Eigenmode steering in spatiotemporal gain-loss acoustic metamaterials

We present a spatiotemporal gain-loss framework for eigenmode steering in coupled acoustic resonators. A cross-coupled gain-loss coefficient links the gain of one resonator to the intensity of its partner, creating nonlinear feedback that conserves total energy while driving the system toward the eigenmode associated with the eigenvalue having the largest imaginary part-a deterministic eigenmode steering. Spatial gain-loss profiles shape the eigenvalue spectrum and attractor landscape, while temporal modulation governs the transition dynamics. When symmetry prevents direct access to a target eigenmode, controlled spatiotemporal perturbations enable otherwise symmetry-forbidden transitions and accelerate convergence. Within this framework, parity-time () symmetry appears as a special case, allowing tunable switching between steering and Rabi-like oscillations near the exceptional point. Full-wave simulations of coupled Helmholtz resonators confirm precise and programmable acoustic energy routing, establishing spatiotemporal gain-loss engineering as a route to reconfigurable wave control and analog information processing.