A Comparative Revisit to the Painleve' Tests for Integrability of Yang Equations, Charap Equations and Their Combinations and Some Unexpected Observations

Abstract

Painleve' test for integrability according to Weiss, Tabor and Carnevale [2] as applied by Chakraborty and Chanda [1,8,9] to the Yang-equations [3], Charap equations [4,5] and the combined Yang–Charap equations [6] have been revisited. Basically two new observations have been reported. (i) From the presentation of Chakraborty and Chanda [9] one could get the impression that for the leading order analysis of the combined equations the prominent role is played by the part that comes from the Yang equations. In the present paper it has been shown that both the Yang equations [3] and the Charap equations [4,5] contribute to the leading order analysis of combined Yang–Charap equations [6]. (ii) The Charap equations have two branches instead of one branch as was reported by Chakraborty and Chanda [8]. For the second branch of the Charap equations [4,5] other than that reported by Chakraborty and Chanda [8] the existence of requisite number of arbitrary functions in the Laurent-like expansion has been investigated. The expansion seems to allow arbitrary functions more than that is required for being a general solution. The similar situations occured to the combined Yang–Charap equations in the work of the Chakraborty and Chanda [9] which could not be completed for the involved nature of calculations.