# Envelope Dynamics and Stability with non-linear Space-Charge Forces

## TL;DR

This paper presents an analytical model for assessing the stability of Gaussian beams under non-linear space-charge forces, incorporating lattice errors and coupling effects, with methods for resonance detection and stability analysis.

## Contribution

It introduces a comprehensive analytical approach to evaluate envelope stability considering non-linear space-charge effects and lattice errors, including resonance identification.

## Key findings

- Space-charge forces significantly influence envelope stability.
- Envelope-lattice and envelope coupling resonances can cause beam blow-up.
- The combined analytical and tracking methods effectively identify stability boundaries.

## Abstract

We developed a model to calculate the stability of Gaussian beam distributions with non-linear space-charge forces in the presence of random and skew-quadrupole errors. The effect of the space-charge force on the beam matrix is calculated analytically including full cross-plane coupling in 4D phase space, which allows us to perform fast parameter studies. For stability analysis, we find the fixed points of the beam including the space-charge forces and construct a Jacobi-matrix by slightly perturbing the periodic solution. The stability of envelope oscillations is inferred by eigenvalue analysis. Furthermore, we employ envelope tracking as a complementary method and compare the results of the eigenvalue analysis with FFT data from the tracked envelope. The non-linearity of the space-charge force in combination with lattice errors and beam coupling opens up for envelope-lattice resonances and envelope coupling resonances. Hitting these resonances leads to envelope blow-up, causing an effective beam mismatch. Therefore, we finally examine the effect of beam mismatch on the envelope tune-shift and its stability.