Collaborative psychiatric management is founded on a person-centred, holistic assessment leading to a diagnostic formulation that guides decision making. Formulation around the individual person, including their unique history and worldview, can be described with presenting, precipitating, predisposing, perpetuating and protective factors as well as the life context for the individual patient. Allied with this, diagnosis – in which the patient’s unique presentation can be evaluated as sharing characteristics and patterns with other patients – can allow for the individual plan to be guided by a wider frame of reference and knowledge. Such diagnostic frameworks have been developed over millennia and across cultures. As well as being important for individual patient care, they are essential for research and service planning. The development of these diagnostic frameworks is discussed with particular reference to the main international classifications of ICD-11 and DSM-5. It is common for people to have more than one diagnosis, and diagnostic hierarchies are considered. Criticisms of the construct of psychiatric diagnosis are reviewed, and an approach to conducting and describing collaborative psychiatric assessment is described.

Psychiatry, according to Johann Christian Reil (1759–1813), the German anatomist who first coined the term, consists of the meeting of two minds, the mind of the patient with the mind of the doctor. As the patient’s story unfolds, the doctor’s task is to recognise the pattern and to do so with compassion. Pattern recognition lies at the heart of the diagnostic process throughout medicine and none more so than in psychiatry, which lacks almost all the special investigations that help clarify diagnosis in other medical specialities. Thus, detailed knowledge of the key features of all the psychiatric disorders, both common and rare, is the core body of information that the psychiatrist will need to acquire during their training years. Because of this, we have provided detailed descriptions of each and every disorder as well as their diagnostic criteria according to DSM-5 and ICD-11.

The last few decades have seen three important developments in nonlinear classical physics, all of which extend across the board of physical disciplines. They have, however, received uneven coverage in the literature.

Perhaps the best known outburst of activity is associated with the soliton, and the most famous development here is the inverse scattering method which has been with us now for over twenty years. There are, however, several other, less known methods for treating solitons. Indeed these compact, single hump wave entities have been known to scientists for over a century and a half (it might be interesting to look through some old ships' log books!). Nevertheless, books on the subject tend to concentrate on the inverse scattering method.

The second much publicized development is a new understanding of some deterministic aspects of chaos as well as the various roads a physical system can take to reach a chaotic state. Established views are being revised and new concepts and indeed even universal constants are being found. These important new developments derive from a realization that complex chaotic behaviour can be described by simple equations. The field has now reached the stage where a summary of basic theory can be given, though applications to specific physical problems are largely at the research stage.

The third development is somewhat less well publicized. Over the last three decades or so, scientists working on fluid dynamics and plasma and solid state theory have developed a multitude of new methods to deal with nonlinear waves.

Introduction

In the earlier chapters of this book the emphasis has been on a study of the existence and stability of nonlinear waves and solitons, that is of coherent structures. Such structures are found in Nature and thus certainly deserve our attention. However, a much more universal type of behaviour is described under the umbrella of turbulence. One envisages turbulence as a phenomenon where some measurable quantity has a rapid space and/or time dependence. For example, in the case of water passing over a weir, the complicated behaviour is apparent in the local velocity of the water. One sees eddies (or vortices) of a range of sizes. They not only move with some background velocity but also interact with one another to produce a continually changing picture.

For another example, consider turbulence in the wake of a cylinder if the water flow is of very high Reynolds number (see Fig. 1.5(c), (d)).

The problem of trying to understand turbulence has been with us for centuries but it still remains a basic unresolved problem. (The beauty and complexity of turbulence was well appreciated by Leonardo da Vinci as is evidenced in his drawings of vortices in water, Fig. 11.1.)

Somewhat ironically, the study of turbulence in plasmas, which themselves are much more complicated media, is more tractable than in water and considerable progress has been made in the last twenty years. However, most theories to date are restricted in that they assume that the energy in the fluctuations is small compared to the kinetic energy of the particles.

Introduction

Some physical equations ask for surgery

Classical physicists usually agree on their equations. In this respect they are very fortunate and can feel rewarded for not working in more fashionable fields such as the frontiers of high energy physics. However, these established equations often compound many different physical effects and can be difficult to solve.

Once we have derived as realistic a set of equations as possible for a given situation, we can try to reduce the number of terms or otherwise simplify by some logical process. Only very good scientists can get away with formulating equations that model chosen phenomena from the start, say by ignoring some physical effects or chopping off terms they consider to be insignificant. Lesser mortals are well advised to develop a systematic scheme for simplifying model equations. To do this one should look for at least one small dimensionless parameter and use it as a surgical tool.

The above remarks concern a theoretical treatment. Computer scientists, on the other hand, will increasingly welcome elaborate mathematical models, embracing more and more rather than less and less physics.

There can be two broad justifications for introducing a small parameter scheme to simplify a system of physical equations (other than being fed up with not being able to solve it). One is that a dimensionless parameter is always small. (An example of such a parameter is the ratio of the centre of mass velocity of a massive heavenly body to c, the velocity of light.

Interest in higher dimensional plasma solitons

As we have seen, quite a lot is now known about one-dimensional plasma waves and solitons. Possibly as a continuation of this effort, or again possibly as a result of most scientists' impatience with a field once it has disclosed some of its secrets, a new discipline of cylindrical and spherical plasma soliton waves has come into being. Investigations began fairly recently (Maxon and Viecelli (1974a,b)), but progress is rapid. Similar phenomena in other fields of classical physics are mentioned briefly at the end in Section 9.7.

In mathematical terms, the cylindrical and spherical plasma (and hydrodynamic) solitons treated here are often described quite well by variants of the model equations that have already appeared in this book. Examples are the Korteweg–de Vries (KdV) cum Boussinesq family (ion acoustic solitons), and the nonlinear Schrödinger family in various geometries and with diverse nonlinearities, the simplest being a cubic term (Langmuir envelope solitons). Both are of course classes of idealized equations. Here we will see how they can be obtained in the higher dimensional, non-Cartesian plasma physics context, what their properties are and also some of the effects they do not describe. Finally, we will see how some of their predictions stand up to laboratory and numerical experiments. A few extensions of the above models will we suggested.

Comparatively little theoretical work has been done on the stability of higher dimensional solitons, existing analyses being either very restricted or incomplete.

Introduction

One of the ironies of wave phenomena is that those waves which are most easily observed, such as waves on the surface of water, are more difficult to analyse theoretically than waves which propagate through a medium, such that their presence has to be inferred indirectly. Surface waves are more easily observed than bulk waves but mathematically it is more difficult to deal with them. The main reason for this is that for surface waves a boundary condition, such as continuity, must be satisfied at the surface of the wave which of course is not generally a simple plane. This was indicated briefly in the Introduction. Fortunately for linearized wave theory, the boundary condition has to be satisfied on the unperturbed surface which is usually planar. If the medium on either side of the boundary is homogeneous, the problem reduces to matching algebraic expressions at the interface. In this manner one obtains an algebraic dispersion relation for surface waves analogous to a bulk dispersion relation. However, in most practical situations the boundary is diffuse and this necessitates solving differential equations, with non-constant coefficients, for the behaviour perpendicular to the boundary. An algebraic dispersion relation is then replaced by an eigenvalue equation with a consequent increase in the difficulty of the problem.

The basic mathematical techniques are illustrated in Section 4.2 by considering the propagation of waves along the surface of a liquid.