PAINLEVE CLASSIFICATION OF ALL SEMILINEAR PARTIAL-DIFFERENTIAL EQUATIONS OF THE 2ND-ORDER .1. HYPERBOLIC-EQUATIONS IN 2 INDEPENDENT VARIABLES

In this paper we give a complete classification of all Painleve-type h yperbolic partial differential equations (PDE's) over the complex doma in of the form u(xy) = F(x, y, u, u(x), u(y)), where F is rational in u, u(x), and u(y), 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 cano nical representative of each class is presented and solved by transfor ming it either to a well-known soliton equation (sine-Gordon, Bullough -Dodd-Mikhailov) or to a linear equation by means of a Backlund corres pondence or simpler change of variables. (The parabolic case, in which 10 more canonical equations are obtained, and semilinear PDE's in thr ee or more variables are treated in the accompanying paper II.) The pr oof that the list is complete involves investigating four sets of nece ssary conditions in turn, each of which has essentially new features p eculiar 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,...,DE-76, accord ing to their Bureau symbols and resonance data, which satisfy those ne cessary conditions for the Painleve property common to each of these f our Painleve classification problems.