Related papers: Modified scattering for nonlinear Schr\"odinger eq…
We consider a Schr{\"o}dinger equation with a nonlinearity which is a general perturbation of a power'' nonlinearity. We construct a profile decomposition adapted to this nonlinearity.We also prove global existence and scattering in a…
Based on the work of Nuttall and Cohen [Phys. Rev. {\bf 188} (1969) 1542] and Resigno et al{} [Phys. Rev. A {\bf 55} (1997) 4253] we present a rigorous formalism for solving the scattering problem for long-range interactions without using…
The long-time asymptotics is analyzed for finite energy solutions of the 1D Schr\"odinger equation coupled to a nonlinear oscillator. The coupled system is invariant with respect to the phase rotation group U(1). For initial states close to…
We consider the asymptotic behavior in time of solutions to the cubic nonlinear Schr"odinger equation with repulsive delta potential (delta-NLS). We shall prove that for a given asymptotic profile, there exists a solution to (delta-NLS)…
We study the asymptotic behaviour in time of solutions and the theory of scattering for the modified Schr"odinger map in two space dimensions. We solve the Cauchy problem with large finite initial time, up to infinity in time, and we…
We construct an isometric modified scattering operator, mapping any sufficiently regular past scattering state, with a small distribution function, to the future one corresponding to forward evolution by the Vlasov-Maxwell system. The main…
We study the long-time behavior of solutions to nonlinear Schroedinger equations with some critical rough potential of inverse square type. The new ingredients are the interaction Morawetz-type inequalities and Sobolev norm property…
In this paper we suggest a new approach for the multichannel Coulomb scattering problem. The Schr\"{o}dinger equation for the problem is reformulated in the form of a set of inhomogeneous equations with a finite-range driving term. The…
In this paper, we consider the asymptotic behaviors of small solutions to the semi-relativistic Hartree equations in two dimension. The nonlinear term is convolved with the Coulomb potential 1/|x|, and it produces the long-range interaction…
We consider the asymptotics of the one-dimensional cubic nonlinear Schr\"odinger equation with an external potential $V$ that does not admit bound states. Assuming that $\jBra{x}^{2+}V(x) \in L^1$ and that $u$ is orthogonal to any…
We study an integrable modification of the focusing nonlinear Schroedinger equation from the point of view of semiclassical asymptotics. In particular, (i) we establish several important consequences of the mixed-type limiting quasilinear…
This paper is devoted to the study of the large-time asymptotics of the small solutions to the matrix nonlinear Schr\"{o}dinger equation with a potential on the half-line and with general selfadjoint boundary condition, and on the line with…
We consider non-linear Schr\"odinger equations with a potential, and non-local non-linearities, that are models in mesoscopic physics, for example of a quantum capacitor, and that also are models of molecular structure. We study in detail…
We consider the following wave guide nonlinear Schr\"odinger equation, \begin{equation} (i\partial \_t+\partial \_{xx}-\vert D\_y\vert )U=\vert U\vert ^2U\ \tag{WS} \end{equation} on the spatial cylinder $\mathbb{R} \_x\times \mathbb{T}…
In this paper, we study the well-posedness theory and the scattering asymptotics for the energy-critical, Schr\"odinger equation with indefinite potential \begin{equation*} \left\{\begin{array}{l} i \partial_t u+\Delta u-V(x)u…
This paper is a continuation of our previous study on the long time behavior of solution to the nonlinear Schr"odinger equation with higher order anisotropic dispersion (4NLS). We prove the long range scattering for (4NLS) with the…
We consider the defocusing nonlinear Schr{\"o}dinger equation in several space dimensions, in the presence of an external potential depending on only one space vari-able. This potential is bounded from below, and may grow arbitrarily fast…
In this work, we study the existence of various classes of standing waves for a nonlinear Schr\"odinger system with quadratic interaction, along with a harmonic or partially harmonic potential. We establish the existence of ground-state…
We consider the scattering theory for the Schrodinger equation with $-\Delta -|x|^{\alpha}$ as a reference Hamiltonian, for $0< \alpha \leq 2$, in any space dimension. We prove that when this Hamiltonian is perturbed by a potential, the…
We consider 3d Schrodinger operator with long-range potential that has short-range radial derivative. The long-time asymptotics of non-stationary problem is studied and existence of modified wave operators is proved. It turns out, the…