Evolution of breathers with spectrally skewed forcing and damping

Slowly modulated nonlinear waves are accurately described in the framework of the conservative nonlinear Schrödinger equation (NLS). However, in many physical systems, the wave evolution is affected by energy gains and losses, therefore requiring introduction of forcing or damping in the NLS framework. Here, we analyze the idealized case in which the forcing term varies linearly with frequency such as the growth rate of positive components matches the decay rate of the negative ones. We reveal that the system is linearly unstable and prone to overall energy growth. The evolution of classical breather solutions is showcased in this framework. The most notable effects are the distortion of the breathers, which speed-up, and in the case of the Kuznetsov-Ma breather results in faster recurrence. Concurrently, spectral asymmetry between low and high frequency components develops.