This chapter discusses finite translative arrangements of a given convex domain in ℝ. For packings, we determine the translative packing density and the asymptotic structure of the optimal packings of n translates for large n. We prove the analogous results for coverings by translates of centrally symmetric convex domain K, and, moreover, the hexagon bound for coverings of any convex shape by at least seven translates of K. The main tools are the Oler inequality for packings and the Fejes Tóth inequality for coverings. In addition, we prove the Bambah inequality for coverings when the convex hull of the centres is covered. Many of these proofs are based on constructing Delone type simplicial complexes. The final part of Chapter 2 discusses problems related to the Hadwiger number of K, which is the maximal number of nonoverlapping translates of K touching K.

Assuming that K is centrally symmetric, let us compare our knowledge of arrangements of translates of K (see this chapter) and of congruent copies of K (see Chapter 1). We have essentially the same results for packings. However, we know much more about coverings by translates because any two homothetic convex domains are noncrossing (see Section 1.5).

About the Minkowski Plane

Convex domains K and C are called homothetic if C = x + λK for some λ > 0. This chapter discusses problems when the geometry of the plane is considered in terms of convex domains homothetic to a given one.

We start this chapter by discussing the Newton number: the maximal number of nonoverlapping unit balls in ℝd touching a given unit ball. The three-dimensional case was the subject of the famous debate between Isaac Newton and David Gregory, and it was probably the first finite packing problem in history.

In the later part of the chapter, optimality of a finite packing of n unit balls means that the volume or some mean projection of the convex hull is minimal. If the dimension d is reasonably large then the packing minimizing the volume of the convex hull is the sausage; namely, the centres are collinear. However, if some mean projection is considered then the convex hull of the balls in an optimal arrangement is essentially some ball for large n in any dimension. For the mean width, we also verify that, in the optimal packing of d + 1 balls, the centres are vertices of a regular simplex.

Concerning optimal coverings of compact convex sets by n unit balls in, mostly conjectures are known; namely, it is conjectured that the optimal coverings are sausage-like (see Section 8.6). However, sound density estimates will be provided when a larger ball is covered by unit balls.

In this chapter only a few proofs are provided because the arguments either use the linear programming bound (6.1) or are presented in Chapter 7 for packings and coverings by congruent copies of a given convex body.

The year 1964 witnessed the publication of two fundamental monographs about infinite packing and covering: László Fejes Tóth's Regular Figures, which focused on arrangements in surfaces of constant curvature, and Claude Ambrose Rogers's Packing and Covering, which discussed translates of a given convex body in higher dimensional Euclidean spaces. This is the finite counterpart of the story told in these works. I discuss arrangements of congruent convex bodies that either form a packing in a convex container or cover a convex shape. In the spherical and the hyperbolic space I only consider packings and coverings by balls. The most frequent quantity to be optimized is the density, which is the ratio of total volume of the congruent bodies over the volume either of the container or of the shape that is covered. In addition, extremal values of the surface area, mean width, or other fundamental quantities are also investigated in the Euclidean case. A fascinating feature of finite packings and coverings is that optimal arrangements are often related to interesting geometric shapes.

The main body of the book consists of two parts, followed by the Appendix, which discusses some important background information and prerequisites. Part 1 collects results about planar arrangements. The story starts with Farkas Bolyai and Axel Thue, who investigated specific finite packings of unit discs in the nineteenth century. After a few sporadic results, the theory of packings and coverings by copies of a convex domain started to flourish following the work of László Fejes Tóth.

In this chapter we review some topics that are needed in the main part of the text. The book uses tools from various branches of mathematics related to convexity; hence, to give the reader a chance to follow the presentation without constantly checking references, this introductory part is more extensive than usual. We present only a few proofs where the exact statement we need is not so easy to access. The chapter is organized in a way that most of the material needed in Part 1 is contained in Sections A.1–A.6.

Before going into details we define the central notions of this book. A set K is called convex if it contains any segment whose endpoints lie in K. In addition, K is a convex body if K is compact and its interior is nonempty. Planar convex bodies are also known as convex domains. A family {Kn} of convex bodies is called a packing if the interiors of any two Ki and Kj, i ≠ j, are disjoint, or, in other words, if Ki and Kj do not overlap. Next, {Kn} is a covering of a set X if the union of the convex bodies contains X. Finally, {Kn} is a tiling of X if each Kn is contained in X, and {Kn} is both a packing and a covering of X.

Some General Notions

As usual ℕ, ℤ, ℝ and denote the family of natural numbers, integers, and real numbers, respectively.

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.