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A search for integrable bilinear equations: The Painlevé approach. (English) Zbl 0729.35133

Summary: The possibility of the existence of new integrable partial differential equations is investigated, using 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 of the third author [J. Math. Phys. 28, 1732-1742, 2094-2101 and 2586-2592 (1987; Zbl 0641.35073, Zbl 0658.35081 and Zbl 0658.35082); 29, No.3, 628-635 (1988; Zbl 0684.35082)] as possessing at least four soliton solutions, are shown to pass the Painlevé test as well, which is a strong indication of their integrability.

MSC:

35Q58 Other completely integrable PDE (MSC2000)
35Q51 Soliton equations
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References:

[1] DOI: 10.1088/0266-5611/3/2/008 · Zbl 0645.35087
[2] DOI: 10.2977/prims/1195182017 · Zbl 0557.35091
[3] DOI: 10.1063/1.528002 · Zbl 0684.35082
[4] DOI: 10.1063/1.528002 · Zbl 0684.35082
[5] DOI: 10.1063/1.524491 · Zbl 0445.35056
[6] DOI: 10.1063/1.525721 · Zbl 0514.35083
[7] DOI: 10.1016/0375-9601(82)90291-2
[8] DOI: 10.1103/PhysRevLett.53.1
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