Painlevé Classification of All Semi linear Partial Differential Equations of the Second Order. I. Hyperbolic Equations in Two Independent Variables

Abstract

In this paper we give a complete classification of all Painlevé‐type hyperbolic partial differential equations (PDE's) over the complex domain of the form uxy = F(x, y, u, ux, uy where F is rational in u, ux, and uy, and locally analytic in x and y. We find exactly 22 equivalence classes of equations (under coordinate changes and Mobius transformations in u), which we denote HS‐I, HS‐II,…, HS‐XXII. A canonical representative of each class is presented and solved by transforming it either to a well‐known soliton equation (sine‐Gordon, Bullough‐Dodd‐Mikhailov) or to a linear equation by means of a Bäcklund correspondence or simpler change of variables. (The parabolic case, in which 10 more canonical equations are obtained, and semilinear PDE's in three or more variables are treated in the accompanying paper II.) The proof that the list is complete involves investigating four sets of necessary conditions in turn, each of which has essentially new features peculiar to the PDE context, as well as familiar features analogous to the corresponding conditions for ordinary differential equations (ODE's) as discussed in the classical literature by Painleve, Gambier, Ince, Bureau, and others. In a setting sufficiently general to embrace ODE's, hyperbolic and parabolic PDE's, and higher dimensional semilinear PDE's, we classify 76 types of O/PDE's, denoted DE‐1,…, according to their Bureau symbols and resonance data, which satisfy those necessary conditions for the Painlevé property common to each of these four Painlevé classification problems.

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

“…This paper completes the Painleve classification of all rational semilinear partial differential equations (PDE's) in two or more variables that was begun in paper I [12] with the treatment of hyperbolic equations in two variables. Our first task will be to complete the Painleve classification of semilinear PDE's in two independent variables by treating the parabolic case, which, after a suitable change of variables, can be put in the form, (1.1 ) where F is rational in u, U x' and u y and locally analytic in x and y, all variables being complex (subscripts denote partial differentiation).…”

The Painlevé Approach to Nonlinear Ordinary Differential Equations

“…In the Lorenz model (1.3), the simultaneous change of variables (x, y, z) → (ξ, η, ζ) and parameters (b, σ, r) → (b, σ, ε) defined by ξ = εx, η = ε 2 σy, ζ = ε 2 σz, ε 2 σr = 1 (5.37) led Robbins [102] to believe to have found a new integrable case, defined by 38) while in fact the new dynamical system is just the simplified of the original one, integrable by elliptic functions. Example 2.…”

Higher‐Order Painlevé Equations in the Polynomial Class II: Bureau Symbol P1

“…This article will demonstrate a variety of different analytic tools to bring the Painlevé classification of the fourth-order polynomial class of differential equations to completion. Bureau symbols were introduced by Bureau in [3] and refined by the author in [4,5,6]. For differential equations in the polynomial class, only Bureau symbols of type Pk occur, where k is a positive integer (except in the case of exotic Painlevé equations [6]).…”