Stabilization of solitons of the multidimensional nonlinear Schrodinger equation: Matter-wave breathers

We demonstrate that stabilization of solitons of the multidimensional Schrodinger 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 which belong to the frontier between expanding and collapsing solutions and to provide an oscillating solitonic structure, some sort of breather-type solution. We obtain precise conditions on the control parameters to achieve the stabilization and compare our results with accurate numerical simulations of the nonlinear Schrodinger equation. Because of the application of these ideas to matter waves these solutions are some sort of matter breathers.