Abstract. We consider the problem of identifying sharp criteria under which radial H 1 (finite energy) solutions to the focusing 3d cubic nonlinear Schrödinger equation (NLS) i∂ t u + ∆u + |u| 2 u = 0 scatter, i.e. approach the solution to a linear Schrödinger equation as t → ±∞. The criteria is expressed in terms of the, where u 0 denotes the initial data, and M [u] and E[u] denote the (conserved in time) mass and energy of the corresponding solution u(t). The focusing NLS possesses a soliton solution e it Q(x), where Q is the ground-state solution to a nonlinear elliptic equation, and we prove that if, then the solution u(t) is globally well-posed and scatters. This condition is sharp in the sense that the soliton solution e it Q(x), for which equality in these conditions is obtained, is global but does not scatter. We further show that if, then the solution blows-up in finite time. The technique employed is parallel to that employed by in their study of the energy-critical NLS.

Abstract. Scattering of radial H 1 solutions to the 3D focusing cubic nonlinear Schrö-dinger equation below a mass-energy thresholdwhere Q is the ground state, was established in Holmer-Roudenko [7]. In this note, we extend the result in [7] to non-radial H 1 data. For this, we prove a non-radial profile decomposition involving a spatial translation parameter. Then, in the spirit of Kenig-Merle [10], we control via momentum conservation the rate of divergence of the spatial translation parameter and by a convexity argument based on a local virial identity deduce scattering. An application to the defocusing case is also mentioned.

We prove local in time well-posedness for the Zakharov system in two space dimensions with large initial data in L 2 × H −1/2 × H −3/2 . This is the space of optimal regularity in the sense that the data-to-solution map fails to be smooth at the origin for any rougher pair of spaces in the L 2 -based Sobolev scale. Moreover, it is a natural space for the Cauchy problem in view of the subsonic limit equation, namely the focusing cubic nonlinear Schrödinger equation. The existence time we obtain depends only upon the corresponding norms of the initial data -a result which is false for the cubic nonlinear Schrödinger equation in dimension two -and it is optimal because Glangetas-Merle's solutions blow up at that time. AMS classification scheme numbers: 35Q551 2 × L 2 × H −1 , one-half a derivative away from the result in Theorem 1.1.One motivation for considering the space L 2 × H −1/2 × H −3/2 is the connection to the cubic nonlinear Schrödinger equation in two spatial dimensions i∂ t u + ∆u + |u| 2 u = 0.(1.2) Consider the Zakharov system with wave speed λ > 0:On the 2d Zakharov system 3 Then formally (1.3) converges to (1.2) as λ → ∞ in the sense that for fixed initial data u λ → u, where (u λ , n λ ) solves (1.3) and u solves (1.2) with the same initial data. Rigorous results of this type in a high regularity setting were obtained by Schochet-Weinstein [21], Added-Added [1], Ozawa-Tsutsumi [20], see also the recent work by Masmoudi-Nakanishi [18] on this issue in 3d.Local well-posedness in L 2 of (1.2) was obtained by Cazenave-Weissler [7]. However, in this version of well-posedness, the time interval of existence depends directly upon the initial data, not just on the L 2 norm of the initial data. Indeed, via the pseudoconformal transformation, it can be shown that a result giving the maximal time of existence in terms of the L 2 norm alone is not possible ‡.Remark 3. Our result gives local well-posedness of (1.3) with a time of existence depending on the L 2 norm of u 0 , but also on the H −1/2 × H −3/2 norm of the wave data (n 0 , n 1 ) as well as the wave speed. Indeed, this claim follows by combining the rescalingand Theorem 1.1. However, note that the lower bound on the maximal time of existence obtain by this method tends to zero as the wave speed goes to infinity.Global well-posedness of (1.1) is known for initial data in the energy space [6,13]; see also [10] regarding bounds on higher order Sobolev norms. Recently, the imposed regularity assumption has been slightly weakened in [11]. Here, Q is the ground state solution for (1.2), i.e. Q is the unique solution to − Q + ∆Q + |Q| 2 Q = 0, Q > 0, Q(x) = Q(|x|), Q ∈ S(R 2 ) (1.4) of minimal L 2 mass. This gives rise to a blow-up solution of (1.2) by the pseudoconformal transformation. This idea is exploited in [14], where Glangetas-Merle construct a family of blow-up solutions for (1.1) of the formfor parameters θ ∈ S 1 , T > 0, and ω ≫ 1, such that P ω ∈ H 1 is smooth and radially symmetric, N ω ∈ L 2 is a radially symmetric Schwartz function, and (P ω , ...

Abstract. We study the Gross-Pitaevskii equation with a repulsive delta function potential. We show that a high velocity incoming soliton is split into a transmitted component and a reflected component. The transmitted mass (L 2 norm squared) is shown to be in good agreement with the quantum transmission rate of the delta function potential. We further show that the transmitted and reflected components resolve into solitons plus dispersive radiation, and quantify the mass and phase of these solitons.

We consider the Cauchy problem for a family of semilinear defocusing Schrödinger equations with monomial nonlinearities in one space dimension. We establish global well-posedness and scattering. Our analysis is based on a four-particle interaction Morawetz estimate giving a priori L 8t,x spacetime control on solutions.2000 Mathematics Subject Classification. 35Q55.