Nonlinear solitary waves with Gaussian tails

We study analytically and numerically the stationary localized solutions of the nonlinear Schrodinger equation (NLSE) with an additional parabolic potential. Such a model occurs in a wide range of physical applications, including plasma physics and nonlinear optics. Bound states with Gaussian tails (these tails appear due to the parabolic linear potential) play an important role in the dynamics of the systems modelled by this equation. We prove the existence of the bound states and describe their properties.