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

distribution properties of compressing sequences derived from primitive sequences over bbz p e let z pe be the integer residue ring with odd prime p and integer e 2 any sequence a over z pe has a unique p adic expansion a a0 a1 p ae 1 pe 1 where ai can be regarded as a sequence over z p for 0 i e 1 let f x be a strongly primitive polynomial over z pe and a b be two primitive sequences generated by f x over z pe assume x0 xe 1 xe 1 x0 xe 2 is an e variable function over z p with the monomial p 1 2 xe 2 p 1 x1 p 1 not pearing in the expression of x0 x1 xe 2 it is shown that if there exists an s z p such that a0 t ae 1 t s if and only if b0 t be 1 t s for all nonnegative t with i 0 where is an m sequence determined by f x and a0 then a b this implies that for compressing sequences derived from primitive sequences generated by f x over z pe single element distribution is unique on all positions t with t 0 in particular when x0 x1 xe 2 0 it is a completion of the former result on the uniqueness of distribution of element 0 in highest level sequences

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

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

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

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

a note on the cluster variation method kikuchi s cluster variation method cvm is reformulated as the truncation of a mobius inversion an attempt is made to explicate and simplify the various approaches to the cvm this formulation makes apparent the connection of the method with other types of cluster approximation an illustration of the procedure is provided

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

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

the cauchy problem for critical and subcritical semilinear parabolic equations in lr ii initial data in critical sobolev spaces abstract by presenting some time space l p l r estimates we will establish the local and global existence and uniqueness of solutions for semilinear parabolic equations with the cauchy data in critical sobolev spaces of negative indices our results contain the complex derivative ginzburg landau equation and the cahn hilliard equation as special cases

compression mappings on primitive sequences over z p sup e let z p sup e be the integer residue ring with odd prime p spl ges 5 and integer e spl ges 2 for a sequence a over z p sup e there is a unique p adic expansion a a sub 0 a spl middot p a sub e 1 spl middot p sup e 1 where each a sub i is a sequence over 0 1 p 1 and can be regarded as a sequence over the finite field gf p naturally let f x be a primitive polynomial over z p sup e and g f x p sup e the set of all primitive sequences generated by f x over z p sup e set spl phi sub e 1 x sub 0 x sub e 1 x sub e 1 sup k spl eta sub e 2 1 x sub 0 x sub 1 x sub e 2 spl psi sub e 1 x sub 0 x sub e 1 x sub e 1 sup k spl eta sub e 2 2 x sub 0 x sub 1 x sub e 2 where spl eta sub e 2 1 and spl eta sub e 2 2 are arbitrary functions of e 1 variables over gf p and 2 spl les k spl les p 1 then the compression mapping spl phi sub e 1 g f x p sup e spl rarr gf p sup spl infin a spl rarr spl phi sub e 1 a sub 0 a sub e 1 is injective that is a b if and only if spl phi sub e 1 a sub 0 a sub e 1 spl phi sub e 1 b sub 0 b sub e 1 for a b spl isin g f x p sup e furthermore if f x is a strongly primitive polynomial over z p sup e then spl phi sub e 1 a sub 0 a sub e 1 spl psi sub e 1 b sub 0 b sub e 1 if and only if a b and spl phi sub e 1 x sub 0 x sub e 1 spl psi sub e 1 x sub 0 x sub e 1 for a b spl isin g f x p sup e

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

global well posedness of the kp i initial value problem in the energy space we prove that the kp i initial value problem begin cases partial tu partial x 3u partial x 1 partial y 2u partial x u 2 2 0 text on mathbb r 2 x y times mathbb r t u 0 phi end cases is globally well posed in the energy space mathbf e 1 mathbb r 2 big phi mathbb r 2 to mathbb r phi mathbf e 1 mathbb r 2 approx phi l 2 partial x phi l 2 big partial x 1 partial y phi big l 2 infty big

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

on the distinctness of binary sequences derived from primitive sequences modulo square free odd integers let m be a square free odd integer and z m the integer residue ring modulo m this paper studies the distinctness of primitive sequences over z m modulo 2 recently for the case of m pq a product of two distinct prime numbers p and q the problem has been almost completely solved as for the case that m is a product of more prime numbers the problem has been quite resistant to proof in this paper a partial proof is given by showing that a class of primitive sequences of order 2n 1 over z m is distinct modulo 2 where n is a positive integer besides as an independent interest this paper also involves two distribution properties of primitive sequences over z m which are related closely to our main results

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 distinctness of modular reductions of maximal length sequences modulo odd prime powers we discuss the distinctness problem of the reductions modulo m of maximal length sequences modulo powers of an odd prime p where the integer m has a prime factor different from p for any two different maximal length sequences generated by the same polynomial we prove that their reductions modulo m are distinct in other words the reduction modulo m of a maximal length sequence is proved to contain all the information of the original sequence

landau lifshitz and higher order nonlinear systems gauge generated from nonlinear schrodinger type equations new landau lifshitz ll and higher order nonlinear systems gauge generated from nonlinear schrodinger ns type equations are presented the consequences of gauge equivalence between different dynamical systems are discussed the gauge connections among various ll and ns equations are found and depicted through a schematic representation

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

coloring random graphs and maximizing local diversity we study a variation of the graph coloring problem on random graphs of finite average connectivity given the number of colors we aim to maximize the number of different colors at neighboring vertices i e one edge distance of any vertex two efficient algorithms belief propagation and walksat are adapted to carry out this task we present experimental results based on two types of random graphs for different system sizes and identify the critical value of the connectivity for the algorithms to find a perfect solution the problem and the suggested algorithms have practical relevance since various applications such as distributed storage can be mapped onto this problem

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

partition function loop series for a general graphical model free energy corrections and message passing equations a loop series expansion for the partition function of a general statistical model on a graph is carried out if the auxiliary probability distributions of the expansion are chosen to be a fixed point of the belief propagation equation the first term of the loop series gives the bethe peierls bp free energy functional at the replica symmetric level of the mean field spin glass theory and corrections are contributed only by subgraphs that are free of dangling edges this result generalizes the early work of chertkov and chernyak on binary statistical models if the belief propagation equation has multiple fixed points a loop series expansion is performed for the grand partition function the first term of this series gives the bp free energy functional at the first step replica symmetry breaking rsb level of the mean field spin glass theory and corrections again come only from subgraphs that are free of dangling edges provided that the auxiliary probability distributions of the expansion are chosen to be a fixed point of the survey propagation equation the same loop series expansion can be performed for higher level partition functions obtaining the higher level rsb bp free energy functionals and the correction terms and message passing equations without using the bp approximation

arithmetic crosscorrelations of feedback with carry shift register sequences an arithmetic version of the crosscorrelation of two sequences is defined generalizing mandelbaum s 1967 arithmetic autocorrelations large families of sequences are constructed with ideal vanishing arithmetic crosscorrelations these sequences are decimations of the 2 adic expansions of rational numbers p q such that 2 is a primitive root module q

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

further result on distribution properties of compressing sequences derived from primitive sequences over bf z p e let p be an odd prime number e an integer greater than 1 and z pe the integer residue ring modulo pe in this paper we obtain an improved result of the previous paper ieee trans inf theory 56 1 2010 555 563 on distribution properties of compressing sequences derived from primitive sequences over z pe it is shown that two primitive sequences and b generated by a strongly primitive polynomial f x over z pe are the same if there exist s z p and k z p such that the distribution of in their compressing sequences ae 1 a0 ae 2 and be 1 b0 be 2 is coincident at the positions t with t k where x0 xe 2 is an e variable polynomial over z p with the coefficient of xe 2p 1 x1p 1x0p 1 not equal to 1 e p 1 2 and is an m sequence over z p determined by f x and a compared with the previous result this gives a more precise characterization on the positions of a compressing sequence i e of the form ae 1 a0 ae 2 derived from a primitive sequence a over z pe that completely determines a in particular the result is also true for the highest level sequence ae 1 by taking x0 xe 2 0

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

minimizing unsatisfaction in colourful neighbourhoods colouring sparse graphs under various restrictions is a theoretical problem of significant practical relevance here we consider the problem of maximizing the number of different colours available at the nodes and their neighbourhoods given a predetermined number of colours in the analytical framework of a tree approximation carried out at both zero and finite temperatures solutions obtained by population dynamics give rise to estimates of the threshold connectivity for the incomplete to complete transition which are consistent with those of existing algorithms the nature of the transition as well as the validity of the tree approximation are investigated

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

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 network source location problem ground state energy entropy and effects of freezing ground state entropy of the network source location problem is evaluated at both the replica symmetric level and one step replica symmetry breaking level using the entropic cavity method the regime that is a focus of this study is closely related to the vertex cover problem with randomly quenched covered nodes the resulting entropic message passing inspired decimation and reinforcement algorithms are used to identify the optimal location of sources in single instances of transportation networks the conventional belief propagation without taking the entropic effect into account is also compared we find that in the glassy phase the entropic message passing inspired decimation yields a lower ground state energy compared to the belief propagation without taking the entropic effect using the extremal optimization algorithm we study the ground state energy and the fraction of frozen hubs and extend the algorithm to collect statistics of the entropy the theoretical results are compared with the extremal optimization results

partial period distribution of fcsr sequences klapper and goresky 1995 introduced feedback with carry shift register fcsr and presented a significant kind of fcsr sequences that is l sequences they showed that the number of 0s and 1s occurring in one of their periods are equal we discuss the partial period distribution of l sequences and show that when the periods become large the proportion of 1s resp 0s occurring in any of their partial periods approximates 50

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

further results on the distinctness of binary sequences derived from primitive sequences modulo square free odd integers this paper studies the distinctness of primitive sequences over z m modulo 2 where m is an odd integer that is composite and square free and z m is the integer residue ring modulo m a new sufficient condition is given for ensuring that primitive sequences generated by a primitive polynomial f x over z m are pairwise distinct modulo 2 such result improves a recent result obtained in our previous paper and consequently the set of primitive sequences over z m that can be proven to be distinct modulo 2 is greatly enlarged

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

distribution properties of compressing sequences derived from primitive sequences modulo odd prime powers let a and b be primitive sequences over z pe with odd prime p and e 2 for certain compressing maps we consider the distribution properties of compressing sequences of a and b and prove that a b if the compressing sequences are equal at the times t such that t k where is a sequence related to a we also discuss the s uniform distribution property of compressing sequences for some compressing maps we have that there exist different primitive sequences such that the compressing sequences are s uniform we also discuss that compressing sequences can be s uniform for how many elements s

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

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

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

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

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

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 lower bound on the linear span of an fcsr we have derived a lower bound on the linear span of a binary sequence generated by a feedback with carry shift register fcsr under the following condition q is a power of a prime such that q r sup e e spl ges 2 and r 2p 1 where both r and p are 2 prime this allows us to design fcsr stream ciphers similar to previously proposed linear feedback shift register lfsr stream ciphers

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