A search for integrable bilinear equations: The Painlevé approach

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Abstract
The possibility of the existence of new integrable partial differential equations is investigated, the tools of singularity analysis. The equations treated are written in the Hirota bilinear formalism. It is shown here how to apply the Painlevé method directly under the bilinear form. Just by studying the dominant part of the equations, the number of cases to be considered can be limited drastically. Finally, the partial differential equations identified in a previous work [J. Hietarinta, J. Meth. Phys. 2 8, 1732, 2096, and 2586 (1987); 2 9, 628 (1988)] as possessing at least four soliton solutions, are shown to pass the Painlevé test as well, which is a strong indication of their integrability.