What is the difference between a linear and a non-linear perturbation? Sometimes you will hear about the stability of certain solutions (black holes, solitons, etc) with respect to perturbations. Often they talk about linear vs. non-linear perturbations.
What is the distinction between a linear and a non-linear perturbation? I assume it has to do with the response of the solution to the perturbation, but how is it determined whether a given perturbation is "non-linear" or "linear"? Or maybe it has to do with the specific method used? (e.g. applying a perturbation to a linearized version of the equations results is a "linear perturbation?)
 A: I would tend to agree with you that this (commonly used) language is somewhat misleading. A perturbation is just a perturbation, "linear" and "non-linear" are words that describe the methods used to understand the perturbation. (To paraphrase a famous physicist, also keep in mind that dividing the world between "linear" and "non-linear" is like dividing the world between "bananas" and "non-bananas".)
Typically the physics of some system is described by some non-linear differential equation. Perhaps we are able to solve that equation in a special case, such as a situation with a lot of symmetry. Then a perturbation describes a deviation of the system from that ideal case we can solve.
If the perturbation is "small" (in some sense that you need to make precise within the context you are working), then you can usually linearize the full differential equation about the ideal solution, and the perturbation will be well-described by solutions to this linearized equation (the linearized equation is a good approximation to the full dynamics). This would be a linear perturbation.
However, if the perturbation is "large", then the linearized differential equation will give a poor description of the behavior of the perturbation (the linearized equations are a poor approximation). So you need to make use of the full non-linear differential equation to understand what is going on, or at least you need to make a better approximation than linearizing. This would be a non-linear perturbation.
