Binary Bell polynomials and Darboux covariant Lax pairs

Extract

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

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