3 - Structure of soliton equations

Summary

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].