THE STRUCTURE OF WTC EXPANSIONS AND APPLICATIONS

We construct generalized Painleve expansions with logarithmic terms fo r a general class of ('non-integrable') scalar equations, and describe their structure in detail. These expansions were introduced without l ogarithms by Weiss-Tabor-Carnevale (WTC). The construction of the form al solutions is shown to involve semi-invariants of binary forms, and tools from invariant theory are applied to the determination of the ty pe of logarithmic terms that are required for the most general singula r series. The structure of the series depends strongly on whether 1 is or is not a resonance. The convergence of these series is obtained as a consequence of the general results of Littman and Kichenassamy. The results are illustrated on a family of fifth-order models for water-w aves, and other examples. We also give necessary and sufficient condit ions for -1 to be a resonance.