Hirota representations of soliton equations have proved very useful. They produced many of the known families of multisoliton solutions, and have often led to a disclosure of the underlying Lax systems and infinite sets of conserved quantities.

A striking feature is the ease with which direct insight can be gained into the nature of the eigenvalue problem associated with soliton equations derivable from a quadratic Hirota equation (for a single Hirota function), such as the KdV equation or the Boussinesq equation. A key element is the bilinear Bäcklund transformation (BT) which can be obtained straight away from the Hirota representation of these equations, through decoupling of a related “two field condition” by means of an appropriate constraint of minimal weight. Details of this procedure have been reported elsewhere. The main point is that bilinear BT's are obtained systematically, without the need of tricky “exchange formulas”. They arise in the form of “Y-systems”, each equation of which belongs to a linear space spanned by a basis of binary Bell polynomials (Y-polynomials).