\(D\) bar dressing and Sato theory (Q1311001)

Soliton and more general integrable partial differential equations have been studied and derived from a number of different points of view and it remains an important task to understand the relations of these as well as their limitations. Sato theory, as first developed by \textit{G. Date, M. Jimbo, M. Kashiwara} and \textit{T. Miwa} [see for example [Nonlinear integrable systems, Proc. RIMS Symp. Kyoto 1981, 39--119 (1983; Zbl 0571.35098)]) has been a powerful tool in the algebraic theory. It interprets the soliton equations as group orbits in an infinite dimensional Grassmannian manifold. It is to be complemented by the more analytic approach of the inverse scattering transform [see for example: ``Theory of solitons'' by \textit{S. P. Novikov, S. V. Manakov, L. P. Pitaevskii}, and \textit{V. E. Zakharov} (1984; Zbl 0598.35002)] which finds its most elegant formulation in the Riemann-Hilbert problem and the \(D\) bar method [\textit{R. Beals} and \textit{R. Coifman}, Proc. Symp. Pure Math. 43, 45--70 (1985; Zbl 0575.35011)]. The chief result of the present paper is a formal derivation of a central result of the Sato theory, the Hirota bilinear identity, from the \(D\) bar method.