Localized Phonon Radiation of Topological Solitons in Graded Bistable Chains

Topological metamaterials offer robust energy localization, yet conventional uniform lattices lack the spatial adaptability to concentrate mechanical energy at designated transducer interfaces. While recent studies have explored random disorder or the nucleation of standing solitons, the deterministic spatial control over moving topological transition waves remains largely unexplored. This paper investigates the dynamics of topological solitons and the resulting phonon radiation in a one-dimensional bistable granular chain subjected to a programmable pre-stress gradient. By combining continuous adiabatic perturbation theory with discrete lattice analysis, we reveal a dual-stage energy concentration mechanism. Macroscopically, the spatial gradient induces a significant variation in the soliton's effective mass, triggering severe nonlinear braking via action invariance. Microscopically, the compressed soliton width exponentially amplifies the local Peierls-Nabarro pinning barrier. Through direct numerical integration, we demonstrate two distinct non-adiabatic radiation regimes: a softening gradient that accelerates the soliton, yielding a continuous, diffuse Cherenkov-like phonon tail; and a hardening gradient that drastically decelerates and pins the soliton, forcing its macroscopic kinetic energy to violently convert into an extreme, highly localized phonon burst at the gradient boundary. This hardening gradient acts as a deterministic spatial energy concentrator, amplifying the local peak phonon energy by orders of magnitude compared to uniform chains. Our findings provide a novel, passive design paradigm for high-efficiency piezoelectric energy harvesting, enabling the spatial programming of mechanical energy deposition without altering the fundamental topology of the chain.