463 research outputs found

### Long Range Scattering and Modified Wave Operators for the Maxwell-Schr"odinger System II. The general case

We study the theory of scattering for the Maxwell-Schr"odinger system in
space dimension 3, in the Coulomb gauge. We prove the existence of modified
wave operators for that system with no size restriction on the Schr"odinger and
Maxwell asymptotic data and we determine the asymptotic behaviour in time of
solutions in the range of the wave operators. The method consists in partially
solving the Maxwell equations for the potentials, substituting the result into
the Schr"odinger equation, which then becomes both nonlinear and nonlocal in
time. The Schr"odinger function is then parametrized in terms of an amplitude
and a phase satisfying a suitable auxiliary system, and the Cauchy problem for
that system, with prescribed asymptotic behaviour determined by the asymptotic
data, is solved by an energy method, thereby leading to solutions of the
original system with prescribed asymptotic behaviour in time. This paper is the
generalization of a previous paper with the same title. However it is entirely
selfcontained and can be read without any previous knowledge of the latter.Comment: latex 96 page

### Long Range Scattering and Modified Wave Operators for some Hartree Type Equations III. Gevrey spaces and low dimensions

We study the theory of scattering for a class of Hartree type equations with
long range interactions in arbitrary space dimension n > or = 1, including the
case of Hartree equations with time dependent potential V(t,x) = kappa t^(mu -
gamma) |x|^{- mu} with 0 < gamma < or =1 and 0 < mu < n.This includes the case
of potential V(x) = kappa |x|^(-gamma) and can be extended to the limiting case
of nonlinear Schr"odinger equations with cubic nonlinearity kappa t^(n- gamma)
u|u|^2.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 (math.AP/9807031 and math.AP/9903073) which covered the range 0 < gamma
< or = 1, but only 0 < mu < or = n-2, and were therefore restricted to space
dimension n>2.Comment: TeX, 96 pages, available http://qcd.th.u-psud.f

### Scattering for the Zakharov system in 3 dimensions

We prove global existence and scattering for small localized solutions of the
Cauchy problem for the Zakharov system in 3 space dimensions. The wave
component is shown to decay pointwise at the optimal rate of t^{-1}, whereas
the Schr\"odinger component decays almost at a rate of t^{-7/6}.Comment: Minor changes and referee's comments include

### Convergence to Scattering States in the Nonlinear Schr\"odinger Equation

In this paper, we consider global solutions of the following nonlinear
Schr\"odinger equation $iu_t+\Delta u+\lambda|u|^\alpha u = 0,$ in $\R^N,$ with
$\lambda\in\R,$ $\alpha\in(0,\frac{4}{N-2})$ $(\alpha\in(0,\infty)$ if $N=1)$
and \linebreak $u(0)\in X\equiv H^1(\R^N)\cap L^2(|x|^2;dx).$ We show that,
under suitable conditions, if the solution $u$ satisfies $e^{-it\Delta}u(t)-u_
\pm\to0$ in $X$ as $t\to\pm\infty$ then $u(t)-e^{it\Delta}u_\pm\to0$ in $X$ as
$t\to\pm\infty.$ We also study the converse. Finally, we estimate
$|\:\|u(t)\|_X-\|e^{it\Delta}u_\pm\|_X\:|$ under some less restrictive
assumptions

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