Abstract
In this chapter we discuss how the Painlevé equations can be written in terms of entire functions, and then in the Hirota bilinear (or multilinear) form. Hirota’s method, which has been so useful in soliton theory, is reviewed, and connections from soliton equations to Painlevé equations through similarity reductions are discussed from this point of view. In the main part we discuss how the singularity structure of the solutions and formal integration of the Painlevé equations can be used to find a representation in terms of entire functions. Sometimes the final result is a pair of Hirota bilinear equations, but forP VI we need also a quadrilinear expression. The use of discrete versions of the Painlevé equations is also discussed briefly. It turns out that with discrete equations one gets better information on the singularities, which can then be represented in terms of functions with a simple zero.
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Hietarinta, J. (1999). Painlevé Equations in Terms of Entire Functions. In: Conte, R. (eds) The Painlevé Property. CRM Series in Mathematical Physics. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-1532-5_11
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DOI: https://doi.org/10.1007/978-1-4612-1532-5_11
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