Asymptotic properties of the dynamics near stationary solutions for some nonlinear schro dinger equations

The present thesis is devoted to the investigation of certain aspects of the large time behavior of the solutions of two nonlinear Schrödinger equations in dimension three in some suitable perturbative regimes. The rst model consist in a Schrödinger equation with a concentrated nonlinearity obtained considering a point (or contact) interaction with strength , which consists of a singular perturbation of the Laplacian described by a selfadjoint operator H , and letting the strength depend on the wave function: i du dt = H u, = (u). It is well-known that the elements of the domain of a point interaction in three dimensions can be written as the sum of a regular function and a function that exhibits a singularity proportional to jx x0j1, where x0 is the location of the point interaction. If q is the so-called charge of the domain element u, i.e. the coe cient of its singular part, then, in order to introduce a nonlinearity, we let the strength depend on u according to the law = jqj , with > 0. This characterizes the model as a focusing NLS with concentrated nonlinearity of power type. In particular, we study orbital and asymptotic stability of standing waves for such a model. We prove the existence of standing waves of the form u(t) = ei!t !, which are orbitally stable in the range 2 (0; 1), and orbitally unstable for 1: Moreover, we show that for 2 (0; p1 2 ) [ p1 2 ; p 3+1 2 p 2 every standing wave is asymptotically stable, in the following sense. Choosing an initial data close to the stationary state in the energy norm, and belonging to a natural weighted Lp space which allows dispersive estimates, the following resolution holds: u(t) = ei!1t+il(t) !1 + Ut 1 + r1, where Ut is the free Schrödinger propagator, !1 > 0 and 1, r1 2 L2(R3) with kr1kL2 = O(tp) as t ! +1, p = 5 4 , 1 4 depending on 2 (0; 1= p 2), 2 (1= p 2; 1), respectively, and nally l(t) is a logarithmic increasing function that appears when 2 (p1 2 ; ), for a certain 2 p1 2 ; p 3+1 2 p 2 i . Notice that in the present model the admitted nonlinearities for which asymptotic stability of solitons is proved, are subcritical in the sense that it does not give rise to blow up, regardless of the chosen initial data. The second model is the energy critical focusing nonlinear Schrödinger equation i du dt = u juj4u. In this case we prove, for any and 0 su ciently small, the existence of radial nite energy solutions of the form u(t; x) = ei (t) 1=2(t)W( (t)x) + ei t + o _H1(1) as t ! +1, where (t) = 0 ln t, (t) = t , W(x) = (1+ 1 3 jxj2)1=2 is the ground state and is arbitrarily small in _H 1.