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.