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

algebraic solutions for all dihedral groups the significant problem with using point groups is the dependence upon extensively tabulated results we present simple algebraic expressions for the primitive characters matrix irreps symmetry adapted functions safs and clebsch gordan coefficients for all dihedral groups dn cnv dnd and dnh those results for arbitrary n and for single and double valued representations have been derived in a simple manner without using group tables previously incomplete tabulated results are now redundant in particular the parity dependence of the safs of the improper dihedral groups is shown analytically simple relations are derived between the safs of the proper and improper dihedral groups so that the safs of the latter can be easily obtained from the safs of dn

inviscid limit behavior of the solution to the one dimensional generalized ginzburg landau equation abstract the cauchy problem for the one dimensional generalized ginzburg landau ggl equation is considered and the inviscid limit behavior of its solution is proved that is the solution of cauchy problem for the ggl equation converges to the solution of cauchy problem for the derivative nonlinear schrodinger equation in the natural space c 0 t h s with s 1 2 for some t 0 if some coefficients tend to zero moreover the convergence holds in c 0 t h 1 for any t 0 if initial data belong to h 2 furthermore the convergence also holds in c 0 t h s for any t 0 if initial data belong to h s s 4 7 with some coefficients conditions

construction of non analytic solutions for the generalized kdv equation abstract in both the periodic and non periodic case we construct non analytic complex valued solutions for the generalized kdv equation with appropriate analytic initial data moreover for the kdv and mkdv we construct real valued non analytic solutions

the unrooted set covering connected subgraph problem differentiating between hiv envelope sequences this paper presents a novel application of operations research techniques to the analysis of hiv env gene sequences aiming to identify key features that are possible vaccine targets these targets are identified as being critical to the transmission of hiv by being present in early transmitted founder sequences and absent in later chronic sequences identifying the key features of env involves two steps first calculating the covariance of amino acid combinations and positions to form a network of related and compensatory mutations and second developing an integer program to identify the smallest connected subgraph of the constructed covariance network that exhibits a set covering property the integer program developed for this analysis labelled the unrooted set covering connected subgraph problem usccsp integrates a set covering problem and connectivity evaluation the latter formulated as a network flow problem the resulting integer program is very large and complex requiring the use of benders decomposition to develop an efficient solution approach the results will demonstrate the necessity of applying acceleration techniques to the benders decomposition solution approach and the effectiveness of these techniques and heuristic approaches for solving the usccsp

interaction equations for short and long dispersive waves abstract we show the time local well posedness for a system of nonlinear dispersive equations for the water wave interaction formula it is shown that for any initial data u0 v0 hs r hs 1 2 r s 0 the solution for the above equation uniquely exists in a subset ofc t t hs c t t hs 1 2 and depends continuously on the data by virtue of a special structure of the nonlinear coupling the solution is stable under a singular limiting process

algebraic solutions for point groups the tetrahedral group for the group chain t d2 c2 a factorization lemma is discovered for the irreducible symmetry operators iso s of a semidirect product group of two abelian groups by means of it simple algebraic expressions of the t d2 single valued irreducible matrices iso s and symmetry adapted functions are derived using the double induced technique the algebraic solutions for the double valued representations of group t in the group chain t c2 are also obtained the simplicity and elegance of the algebraic solution of the point group can be compared with that of the analytic solution of the rotation group

on global existence and scattering for the wave maps equation we prove global existence and scattering for the wave maps equation in n 1 dimensions n 2 3 for initial data which is small in the scale invariant homogeneous besov space 2 1 n 2 x 2 1 n 2 1 this result was proved in an earlier paper by the author for n 4

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 non homogeneous boundary value problem for the korteweg de vries equation posed on a finite domain ii abstract studied here is an initial and boundary value problem for the korteweg de vries equation posed on a bounded interval with nonhomogeneous boundary conditions this particular problem arises naturally in certain circumstances when the equation is used as a model for waves and a numerical scheme is needed it is shown here that this initial boundary value problem is globally well posed in the l 2 based sobolev space h s 0 1 for any s 0 in addition the mapping that associates to appropriate initial and boundary data the corresponding solution is shown to be analytic as a function between appropriate banach spaces

local smoothing properties of dispersive equations is it possible for time evolution partial differential equations which are reversible and conservative to smooth locally the initial data for the linear wave equation for instance the answer is no however in 10 t kato found a local smoothing property of the korteweg de vries equation the solution of the initial value problem is locally one derivative smoother than the initial datum kato s proof uses in a curcial way the algebraic properties of the symbol for the korteweg de vries equation and the fact that the underlying spatial dimension is one actually judging from the way several integrations by parts and cancellations conspire to reveal a smoothing effect one would be inclined to believe this was a special property of the k dv equation this is not however the case in this paper we attempt to describe a general local smoothing effect for dispersive equations and systems the smoothing effect is due to the dispersive nature of the linear part of the equation all the physically significant dispersive equations and systems known to us have linear parts displaying this local smoothing property to mention only a few the k dv benjamin ono intermediate long wave various boussinesq and schrodinger equations are included we study thus equations and systems of the form

on the cauchy problem for the generalized shallow water wave equation abstract in this paper we mainly study the cauchy problem for the generalized shallow water wave equation in the sobolev space h s of lower order s using the crucial bilinear estimates in the fourier transform restriction spaces related to the shallow water wave equation we establish local well posedness in h s with any s 11 16

pseudodifferential operator with a nonanalytic symbol on a half line we study the initial boundary value problem for the nonlinear pseudodifferential equations with nonanalytic symbol we study traditionally important problems of a theory of nonlinear partial differential equations such as global in time existence of solutions to the initial boundary value problem and the asymptotic behavior of solutions for large time

almost optimal local well posedness for the 3 1 dimensional maxwell klein gordon equations this paper contains a detailed study of the local in time regularity properties of the maxwell klein gordon mkg equations the mkg equations represent a physical model for the interaction of a spin 0 particle sometimes referred to as a higgs or scalar field with the classical equations of electrodynamics i e maxwell s equations in terms of the calculus of variations the mkg system consists of the euler lagrange equations associated to the density l 4 f f 1 2 d d where f a a is the electro magnetic field associated to the set a of real valued potentials on the minkowski space d 1a is the corresponding covariant derivative with respect to those potentials and r c is a complex scalar field in terms of the field f the euler lagrange equations read f d 1a d d 0 1b

local well posedness of the ostrovsky stepanyams and tsimring equation in sobolev spaces of negative indices abstract in this paper we prove by using the method of bilinear estimate in the bourgain type spaces that the initial value problem of the ost equation u t u x x x h u x h u x x x u u x 0 x r t 0 where 0 and h denotes the usual hilbert transformation is locally well posed in the sobolev space h s r when s 1

a regularized decomposition method for minimizing a sum of polyhedral functions a problem of minimizing a sum of many convex piecewise linear functions is considered in view of applications to two stage linear programming where objectives are marginal values of lower level problems it is assumed that domains of objectives may be proper polyhedral subsets of the space of decision variables and are defined by piecewise linear induced feasibility constraints we propose a new decomposition method that may start from an arbitrary point and simultaneously processes objective and feasibility cuts for each component the master program is augmented with a quadratic regularizing term and comprises an a priori bounded number of cuts the method goes through nonbasic points in general and is finitely convergent without any nondegeneracy assumptions next we present a special technique for solving the regularized master problem that uses an active set strategy and qr factorization and exploits the structure of the master finally some numerical evidence is given

numerical simulation of the stochastic korteweg de vries equation abstract in this work we numerically investigate the influence of a homogeneous noise on the evolution of solitons for the korteweg de vries equation our numerical method is based on finite elements and least squares we present numerical experiments for different values of noise amplitude and describe different types of behaviours

global existence for 2d nonlinear schrodinger equations via high low frequency decomposition method abstract we study global existence of solutions for the cauchy problem of the nonlinear schrodinger equation i u t u u 2 m u in the 2 dimension case where m is a positive integer m 2 using the high low frequency decomposition method we prove that if 10 m 6 10 m 5 s 1 then for any initial value h s r 2 the cauchy problem has a global solution in c r h s r 2 and it can be split into u t e i t y t with y c r h 1 r 2 satisfying y t h 1 1 t 2 1 s 10 m 5 s 10 m 6 where is an arbitrary sufficiently small positive number

analyticity and smoothing effect for the korteweg de vries equation with a single point singularity we show that a solution of the cauchy problem for the kdv equation cases partial tv partial 3 x v partial x v 2 0 qquad t in t t x in bbb r v 0 x phi x endcases has a drastic smoothing effect up to real analyticity if the initial data only have a single point singularity at x 0 it is shown that for h s mathbb r s 3 4 data satisfying the condition sum k 0 infty frac a 0 k k x partial x k phi h s infty the solution is analytic in both space and time variable the above condition allows us to take as initial data the dirac delta measure or the cauchy principal value of 1 x the argument is based on the recent progress on the well posedness result by bourgain 2 and kenig ponce vega 20 and a systematic use of the dilation generator 3t partial t x partial x

exact boundary controllability of the korteweg de vries equation we consider boundary control of the distributed parameter system described by the korteweg de vries kdv equation posed on a finite interval alpha leq x leq beta left begin array l u t u x uu x u xxx 0 u alpha t h 1 t qquad u beta t h 2 t qquad u x beta t h 3 t u end array right qquad for t geq 0 it is shown that by choosing appropriate control inputs h j t j 1 2 3 one can always guide the system from a given initial state phi in h s alpha beta to a given terminal state psi in h s alpha beta in the time period 0 t so long as phi and psi satisfy phi cdot w cdot 0 h s alpha beta leq delta quad mbox and quad psi cdot w cdot t h s alpha beta leq delta for some delta 0 independent of phi and psi where s geq 0 and w equiv w x t is a given smooth solution of the kdv equation this exact boundary controllability is established by considering a related initial value control problem of the kdv equation posed on the whole line r various recently discovered smoothing properties of the kdv equation have played important roles in our approach

a korteweg de vries type of fifth order equations on a finite domain with point dissipation abstract the paper discusses initial value problem of a korteweg de vries type of fifth order equation w t w x x x w x x x x x j 1 n a j w j w x 0 w x 0 w 0 x posed on a periodic domain x 0 2 with periodic boundary conditions w i x 0 t w i x 2 t i 0 2 3 4 and an l 2 stabilizing feedback control law w x 2 t w x 0 t 1 w x x x 0 t where n is a fixed positive integer a j j 1 2 n are real constants and 1 it is shown that for w 0 x h 1 0 2 with the boundary conditions described above the problem is locally well posed for w c 0 t h 1 0 2 with a conserved volume of w w 0 2 w x t d x moreover the solution with small initial condition exists globally and approaches to w 0 x 2 as t

a microscopic derivation of time dependent correlation functions of the 1d cubic nonlinear schrodinger equation abstract we give a microscopic derivation of time dependent correlation functions of the 1d cubic nonlinear schrodinger equation nls from many body quantum theory the starting point of our proof is 11 on the time independent problem and 16 on the corresponding problem on a finite lattice an important new obstacle in our analysis is the need to work with a cutoff in the number of particles which breaks the gaussian structure of the free quantum field and prevents the use of the wick theorem we overcome it by means of complex analytic methods our methods apply to the nonlocal nls with bounded convolution potential in the periodic setting we also consider the local nls arising from short range interactions in the many body setting to that end we need the dispersion of the nls in the form of periodic strichartz estimates in x s b spaces

well posedness of the cauchy problem for ostrovsky stepanyams and tsimring equation with low regularity data abstract in this paper we prove that the initial value problem of the ost equation u t u x x x h u x h u x x x u u x 0 x r t 0 where 0 and h denotes the usual hilbert transformation is locally well posed in the sobolev space h s r when s 3 4 and globally well posed in h s r when s 0

low regularity solutions of non homogeneous boundary value problems of a higher order boussinesq equation in a quarter plane abstract in this article we study an initial boundary value problem of the sixth order boussinesq equation on a half line with nonhomogeneous boundary conditions u t t u x x u x x x x u x x x x x x u 2 x x 0 x 0 t 0 u x 0 x u t x 0 x x 0 u 0 t h 1 t u x x 0 t h 2 t u x x x x 0 t h 3 t t 0 where 1 it is shown that the problem is locally well posed in c r l o c h s r with initial condition h s r h s 1 r and boundary condition h 1 h 2 h 3 in the product space h s 1 3 r h s 1 3 r h s 3 3 r for 1 2 s 0

on the stochastic korteweg de vries equation abstract we consider a stochastic korteweg de vries equation forced by a random term of white noise type this can be a model of water waves on a fluid submitted to a random pressure we prove existence and uniqueness of solutions inh1 r in the case of additive noise and existence of martingales solutions inl2 r in the case of multiplicative noise copyright 1998 academic press nous etudions une equation de korteweg de vries stochastique comportant une force aleatoire de type bruit blanc cette equation peut decrire les ondes de surface sur un fluide soumis a un champ de pression aleatoire nous montrons l existence et l unicite de solutions dansh1 r lorsque le bruit est additif puis dans le cas du bruit multiplicatif nous etablissons l existence de solutions martingales dansl2 r

well posedness and blow up solutions for an integrable nonlinearly dispersive model wave equation abstract we establish local well posedness in the sobolev space hs with any s 3 2 for an integrable nonlinearly dispersive wave equation arising as a model for shallow water waves known as the camassa holm equation however unlike the more familiar korteweg devries model we demonstrate conditions on the initial data that lead to finite time blow up of certain solutions

nonlinear evolution equations with a fractional derivative on a half line abstract we study the initial boundary value problem on a half line for the nonlinear evolution equations with a fractional derivative we study traditionally important problems of a theory of nonlinear partial differential equations such as global in time existence of solutions to the initial boundary value problem and the asymptotic behavior of solutions for large time

well posedness for the stochastic higher order modified camassa holm equation abstract in this article the stochastic higher order modified camassa holm equation with additive noise is considered using the fourier restriction norm method and bilinear estimates the local existence and uniqueness result is proved by the contraction mapping principle the global existence of the h 1 r solution of this problem is also obtained by energy inequality

analyticity of the cauchy problem for an integrable evolution equation in this paper we consider the periodic cauchy problem for the camassa holm equation with analytic initial data and prove that its solutions are analytic in both variables globally in space and locally in time

the limit behavior of the solutions to a class of nonlinear dispersive wave equations in this paper we study the limit behavior of the solutions to a class of nonlinear dispersive wave equations we also demonstrate that the solutions to eq 1 1 converge to the solution to the corresponding bbm equation as the parameter converges to zero and the convergence of solutions to eq 1 1 as 2 0 is studied in hs r s 32 given the discussion of the parameters in the nonlinear dispersive wave equation 1 1 one will obtain some conditions in which compacton and peakon occur

smoothing properties and retarded estimates for some dispersive evolution equations smoothing properties in the form of space time integrability properties play an important role in the study of dispersive evolution equations a number of them follow from a combination of general arguments and specific estimates we present a general formulation which makes the separation between the two types of ingredients as clear as possible and we illustrate it with the examples of the schrodinger equation of the wave equation and of a class of 1 1 dimensional equations related to the benjamin ono equation of special interest for the cauchy problem are retarded estimates expressed in terms of those properties we derive a number of such estimates associated with the last example and we mention briefly an application of those estimates to the cauchy problem for the generalized benjamin ono equation

low regularity solutions of the periodic general degasperis procesi equation abstract this paper studies low regularity solutions of the periodic general degasperis procesi equation with an initial value the existence and the uniqueness of solutions are proved the results are illustrated by considering the periodic peakons of the periodic general degasperis procesi equation

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

exact controllability and stabilizability of the korteweg de vries equation in this paper we consider distributed control of the system described by the korteweg de vries equation i atu uaxu e3 u f on the interval 0 0 with periodic boundary conditions ii au 27r t ku o t k 0 1 2 where the distributed control f f x t is restricted so that the volume f u x t dx of the solution is conserved both exact controllability and stabilizibility questions are studied for the system in the case of open loop control if the control f is allowed to act on the whole spatial domain 0 27r it is shown that the system is globally exactly controllable i e for given t 0 and functions q x x with the same volume one can alway find a control f so that the system i ii has a solution u x t satisfying u x 0 0 x u x t x if the control f is allowed to act on only a small subset of the domain 0 27r then the same result still holds if the initial and terminal states b and 0 have small amplitude in a certain sense in the case of closed loop control the distributed control f is assumed to be generated by a linear feedback law conserving the volume while monotonically reducing f u x t 2dx the solutions of the resulting closed loop system are shown to have uniform exponential decay to a constant state as in the open loop control case a small amplitude assumption is needed if the control is allowed to act on only a small subdomain the smoothing property of the periodic linear kdv equation discovered recently by bourgain has played an important role in establishing the exact controllability and stabilizability results presented in this paper

asymptotic smoothing and the global attractor of a weakly damped kdv equation on the real line abstract the existence of the global attractor of a weakly damped forced korteweg de vries equation in the phase space l 2 r is proved an optimal asymptotic smoothing effect of the equation is also shown namely that for forces in l 2 r the global attractor in the phase space l 2 r is actually a compact set in h 3 r the energy equation method is used in conjunction with a suitable splitting of the solutions the dispersive regularization properties of the equation in the context of bourgain spaces are extensively exploited

strichartz estimates for dispersive equations and solvability of the kawahara equation abstract we first establish a series of strichartz estimates for a general class of linear dispersive equations by applying the theory of oscillatory integrals established by kenig ponce and vega next we use such estimates to study solvability of the cauchy problem of the kawahara equation t u a u x u x 3 u x 5 u 0 in the class c r h s r local existence is proved for s 1 4 and global existence is proved for s 2

taylor series expansion for solutions of the korteweg devries equation with respect to their initial values abstract the initial value problem ivp for the korteweg devries kdv equation t u u x u 3 x u 0 u x 0 x for x r t r establishes a nonlinear map k from the space hs r to the space c t t hs r it has been known for many years that this map k is continuous bona and smith 1975 kato 1983 and it was proved recently that this map is lipschitz continuous kenig et al 1993 it is shown in this paper that the nonlinear map k is infinitely many times frechet differentiable from hs r to c t t hs r furthermore it is proved that k has a taylor series expansion at any given hs r i e formula where kn the nth derivative of k at is an n linear map from the n fold space of hs r to c t t hs r and the series converges in the space c t t hs r uniformly for h for some 0 each term yn kn hn in the series solves a linearized kdv equation thus any solutions of the ivp can be obtained by solving a series of linear problems by contrast the corresponding map kp established by the initial value problem for the periodic kdv equation t u u x u 3 x u 0 u x 0 x for x s t r where s is the unit length circle in the plane is known to be continuous only from hs s to c t t hs s this is due to the lack of smoothing effects for solutions of the periodic kdv equation in this paper it is shown that kp is lipschitz continuous from hs 1 s to c t t hs s and is n times frechet differentiable from hs n 1 s to c t t hs s for any n 1 the method developed in this paper applies to other nonlinear dispersive wave equations

white noise driven korteweg de vries equation abstract we consider a stochastic korteweg de vries equation on the real line the noise is additive we use function spaces similar to those introduced by bourgain to prove well posedness results for the korteweg de vries equation in l 2 r we are able to handle a noise which is locally white in space and time more precisely it is a space time white noise multiplied by an l 2 function of the space variable due to the lack of a priori estimates we can only get a local existence result in time however we obtain the global existence of l 2 r solutions when the covariance operator of the noise is hilbert schmidt in l 2 r

algebraic expressions of the clebsch gordon coefficients of the point group t a method for finding algebraic expressions of the clebsch gordan cg coefficients of point groups is proposed and applied to the tetrahedral group it is shown that in constructing the cg coefficients the irreducible symmetry operator iso of a double group g can be replaced by an effective iso which is much simpler than the usual iso the effective iso for the group chain t c3 is p 3d c2z c2z where d c2z is the matrix of c2z in the irrep of t with this effective iso and the algebraic expression of d c2z the algebraic expressions are derived for the real cg coefficients of t in the group chain t c3 the algebraic expressions for the complex real cg coefficients of the group chain t d2 c2 t c2 have also been obtained

well posedness of the initial value problem for the korteweg de vries equation 1 1 itu axu u1xu o x t e r u x 0 uo x the kdv equation which was first derived as a model for unidirectional propagation of nonlinear dispersive long waves 21 has been considered in different contexts namely in its relation with the inverse scattering method in plasma physics and in algebraic geometry see 24 and references therein our purpose is to study local and global well posedness of the ivp 1 1 in classical sobolev spaces hs r we shall say that the ivp 1 1 is locally resp globally well posed in the function space x if it induces a dynamical system on x by generating a continuous local resp global flow it was established in the works of bona and smith 3 bona and scott 2 saut and temam 30 and kato 1 5 that the ivp 1 1 is locally resp globally well posed in hs with s 3 2 resp s 2 roughly speaking global well posedness in hs depends on the available local theory and on the conservation laws satisfied by solutions of 1 1 namely

accelerating benders decomposition algorithmic enhancement and model selection criteria this paper proposes methodology for improving the performance of benders decomposition when applied to mixed integer programs it introduces a new technique for accelerating the convergence of the algorithm and theory for distinguishing good model formulations of a problem that has distinct but equivalent mixed integer programming representations the acceleration technique is based upon selecting judiciously from the alternate optima of the benders subproblem to generate strong or pareto optimal cuts this methodology also applies to a much broader class of optimization algorithms that includes dantzig wolfe decomposition for linear and nonlinear programs and related cutting plane type algorithms that arise in resource directive and price decomposition when specialized to network location problems this cut generation technique leads to very efficient algorithms that exploit the underlying structure of these models in discussing the proper formulation of mixed integer programs we suggest criteri

the cauchy problem for the fourth order nonlinear schrodinger equation related to the vortex filament abstract the cauchy problem of one dimensional fourth order nonlinear schrodinger equation related to the vortex filament is studied local well posedness for initial data in h s r s 1 2 is obtained by the fourier restriction norm method under certain coefficient condition

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

refined energy inequality with application to well posedness for the fourth order nonlinear schrodinger type equation on torus we consider the time local and global well posedness for the fourth order nonlinear schrodinger type equation 4nls on the torus the nonlinear term of 4nls contains the derivatives of unknown function and this prevents us to apply the classical energy method to overcome this difficulty we introduce the modified energy and derive an a priori estimate for the solution to 4nls

global solutions of the derivative schrodinger equation in a class of functions analytic in a strip lower bounds on the rate of decrease in time of a uniform radius of spatial analyticity for solutions of the derivative schrodinger equation are derived the bounds depend algebraically on time they are valid as long as the initial datum is of order one in l2 norm and satisfies suitable spatial decay requirements on the real axis

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

global solutions of the klein gordon schrodinger system with rough data in r 2 1 abstract in this paper we consider the klein gordon schrodinger system with quadratic yakuwa coupling and cubic autointeractions in r 2 1 and prove the existence and uniqueness of global solution for rough data the techniques to be used are adapted from a general scheme originally introduced by j bourgain to split the data into the low and high frequency parts

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

gevrey regularity for a class of water wave models abstract local well posedness for a class of higher order nonlinear dispersive partial differential equations is obtained in spaces of functions analytic on a strip around the real axis the proof relies on estimates in space time norms adapted to the linear part of the equation the class of equations in view contains a number of equations arising in the modeling of waves in fluids and also in other applications