THE PAINLEVE ANALYSIS AND EXACT TRAVELING-WAVE SOLUTIONS TO NONLINEAR PARTIAL-DIFFERENTIAL EQUATIONS

A partial differential equation (PDE) has the Painleve property when the solutions of the PDE are 'single-valued'' about the movable ''singular'' manifold. If u(z1,...,z(n)) is a solution of the PDE then [GRAPHICS] where phi = phi(z1,..., z(n)) and u(j) = u(j) (z1,...,z(n)) are analytic functions of (z1,...,z(n)) in a neighbourhood of the manifold phi(z1,...,z(n)) = 0 and alpha is a negative integer. Substitution of (1) into the PDE determines the possible values of alpha and defines a set of recursion relations for the u(j). Values of j for which these recursion relations are not satisfied are called the ''resonances'' of the recursion relation. These resonances introduce an arbitrary function u(j) and a ''compatibility condition'' on the functions (phi,u0,...,u(j-1)). The PDE possesses the Painleve property when the compatibility condition is satisfied at these resonances. Further, it is possible to define a Backlund transformation by truncating the series (1) at j = -alpha.