Abstract
Precedently, we have applied the Painlevé analysis introduced by Weiss, Tabor and Carnevale in 1983 [1] to the finding of solutions of partially integrable nonlinear partial differential equations (PDE’s) [2,3]. Here, we use this analysis for determining Lax pairs when the PDE passes the Painlevé test and is presumably integrable. The starting point is the set of Painlevé-Darboux (P-D) equations which comes out of the Painlevé analysis as a condition for a Laurent series truncated at the constant level term in its expansion variable to represent a solution of the PDE. Those equations, depending on the singular manifold φ, are simpler to analyze than the PDE itself. Moreover, Weiss [4] has shown that it is enough to consider only one equation (the singular manifold equation) deduced from the P-D set to determine the Lax pair. His method has been successfully applied to many nonlinear PDE’s but his presentation has the main drawback not to be algorithmic.
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Musette, M. (1992). Insertion of the Darboux Transformation in the Invariant Painlevé Analysis of Nonlinear Partial Differential Equations. In: Levi, D., Winternitz, P. (eds) Painlevé Transcendents. NATO ASI Series, vol 278. Springer, Boston, MA. https://doi.org/10.1007/978-1-4899-1158-2_13
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DOI: https://doi.org/10.1007/978-1-4899-1158-2_13
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