Existence and Stability of Solitary Waves for NLS with Defects

Abstract

The main focus of this dissertation is to investigate the existence and stability of solitary waves to dispersive partial differential equations and, in particular to nonlinear Schr\"odinger equations with defects nonlinearity. First we identify necessary and sufficient conditions for the existence of appropriately localized waves for the inhomogeneous semi-linear Schr\"odinger equation driven by the subLaplacian dispersion operators $(-\De)^s, 0\f n2$. We construct the solitary waves explicitly, in an optimal range of the parameters, so that they belong to the natural energy space $H^s$. Next, we provide a complete classification of their spectral stability. Finally, we show that the waves are non-degenerate and consequently orbitally stable, whenever they are spectrally stable. Incidentally, our construction shows that the soliton profiles for the concentrated NLS are in fact exact minimizers of the Sobolev embedding $H^s(\rn)\hookrightarrow L^\infty(\rn)$, which provides an alternative calculation and justification of the sharp constants in these inequalities. Lastly, we consider the degenerate semi-linear Schr\"odinger and Korteweg-de Vries equations in one spatial dimension. We construct special solutions of the two models, namely standing wave solutions of NLS and traveling waves, which turn out to have compact support and are thus known as compactons. We show that the compactons are unique bell-shaped solutions of the corresponding PDE's and for appropriate variational problems as well. We provide a complete spectral characterization of such waves, for all values of $p$. Namely, we show that all waves are spectrally stable for $28$. This extends previous work of Germain, Harrop-Griffiths and Marzuola, \cite{GM1}, who have previously established orbital stability for some specific waves, in the range $p<8$.