Develop a theoretical framework for the interplay between particle resonances and parametric instabilities

Develop a theoretical framework that explicitly describes the interplay between particle resonances (incoherent resonances represented via Hamiltonian formalism for non-Kapchinskij–Vladmirskij distributions) and parametric instabilities (coherent eigenmode instabilities derived from Vlasov–Poisson equations) in high-intensity linear accelerators, so that the relationship between these two families of mechanisms is theoretically characterized.

Background

The paper distinguishes two families of halo mechanisms in high-intensity linear accelerators: particle resonances and parametric instabilities, each with its own theoretical framework (Hamiltonian formalism for particle resonances and Vlasov–Poisson equations for parametric instabilities). Despite empirical and simulation-based insights into their manifestations and dominance in various regimes, the authors note that no general theory currently captures their interplay.

This absence of a unified framework is highlighted as a gap, motivating the need for a formal theory to connect and explain when and how these mechanisms interact.