Introduction

Mikio Sato [12, 13] was the first to discover that the KP (Kadomtsev—Petviashvili) equation is the most fundamental among the many soliton equations. Sato discovered that polynomial solutions of the bilinear KP equation are equivalent to the characteristic polynomials of the general linear group. Later, he found a Lax pair for a hierarchy of KP-like equations by means of a pseudo-differential operator, and came to the conclusion that the KP equation is equivalent to the motion of a point in a Grassmanian manifold and its bilinear equation is nothing but a Plücker relation. Also, Junkichi Satsuma [37] had discovered before Sato that the soliton solutions of the KdV equation could be expressed in terms of wronskian determinants. Later, in 1983, Freeman and Nimmo [38, 39] found that the KP bilinear equation could be rewritten as a determinantal identity if one expresses its soliton solutions in terms of wronskians. In this chapter, we develop the above results and show that some bilinear soliton equations having solutions expressed as pfaffians (or as determinants) are nothing but pfaffian identities.

Remark

The KdV equation is a 1+1-dimensional equation describing shallow water waves. The KP equation was introduced in order to discuss the stability of these waves to perpendicular horizontal perturbations [40].

No sooner had the author started to describe an application of the direct method that he realized that he had used up the allotted space on the fundamentals. Even though he thought of shortening some of the detailed explanations, he remembered that he had suffered reading difficult mathematics books because of their terse style, and so decided to retain the seemingly superfluous remarks.

Let us briefly mention some topics not discussed in the book.

1. Fundamental soliton equations such as the KP, BKP and Toda equations and their Bäcklund transformation formulae may be regarded as ‘atoms’ for constructing various kinds of soliton equations. Combination of these equations generate many other soliton equations and their solutions. Modern science has been able to understand the properties of materials by decomposing them into their constituents, or atoms, and has managed to create new materials by combining different atoms. It is a pity that lack of space prevented the author explaining how to construct new soliton equations from the above atoms. For example, the KP equation and its Bäcklund transformation formula may also be considered as the bilinear form of the nonlinear Schrödinger equation. In this way, we can construct the dromion solution (two-dimensionally localized soliton) for the Davey—Stewartson equation. It should also be noted that apparently different nonlinear partial differential equations are frequently transformed into the same bilinear form.

2. […]

The second half of the twentieth century saw a resurgence in the study of classical physics. Scientists began paying particular attention to the effects caused by the nonlinearity in dynamical equations. This nonlinearity was found to have two interesting manifestations of opposite nature: chaos, that is the apparent randomness in the behaviour of perfectly deterministic systems, and solitons, that is localized, stable moving objects that scattered elastically. Both of these topics have now been developed into paradigms, with solid mathematical background and with a wide range of physical observations and concrete applications.

This book is concerned with a particular method used in the study of solitons. There are many ways of studying the integrable nonlinear evolution equations that have soliton solutions, each method having its own assumptions and areas of applicability. For example, the inverse scattering transform (IST) can be used to solve initial value problems, but it uses powerful analytical methods and therefore makes strong assumptions about the nonlinear equations. On the other hand, one can find a travelling wave solution to almost all equations by a simple substitution which reduces the equation to an ordinary differential equation. Between these two extremes lies Hirota's direct method. Although the transformation was, at its heart, inspired by the IST, Hirota's method does not need the same mathematical assumption and, as a consequence, the method is applicable to a wider class of equations than the IST.

Introduction

In Chapter 1, we discussed transformations between soliton equations and bilinear equations and the solution of such equations. But what is a bilinear equation, or, more concretely, what mathematical structures are characteristic of bilinear equations? One answer to this question is the existence of groups (related to affine Lie algebras) which act on bilinear equations. In fact, a collective understanding of soliton equations has developed from this viewpoint, and many new soliton equations have been found using this group-theoretical method. This approach, however, calls for a deep knowledge of algebra, and, even when this has been attained, it is difficult to apply. Soon after the birth of quantum mechanics, group theory became a great craze (called Gruppenpest), where many people only studied group theory and never managed to apply it.

Since this book is aimed at students of science and technology, we omit most of the group theory.

A new viewpoint, discovered by Mikio Sato [12, 13], is to regard bilinear equations as equivalent to Plücker relations in a Grassmann manifold. This interpretation of soliton equations, based on a deep knowledge of mathematics, is admirable and beautiful, and has had a strong influence on the author. However, its later development has been so abstract that the author has not been able to understand it completely.

A soliton is a particular type of solitary wave, which is not destroyed when it collides with another wave of the same kind. Such behaviour is suggested by numerical simulation, but is it really possible that the soliton completely recovers its original shape after a collision? In detailed analysis of the results of such numerical simulations, some ripples can be observed after a collision, and it therefore seems that the original shape is not completely recovered. Therefore, in order to clarify whether or not solitons are destroyed through their collisions, it is necessary to find exact solutions of soliton equations.

Generally, it is a very hard task to find exact solutions of nonlinear partial differential equations, including soliton equations. Moreover, even if one manages to find a method for solving one nonlinear equation, in general such a method will not be applicable to other equations. Does there exist any successful and universal tool enabling one to solve many types of nonlinear equations which does not require a deep understanding of mathematics? For this purpose, a direct method has been investigated.

In Chapter 1, we discuss in an intuitive way the conditions under which a solitary wave is formed and we show that a nonlinear solitary wave cannot be made by the superposition of linear waves.