Exact soliton solutions and their stability control in the nonlinear Schrodinger equation with spatiotemporally modulated nonlinearity

We put forward a generic transformation which helps to find exact soliton solutions of the nonlinear Schrodinger equation with a spatiotemporal modulation of the nonlinearity and external potentials. As an example, we construct exact solitons for the defocusing nonlinearity and harmonic potential. When the soliton's eigenvalue is fixed, the number of exact solutions is determined by energy levels of the linear harmonic oscillator. In addition to the stable fundamental solitons, stable higher-order modes, describing array of dark solitons nested in a finite-width background, are constructed too. We also show how to control the instability domain of the nonstationary solitons.