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## Über dieses Buch

This is the first book to treat combinatorial and geometric aspects of two-dimensional solitons. Based on recent research by the author and his collaborators, the book presents new developments focused on an interplay between the theory of solitons and the combinatorics of finite-dimensional Grassmannians, in particular, the totally nonnegative (TNN) parts of the Grassmannians.

The book begins with a brief introduction to the theory of the Kadomtsev–Petviashvili (KP) equation and its soliton solutions, called the KP solitons. Owing to the nonlinearity in the KP equation, the KP solitons form very complex but interesting web-like patterns in two dimensions. These patterns are referred to as soliton graphs. The main aim of the book is to investigate the detailed structure of the soliton graphs and to classify these graphs. It turns out that the problem has an intimate connection with the study of the TNN part of the Grassmannians. The book also provides an elementary introduction to the recent development of the combinatorial aspect of the TNN Grassmannians and their parameterizations, which will be useful for solving the classification problem.

This work appeals to readers interested in real algebraic geometry, combinatorics, and soliton theory of integrable systems. It can serve as a valuable reference for an expert, a textbook for a special topics graduate course, or a source for independent study projects for advanced upper-level undergraduates specializing in physics and mathematics.

## Inhaltsverzeichnis

### Chapter 1. Introduction to KP Theory and KP Solitons

Abstract
We begin with a study of the Burgers equation, which is the simplest equation combining both nonlinear propagation effects and diffusive effects, and can be used to describe a weak shock phenomena in gas dynamics (see e.g. [131]). The Burgers equation can be linearized by a nonlinear transformation, known as the Cole-Hopf transformation. The linearization then shows that the Burger equation has an infinite number of symmetries, and the set of those symmetries defines the Burgers hierarchy. The linearization enables us to construct several exact solutions such as multi-shock solutions. It turns out that the set of those exact solutions forms a subclass of the solutions of the KP equation, and multi-shock solutions give examples of the resonant interactions in the KP solutions. We then extend the Cole-Hopf transformation to construct a multi-component Burgers hierarchy, and introduce the $$\tau$$-function, which generates a large class of exact solutions of the KP equation, referred to as KP solitons. Based on the study of this multi-component Burgers hierarchy, we explain its connection to the Sato theory, which provides a mathematical foundation of the KP hierarchy in terms of an infinite dimensional Grassmann variety called the Sato Grassmannian [112, 113]. In this book, we consider a finite dimensional version of the Sato Grassmannian and construct each KP soliton from a point of this Grassmannian.
Yuji Kodama

### Chapter 2. Lax-Sato Formulation of the KP Hierarchy

Abstract
In this chapter, we briefly review the Lax formulation of the KP hierarchy, which consists of an infinite set of linear equations whose compatibility conditions give rise to the flows corresponding to the KP hierarchy. The main purpose of this section is to highlight the basic framework of integrability underlying the KP theory. Then we show that the multi-component Burgers hierarchy discussed in the previous chapter appears as a finite reduction in the Sato theory. In particular, we emphasize the importance of the $$\tau$$-function and explain the central role of the $$\tau$$-function in the KP hierarchy. The materials discussed in this chapter can also be found in [27, 35, 37, 91, 100, 111–113, 132].
Yuji Kodama

### Chapter 3. Two-Dimensional Solitons

Abstract
In Chap. 2, we have shown that the KP hierarchy admits particular solutions, called the KP solitons, the main subject of this book, which are expressed by the Wronskian form. In this chapter, we show that this determinant structure is common for other two-dimensional integrable systems generated by several reductions of the modified bilinear identity proposed by Ueno-Takasaki [128] (see [123] for a further generalization of the bilinear identity). In addition to the KP hierarchy, these integrable systems also include the two-dimensional Toda lattice hierarchy and the Davey-Stewartson hierarchy. Here we construct their soliton solutions in the determinant form and show that their wave parameters for these solutions are chosen from conic curves, that is, the KP soliton from the parabola, the two-dimensional Toda soliton from the hyperbola, and the Davey-Stewartson soliton from the circle.
Yuji Kodama

### Chapter 4. Introduction to the Real Grassmannian

Abstract
This chapter gives a brief introduction to the real Grassmannian $$\text {Gr}(N, M)$$, the set of N-dimensional subspaces in $$\mathbb {R}^M$$, which provides a foundation of a classification of the KP solitons. A point of $$\text {Gr}(N, M)$$ can be represented by an $$N\times M$$ matrix of full rank. We introduce the Schubert decomposition of $$\mathrm{Gr}(N, M)$$ and label each Schubert cell using a Young diagram and a permutation in the symmetric group $$S_M$$. We also introduce a combinatorial tool called the pipedream over the Young diagram, which gives a graphical interpretation of the permutation [103]. The pipedream will be useful to describe the spatial structure of the KP soliton as we will see in the later chapters. (See, for example, [15, 44, 45, 49] for the general information on the Grassmannian, the Young diagram and the symmetric group of permutations.)
Yuji Kodama

### Chapter 5. The Deodhar Decomposition for the Grassmannian and the Positivity

Abstract
In this chapter, we start to review the flag variety $$G/B^+$$ for $$G=\mathrm {SL}_M(\mathbb R)$$ and the Borel subgroup $$B^+$$ of upper triangular matrices, and introduce the Deodhar decomposition of $$G/B^+$$ [33, 34]. Then we give a refinement of the Schubert decomposition of $$\mathrm{Gr}(N, M)$$ as a projection of the Deodhar decomposition, and parametrize each component of the refinement by introducing Go-diagram, which is a Young diagram decorated with black and white stones. In particular, if the Go-diagram has only white stones, it represents a projected Deodhar component of the totally nonnegative (TNN) Grassmannian $$\mathrm{Gr}(N, M)_{\ge 0}$$. The Go-diagram in this case is the -diagram introduced by Postnikov in [103]. We also construct an explicit form of matrix $$A\in \mathrm{Gr}(N, M)_{\ge 0}$$ as a point of the projected Deodhar component by using the parameterizations of the flag variety due to Marsh and Rietsch [83]. We conclude this section to give an algorithm to compute an explicit form of the matrix A and to discuss the positivity of A. Most of the materials presented here can be also found in [72].
Yuji Kodama

### Chapter 6. Classification of KP Solitons

Abstract
This chapter concerns with a classification problem of the asymptotic spatial structure of “regular” KP solitons for large |y| using the Deodhar decomposition of the Grassmannian. More precisely, we consider the soliton solution from a point A represented by an $$N\times M$$ matrix on a Deodhar component $$\mathscr {P}_{v, w}^{>0}$$ of the TNN Grassmannian $$\mathrm{Gr}(N, M)_{\ge 0}$$ and classify the asymptotic structure of the solution in the xy-plane. It turns out that the asymptotic structure of the soliton solutions from the matrix A is completely characterized by a derangement $$\pi =vw^{-1}$$ of the Deodhar component $$\mathscr {P}_{v, w}^{>0}$$, such that the line-solitons for $$y\gg 0$$ are of $$[i,\pi (i)]$$-types with $$i<\pi (i)$$ and those for $$y\ll 0$$ are of $$[\pi (j), j]$$-types with $$\pi (j)<j$$.
Yuji Kodama

### Chapter 7. KP Solitons on

Abstract
In this chapter, we will discuss the reflective symmetry $$(x, y, t) \leftrightarrow (-x,-y,-t)$$ of the KP equation, and show that this symmetry is a consequence of the duality between the Grassmannian $$\mathrm{Gr}(N, M)$$ and $$\mathrm{Gr}(M-N, M)$$ in terms of the KP solitons. Using this duality, we construct the KP solitons for $$\mathrm{Gr}(M-N, M)_{\ge 0}$$ from those for $$\mathrm{Gr}(N, M)_{\ge 0}$$. We then consider a special class of KP solitons for $$\mathrm{Gr}(N, 2N)_{\ge 0}$$, which consists of the same set of the asymptotic solitons at both $$y\ll 0$$ and $$y\gg 0$$, i.e. $$\mathscr {S}_+=\mathscr {S}_-$$. The soliton solutions of this type are referred to as N-soliton solutions. The simplest ones of these solutions consist of N line-solitons having non-resonant interactions among those solitons and are generated from the points in an irreducible component of the lowest dimension, N, of $$\text {Gr}(N, 2N)_{\ge 0}$$. We also discuss some combinatorial properties of those solutions. For the simplest cases of non-resonant interactions, the total number of such N-soliton solutions is given by a Catalan number $$C_N=\frac{1}{N+1}\left( {\begin{array}{c}2N\\ N\end{array}}\right)$$.
Yuji Kodama

### Chapter 8. Soliton Graphs

Abstract
In this chapter, we explicitly construct the soliton graph $$\mathscr {C}_{t}({\mathscr {M}}(A))$$ for an irreducible matrix A from $$\mathscr {P}_{v, w}^{>0}\subset \mathrm{Gr}(N, M)_{\ge 0}$$. In particular, we provide an algorithm that constructs the soliton graphs for the matrix A and give coordinates for all of the trivalent vertices, which then allows one to completely describe the soliton graph. Most of this chapter will be devoted to the case when $$t<0$$, with the final section explaining how the same ideas can be applied to the case when $$t>0$$.
Yuji Kodama

### Backmatter

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