Jean Ginibre

On a Class of non Linear Schrödinger Equations with non Local Interaction.

As regards the first question we prove existence and uniqueness of the solutions of the Cauchy problem with finite initial time; as regards the second one, we prove the existence of solutions of the Cauchy problem with infinite initial time, which implies the existence of the wave operators, and we prove asymptotic completeness for a class of repulsive interactions. The main reason for this investigation is that the Eq. (0.1) can be considered as the classical limit, in the field sense, of the field equation describing a

Long Range Scattering and Modified Wave Operators for some Hartree Type Equations II

Abstract. We study the theory of scattering for a class of Hartree type equations with long range interactions in space dimension n≥ 3, including Hartree equations with potential V(x) =

Long Range Scattering and Modified Wave Operators for the Wave-Schrödinger System II

\lambda \vert x \vert ^{-\gamma}

Long Range Scattering and Modified Wave Operators for the Maxwell–Schrödinger System II. The General Case

. For 0 <

The global Cauchy problem for the nonlinear Schrodinger equation revisited

\gamma

The global Cauchy problem for the non linear Klein-Gordon equation

≤ 1 we prove the existence of modified wave operators 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 a previous paper which covered the range 1/2 <

Long Range Scattering and Modified Wave Operators for some Hartree Type Equations, III. Gevrey Spaces and Low Dimensions

\gamma

Existence and uniqueness of solutions for the generalized Korteweg de Vries equation

< 1.

QUADRATIC MORAWETZ INEQUALITIES AND ASYMPTOTIC COMPLETENESS IN THE ENERGY SPACE FOR NONLINEAR SCHRÖDINGER AND HARTREE EQUATIONS

Abstract. We study the theory of scattering for the system consisting of a Schrödinger equation and a wave equation with a Yukawa type coupling in space dimension 3. We prove in particular the existence of modified wave operators for that system with no size restriction on the data and we determine the asymptotic behaviour in time of solutions in the range of the wave operators. The method consists in solving the wave equation, substituting the result into the Schrödinger equation, which then becomes both nonlinear and nonlocal in time, and treating the latter by the method previously used for a family of generalized Hartree equations with long range interactions.

Long Range Scattering for the Maxwell-Schrödinger System with Large Magnetic Field Data and Small Schrödinger Data

Abstract.We study the theory of scattering for the Maxwell–Schrödinger 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 Schrödinger 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 Schrödinger equation, which then becomes both nonlinear and nonlocal in time. The Schrödinger 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 self contained and can be read without any previous knowledge of the latter.