Finite - genus solutions for the Hirota ’ s bilinear difference equation

The finite-genus solutions for the Hirota's bilinear difference equation are constructed using the Fay's identities for the θ-functions of compact Riemann surfaces. In the present work I want to consider once more the question of constructing the finite-genus solutions for the famous Hirota's bilinear difference equation (HBDE) [1] which has been solved in [2] using the so-called algebraic-geometrical approach. This method, which is the most powerful method for deriving the quasi-periodic solutions (QPS) and which has been developed for almost all known integrable systems, exploits some rather sophisticated pieces of the theory of functions of complex variables and is based on some theorems determining the number of functions with prescribed structure of singularities on the Riemann surfaces (see [3] for review). However, in some cases the QPS can be found with less efforts, in a more straightforward way, using the fact that the finite-genus QPS (and namely they are the subject of this note) of all integrable equations possess similar and rather simple structure: up to some phases they are 'meromorphic' combinations of the θ-functions associated with compact Riemann surfaces [4] (the situation resembles the pure soliton case, where solutions are rational functions of exponents). Thus, to construct these solutions we only have to determine some constant parameters. This, as in the pure soliton case, can be done directly, using the well-known properties of the θ-functions of compact Riemann surfaces. In [5] such approach was developed for the Ablowitz-Ladik hierarchy, where the finite-genus solutions were 'extracted' from the so-called Fay's formulae [6, 4]. The fact, that the HBDE is closely related to the Fay's identities is not new and was mentioned, e.g., in [2] (see Remark 2.7), but contrary to this work I will use these identities as a starting point and will show how one can derive from them, by rather short and very simple calculations