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

Abstract

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.

Keywords:

• partial difference equations
• integrability
• Hirota bilinear method
• lattice equations

Acknowledgements

This project is partially supported by the NSF of China (No. 11071157), SRF of the DPHE of China (No. 20113108110002) and Shanghai Leading Academic Discipline Project (No. J50101).

Notes

1. [email protected].

2. In the literature one sometimes finds examples of over-bilinearization, in which a multi-linear equation is broken down into too many bilinear equations. One case study can be found in [7].

3. In REDUCE such rewrite rules are applied repeatedly until the expression no longer contains the monomial on the LHS of the rule. In this particular case the final expression will only contain first powers of the pp(n).

4. For these computations it is convenient to use the rational parametrization and correspondingly , , where w, z are the two new free parameters.

5. This is also the type of limit by which Miwa's equation reduces to the Hirota's equations.

Additional information

Notes on contributors

Da-jun Zhang

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