Dissipative Dynamics and Active Stabilization of Linear and Nonlinear Waves in Non-PT-Symmetric Harmonic Traps

We investigate the dissipative dynamics of linear and nonlinear waves in harmonic traps by means of engineered complex non-Hermitian potentials. By combining an analytical mapping between real and complex Schr ¨odinger equations with direct numerical simulations, we show that while in the linear case the damped motion leads to the formation of a stationary state at the trap center, in the nonlinear case a static potential design alone is insufficient to ensure long-term stability. Instead, the system relaxes toward a long-lived metastable configuration that eventually undergoes decay or collapse. To overcome this limitation, we introduce a time-dependent modulation of the nonlinearity that effectively converts these metastable states into robust non- equilibrium stationary states. This approach establishes a general strategy for controlling nonlinear waves in non-Hermitian systems, with potential applications in photonics and Bose–Einstein condensates.