Abstract: In this talk, I will present analytical studies of standing waves in three NLS models. We first consider the spectral stability of ground states of semi-linear Schrodinger and Klein-Gordon equations with fractional dispersion. We use Hamiltonian index counting theory, together with the information from a variational construction to develop sharp conditions for spectral stability for these waves. The second type we consider is a nonlocal NLS which comes from modeling nonlinear waves in Parity-time symmetric systems. We investigate the spectral stability of standing waves of its $\mathcal{PT}$ symmetric solutions. The third case is about the existence and the stability of the vortices for the NLS in higher dimensions. We extend the existence and stability results of Mizumachi from two-space dimensions to $n$ space dimensions.
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