a non-linear equation for entanglement entropy: analytical and numerical approaches

In this work, we investigate a non-linear differential equation relevant to the computation of entanglement entropy in interacting quantum systems. we do this, inspired by field-theoretic and holographic models, we introduce a simplified equation to describe the entropy as a function of subsystem size and coupling strength. by comparing analytical approximations with numerical solutions, we uncover a crossover behavior between area-law and volume-law regimes, which are key features of entanglement dynamics in quantum systems. this minimal model captures the essential non-linear structure underlying entanglement growth, offering insights into the interplay between subsystem size and interaction strength. our findings provide a framework for understanding the transition between distinct entanglement scaling laws, bridging theoretical predictions and numerical observations. this work highlights the utility of simplified models in exploring complex quantum phenomena and contributes to the broader understanding of entanglement dynamics in interacting systems, with potential applications in quantum information and condensed matter physics.