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.

Introduction

In this chapter we will deal in some detail with the mathematical methods that can be used to treat fully nonlinear wave problems.

The best known single development of the last thirty years is the discovery of a method for solving the initial value problem for a limited class of nonlinear partial differential equations. This is the inverse scattering method (ISM). This method, however, proves difficult to extend to general initial conditions and is more useful for solitons than for waves. It will be presented in Chapter 7. There are, however, several other methods for dealing with nonlinear waves (often solitons also). They deserve notice in their own right. Some of them have been developed fairly recently. Few are limited to equations solvable by ISM. This chapter will concentrate on these methods as applied to nonlinear waves and solitons. We hope to give an idea of how rich the family of known nonlinear waves now is.

Thus this chapter and the next will concentrate on methods as illustrated by simple plasma physics, fluid dynamics and other problems. We will find the shapes of nonlinear waves and solitons without in principle assuming small amplitude. Where we do restrict considerations to small amplitude (Section 6.5) it will be done in the hope of extracting more information out of an exact method (Lagrangian description of fluid flow) than would otherwise be forthcoming. Yes, by restricting a formalism (small in place of general amplitude) we will learn more.

The next step in investigating soliton behaviour

In the previous chapters, we introduced a wide variety of solitons. We started with onedimensional ones, depending on x - vt, or else k · x - ωt when embedded in a higher dimensional space. Next, we saw how two or more such entities could combine to produce X shaped solitons, still behaving like their one-dimensional components far from the intersections. Finally, in Chapter 9, we looked at fully two- and three-dimensional compact entities produced both experimentally and theoretically from model equations. Stability was considered in some detail, especially in Chapter 8. When so doing, we performed small perturbation analyses, linearizing around the soliton structure (often treated as a limit of a nonlinear wavetrain). The reader should be warned against being seduced into thinking that linearization tells the whole story. Linear instability, leading to exponential growth, cannot persist. Some possible subsequent scenarios were outlined in Chapter 8. More will come in Chapter 11. Here we will focus on still another fate, that of soliton metamorphisis. If, say, both one- and two-dimensional solitons are known to exist in a given medium and, furthermore, the one-dimensional version is linearly unstable, might it not break up into an array of two-dimensional solitons? Could not two-dimensional solitons likewise produce three-dimensional decay products after a while? Can decay proceed directly from ID to 3D? In a series of papers, it was shown that one-dimensional Zakharov–Kuznietsov solitons, discussed in Chapter 8, can indeed break up into two-dimensional, cylindrical entities, functions of (x - ωt)2 + y2.

Occurrence of nonlinear waves and instabilities in Nature

This book is concerned with the propagation of waves and instabilities both linear and nonlinear, but concentrating on the latter.

The main advances in this subject have quite naturally come from studies involving fluids and more recently plasmas. The latter primarily because of the possibility of ‘cheap, unlimited’ (and hopefully safe) power from thermonuclear reactions. Everyone is of course familiar with waves on water if only being aware of the many instances where they provide examples of natural beauty. It is not so obvious that very similar waves can exist in a plasma which, to a good approximation, is usually a very dilute assembly of ions and electrons. We shall see later in this chapter that this is indeed so and fluids and plasmas have much in common. However, plasmas also show a much wider range of phenomena basically because they are composed of two or more components and also can be made strongly anisotropic by the introduction of magnetic fields, something that is not possible for simple fluids. This richer variety of phenomena has also been a reason why plasmas have had more than their share of attention.

The above remarks notwithstanding, there are numerous media other than plasmas and fluids which can support waves and/or propagate instabilities. As we will see, some of these are more intriguing than others.

Introduction

In Chapter 2 a linear theory of waves and instabilities was presented for the case of the propagation of waves in infinite and uniform media. Basically one considers the wave disturbance (in one dimension) to be proportional to exp(ikx - iωt) with ω, k satisfying the dispersion relation D(ω, k) = 0. In many problems, for example infinite media, one must take k to be real, in which case one distinguishes between stable (Imω(k) ≤ 0 and unstable disturbances for a particular k value Imω(k) ≤ 0).

Early experiments suggested that such an approach was not always sufficient. For example, experiments involving the interaction of charged particle beams with stationary plasmas consistently showed little sign of being unstable. On the other hand, all theoretical models based on the ideas outlined above unequivocably suggested the system to be unstable. In the case of plasmabeam experiments, the explanation which resolved this difficulty was given independently by Sturrock (1958) and by Fainberg, Kurilko and Shapiro (1961). The Soviet authors based their method on earlier work by Landau and Lifshitz (1959) Chapter 3. These latter authors were concerned with problems in fluid mechanics.

The resolution of this problem is based on the fact that it is not sufficient to treat the time development of a system by considering just a single k mode, but rather it is necessary to consider a spatial pulse or wave packet which is composed of a range of k values. Then unstable media can be classified into two distinct types.

A brief historical survey of large amplitude nonlinear wave studies

This chapter singles out some nonlinear wave phenomena for detailed treatment. Although we believe these phenomena to be important, we also feel we owe the reader a broader view of various research areas that have developed over the years. Of course, the survey of this section is still somewhat selective, but it does offer the opportunity of reaching much of the generic work on the subject.

During the 1960s the limitations of treating nonlinear waves by expanding in powers of their amplitude A became embarrassingly evident. Fluid dynamists were the first to realize the need for new methods, and they were closely followed by those working in plasma and condensed matter physics. Although, as we saw in Chapter 5, quite a lot of physics can be introduced through the back door of A2 theory, some large amplitude effects cannot. A few examples, some of which have been known to navigators and fluid dynamists for quite a long time as observed phenomena, are:

1. The formation of sharp crests on steady profile waves.

2. The formation of ordinary (single hump) and envelope solitons on the water surface and their dynamics.

3. Wave breaking. (One way this can happen is when a large amplitude wave propagates into a region of shallow water. This is familiar from summer holidays on the beach.)

4. Nonlinear waves evolving into soliton trains.

When this book was first published in 1990, it became more popular than we expected. A book club chose it as its Book of the Month. It was reprinted in 1992. Eight years have now gone by and we feel it is time for a proper revision. New results and references have been added. On the other hand, some chapters remain largely as they were, since we feel that the presentation of the basic ideas to be found there remains valid. Chapter 11 (Chapter 10 in the first edition) on chaotic phenomena is an example of this.

The only criticism anyone made to our faces was that we leaned too heavily on plasma physics and hydrodynamics for examples, whereas most phenomena and methods we consider have wider applications. These include optics, biology, solid state physics and other fields. This shortcoming has now been rectified to a certain extent. Also, a new chapter on soliton metamorphosis, including some colour plates, has been added (Chapter 10).

However, much of the text has been left as it was. Thus ‘recent’ should be read as recent in 1990. Some printing errors have been corrected. Once again, Dr Simon Capelin of the Cambridge University Press has been patient and helpful. Ms Lenkowska-Czerwihska spent a large portion of her time in Warwick helping us organize our material.