Introduction

Linearization gives rise to such simplification that in many cases it is pushed to its limits and sometimes beyond in the hope that by understanding the linear problem we may gain some insight into the non-linear physics. Perhaps the clearest example of the progress that can be made by analysing linearized equations is in cold plasma wave theory, but linearization, in one form or another, is almost universally applied. For instance, the drift velocities of particle orbit theory are of first order in the ratio of Larmor radius to inhomogeneity scale length. In kinetic theory it is invariably assumed that the distribution function is close to a local equilibrium distribution.

A question of fundamental importance is then, ‘How realistic and relevant are linear theories?’ Some problems are essentially non-linear in that there is no useful small parameter to allow linearization. Examples of these are sheaths, discussed in Chapter 11, and shock waves. Primarily, our intention is to address the subsidiary question: ‘Given that there is a valid linear regime, to what extent need we concern ourselves with non-linear effects?’

Of course, if the linear solution predicts instability then we know that, in time, it will become invalid because the approximation on which the linearization is based no longer holds good. In such cases the aim might be to identify and investigate non-linear processes that come into play and quench the instability. However, an unstable linear regime is emphatically not a pre-requisite for an interest in nonlinear phenomena. There are many situations in which the linear equations give only stable solutions but the non-linear equations are secular, i.e. under certain conditions some solutions grow with time.