Higher-order Painleve equations in the polynomial class II: Bureau symbol P1

摘要

In this article, we complete the Painleve classification of fourth-order differential equations in the polynomial class that was begun in paper I, where the subcase having Bureau symbol P2 was treated. This article treats the more difficult subcase having Bureau symbol P1. Some of the calculations involve the use of computer searches to find all cases of integer resonances. Other cases are better handled with the Conte-Fordy-Pickering test for negative resonances. The final list consists of 19 equations denoted F-I, F-II, ... , F-XIX, 17 of which have the Painleve property while 2 (F-II, F-XIX) have Painleve violations but are nevertheless very interesting from the point of view of Painleve analysis. The main task of this article is to prove that the 17 Painleve-type equations and the equivalence classes that they generate provide the complete classification of the fourth-order polynomial class. Equations F-V, F-VI, F-XVII, and F-XVIII define higher-order Painleve transcendents. Of these, F-VI was new in paper I while the other three are group-invariant reductions of the KdV5, the modified KdV5, and the modified Sawada-Kotera equations, respectively. Seven of the 19 equations involve hyperelliptic functions of genus 2. Partial results on the fourth-order classification problem have been obtained previously by Bureau, Exton, and Martynov, the latter author finding all but four of the relevant reduced equations. Complete solutions are given except in the cases that define the aforementioned higher-order transcendents.