463 research outputs found

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

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    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

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    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

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    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

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    In this paper, we consider global solutions of the following nonlinear Schr\"odinger equation iut+Δu+λuαu=0,iu_t+\Delta u+\lambda|u|^\alpha u = 0, in RN,\R^N, with λR,\lambda\in\R, α(0,4N2)\alpha\in(0,\frac{4}{N-2}) (α(0,)(\alpha\in(0,\infty) if N=1)N=1) and \linebreak u(0)XH1(RN)L2(x2;dx).u(0)\in X\equiv H^1(\R^N)\cap L^2(|x|^2;dx). We show that, under suitable conditions, if the solution uu satisfies eitΔu(t)u±0e^{-it\Delta}u(t)-u_ \pm\to0 in XX as t±t\to\pm\infty then u(t)eitΔu±0u(t)-e^{it\Delta}u_\pm\to0 in XX as t±.t\to\pm\infty. We also study the converse. Finally, we estimate u(t)XeitΔu±X|\:\|u(t)\|_X-\|e^{it\Delta}u_\pm\|_X\:| under some less restrictive assumptions
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