ON THE BILINEAR FORMS OF PAINLEVé'S 4TH EQUATION

Abstract

Equations in Hirota's bilinear form appear naturally whenever one needs to write integrable differential equations in terms of "nice" functions. In the case of soliton equations nice typically means polynomials of exponentials with lin- ear exponents, whereas in the context of Painleve equations nice often means entire functions. The natural or original dependent variables of the equation are usually not the best in this respect and a change of the dependent vari- ables is necessary. This dependent variable transformation can be somewhat involved. Since the solutions of Painleve equations are meromorphic by definition (the movable singularities can be at worst poles) it is natural to express them as ratios of entire functions, and this leads to homogeneous equations and often to equations in Hirota's bilinear form. Indeed, bilinear forms were already derived by Painleve himself (1). It should be noted, however, that the converse of the above is not necessarily true: a bilinear form does not by itself imply that the independent functions are regular in any sense, in fact bilinear forms exist even for non integrable equations. In the field of integrable partial differential equations, the so-called τ - functions play a major role. These functions actually provide a huge pool of such "nice" functions, in both of the aforementioned interpretations, and they also possess an extremely rich algebraic and algebro-geometric structure. This is the main theme of Sato-theory, one of the great unifying theories for the description of integrable systems. The relevance of τ -functions for the study of the Painleve equations has been recognized ever since the seminal work of Jimbo and Miwa on isomonodromy deformations (2) and in particular since the results obtained by Okamoto, regarding the algebro-geometric properties of the Painleve equations (3).