Discrete Boussinesq-type equations

Abstract

<p>We present a comprehensive review of the discrete Boussinesq equations based<br>on their three-component forms on an elementary quadrilateral. These equa-<br>tions were originally found by Nijhoff et al using the direct linearization method<br>and later generalized by Hietarinta using a search method based on multidi-<br>mensional consistency. We derive from these three-component equations their<br>two- and one-component variants. From the one-component form we derive two<br>different semi-continuous limits as well as their fully continuous limits, which<br>turn out to be PDE’s for the regular, modified and Schwarzian Boussinesq<br>equations. Several kinds of Lax pairs are also provided. Finally we give their<br>Hirota bilinear forms and multi-soliton solutions in terms of Casoratians.<br></p>