2012/11/29 by Jarmo Hietarinta, Da-jun Zhang, Da‐jun Zhang · 2 citations
Mathematics · Physics and Astronomy · #Fractional Differential Equations Solutions #Nonlinear Photonic Systems #Nonlinear Waves and Solitons
paper · doi:10.1080/10236198.2012.740026
openalex publication_date 2012/11/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/29
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 3 × 3 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.