# Perturbations against a Q-ball. II. Contribution of nonoscillation modes

## TL;DR

This paper extends the analysis of Q-ball perturbations by demonstrating the importance of nonlinear equations for nonoscillation modes and confirming charge and energy additivity in complex perturbations.

## Contribution

It introduces the necessity of nonlinear equations for accurately describing nonoscillation modes in Q-ball perturbations and confirms additivity of charge and energy for combined modes.

## Key findings

- Nonoscillation modes require nonlinear equations for proper description.
- Charge and energy are additive for combined oscillation and nonoscillation modes.
- Nonlinear perturbation analysis extends previous linear approaches.

## Abstract

In the present paper, discussion of perturbations against a Q-ball solution is continued. It is shown that in order to correctly describe perturbations containing nonoscillation modes, it is also necessary to consider nonlinear equations of motion for the perturbations, like in the case of oscillation modes only. It is also shown that the additivity of the charge and the energy of different modes holds for the most general nonlinear perturbation consisting of oscillation and nonoscillation modes.