The structure of WTC expansions and applications

Abstract

We construct generalized Painleve expansions with logarithmic terms for a general class of ('non-integrable') scalar equations, and describe their structure in detail. These expansions were introduced without logarithms by Weiss-Tabor-Carnevale (WTC-1983). The construction of the formal solutions is shown to involve semi-invariants of binary forms, and tools from invariant theory are applied to the determination of the type of logarithmic terms that are required for the most general singular 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 (1993). The results are illustrated on a family of fifth-order models for water-waves, and other examples. We also give necessary and sufficient conditions for -1 to be a resonance.