Hirota's method and the search for integrable partial difference equations. 1. Equations on a 3x3 stencil

Abstract: Hirota's bilinear method ("direct method") has been very effective in constructing soliton solutions to many integrable equations. The construction of one- and two-soliton solutions is possible even for non-integrable bilinear equations, but the existence of a generic three-soliton solution imposes severe constraints and is in fact equivalent to integrability. This property has been used before in searching for integrable partial differential equations, and in this paper we apply it to two dimensional partial difference equations defined on a 3x3 stencil. We also discuss how the obtained equations are related to projections and limits of the three-dimensional master equations of Hirota and Miwa, and find that sometimes a singular limit is needed.