Solitons in a medium with linear dissipation and localized gain
  We present a variety of dissipative solitons and breathing modes in a medium with localized gain and homogeneous linear dissipation. The system posses a number of unusual properties, like exponentially localized modes in both focusing and defocusing media, the existence of modes in focusing media at negative propagation constant values, the simultaneous existence of stable symmetric and anti-symmetric localized modes when the gain landscape possesses two local maxima, as well as the existence of stable breathing solutions. 
Dmitry A. Zezyulin, Yaroslav V. Kartashov, Vladimir V. Konotop
physics.optics nlin.PS