltonians

global stabilization of the generalized korteweg de vries equation posed on a finite domain this paper is concerned with the internal stabilization of the generalized korteweg de vries equation on a bounded domain the global well posedness and the exponential stability are investigated when the exponent in the nonlinear term ranges over the interval 1 4 the global exponential stability is obtained whatever the location where the damping is active confirming positively a conjecture of perla menzala vasconcellos and zuazua quart appl math 60 2002 pp 111 129

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

a transformation to avoid solutions at infinity for polynomial systems in finding all solutions to polynomial systems the existence of solutions at infinity makes the problem more difficult particularly when a continuation method is being used as the solution technique systems with solutions at infinity do arise in applications for example in computer graphics and geometric modeling in this paper a simple transformation of the system is given which eliminates solutions at infinity

global well posedness and nonsqueezing property for the higher order kdv type flow abstract in this paper we prove that the periodic higher order kdv type equation t u 1 j 1 x 2 j 1 u 1 2 x u 2 0 t x r t u 0 x u 0 x u 0 h s t is globally well posed in h s for s j 2 j 3 the proof is based on i method introduced by colliander et al 4 we also prove the nonsqueezing property of the periodic higher order kdv type equation the proof relies on gromov s nonsqueezing theorem for the finite dimensional hamiltonian system and an approximation argument for the solution flow more precisely after taking the frequency truncation to the solution flow we apply the nonsqueezing theorem by using the approximation argument we extend this result to the infinite dimensional system this argument was introduced by kuksin 14 and made concretely by bourgain 2 for the 1d cubic nls flow and colliander et al 5 for the kdv flow one of our observations is that the higher order kdv type equation has the better modulation effect from the non resonant interaction than that the kdv equation has hence unlike the work of colliander et al 5 we can get the nonsqueezing property for the solution flow without the miura transform

infinite dimensional symplectic capacities and a squeezing theorem for hamiltonian pde s we study partial differential equations of hamiltonian form and treat them as infinite dimensional hamiltonian systems in a functional phase space ofx dependent functions in this phase space we construct an invariant symplectic capacity and prove a version of gromov s non squeezing theorem we give an interpretation of the theorem in terms of the energy transition to high frequencies problem

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

numerical linear algebra aspects of globally convergent homotopy methods probability one homotopy algorithms are a class of methods for solving nonlinear systems of equations that are globally convergent with probability one these methods are theoretically powerful and if constructed and implemented properly are robust numerically stable accurate and practical the concomitant numerical linear algebra problems deal with rectangular matrices and good algorithms require a delicate balance not always achieved of accuracy robustness and efficiency in both space and time the author s experience with globally convergent homotopy algorithms is surveyed here and some of the linear algebra difficulties for dense and sparse problems are discussed

stability analysis of solitary wave solutions for the fourth order nonlinear boussinesq water wave equation abstract in the present study the nonlinear boussinesq type equation describe the bi directional propagation of small amplitude long capillary gravity waves on the surface of shallow water by using the extended auxiliary equation method we obtained some new soliton like solutions for the two dimensional fourth order nonlinear boussinesq equation with constant coefficient these solutions include symmetrical non symmetrical kink solutions solitary pattern solutions jacobi and weierstrass elliptic function solutions and triangular function solutions the stability analysis for these solutions are discussed

scattering for the quartic generalised korteweg de vries equation abstract we show that the quartic generalised kdv equation u t u x x x u 4 x 0 is globally well posed for data in the critical scale invariant space h x 1 6 r with small norm and locally well posed for large norm improving a result of grunrock a grunrock a bilinear airy estimate with application to gkdv 3 differential integral equations 18 12 2005 1333 1339 as an application we obtain scattering results in h x 1 r h x 1 6 r for the radiation component of a perturbed soliton for this equation improving the asymptotic stability results of martel and merle y martel f merle asymptotic stability of solitons for subcritical generalized kdv equations arch ration mech anal 157 3 2001 219 254

on the low regularity of the benney lin equation abstract we consider the low regularity of the benney lin equation u t u u x u x x x u x x u x x x x u x x x x x 0 we established the global well posedness for the initial value problem of benney lin equation in the sobolev spaces h s r for 0 s 2 improving the well posedness result of biagioni and linares h a biaginoi f linares on the benney lin and kawahara equation j math anal appl 211 1997 131 152 for s 2 we also prove some ill posedness issues

global well posedness in l 2 for the periodic benjamin ono equation we prove that the benjamin ono equation is globally well posed in h s bbb t for s ge 0 moreover we show that the associated flow map is lipschitz on every bounded set of h s 0 bbb t s ge 0 and even real analytic in this space for small times this result is sharp in the sense that the flow map if it can be defined and coincides with the standard flow map on h 0 infty bbb t cannot be of class c 1 alpha alpha 0 from h 0 s bbb t into h 0 s bbb t as soon as s 0

frequency uniform decomposition method for the generalized bo kdv and nls equations abstract sufficient and necessary conditions for the embeddings between besov spaces b p q s 1 and modulation spaces m p q s 2 are obtained moreover using the frequency uniform decomposition method we study the cauchy problem for the generalized bo kdv and nls equations for which the global well posedness of solutions with the small rough data in certain modulation spaces m 2 1 s is shown

low regularity global solutions for nonlinear evolution equations of kirchhoff type we study the global solvability of the cauchy dirichlet problem for two second order in time nonlinear integro differential equations 1 the extensible beam plate equation utt 2u m u 2dx u 0 x t r 2 a special case of the kirchhoff equation utt a b u 2dx 2 u 0 x t r by exploiting the i method of j colliander m keel g staffilani h takaoka t tao we prove that both equations admit global in time infinite energy solutions in case 1 the energy is the mechanical energy in case 2 it is a second order invariant introduced by s i pokhozhaev for the extensible beam equation 1 we also consider the effect of linear dissipation on such low regularity solutions and we prove their exponential decay as t

global well posedness for schrodinger equation with derivative in h 1 2 r abstract in this paper we consider the cauchy problem of the cubic nonlinear schrodinger equation with derivative in h s r this equation was known to be the local well posedness for s 1 2 takaoka 1999 27 ill posedness for s 1 2 biagioni and linares 2001 1 etc and global well posedness for s 1 2 i team 2002 10 in this paper we show that it is global well posedness in the endpoint space h 1 2 r which remained open previously the main approach is the third generation i method combined with a new resonant decomposition technique the resonant decomposition is applied to control the singularity coming from the resonant interaction

global well posedness and inviscid limit for the korteweg de vries burgers equation abstract considering the cauchy problem for the korteweg de vries burgers equation u t u x x x x 2 u u 2 x 0 u 0 where 0 1 and u is a real valued function we show that it is globally well posed in h s s s and uniformly globally well posed in h s s 3 4 for all 0 1 moreover we prove that for any t 0 its solution converges in c 0 t h s to that of the kdv equation if tends to 0

solitary wave solutions of 3 1 dimensional extended zakharov kuznetsov equation by lie symmetry approach abstract in this paper the solitary wave solutions of 3 1 dimensional extended zakharov kuznetsov ezk equation are constructed which appear in the magnetized two ion temperature dusty plasma and quantum physics lie group of transformation method is proposed to investigate the solution of 3 1 dimensional ezk equation via lie symmetry method the optimal system of one dimensional lie subalgebra is constructed by using lie point symmetries the three dimensional ezk equation reduced into number of ordinary differential equations odes by applying similarity reductions consequently solutions so extracted are more general than erstwhile known results we have obtained twenty one solutions in the explicit form some of them are likewise general and some are new for the best study of us eventually single soliton quasi periodic soliton multisoliton lump type soliton traveling wave and solitary wave interaction behavior are illustrated graphically through numerical simulation for physical affirmation of the results please check whether the affiliations are correct

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

continuity of the flow of kdv with regard to the wasserstein metrics and application to an invariant measure abstract in this paper we prove the continuity of the flow of kdv on spaces of probability measures with respect to a combination of wasserstein distances on h s s 0 and l 2 we are motivated by the existence of an invariant measure belonging to the spaces onto which these distances are defined

generalized definition of time delay in scattering theory we advocate for the systematic use of a symmetrized definition of time delay in scattering theory in two body scattering processes we show that the symmetrized time delay exists for arbitrary dilated spatial regions symmetric with respect to the origin it is equal to the usual time delay plus a new contribution which vanishes in the case of spherical spatial regions we also prove that the symmetrized time delay is invariant under an appropriate mapping of time reversal these results are also discussed in the context of classical scattering theory

well posedness for the cauchy problem associated to the hirota satsuma equation periodic case abstract we consider a system of korteweg de vries kdv equations coupled through nonlinear terms called the hirota satsuma system we study the initial value problem ivp associated to this system in the periodic case for given data in sobolev spaces h s h s 1 with regularity below the one given by the conservation laws using the fourier transform restriction norm method we prove local well posedness whenever s 1 2 also with some restriction on the parameters of the system we use the recent technique introduced by colliander et al called i method and almost conserved quantities to prove global well posedness for s 3 14

global analysis of a model for capillary water waves in two dimensions in this paper we prove a global regularity result for a quadratic quasilinear model associated to the water waves system with surface tension and no gravity in dimension 2 the capillary waves system the model we consider here retains most of the difficulties of the full capillary water waves system including the delicate time resonance structure and modified scattering it is slightly simpler however at the technical level and our goal here is to present our method in this simplified situation the analysis of the full system is presented in a preprint by the authors 2014 2016 wiley periodicals inc

a parallel algorithm for simple roots for polynomials abstract a method for finding simple roots of arbitrary polynomials based on divided differences is discussed theoretical background is presented for the case of simple roots numerical results are presented which show the algorithm finds simple and multiple zeros to an accuracy usually limited by the accuracy of polynomial evaluation the method is designed for a simd parallel computer the algorithm is compared to two other frequently used polynomial root finders the jenkins traub algorithm and laguerre s method

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 power series method for nonlocal and nonlinear evolution equations abstract the initial value problem for a 4 parameter family of nonlocal and nonlinear evolution equations with data in a space of analytic functions is solved by using a power series method in abstract banach spaces in addition to determining the power series expansion of the solution this method also provides an estimate of the analytic lifespan expressed in terms of the norm of the initial data thus establishing an abstract cauchy kovalevsky type theorem for these equations

global cauchy problem for the ostrovsky equation abstract in this article we consider the initial value problem for the ostrovsky equation t u x 3 u x 1 u u x u 0 x r t r u x 0 u 0 x with initial data in sobolev spaces h s r using bourgain spaces we prove that the problem is globally well posed for s 3 10 for both signs in the equation

compactons and solitary patterns solutions to fifth order kdv like equations abstract new ansatze that involve hyperbolic functions are introduced to handle modified fifth order kdv equations a set of entirely new compactons solitary patterns and periodic solutions is constructed the study introduces new approaches and presents useful guidance for other related problems

variational principle of the zakharov shabat equations abstract we put forth a variational principle for the zakharov shabat zs equations which is the basis of the inverse scattering transform of a number of important nonlinear pde s using the variational representation of the zs equations we develop an approximate analytical technique for finding discrete eigenvalues of the complex spectral parameter in the zs equations for a given pulse shaped potential which is equivalent to the physically important problem of finding the soliton content of the given initial pulse we apply the technique to several particular shapes of the pulse and demonstrate that the simplest version of the variational approximation based on trial functions with one or two free parameters proves to be fully analytically tractable and it yields threshold conditions for the appearance of the first soliton or of the first soliton pair which are in a fairly good agreement with available numerical results however more free parameters are necessary to allow prediction of additional solitons produced by the given pulse

the global attractor for a weakly damped forced shallow water equation in sobolev spaces of low regularity abstract we consider the global attractor for the weakly damped forced shallow water equation t u x 3 u 1 2 x u 2 1 x 2 1 x u 2 1 2 x u 2 u f in sobolev spaces h s t for s 1 under the assumption that f h 1 t we prove the existence of the global attractor for the equation in h s t for 2 3 s 1 which is identical to that in h 1 t our method is a combination of the i method and decomposing the solution into two parts one of which is uniformly bounded in h 1 t while the other decays exponentially in h s t in the argument we establish some bilinear estimates involving the operator i to make the index s as low as possible

local well posedness for the dispersion generalized periodic kdv equation abstract in this paper we set up the local well posedness of the initial value problem for the dispersion generalized periodic kdv equation t u x d x u x u 2 u 0 for 2 s 4 and h s t and we show that the 4 is a lower endpoint to obtain the bilinear estimates 1 2 and 1 3 which are the crucial steps to obtain the local well posedness by picard iteration the case 2 was studied in kenig et al 1996 10

analytic well posedness of periodic gkdv abstract in the periodic case it is proved that the cauchy problem for the generalized korteweg de vries equation gkdv is locally well posed in a class of analytic functions that can be extended holomorphically in a symmetric strip of the complex plane around the x axis thus the uniform analyticity radius of the solution does not change as time progresses also information about the regularity of the solution in the time variable is provided

abundant solitons solutions for several forms of the fifth order kdv equation by using the tanh method in this work we use the tanh method for solving several forms of the fifth order nonlinear kdv equation the forms include the lax sawada kotera sk kaup kupershmidt kk and the ito forms and other related special cases abundant solitons solutions are derived two necessary criteria are established to build up reliable strategies that govern the relation between the parameters of the equation previously known solutions are recovered and entirely new bell shaped solitons are determined

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

on the regularization mechanism for the periodic korteweg de vries equation in this paper we develop and use successive averaging methods for explaining the regularization mechanism in the the periodic korteweg de vries kdv equation in the homogeneous sobolev spaces s for s 0 specifically we prove the global existence uniqueness and lipschitz continuous dependence on the initial data of the solutions of the periodic kdv for the case where the initial data is in l2 we also show the lipschitz continuous dependence of these solutions with respect to the initial data as maps from s to s for s 1 0 2010 wiley periodicals inc

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

well posedness for stochastic camassa holm equation abstract the camassa holm equation describes the unidirectional propagation of waves at the free surface of shallow water under the influence of gravity due to uncertainty in the modelling and external environment this modelling could be subject to random fluctuations in this article the stochastic camassa holm equation with additive noise is considered using regularization a local existence and uniqueness result in the sobolev space h s r with s 3 2 of stochastic camassa holm equation is obtained with the help of priori estimates the local solution will blow up in h q r in finite time for any q 3 2 when initial value satisfies some conditions

riccati map on l02 t and its applications abstract this paper is concerned with the spectral properties of the schrodinger operator l q def d 2 d x 2 q with periodic potential q from the sobolev space h 1 t we obtain asymptotic formulas and a priori estimates for the periodic and dirichlet eigenvalues which generalize known results for the case of potentials q l 2 t the key idea is to reduce the problem to a known one the spectrum of the impedance operator via a nonlinear analytic isomorphism between l 0 2 t and the sobolev space h 0 1 t

rough blowup solutions to the l 2 critical nls we study the singularity formation for the cubic focusing l 2 critical nonlinear schrodinger equation on mathbb r 2 in a series of recent works merle and raphael have completely described the so called log log blowup regime and proven its stability in the energy space h 1 our aim in this paper is to investigate the stability of this blowup regime under rough perturbations in the direction of developing a theory at the level of the critical space l 2 by blending the merle raphael techniques with the quantitative i method developed by colliander keel staffilani takaoka and tao for the study of the cauchy problem for rough data we obtain the stability of the log log regime in h s for all s 0

fractional solitary wave solutions of the nonlinear higher order extended kdv equation in a stratified shear flow the problem formulations of models for internal solitary waves in a stratified shear flow with a free surface are presented solitary waves solutions are generated by deriving the nonlinear higher order of extended kdv equations for the free surface displacement all coefficients of the nonlinear higher order extended kdv equation are expressed in terms of integrals of the modal function for the linear long wave theory the electric field potential and the fluid pressure in the form of traveling wave solutions of the extended kdv equation are obtained the stability of the obtained solutions and the movement role of the waves by making the graphs of the exact solutions are discussed and analyzed

a priori estimates and weak solutions for the derivative nonlinear schrodinger equation on torus below h 1 2 abstract we propose a priori estimates for a weak solution to the derivative nonlinear schrodinger equation dnls on torus with small l 2 norm datum in low regularity sobolev spaces these estimates allow us to show the existence of solutions in h s t with some s 1 2 in a relatively weak sense furthermore we make some remarks on the error estimates arising from the finite dimensional approximation solutions of dnls using the fourier lesbesgue type as auxiliary spaces despite the fact that nahmod oh rey bellet and staffilani 12 have already seen them

a homotopy for solving polynomial systems this paper presents a continuation method for finding all solutions to polynomial systems it features a simpler homotopy than has been previously published

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

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

self spreading and strength of singularities for solutions to semilinear wave equations algebres microlocales propagation de la regularite solutions lineaires conduisant a l etalement auto etalement des singularites

traveling wave solutions of the degasperis procesi equation we classify all weak traveling wave solutions of the degasperis procesi equation in addition to smooth and peaked solutions the equation is shown to admit more exotic traveling waves such as cuspons stumpons and composite waves

normal form approach to near linear dynamics of modified kdv equation abstract in this paper we implement normal form reduction to the periodic modified korteweg de vries mkdv equation to investigate the behavior of a solution when a subtle high frequency initial data is given we use differentiation by parts to decompose the equation into resonant and non resonant parts and provide some nonlinear estimates for each term if a subtle high frequency initial data is given a solution of the mkdv equation can be approximated by a solution of the linearized mkdv equation for large times

on the hamiltonian interpolation of near to the identity symplectic mappings with application to symplectic integration algorithms we reconsider the problem of the hamiltonian interpolation of symplectic mappings following moser s scheme we prove that for any mapping e analytic and e close to the identity there exists an analytic autonomous hamiltonian system he such that its time one mapping he differs from e by a quantity exponentially small in 1 e this result is applied in particular to the problem of numerical integration of hamiltonian systems by symplectic algorithms it turns out that when using an analytic symplectic algorithm of orders to integrate a hamiltonian systemk one actually follows exactly namely within the computer roundoff error the trajectories of the interpolating hamiltonian he or equivalently of the rescaled hamiltonian ke e 1he which differs fromk but turns out to be e5 close to it special attention is devoted to numerical integration for scattering problems

quadratic dispersive generalized benjamin ono equation abstract this paper is concerned with quadratic dispersive generalized benjamin ono equation 1 1 1 2 we prove sharp local well posedness in h s s 1 4 with a solution map that is analytic from h s r to c 0 t h s r and the ill posed result in h s r as soon as s s in the sense that the flow map u 0 u t cannot be continuous

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

a globally convergent parallel algorithm for zeros of polynomial systems certain classes of nonlinear systems of equations such as polynomial systems have properties that make them particularly amenable to solution on distributed computing systems some algorithms considered unfavorably on a single processor serial computer may be excellent on a distributed system this paper considers the solution of polynomial systems of equations via a globally convergent homotopy algorithm on a hypercube some computational results are reported

global well posedness results for the 2d schrodinger debye system abstract in this work we study the well posedness of the schrodinger debye system i t u 1 2 u u v t 0 x r 2 t v v u 2 0 1 u 0 x u 0 x v 0 x v 0 x in the context of sobolev spaces of type h k 1 r 2 h k 2 r 2 where k 1 k 2 are integers we prove some a priori estimates and therefore global well posedness using nonlinear interpolation of operators we prove some global well posedness results when u 0 v 0 h s r 2 h l r 2 where s l r are some convex combinations of k 1 and k 2

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

periodic fourth order cubic nls local well posedness and non squeezing property abstract in this paper we consider the cubic fourth order nonlinear schrodinger equation 4nls under the periodic boundary condition we prove two results one is the local well posedness in h s t with 1 3 s 0 for the cauchy problem of the wick ordered 4nls the other one is the non squeezing property for the flow map of 4nls in the symplectic phase space l 2 t to prove the former we used the ideas introduced in 36 and 27 and to prove the latter we used the ideas in 8

decay of regular solutions for the critical 2d zakharov kuznetsov equation posed on rectangles an initial boundary value problem for the critical generalized 2d zakharov kuznetsov equation posed on rectangles is considered the existence uniqueness and exponential decay rate of global regular solutions for small initial data are established

a formula relating sojourn times to the time of arrival in hamiltonian dynamics we consider on a manifold m equipped with a poisson bracket a hamiltonian h with complete flow and a family 1 d of abstract position observables satisfying the condition j h h 0 for each j under these assumptions we prove a new formula relating sojourn times in dilated regions defined in terms of to the time of arrival of classical orbits the correspondence between this formula and a formula established recently in the framework of quantum mechanics is put into evidence among other examples our theory applies to stark hamiltonians homogeneous hamiltonians purely kinetic hamiltonians the repulsive harmonic potential central force systems the poincar ball model the wave equation the nonlinear schr dinger equation the korteweg de vries equation and quantum hamiltonians defined via expectation values

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

the cauchy problem for a 4 parameter family of equations with peakon traveling waves abstract the initial value problem for a novel 4 parameter family of evolution equations which are nonlinear and nonlocal and possess peakon traveling wave solutions is studied on both the line and the circle it is proved that this family of equations is well posed in the sense of hadamard when the initial data belong to the sobolev spaces h s with s 5 2 also it is shown that the data to solution map is not uniformly continuous however if h s s 5 2 is equipped with a weaker h r norm 0 r s then the solution map becomes holder continuous