In Chapter 1 we introduced some particular functions of a complex variable, such as powers and the exponential, that were needed for the chapters to follow. We now take up the subject again and develop a general theory of functions of a complex variable, including integration in the complex plane. Since the plane has no physical meaning it might seem that we are embarking on a study that has no relevance to an engineer, but a familiarity with complex-variable techniques is important in all branches of engineering. In particular, they are essential for a proper appreciation of the Fourier- and inverse Laplace-transform operations discussed in later chapters, and this is the reason for their consideration now.

General properties

We consider complex functions of the complex variable z = x + iy, where the definition of a function is analogous to that in real variable theory:

A domain is simply an open region of the complex plane bounded by a closed contour, and we shall use such terms with no more precise definition than geometrical intuition requires.

Having now arrived at the heart of our subject, we show how the mathematical techniques that have been discussed can be used to solve some of the problems associated with linear systems.

Basic concepts

In its most general form a system is any physical device that when stimulated or excited, produces a response. The system could be as complicated as (a model of) the human brain or as simple as an electrical circuit with a lumped resistance, and the problem the engineer could face is to determine the response or output of the system resulting from a known excitation or input. To develop procedures that are applicable to a variety of systems, it is necessary to restrict the type of system considered.

Physical description

For the systems of concern to us it is assumed that the input and output are functions of a single real variable t, which we shall speak of as time, and that a causal relationship exists between the two. The term causal implies that the output is a function of the input alone, and because of this, the system is “nonanticipatory” that is, there is no output until the input is applied. This is an attribute of all physically realizable systems. For convenience we shall refer to the input and output as signals, and if these are denoted by f(t) and x(t), respectively, a system can be depicted as shown in Fig. 3.1.

For the past decade and more, all students in electrical and computer engineering at the University of Michigan have been required to take a course concerned with the mathematical methods for the solution of linear-systems problems. The course is typically taken in the junior year after completion of a basic four-semester mathematics sequence covering analytic geometry, matrices and determinants, differential and integral calculus, elementary differential equations, and so forth. Because it is the last course in mathematics that all must take for a bachelor's degree, a number of topics are included in addition to those customary in an introductory systems course, for example, functions of a complex variable with particular reference to integration in the complex plane, and Fourier series and transforms.

Some of the courses that make use of this material can be taken concurrently thereby constraining the order in which the topics are covered. It is, for example, necessary that Laplace transforms be introduced early and the treatment carried to such a stage that the student is able to use the transform to solve initial-value problems. Fourier series must also be covered in the first half of the course, and the net result is an ordering that is different from the mathematically natural one, but that is quite advantageous in practice. Thus, the relatively simple material dealing with the Laplace transform and its applications comes at the beginning and provides the student with a sense of achievement prior to the introduction of the more abstract material on functions of a complex variable.

INTRODUCTION.

The spectral transform was first used by Gardner et al. (1967) to solve the Korteweg-de Vrles equation

Their method was to solve the Schrödinger equation

and extract the “spectral data”, S from this solution. Then they discovered that if the “potential” q(x) evolved with respect to t according to (1.1) each item of spectral data evolved as the solution of a first order linear ordinary differential equation with t as independent variable. Thus they were able to find S at a later time and by using the Gel'fand- Levitan-Marchenko equation to reconstruct q(x) at this later time.

For some years it was thought that this was a freak result until Zakharov and Shabat (1971) solved the nonlinear Schrödinger equation

in a similar way using the spectral problem

This was followed by the famous “AKNS” paper (1974) which firmly established the spectral transform (or inverse scattering method) as a very powerful weapon in the armoury of mathematicians and physicists who were studying the “soliton” (Bullough and Caudrey, 1980).

Matrix generalizations of (1.2) and (1.4) greatly extended their scope (Wadati, 1980; Calogero and Degasperis, 1980), but it was 1980 before a third order spectral transform was discovered (Kaup, 1980; Caudrey, 1980). Higher order ones have been found since then (Mikhailov, 1981).

This paper describes a fairly general nth order spectral transform (Caudrey, 1982) and is organized as follows. Section 2 describes the spectral problem and shows how it can be solved. Sections 3 and 4 give the spectral transform and its inverse. Section 5 shows that the original spectral transform of Gardner et al. is just a special case of this general transform (as is that of Zakharov and Shabat). Finally, Section 6 shows how a system of nonlinear Klein-Gordon equations can be solved.

THE SPECTRAL PROBLEM.

A general nthorder spectral problem can be written in the form of a set of first order equations.

INTRODUCTION.

In an unmagnetized plasma, three kinds of waves can propagate. These are the ion-acoustic, electron plasma, and light waves. The linear dispersion relations of these waves are, respectively, given by

where cs is the sound speed, ƛDis the electron Debye length, is the s 1) electron plasma frequency, and c is the velocity of light.

The most important role of nonlinear effects is to cause steepening of the leading edge of a wave. However, it is frequently found that the dispersion effects become significant as the steepness of the front increases. The competition between nonlinearity and dispersion leads to localized waves often called “solitons”.

In the present article, we review theories of solitons in an unmagnetized plasma. To understand the physics, we have chosen three simple waves (as given above) and have worked out nonlinear theories for them. As is well-known (Sagdeev, 1966), supersonic compresslonal ion-acoustic solitons have a maximum potential ∼ 1.3 Te/e, and the corresponding speed is about 1.6 cs. On the other hand, finite amplitude envelope Langmuir and light wave solitons are obtained by incorporating a fully nonlinear analysis of the low-frequency plasma motion. It is found that under certain circumstances, equations governing the stationary solitons are exactly integrable and can be written in terms of the energy integral of a classical particle. By analogy with the motion of an oscillator in a potential well, we analyze the conditions under which localized solutions exist. Our theory of Langmuir wave envelope solitons compares favorably with laboratory experiments. In the small amplitude limit, a perturbation theory has been introduced which enables us to derive a set of nonlinear evolution equations describing Langmuir soliton turbulence. A new kind of electromagnetic soliton emerges when the relativistic corrections to the nonlinear current density and the ponderomotive force are taken into account. Here, the forced electron density perturbation contains a depression at the center, together with shoulders of density excess on the sides. Such kinds of nonlinear entities can effectively accelerate electrons in plasma.

INTRODUCTION

The basis of the theory of the development of wave motion in classical physics is to treat the wave as a perturbation on an equilibrium or steady state, and to linearize the governing equations in the perturbation variables. This familiar technique has been fully exploited in the theory of the propagation of electromagnetic waves in plasmas, where it has produced a wealth of detailed analysis which has application to a wide range of physical phenomena.

On the other hand, waves in plasmas are subject to nonlinear effects which manifest themselves quite readily. Typical is the ‘Luxembourg effect, observed in the early years of broadcasting, in which the field of a ‘disturbing’ wave so influences the ionospheric environment through which a ‘wanted’ wave is propagated that the latter is noticeably modulated. Many other kinds of ‘wave-interaction’ are possible, and a variety of studies of the interaction between two or more waves probably forms the bulk of the literature on nonlinear plasma waves. The emphasis in such work is mainly on obtaining the first order corrections to linear theory, and investigating the conditions most favourable for interaction.

Another corner of nonlinear theory, which may perhaps have more attraction for the mathematician, owes its stimulus to comparatively recent additions to our physical understanding. It is recognized that both the laser and the pulsar are capable of generating an electromagnetic wave that, at its specific frequency, is so strong that its passage through a plasma constitutes a finite-amplitude, maybe even a largeamplitude, wave, whose description is quite outside the scope of linear theory. This poses the problem of finding solutions of the exact (nonlinear) governing equations. Specifically, it is of interest to seek periodic solutions, with the reasonable expectation that there will be those that recover the familiar waves of linear theory in the small amplitude limit. The earliest work of this kind, Akhiezer & Polovin (1956), in fact predated the appreciation of the applications mentioned, and did not apparently attract much attention until comparatively recently. A comprehensive review of developments up to a few years ago is available in Decoster (1978), and some more recent work is covered in the present paper.

INTRODUCTION.

Analytical methods of modelling water waves of small but finite height are based on the linear theory and improved with weakly nonlinear theories (West, 1981). An alternative is to develop, with computer assistance, water wave models which are nonlinear in their lowest approximation and are valid for a range of heights up to the onset of wave breaking (Schwarz & Fenton, 1982). The present approach falls into the latter category, and is concerned with investigating wave geometries which occur locally in deep water.

Water waves propagating from a surface disturbance are subject to dispersion modified by nonlinear wave interactions. This property suggests that the numerical resolution into Fourier components of the nonlinear equations governing the evolution of a water wave system models the dispersion and its modification, and is therefore a natural method for investigating water wave properties. Fornberg S Whitham (1978) used this approach in studying certain nonlinear model equations for wave phenomena. It is applied here to Laplace's equation with the nonlinear boundary conditions appropriate to irrotational gravity wave propagation in deep water.

Analytical solutions in the form of perturbation expansions exist for two dimensional water waves of permanent shape in deep water (Stokes waves) for which the dispersion and nonlinear modification are in balance. A number of computer-based methods have been used (Schwartz S Fenton, 1982, §2) to extend the calculations up to the highest waves of permanent form. The present method is demonstrated first (§2) for the calculation of two dimensional permanent waves. Three dimensional permanent waves have been found recently as perturbations to two dimensional permanent waves. The present method allows calculations of three dimensional waves independently of two dimensional waves, and one such example is presented below (§3).

Waves on the ocean surface often occur locally as a wave group with an envelope that changes slowly as the waves propagate. Analytical solutions exist for weakly nonlinear wave groups of permanent envelope in two and three dimensions. The present method is applied to the calculation of wave groups of permanent envelope in two dimensions (§4) and in three dimensions (§5), in both cases without the restrictions on wave height which are needed for the analytical solutions.

INTRODUCTION.

Till recently, one notable hiatus in the theory of surface waves was the absence of any satisfactory analysis to describe an overturning wave. In this category we include both the well-known “plunging breaker” and also any standing or partially reflected wave which produces a symmetric or an asymmetric jet, with particle velocities sometimes much exceeding the linear phase-speed.

A first attempt to describe the jet from a two-dimensional standing wave was made by Longuet-Higgins (1972), who introduced the “Dirichlet hyperbola”, a flow in which any cross-section of the free surface takes the form of a hyperbola with varying angle between the asymptotes. Numerical experiments by Mclver and Peregrine (1981) have shown this solution to fit their calculations quite well. The solution was further analysed in a second paper (Longuet-Higgins, 1976) where a limiting form, the “Dirichlet parabola”, was shown to be a member of a wider class of selfsimilar flows in two and three dimensions. Using a formalism introduced by John (1953) for irrotational flows in two dimensions, the author also showed the Dirichlet parabola to be one of a more general class of selfsimilar flows having a time-dependent free surface.

All the above flows were gravity-free, that is to say they did not involve g explicitly; they are essentially descriptions of a rapidly evolving flow seen in a frame of reference which itself is in free-fall.

A useful advance came with the development of a numerical time-stepping technique for unsteady gravity waves by Longuet-Higgins and Cokelet (1976, 1978). As later refined and modified by Vinje and Brevig (1981), Mclver and Peregrine (1981) and others, this has given accurate and reproducible results for overturning waves, with which analytic expressions can be compared.

A further advance on the analytic front came with the introduction by Longuet-Higgins (1980a) of a general technique for describing free-surface flows, that is flows satisfying the two boundary conditions

at a free surface. Particular attention was paid to the parametric representation of the flow in a form

where both the complex coordinate z = x + iy and the velocity potential x are expressed as functions of the intermediate complex variable u and the time t.

INTRODUCTION.

The goal of this paper is to present a recently developed formalism for a) solving inverse problems in the plane for potentials decaying at infinity (i.e. given appropriate scattering data reconstruct the potential q(x,y); b) solving the initial value problem (for appropriately decaying initial data) of certain nonlinear evolution equations in two spatial and one temporal dimensions (i.e. given q(x,y,o) find q(x,y,t)). Several results obtained by this formalism are also summarized.

This formalism has been developed in a series of papers by Fokas and Ablowitz (1982a,b,c; 1983), Ablowitz et al (1982), Fokas (1982), where several inverse problems related to physically significant multidimensional equations have been formally solved. The inverse problem associated with a certain differential Riemann-Hilbert problem (in the complex x-plane), which is related to the Benjamin-Ono (BO) equation was considered in Fokas and Ablowitz (1982a). The BO equation, although an equation in 1+1 (i.e. in one space and one time dimension) has many features similar to problems in 2+1 (this results from its nonlocal character). In this sense, BO acts as a pivot from 1 + 1 to 2 + 1. The inverse problem associated with the “time“-dependent Schrb'dinger equation (see Dryuma (1974))

as well as the initial value problem of the related Kadomtsev-Petviashvili (KP)I (1970)

were considered in Fokas and Ablowitz (1982a,b). The inverse problem associated with

and the related KPII were considered in Ablowitz et al (1982). The inverse problem associated with the matrix equation

was considered by Fokas (1983) and Fokas and Ablowitz (1983). In Equation (1.5), B is a constant n x n diagonal matrix, J is a constant n x n diagonal matrix with elements either all real (hyperbolic case) or all purely imaginary (elliptic case), and q(x,y,t) is a n x n off-diagonal matrix containing the potentials (or field variables). Equation (1.5) can be used to solve several physical nonlinear equations in 2 + 1 (Ablowitz and Haberman, 19 7 5). Among them are the n-wave interaction (Ablowitz and Segur, 1981), variants of the so-called Davey-Stewartson (DS) equation (1974) (which is the long wave limi of the Benney-Eoskes equation (1969)) and the modified KP (MKP) equation.

INTRODUCTION.

The last decade and more has produced an altogether unlooked-for impetus in the study of certain partial differential equations by use of the inverse scattering transform. Two of the (now) standard equations which are susceptible to this technique arise quite naturally in the study of water waves: the Korteweg-de Vries (KdV) equation and the nonlinear Schrodinger (NLS) equation. This suggests the possibility that there exist other equations of a similar character which are also relevant in water wave theory. The similarity may merely be that the equations are generalisations (more terms, variable coefficients, etc.) which convert the problem into a non-integrable one. On the other hand a conceivable result is that we generate other integrable equations which are extensions of the classical equations to different - possibly higher dimensional - co-ordinate systems. The overall picture is that of a number of diverse equations which describe various aspects of the same underlying problem. This has the virtue that we can more readily compare and contrast the equations, and in some cases specifically relate one to another.

In this paper we shall collect together many of the varied forms of KdV (and to a lesser extent NLS) equations which arise in water wave theory. To emphasise the connecting themes the same variables and parameters will be used throughout. We shall start from an appropriate set of basic equations and thence develop both KdV and NLS equations for one spatial dimension. Since both equations describe alternative aspects of the same problem (by employing different limits in parameter space), it should be possible to match these two equations: this is readily demonstrated. We then turn to the two-dimensional problems which correspond to both the KdV and NLS equations. Some properties of the relevant similarity solutions in one and two dimensions are mentioned, together with matching to the near-field, i.e. to initial data. Finally, we briefly comment on a few other more involved equations of KdV-type which describe the rôle of other physical processes (such as viscous dissipation).