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We study the scattering behavior of global solutions to stochastic nonlinear Schr\"odinger equations with linear multiplicative noise. In the case where the quadratic variation of the noise is globally finite and the nonlinearity is…
We investigate the global well-posedness and modified scattering for the one-dimensional Schr\"odinger equation with gauge-invariant polynomial nonlinearity. For small localized initial data of finite energy in a low-regularity class, we…
This article concerns the long-time dynamics of quantum particles in the semi-classical regime. First, we show that for the nonlinear Hartree equation with short-range interaction potential, small-data solutions obey dispersion bounds and…
We review some recent results on nonlinear Schrodinger equations with potential, with emphasis on the case where the potential is a second order polynomial, for which the interaction between the linear dynamics caused by the potential, and…
We consider the nonlinear Schrodinger equation under a partial quadratic confinement. We show that the global dispersion corresponding to the direction(s) with no potential is enough to prove global in time Strichartz estimates, from which…
The Nonlinear Schroedinger Equation (NLSE) with a random potential is motivated by experiments in optics and in atom optics and is a paradigm for the competition between the randomness and nonlinearity. The analysis of the NLSE with a…
We consider a nonlinear Schr\"odinger equation in $\R^3$ with a bounded local potential. The linear Hamiltonian is assumed to have two bound states with the eigenvalues satisfying some resonance condition. Suppose that the initial data is…
Our aim in this work is to give some quantitative insight on the dispersive effects exhibited by solutions of a semiclassical Schr{\"o}dinger-type equation in R d. We describe quantitatively the localisation of the energy in a long-time…
We prove a global well-posedness result for defocusing nonlinear Schrodinger equations with time dependent potential. We then focus on time dependent harmonic potentials. This aspect is motivated by Physics (Bose--Einstein condensation),…
We study the nonlinear Schr\"odinger equation (NLS) with bounded initial data which does not vanish at infinity. Examples include periodic, quasi-periodic and random initial data. On the lattice we prove that solutions are polynomially…
We study the asymptotic behaviour of solutions to semi-classical nonlinear Schrodinger equations with a potential, for concentrating and oscillating initial data, when the nonlinearity is repulsive and the potential is a polynomial of…
We consider a nonlinear Schr\"odinger equation with a bounded local potential in $R^3$. The linear Hamiltonian is assumed to have two bound states with the eigenvalues satisfying some resonance condition. Suppose that the initial data are…
We consider Schr\"{o}dinger equations with linearly energy-depending potentials which are compactly supported on the half-line. We first provide estimates of the number of eigenvalues and resonances for such complex-valued potentials under…
We consider the Schroedinger equation with a general interaction term, which is localized in space. The interaction may be x, t dependent and non-linear. Purely non-linear parts of the interaction are localized via the radial Sobolev…
A nonlocal-in-time problem for the abstract Schr\"odinger equation is considered. By exploiting the linear nature of nonlocal condition we derive an exact representation of the solution operator under assumptions that the spectrum of…
We consider a class of biharmonic nonlinear Schr\"odinger equations with a focusing inhomogeneous power-type nonlinearity \[ i\partial_t u -\Delta^2 u+\mu\Delta u +|x|^{-b} |u|^\alpha u=0, \quad \left. u\right|_{t=0}=u_0 \in…
We investigate the asymptotic behavior of solutions to semi-classical Schroedinger equations with nonlinearities of Hartree type. For a weakly nonlinear scaling, we show the validity of an asymptotic superposition principle for slowly…
We study the large-time behavior of global energy class ($H^1$) solutions of the one-dimensional nonlinear Schr\"odinger equation with a general localized potential term and a defocusing nonlinear term. By using a new type of interaction…
This paper is devoted to a comprehensive study of the nonlinear Schr\"odinger equations with combined nonlinearities of the power-type and Hartree-type in any dimension n\ge3. With some structural conditions, a nearly whole picture of the…
In this paper we study the asymptotic behavior of a quadratic Schr\"{o}dinger equation with electromagnetic potentials. We prove that small solutions scatter. The proof builds on earlier work of the author for quadratic NLS with a non…