Stabilization of solitons of the multidimensional nonlinear Schrِdinger equation: matter-wave breathers

We demonstrate that stabilization of solitons of the multidimensional Schrِdinger equation with a cubic nonlinearity may be achieved by a suitable periodic control of the nonlinear term. The effect of this control is to stabilize the unstable solitary waves into a breather-type solution. Using an asymptotic method we obtain precise conditions on the parameters which allow to achieve the stabilization and compare our results with accurate numerical simulations of the nonlinear Schrِdinger equation.