Quantization-locked attractors and deterministic stabilization of localized wave packets in discrete nonlinear systems

<title>Abstract</title> <p>We introduce the concept of quantization-locked equilibria for localized wave packets in digitized nonlinear dynamical systems. Continuous models assume an infinite-precision parameter manifold, an idealization that breaks down when control and measurement are performed on finite quantized grids. By treating the envelope dynamics as a second-order ordinary differential equation, we show that discretization fractures the continuous existence curve into a deterministic hierarchy of discrete fixed points. A residual quantization gap drives drift unless the discrete control lattice resonates with a nonlinear invariant of the system. At these resonant coordinates the effective acceleration vanishes and the system locks into a quantization-locked state with suppressed drift. For the primary rank equilibrium, we demonstrate stable propagation extended by more than two orders of magnitude compared to non-resonant parameters. We illustrate the framework with a dark vortex beam in a negative saturable medium governed by a paraxial envelope equation. The results provide a general analytical bridge between digital discretization and continuous nonlinear evolution, with relevance to any system described by a second-order envelope equation with quantized control parameters.</p>