Two-dimensional solitary waves in media with quadratic and cubic nonlinearity

We determine the existence and stability regimes of bright $2+1$-dimensional spatial solitary waves in media with quadratic (or ${\chi }^{\mathrm{}\left(2\right)\mathrm{}}$) and focusing cubic nonlinearities. We derive a necessary criterion for linear stability of these solitons, and use it to show that the quadratic nonlinearity enables stable solitons to exist when the cubic nonlinearity is sufficiently weak. We discuss why the Vakhitov-Kolokolov criterion for stability in ${\chi }^{\mathrm{}\left(2\right)\mathrm{}}$ systems is only a necessary criterion, and show an example where it fails. We further derive and study a simple adiabatical model for the soliton dynamics close to the instability threshold. Finally, we study the interesting dynamics of the solitons in the unstable regime, where we demonstrate the existence of two different limits described by nonlinear Schrödinger equations.