J. Funct. Anal. 203 (2003), no. 2, 453-493We consider a nonlinear semi-classical Schrödinger equation for which it is known that quadratic oscillations lead to focusing at one point, described by a nonlinear scattering operator. If the initial data is an energy bounded sequence, we prove that the nonlinear term has an effect at leading order only if the initial data have quadratic oscillations; the proof relies on a linearizability condition (which can be expressed in terms of Wigner measures). When the initial data is a sum of such quadratic oscillations, we prove that the associate solution is the superposition of the nonlinear evolution of each of them, up to a small remainder term. In an appendix, we transpose those results to the case of the nonlinear Schrödinger equation with harmonic potential

We consider the propagation of wave packets for the nonlinear Schrödinger equation, in the semi-classical limit. We establish the existence of a critical size for the initial data, in terms of the Planck constant: if the initial data are too small, the nonlinearity is negligible up to the Ehrenfest time. If the initial data have the critical size, then at leading order the wave function propagates like a coherent state whose envelope is given by a nonlinear equation, up to a time of the same order as the Ehrenfest time. We also prove a nonlinear superposition principle for these nonlinear wave packets.This work was supported by the French ANR project R.A.S. (ANR-08-JCJC-0124-01).

We consider u 1 and u 2 , two blow-up solutions ofwhere p > 1, (N − 2)p < N + 2 and either u(0) 0 or (3N − 4)p < 3N + 8. We assume that u 1 and u 2 blow-up at the same time T , at the same unique point a ∈ R N and that they have the same (generic) profile. We then obtain a sharp estimate onIn particular, we show that, up to a scaling, this difference is uniformly bounded and goes to zero as (x, t) → (a, T ), provided that N = 1 and p 3. As an application of our result, we show the stability of the considered profile in N dimensions.

We consider composite quantum-dynamical systems that can be partitioned into weakly interacting subsystems, similar to system–bath type situations. Using a factorized wave function ansatz, we mathematically characterize dynamical scale separation. Specifically, we investigate a coupling régime that is partially flat, i.e. slowly varying with respect to one set of variables, for example, those of the bath. Further, we study the situation where one of the sets of variables is semiclassically scaled and derive a quantum–classical formulation. In both situations, we propose two schemes of dimension reduction: one based on Taylor expansion (collocation) and the other one based on partial averaging (mean-field). We analyze the error for the wave function and for the action of observables, obtaining comparable estimates for both approaches. The present study is the first step towards a general analysis of scale separation in the context of tensorized wavefunction representations.