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

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

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

price dynamics of the financial markets using the stochastic differential equation for a potential double well abstract we have used the ito s stochastic differential equation for the double well with additive white noise as a mathematical model for price dynamics of the financial market we have presented a model which allows us to test within the same framework the comparative explanatory power of rational agents versus irrational agents with respect to the facts of financial markets we have obtained the mean price in terms of the parameter that represents the force of the randomness term of the model

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

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

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

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

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

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

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

a numerical solution for the simultaneous disturbance decoupling and row by row decoupling problem the numerical computation of the simultaneous disturbance decoupling and row by row decoupling problem is considered based on a condensed form for the system matrices the solvability of the simultaneous disturbance decoupling and row by row decoupling problem is reduced to the verification of an integer equality and the nonsingularity of a lower dimensional constant matrix the condensed form we use is computed using only orthogonal transformations which can be implemented in a numerically stable way hence it leads directly to an effective numerical method for solving the disturbance decoupling and row by row decoupling problem using existing tools such as lapack and matlab our result complements existing geometric and structural approaches which are of a theoretical nature and are not suitable for numerical computation