Some Remarks on the Hirota Bilinear Identity

Abstract

The Hirota bilinear identity (H) ∫C w(x, k)w*(y, k)dk = 0, as a residue calculation with C a circle at “∞”, is an important ingredient in soliton mathematics (cf. [ 1;2;3;4;5;7;10;13]). Along with the resulting Hirota equations this has deep geometrical meaning and we indicate below also the nature of its conceptual connection to completeness for wave functions (cf. [1;3;4; 10;13]). Given the recent geometrical development of scattering for KdV from the Grassmannian viewpoint in [9] it seems to be appropriate to discuss the nature of (H) in the context of [9] where the residue calculations do not make sense; the article should serve as a complement to [ 1;2;7;9;10;l3] and full details will appear in [4].