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

: Jarmo Hietarinta, Da-Jun Zhang

Publisher: TAYLOR & FRANCIS LTD

: 2013

: Journal of Difference Equations and Applications

: JOURNAL OF DIFFERENCE EQUATIONS AND APPLICATIONS

: J DIFFER EQU APPL

: 8

: 19

: 8

: 1292

: 1316

: 25

: 1023-6198

DOI: https://doi.org/10.1080/10236198.2012.740026(external)

Hirota's bilinear method (direct method') has been very effective for constructing soliton solutions to many integrable equations. The construction of one-soliton solution (1SS) and two-soliton solution (2SS) is possible even for non-integrable bilinear equations, but the existence of a generic three-soliton solution (3SS) 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 (2D) partial difference equations defined on a 3x3 stencil. We also discuss how the obtained equations are related to projections and limits of the 3D master equations of Hirota and Miwa, and find that sometimes a singular limit is needed.