Dynamic power balance for nonlinear waves in unbalanced gain and loss landscapes

The presence of losses in nonlinear photonic structures is a crucial issue for modern applications. Active parts are introduced for wave power compensation resulting in unbalanced gain and loss landscapes where localized beam propagation is, in general, dynamically unstable. Here we provide generic sufficient conditions for the relation between the gain-loss and the refractive index profiles in order to ensure efficient wave trapping and stable propagation for a wide range of beam launching conditions such as initial power, angle of incidence and position. The stability is a consequence of an underlying dynamic power balance mechanism related to a conserved quantity of wave dynamics.