smoothing effect for some schrodinger equations abstract we study the cauchy problem for the following schrodinger equation in r n n n i t u 1 2 u v 1 u v 2 u 2 u t x r r n u 0 x r n where v 1 v 1 x 1 x 1 v 2 v 2 x k 2 3 k x k k r 1 k 3 0 1 min 2 n 2 0 2 3 min 2 n we prove the existence uniqueness and smoothing effect of global solutions of with not necessarily in the sobolev space h1 rn

asymptotic behavior for solutions to the dissipative nonlinear schrodinger equations with the fractional sobolev space we study the cauchy problem for the nonlinear schrodinger equations with dissipative nonlinearity in particular we show the time decay estimate for global solutions in lebesgue space of square integrable functions main results of this paper are the extension of results that have been proved in the work of hayashi et al adv math phys 5 3702738 2016 we study the cauchy problem for the nonlinear schrodinger equations with dissipative nonlinearity in particular we show the time decay estimate for global solutions in lebesgue space of square integrable functions main results of this paper are the extension of results that have been proved in the work of hayashi et al adv math phys 5 3702738 2016

nonexistence of asymptotically free solutions for a nonlinear schrodinger equation let u be a nontrivial smooth solution to iut u u p 1u if n 1 and 2 p 3 then there does not exist any finite energy free solution v such that u t v t 2 0 as t this extends a theorem of strauss in which the same result was proved for 1 p 2

nonexistence of the usual scattering states for the generalized ostrovsky hunter equation we consider the cauchy problem for the generalized ostrovsky hunter equation utx u u 1u xx we prove the nonexistence of the usual scattering states

modified scattering for the critical nonlinear schrodinger equation abstract we consider the nonlinear schrodinger equation i u t u u 2 n u in all dimensions n 1 where c and 0 we construct a class of initial values for which the corresponding solution is global and decays as t like t n 2 if 0 and like t log t n 2 if 0 moreover we give an asymptotic expansion of those solutions as t we construct solutions that do not vanish so as to avoid any issue related to the lack of regularity of the nonlinearity at u 0 to study the asymptotic behavior we apply the pseudo conformal transformation and estimate the solutions by allowing a certain growth of the sobolev norms which depends on the order of regularity through a cascade of exponents

global existence of solutions for a subcritical nonlinear schrodinger equation abstract we consider the one dimensional nonlinear schrodinger equation i u t 1 2 u x x n u u x r t 1 u 1 x u 0 x x r where the nonlinearity n u u u 2 u 3 u 3 2 u 2 the exponent 0 is sufficiently small our purpose in this paper is to prove a global existence in time of small solutions

asymptotic expansion of the solution to the nonlinear schrodinger equation with nonlocal interaction this paper deals with the equationiut 1 2 u x 1 u 2 u u 0 x u0 x here u is a complex valued function of t x r rn n 2 and is a real number if u0 is small in l2 s with s n 2 2 then the solution u t behaves asymptotically asu t x it n 2exp i x 2 2t is t x t x t t 1 j 02 1 j x t logt j o t n 2 1 uniformly in rn as t here is a suitable function called the modified scattering state and the functions s 1 j j 0 1 2 are represented explicitly by using

asymptotics of odd solutions for cubic nonlinear schrodinger equations we consider the cauchy problem for a cubic nonlinear schrodinger equation in the case of an odd initial data from h2 h0 2 we prove the global existence in time of solutions to the cauchy problem and construct the modified asymptotics for large values of time

probabilistic analysis of bin packing heuristics in this paper we present both a probabilistic and statistical analysis of several bin packing heuristics with mild conditions on the distribution of problem data we show that the ratio of the expected number of bins packed for these heuristics to the number of elements converges to some constant this constant can either be derived analytically or estimated through simulation we analyze the variance in performance of these heuristics and we further show that as the number of elements n increases the probability that the number of bins required for these heuristics exceeds any fixed percentage of their expected performance goes to zero

asymptotic behavior of the nonlinear schrodinger equation with harmonic trapping we consider the cubic nonlinear schrodinger equation with harmonic trapping on d 1 d 5 in the case when all directions but one are trapped aka cigar shaped trap we prove modified scattering and construct modified wave operators for small initial and final data respectively the asymptotic behavior turns out to be a rather vigorous departure from linear scattering and is dictated by the resonant system of the nls equation with full trapping on d 1 in the physical dimension d 3 this system turns out to be exactly the cr equation derived by faou germain and the first author as the large box limit of the resonant nls equation in the homogeneous zero potential setting the special dynamics of the latter equation combined with the above modified scattering results allow us to justify and extend some physical approximations in the theory of bose einstein condensates in cigar shaped traps 2015 wiley periodicals inc

inhomogeneous dirichlet boundary value problem for one dimensional nonlinear schrodinger equations via factorization techniques abstract we consider the inhomogeneous dirichlet boundary value problem for the cubic nonlinear schrodinger equations on the half line we present sufficient conditions of initial and boundary data which ensure asymptotic behavior of small solutions to equations by using the classical energy method and factorization techniques

asymptotic stability of solitons for mkdv abstract we prove a full asymptotic stability result for solitary wave solutions of the mkdv equation we consider small perturbations of solitary waves with polynomial decay at infinity and prove that solutions of the cauchy problem evolving from such data tend uniformly on the real line to another solitary wave as time goes to infinity we describe precisely the asymptotics of the perturbation behind the solitary wave showing that it satisfies a nonlinearly modified scattering behavior this latter part of our result relies on a precise study of the asymptotic behavior of small solutions of the mkdv equation

sharp l r asymptotics of the small solutions to the nonlinear schrodinger equations this paper studies the large time behavior of the small solution to the nonlinear schrodinger equation with power type nonlinearity if the power is large enough then it is well known that the nonlinear solution asymptotically behaves like a linear solution as t see e g j funct anal 32 1979 1 j math pures appl 64 1985 363 our concern at the present work is to determine the sharp decay rate of the difference between the nonlinear solution and the linear solution in lr 2 r spaces

nonlinear fractional schrodinger equations in one dimension abstract we consider the question of global existence of small smooth and localized solutions of a certain fractional semilinear cubic nls in one dimension i t u u c 0 u 2 u c 1 u 3 c 2 u u 2 c 3 u 3 x x 1 2 where c 0 r and c 1 c 2 c 3 c this model is motivated by the two dimensional water wave equation which has a somewhat similar structure in the eulerian formulation in the case of irrotational flows we show that one cannot expect linear scattering even in this simplified model more precisely we identify a suitable nonlinear logarithmic correction and prove global existence and modified scattering of solutions

fractional nonlinear schrodinger equation of order α 0 1 abstract we consider the cauchy problem for the fractional nonlinear schrodinger equation i t u 1 x u u 2 u t 0 x r u 0 x u 0 x x r where r the order of the fractional derivative 0 1 we obtain the large time asymptoitic behavior of solutions which has a logarithmic phase modifications for large time comparing with the linear problem

long range scattering for nonlinear schrodinger equations in one space dimension we consider the scattering problem for the nonlinear schrodinger equation in 1 1 dimensions open image in new window where x r 0 r p 3 we show that modified wave operators for exist on a dense set of a neighborhood of zero in the lebesgue spacel2 r or in the sobolev spaceh1 r the modified wave operators are introduced in order to control the long range nonlinearity u 2u

multidimensional inhomogeneous mixed initial boundary value problem for the nonlinear schrodinger equation abstract we consider the inhomogeneous mixed initial boundary value problem for the nonlinear multidimensional schrodinger equation formulated on upper right quarter plane we study traditionally important problems of the 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 also we are interested in the study of the influence of the mixta boundary data on the asymptotic behavior of solutions our approach to get well posedness of nonlinear problems is based on the studying a linear theory and then using the fixed point argument to get a linear theory for the multidimensional model we propose general method based on laplace approach and theory cauchy type integral equations to get smooth solutions in l we modify a method based on the factorization for the free schrodinger evolution group the advantage of our approach is that it can also be applied to non integrable equations and arbitrary boundary conditions this approach is new and it is not standard we believe that the results of this paper could be applicable to study a wide class of dissipative multidimensional nonlinear equations by the use of techniques of nonlinear analysis

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

quadratic nonlinear klein gordon equation in one dimension we study the initial value problem for the quadratic nonlinear klein gordon equation vtt v vxx v2 t r x r with initial conditions v 0 x v0 x vt 0 x v1 x x r where v0 and v1 are real valued functions r using the method of normal forms of shatah normal forms and quadratic nonlinear klein gordon equations commun pure appl math 38 685 696 1985 we obtain a sharp asymptotic behavior of small solutions without the condition of a compact support on the initial data which was assumed in the previous work of j m delort existence globale et comportement asymptotique pour l equation de klein gordon quasi lineaire a donnees petites en dimension 1 ann sci ec normale super 34 4 1 61 2001

long range scattering and modified wave operators for some hartree type equations iii gevrey spaces and low dimensions abstract we study the theory of scattering for a class of hartree type equations with long range interactions in arbitrary space dimension n 1 including the case of hartree equations with time dependent potential v t x t x with 0 1 and 0 n this includes the case of potential v x x and can be extended to the limiting case of nonlinear schrodinger equations with cubic nonlinearity t n u 2 u using gevrey spaces of asymptotic states and solutions we prove the existence of modified local wave operators at infinity with no size restriction on the data and we determine the asymptotic behaviour in time of solutions in the range of the wave operators thereby extending the results of previous papers which covered the range 0 1 but only 0 n 2 and were therefore restricted to space dimension n 3

a system of quadratic nonlinear klein gordon equations in 2d abstract we consider the system of quadratic nonlinear klein gordon equations t 2 v 1 v 1 m 1 2 v 1 v 1 v 2 t 2 v 2 v 2 m 2 2 v 2 v 1 2 in two space dimensions under the mass condition such that 2 m 1 m 2 we construct the scattering operator in an almost natural weighted sobolev class h 1 1 or the scattering problem in lower order sobolev class h 4 3 1 h 1 where 0 is small

inhomogeneous neumann boundary value problem for one dimensional nonlinear schrodinger equations via factorization techniques we consider the inhomogeneous neumann boundary value problem for the nonlinear schrodinger equations on the half line we present sufficient conditions of initial and boundary data which ensure asymptotic behavior of small solutions to the equations by using the classical energy method and factorization techniques we consider the inhomogeneous neumann boundary value problem for the nonlinear schrodinger equations on the half line we present sufficient conditions of initial and boundary data which ensure asymptotic behavior of small solutions to the equations by using the classical energy method and factorization techniques

final state problem for the cubic nonlinear klein gordon equation we study the final state problem for the nonlinear klein gordon equation utt u uxx u3 t r x r where r we prove the existence of solutions in the neighborhood of the approximate solutions 2 re u t w t where u t is the free evolution group defined by u t f 1e it f x 1 x2 f and f 1 are the direct and inverse fourier transformations respectively and w t x f 1 u e 3 2 i 2 u 2log t with a given final data u is a real valued function and 3 i u l is small

inhomogeneous dirichlet boundary value problem for nonlinear schrodinger equations with a power nonlinearity on the upper half plane abstract we consider the inhomogeneous dirichlet boundary value problem for nonlinear schrodinger equations with a power nonlinearity in the upper half plane we present sufficient conditions of initial and boundary data which ensure asymptotic behavior of small solutions to the problem by using the classical energy method and factorization techniques

critical nonlinear schrodinger equations with data in homogeneous weighted l2 spaces abstract we consider the initial value problem for nonlinear schrodinger equations with the critical nonlinearities 1 u 2 n u where im 1 0 when the space dimension n 1 2 we prove the global existence of small solutions in homogeneous weighted l 2 r n spaces it is shown that the small solutions decay uniformly like t n 2 for t 1 if im 1 0 the higher uniform time decay rates t n 2 log t n 2 for t 1 are obtained if im 1 0

on asymptotic behavior of solutions to korteweg de vries type equations related to vortex filament with axial flow we study the global existence and asymptotic behavior in time of solutions to the korteweg de vries type equation called as hirota equation this equation is a mixture of cubic nonlinear schrodinger equation and modified korteweg de vries equation we show the unique existence of the solution for this equation which tends to the given modified free profile by using the two asymptotic formulae for some oscillatory integrals

asymptotics for large time of solutions to the nonlinear schrodinger and hartree equations we study the asymptotic behavior in time of solutions to the cauchy problems for the nonlinear schrodinger equation with a critical power nonlinearity and the hartree equation we prove the existence of modified scattering states and the sharp time decay estimate in the uniform norm of solutions to the cauchy problem with small initial data this estimate is very important for the proof of the existence of modified scattering states to the nonlinear schrodinger equations with a critical nonlinearity and the hartree equation in order to derive the desired estimates we introduce a certain phase function since the previous methods based solely on a priori estimates of the operator x it acting on the solution without specifying any phase function do not work for the critical case under consideration the well known nonexistence of the usual l 2 scattering states shows that our result is sharp

long range scattering for non linear schrodinger and hartree equations in space dimension n 2 we consider the scattering problem for the non linear schrodinger nls equation with a power interaction with critical powerp 1 2 n in space dimensionsn 2 and 3 and for the hartree equation with potential x 1 in space dimensionn 2 we prove the existence of modified wave operators in thel2 sense on a dense set of small and sufficiently regular asymptotic states

modulation spaces and non linear hartree type equations abstract we study the cauchy problem for hartree equation with cubic convolution nonlinearity f u k u 2 u under a specified condition on potential k with cauchy data in modulation spaces m p q r n we establish global well posedness results in m p p r n with 1 p 2 n n when k x x r 0 m i n 2 n 2 in m p q r n with 1 q p 2 when k m 1 r n

asymptotic behavior of solutions to the short pulse equation with critical nonlinearity abstract we consider the cauchy problem for the short pulse equation u t x u u 3 x x x r t 0 u 0 x u 0 x x r where u 0 is a real valued function we prove the global existence of small solutions to the short pulse equation moreover we give the l time decay estimate u t l c 1 t 1 2 and the asymptotic behavior of solutions

sharp asymptotic behavior of solutions for cubic nonlinear schrodinger equations with a potential we consider the cubic nonlinear schrodinger equation with a potential we obtain a sharp time decay for the global in time solution and we get the large time asymptotic profile of small solutions to the cauchy problem

nonlinear schrodinger equations with exceptional potentials abstract we consider the cubic nonlinear schrodinger equation with an exceptional potential we obtain a sharp time decay for the global in time solution and we get the large time asymptotic profile of small solutions we prove the existence of modified scattering for this model that is linear scattering modulated by a phase our approach is based on the spectral theorem for the perturbed linear schrodinger operator and a factorization technique that allows us to control the resonant nonlinear term we make some parity assumptions in order to control the small energy behavior of the scattering coefficients and of the wave functions

piecewise linear system modeling based on a continuous threshold decomposition the continuous threshold decomposition is a segmentation operator used to split a signal into a set of multilevel components this decomposition method can be used to represent continuous multivariate piecewise linear pwl functions and therefore can be employed to describe pwl systems defined over a rectangular lattice the resulting filters are canonical and have a multichannel structure that can be exploited for the development of rapidly convergent algorithms the optimum design of the class of pwl filters introduced in this paper can be postulated as a least squares problem whose variables separate into a linear and a nonlinear part based on this feature parameter estimation algorithms are developed first a block data processing algorithm that combines linear least squares with grid localization through recursive partitioning is introduced second a time adaptive method based on the combination of an rls algorithm for coefficient updating and a signed gradient descent module for threshold adaptation is proposed and analyzed a system identification problem for wave propagation through a nonlinear multilayer channel serves as a comparative example where the concepts introduced are tested against the linear volterra and neural network alternatives

modified scattering and beating effect for coupled schrodinger systems on product spaces with small initial data in this paper we study a coupled nonlinear schrodinger system with small initial data in a product space we establish a modified scattering of the solutions of this system and we construct a modified wave operator the study of the resonant system which provides the asymptotic dynamics allows us to highlight a control of the sobolev norms and interesting dynamics with the beating effect the proof uses a recent work of hani pausader tzvetkov and visciglia for the modified scattering and a recent work of grebert paturel and thomann for the study of the resonant system