Reflection and Refraction at Nonlinear Temporal Boundaries in Synthetic Lattices

Temporal boundaries in time-modulated media provide a powerful route toward wave manipulation beyond conventional spatial boundaries. Here, we investigate nonlinear temporal boundaries generated by interaction quenches in a synthetic lattice with exactly solvable interacting dynamics. Unlike conventional temporal boundaries arising from abrupt changes of single-particle dispersion, the present system realizes a self-induced temporal medium in which the propagating wave packet dynamically determines its own effective dispersion and transport properties. By solving the nonlinear Schr\"odinger dynamics analytically, we show that the interaction generates an emergent wave-packet-dependent band structure and a state-dependent temporal refractive response while preserving fully controllable evolution. Based on this framework, we establish a nonlinear temporal-scattering picture and uncover phenomena including amplitude-dependent temporal reflection/refraction and nonlinear temporal birefringence. Furthermore, we demonstrate that gradient-induced Bloch oscillations suppress wave-packet diffusion and enable coherent periodic transport with exact state reconstruction. Our results extend temporal reflection and refraction from dispersion-quenched linear systems to interaction-quenched nonlinear media and provide a tractable framework for nonlinear wave manipulation in synthetic lattices.