Multistable dissipative structures pinned to dual hot spots

We analyze the formation of one-dimensional localized patterns in a nonlinear dissipative medium including a set of two narrow hot spots (HSs), which carry the linear gain, local potential, cubic self-interaction, and cubic loss, while the linear loss acts in the host medium. This system can be realized as a spatial-domain one in optics and also in Bose-Einstein condensates of quasiparticles in solid-state settings. Recently, exact solutions were found for localized modes pinned to the single HS represented by the delta function. The present paper reports analytical and numerical solutions for coexisting two-and multipeak modes, which may be symmetric or antisymmetric with respect to the underlying HS pair. Stability of the modes is explored through simulations of their perturbed evolution. The sign of the cubic nonlinearity plays a crucial role: in the case of the self-focusing, only the fundamental symmetric and antisymmetric modes, with two local peaks tacked to the HSs, and no additional peaks between them, may be stable. In this case, all the higher-order multipeak modes, being unstable, evolve into the fundamental ones. Stability regions for the fundamental modes are reported. A more interesting situation is found in the case of the self-defocusing cubic nonlinearity, with the HS pair giving rise to a multistability, with up to eight coexisting stable multipeak patterns, symmetric and antisymmetric ones. The system without the self-interaction, the nonlinearity being represented only by the local cubic loss, is investigated too. This case is similar to those with the self-focusing or defocusing nonlinearity, if the linear potential of the HS is, respectively, attractive or repulsive. An additional feature of the former setting is the coexistence of the stable fundamental modes with robust breathers.