Resonant wave-particle interaction near the nonlinear threshold: Type I Painleve equation based model

Wave-particle resonant interactions play a crucial role in the dynamics of energetic particle fluxes across various space plasma systems, including the near-Earth plasma environment. When waves are sufficiently intense, these interactions become strongly nonlinear, with effects of particle phase bunching and trapping. This paper examines several examples of nonlinear resonant interactions involving energetic electrons, intense whistler-mode waves, electromagnetic ion cyclotron waves, and kinetic Alfven waves within the Earth’s inner magnetosphere. We focus on a specific scenario when resonant interactions occur near the threshold of the nonlinear regime. We demonstrate that, for such threshold wave amplitude, the classical model of resonant wave particle interactions, represented by the pendulum equation, can be reduced to Type I Painleve equation. Additionally, we derive an equation describing the scaling of electron momentum (and thus energy) change with wave amplitudes for this regime. We discuss the importance of this scaling in explaining observed energetic electron losses in Earth’s inner magnetosphere.