Abstract. We present a set of bilinear matrix identities that generalize the ones that have been used to construct the bright soliton solutions for various models. As an example of an application of these identities, we present a simple derivation of the N -bright soliton solutions for the Ablowitz-Ladik hierarchy.
Introduction.This paper is a continuation of the work initiated in [1] where we presented the set of the soliton Fay identities. The main aim of [1] was to summarize the results of a number of works devoted to the soliton solutions for various integrable models. We studied some properties of the matrices that had been used in [2,3,4,5,6,7] to construct the dark solitons and presented some identities that are satisfied by these matrices. These identities, which may be viewed as soliton analogues of the Fay identities [8,9], can be used as a starting point for the so-called direct approach to the integrable equations.The classical Fay identities [8,9] have been derived for the theta-functions associated with compact Riemann surfaces of the finite genus. It is a known fact that the so-called finite-gap solutions for integrable equations (which are built of these theta-functions) can be transformed into soliton ones by degenerating the corresponding Riemann surface (see [10,11] and references therein). Thus, the soliton identities of [1] and of this paper can be, in principle, obtained by limiting procedure form the classical Fay identities. However, we do not use such approach, since our identities can be derived by elementary calculations without invoking the much more difficult algebro-geometric constructions of [8]. In other words, we use the term 'Fay identity' to refer to bilinear identities that are derived from the properties of the involved objects (theta-functions for classical Fay identities, and soliton matrices for our case).Soliton Fay identities. II. Bright soliton case.