Many nonlinear equations are said to belong to the two-dimensional KP hierarchy. The solution can be expressed by elliptic functions. The purpose of this book is to apply several extensions of the Sato theory, like Grassmannians, of the author and others, to the study of the special KP equation for \(u( t , x,y )\): \[ (-4u_{t}+u_{xxx}+6uu_x )_x+3u_{yy}=0, \] and similar equations. Also, the Burger equation and the Davey-Stewartson equation are studied. The Hirota bilinear equation for the \(\tau\) function, a solution of the KP equation based on the residue theory, outlined in the book, is related to the Sato theory. Several notations for diagrams, which represent the symmetries under consideration are introduced; many of these diagrams are equivalent. For example, the derivatives of the \(\tau\) function can be computed by Young and Maya diagrams. Later, asymptotics are used to classify so-called KP solitons and \(q\)-Catalan numbers and \(q\)-Hermite polynomials are introduced for the benefit of \(N\)-soliton solutions. Finally, \(q\)-Rogers-Ramanujan continued fractions are presented by considering so-called chord diagrams.
An index would have been highly useful, since many different concepts are introduced in this very interesting book.