well posedness for nonlinear schrodinger equations with boundary forces in low dimensions by strichartz estimates abstract in this paper we study the well posedness of solutions for nonlinear schrodinger equations on one and two dimensional domains with boundary where the boundary is disturbed by an external inhomogeneous type of dirichlet or neumann force we first prove the local existence of solutions at the energy level for quadratic and superquadratic sources using the strichartz estimates on domains secondly we obtain conditional uniqueness and local stability then we prove the boundedness of solutions in the energy space to pass from the local theory to the global theory regarding subquadratic sources we appeal to classical methods and trudinger s inequality to prove the uniqueness which combined with the existence of weak energy solutions mass and energy inequalities eventually implies the continuity of solutions in time

monotonicity method applied to the complex ginzburg landau and related equations abstract global existence of unique strong solutions is established for the complex ginzburg landau equation tu i u i u p 1u u 0 where 0 0 r p 1 and 1 2 p p 1 the key is a new inequality in monotonicity methods it is based on the sectorial estimates of in lp 1 and the nonlinear operator u u p 1u appearing in the equation the key inequality also yields the global existence of unique strong solutions of the nonlinear schrodinger type equation with monotone nonlinearity tu i u u p 1u 0 for all p 1

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

strichartz inequalities and the nonlinear schrodinger equation on compact manifolds kklwe prove strichartz estimates with fractional loss of derivatives for the schrodinger equation on any riemannian compact manifold as a consequence we infer low regularity local well posedness results in any dimension as well as global existence results for the cauchy problem of nonlinear schrodinger equations on surfaces in the case of defocusing polynomial nonlinearities and on three manifolds in the case of cubic defocusing nonlinearities we also discuss the optimality of these strichartz estimates on spheres

global existence and open loop exponential stabilization of weak solutions for nonlinear schrodinger equations with localized external neumann manipulation abstract in this paper we consider the nonlinear schrodinger equations nls with focusing defocusing interior source and possibly nonlinear damping on a bounded regular domain in the euclidean space moreover it is assumed that the solutions are subject to external neumann boundary manipulation on one portion of the boundary our aim is to obtain global existence of the weak solutions under various assumptions on the sign of the source and powers of the nonlinearities in the case of a linear damping we also prove that solutions decay exponentially under the assumption that the neumann type control at the boundary decays in a similar manner

well posedness and energy decay estimates in the cauchy problem for the damped defocusing schrodinger equation abstract in this paper we study the existence at the h 1 level as well as the stability for the damped defocusing schrodinger equation in r d the considered damping coefficient is time dependent and may vanish at infinity to prove the existence we employ the method devised by ozsari kalantarov and lasiecka 27 which is based on monotone operators theory in particular when d 1 or d 2 we obtain the uniqueness decay estimates for the l 2 level and h 1 l p 2 level energies are established with the help of direct multipliers method coupled with refined energy estimates and a lower semi continuity argument

on smooth solutions to the initial boundary value problem for the nonlinear schro dinger equation in two space dimensions on considere le probleme aux valeurs limites et initiales iu t u f u 2 u 0 x t 0 u x 0 u 0 x x u 0 t 0 ou est un domaine borne de r 2 a frontiere lisse

exact solutions of nonlinear diffusion equations by variational iteration method in this paper he s variational iteration method is applied to obtain exact solutions of some nonlinear diffusion equations the variational iteration method is used to construct correction functionals using general lagrange multipliers identified optimally via the variational theory and the initial approximations can be freely chosen with unknown constants the solutions obtained are compared with those obtained by the adomian decomposition method and homotopy perturbation method showing excellent agreement but the variational iteration method is more effective he s variational iteration method can be introduced to overcome the difficulties arising in calculating adomian polynomials

on the topology of the lorenz system we present a new paradigm for three dimensional chaos and specifically for the lorenz equations the main difficulty in these equations and for a generic flow in dimension 3 is the existence of singularities we show how to use knot theory as a way to remove the singularities specifically we claim i for certain parameters the lorenz system has an invariant one dimensional curve which is a trefoil knot the knot is a union of invariant manifolds of the singular points ii the flow is topologically equivalent to an anosov flow on the complement of this curve and moreover to a geodesic flow iii when varying the parameters the system exhibits topological phase transitions i e for special parameter values it will be topologically equivalent to an anosov flow on a knot complement different knots appear for different parameter values and each knot controls the dynamics at nearby parameters

weakly damped focusing nonlinear schrodinger equations with dirichlet control abstract in this article we consider the weakly damped focusing nonlinear schrodinger equations on bounded domains at the natural h 1 energy level with dirichlet control acting on a portion of the boundary we introduce the dynamic extension method for homogenizing the inhomogeneous boundary input then we construct approximate solutions using monotone operator theory a hidden trace regularity is proved to control the norm of the solutions in a global sense this allows the use of compactness techniques by which we prove the existence of weak solutions finally using multiplier techniques we prove the exponential decay of solutions under the assumption that the boundary control also decays in a similar fashion

qualitative properties of solutions for nonlinear schrodinger equations with nonlinear boundary conditions on the half line in this paper we study the interaction between a nonlinear focusing robin type boundary source a nonlinear defocusing interior source and a weak damping term for nonlinear schrodinger equations posed on the infinite half line we construct solutions with negative initial energy satisfying a certain set of conditions which blow up in finite time in the h1 sense we obtain a sufficient condition relating the powers of nonlinearities present in the model which allows construction of blow up solutions in addition to the blow up property we also discuss the stabilization property and the critical exponent for this model

qualitative aspects for the cubic nonlinear schrodinger equations with localized damping exponential and polynomial stabilization abstract results of exponential polynomial decay rates of the energy in l 2 level related to the cubic nonlinear schrodinger equation with localized damping posed on the whole real line will be established in this paper we use kato s theory and a priori estimates to obtain the result of global well posedness and we determine exponential polynomial stabilization combining the ideas of unique continuation due to zhang semigroup property and komornik s approach

an inhomogeneous boundary value problem for nonlinear schrodinger equations abstract we consider a nonlinear schrodinger equation in a domain r n with the inhomegeneous dirichlet boundary condition u q where q is a given smooth function the nonlinear term contributes a positive term to the energy we prove the existence of global solutions of finite energy

stabilization and control for the nonlinear schrodinger equation on a compact surface in this paper we study the stabilization property and the exact controllability for the nonlinear schrodinger equation on a two dimensional compact riemannian manifold without boundary we use a pseudo differential dissipation the proofs are based on a result of propagation of singularities and on recent dispersion estimates strichartz type inequalities due to n burq p gerard and n tzvetkov

exact boundary controllability of the nonlinear schrodinger equation abstract this paper studies the exact boundary controllability of the semi linear schrodinger equation posed on a bounded domain r n with either the dirichlet boundary conditions or the neumann boundary conditions it is shown that if s n 2 or 0 s n 2 with 1 n 2 2 s or s 0 1 with n 2 then the systems are locally exactly controllable in the classical sobolev space h s around any smooth solution of the cubic schrodinger equation

uniform decay rate estimates for schrodinger and plate equations with nonlinear locally distributed damping on a compact n dimensional riemannian manifold m g we establish uniform decay rate estimates for the linear schrodinger and plate equations subject to an internal nonlinear damping locally distributed on the manifold our approach can be also employed for other equations provided that inverse inequality for the linear model occurs in the particular case of the wave equation where the well known geometric control condition gcc is equivalent to the observability inequality our method generalizes the results due to cavalcanti et al 2010 2009 9 10 regarding the optimal choice of dissipative regions

exact traveling solutions of some nonlinear evolution equation by g g expansion method in this paper we establish exact solutions for some nonlinear evolution equations the g g expansion method was used to construct traveling wave solutions of ramani s equations in this method we take the advantages of the general solutions of the second order lode to solve effectively many nonlinear evolution equations the g g expansion method presents a wider applicability for handling nonlinear wave equations

existence of solutions for schrodinger evolution equations we study the existence uniqueness and regularity of the solution of the initial value problem for the time dependent schrodinger equationi u t 1 2 u v t x u u 0 u0 we provide sufficient conditions onv t x such that the equation generates a unique unitary propagatoru t s and such thatu t s u0ec1 l2 c0 h2 n foru0eh2 n the conditions are general enough to accommodate moving singularities of type x 2 n 4 or x n 2 n 3

on global solutions to the initial boundary value problem for the damped nonlinear schrodinger equations where a 0 and q is a bounded domain in rn with smooth boundary x when a 0 undamped case the global existence of smooth solutions to the initial boundary value problem l 3 has been proved only when n 2 and f lu12 w lulp with p 2 brezis and gallouet cl with 2 i p i3 m tsutsumi 9 higher dimensional cases are very difficult pribl s and remain unproved as far as the author knows if we add a linear damping term iau a 0 then the mass ilull iztr1 and the energy ivu x t l dx j f lu x t l dx decay exponentially as t cc as is seen below which may assert the global existence of small smooth solutions even if the nonlinear term f lul u does not have a proper sign it is our purpose to establish the global existence and decay estimates of smooth solutions of the problem l 3 with small data there are a number of works on the nonlinear evolution equations see 3 5 71 and their references in the following spirit energy and decay estimates give global existence of solutions if the damping term is not added there is no hope that the decay estimates of solutions in some f q p 2 2 can be obtained since

a simple approach to the cauchy problem for complex ginzburg landau equations by compactness methods abstract this paper is concerned with the cauchy problem cgl in l 2 r n for complex ginzburg landau equations with laplacian and nonlinear term u q 2 u multiplied by the complex coefficients i and i respectively q 2 0 0 r the global existence of strong solutions to cgl is established without any upper restriction on q 2 but with some restriction on and the result corresponds to ginibre and velo 1996 3 proposition 5 1 which is technically proved by combining convolution regularizing methods with compactness localizing methods while our proof here is fairly simplified the key to our proof is the cauchy problem cgl r which is cgl with replaced with v r where v r x x r 2 x r v r x 0 x r the solvability of cgl r is a direct consequence of okazawa and yokota 2002 16 theorem 4 1 taking the limit of global strong solutions to cgl r as r yields a global strong solution to cgl the result gives also an unbounded version of okazawa and yokota 2002 16 theorem 1 1 with p 2 for the initial boundary value problem on bounded domains

uniform decay rates for the energy of weakly damped defocusing semilinear schrodinger equations with inhomogeneous dirichlet boundary control abstract in this paper we study the open loop stabilization as well as the existence and regularity of solutions of the weakly damped defocusing semilinear schrodinger equation with an inhomogeneous dirichlet boundary control first of all we prove the global existence of weak solutions at the h 1 energy level together with the stabilization in the same sense it is then deduced that the decay rate of the boundary data controls the decay rate of the solutions up to an exponential rate secondly we prove some regularity and stabilization results for the strong solutions in h 2 sense the proof uses the direct multiplier method combined with monotonicity and compactness techniques the result for weak solutions is strong in the sense that it is independent of the dimension of the domain the power of the nonlinearity and the smallness of the initial data however the regularity and stabilization of strong solutions are obtained only in low dimensions with small initial and boundary data