Analytical and Numerical Results for Some Classes of Nonlinear Schrödinger Equations

This thesis is devoted to the study of nonlinear dispersive partial differential equations of Schrödinger type. The main questions we investigate are long-time behavior or occurrence of a finite time singularity, as well as stability properties of solitary wave solutions. The derivative nonlinear Schrödinger (DNLS) equation is a nonlinear dispersive model that appears in the description of wave propagation in plasmas. The first part of this thesis concerns a DNLS equation with a generalized nonlinearity (gDNLS). We first investigate numerically the possible occurrence of singularities. We show that, in the L2-supercritical regime, singularities can occur. We obtain a precise description of the local structure of the solution in terms of the blowup rate and asymptotic profile, in a form similar to that of the nonlinear Schrödinger equation (NLS) with supercritical power law nonlinearity. We also show that the gDNLS equation possesses a two-parameter family of solitary wave solutions and study their stability. We fully classify their orbital stability or orbital instability properties according to the strength of the nonlinearity and, in some instances, their velocity. In linear quantum mechanical scattering theory, the phenomenon of resonant tunneling refers to the situation where incoming waves are fully transmitted through potential barriers at certain energies. In the second part of this thesis, we consider the one-dimensional cubic NLS equation with two classes of external potentials, namely the ‘box’ potential and a repulsive 2-delta potential. We demonstrate numerically that resonant tunneling may occur in a nonlinear setting: Taking initial condition as a slightly perturbed, fast moving NLS soliton, we show that, under a certain resonant condition, the incoming soliton is almost fully transmitted. As the velocity of the incoming soliton increases, the transmitted mass of the soliton converges to the total mass.Ph