Painlevé Classification of All Semilinear Partial Differential Equations of the Second Order. II. Parabolic and Higher Dimensional Equations

This paper extends the work of the previous paper (I) on the Painlevé classification of second‐order semilinear partial differential equations to the case of parabolic equations in two independent variables, u xx = F ( x , y , u , u x , u y ), and irreducible equations in three or more independent variables of the form, Σ ij R ij ( x 1 ,…, x n ) u , ij = F ( x 1 ,…, x n ; u , 1 ,…, u , n ). In each case, F is assumed to be rational in u and its first derivatives and no other simplifying assumptions are made. In addition to the 22 hyperbolic equations found in paper I, we find 10 equivalence classes of parabolic equations with the Painlevé property, denoted PS‐I, PS‐I1,…, PS‐X, equation PS‐II being a generalization of Burgers' equation denoted the Forsyth‐Burgers equation, and 13 higher‐dimensional Painlevé equations, denoted GS‐I, GS‐II,…, GS‐XIII. The lists are complete up to the equivalence relation of Möbius transformations in u and arbitrary changes of the independent variables. In order to avoid repetition, the proofs are sketched very briefly in cases where they closely resemble those for the corresponding hyperbolic problem. Every equation is solved by transforming to a linear partial differential equation, from which it follows that there are no non trivial soliton equations among the two classes of Painlevé equations treated in this paper.